PHASE DIAGRAMS AND
THERMODYNAMIC
MODELING OF
SOLUTIONS
PHASE DIAGRAMS AND
THERMODYNAMIC
MODELING OF
SOLUTIONS
ARTHUR D. PELTON
Dep’t Chemical Engineering, Centre de Recherche en
Calcul Thermochimique, Ecole Polytechnique de Montreal
Elsevier
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Cover credit: A.D. Pelton, “Thermodynamics and Phase Diagrams” in “Physical Metallurgy
5th edition”, D. E. Laughlin and K. Hono (eds.), Elsevier, pp. 203-303 (2014)
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Dedication
To Brenda
ACKNOWLEDGMENTS
This book is the result of over 40 years of invaluable collaboration and discussions with a long list of students and colleagues.
It would have been impossible to write this book without the
FactSage software and databases.
I have collaborated with Chris Bale for 45 years and with
Gunnar Eriksson for 40 years on the development of FactSage.
I owe them an immense debt of gratitude.
I am particularly indebted to my mentor Milton Blander for his
inspiration and deep insights into the mysteries of the thermodynamics of solutions.
Special thanks are due to Evguenia Sokolenko for her skillful
and essential assistance in preparing the figures and formatting
the text and equations.
Finally, I wish to acknowledge the Natural Sciences and
Engineering Research Council of Canada for financial assistance
over many years.
xv
INTRODUCTION
An understanding of phase diagrams is fundamental and
essential to the study of materials science, and an understanding
of thermodynamics is fundamental to an understanding of phase
diagrams. A knowledge of the equilibrium state of a system under
a given set of conditions is the starting point in the description of
many phenomena and processes.
The theme of Part I of this book is the relationship between
phase diagrams and thermodynamics. To understand phase diagrams properly, it has always been necessary to understand their
thermodynamic basis. However, in recent years, the relationship
between thermodynamics and phase diagrams has taken on a
new and important practical dimension. With the development
of large evaluated databases of the thermodynamic properties
of thousands of compound and solutions and of software to calculate the conditions for chemical equilibrium (by minimizing the
Gibbs energy), it is possible to rapidly calculate and plot desired
phase diagram sections of multicomponent systems. Most of
the phase diagrams shown in this book were calculated thermodynamically with the FactSage software and databases (see FactSage in the list of websites).
Several large integrated thermodynamic database computing
systems have been developed in recent years. (See list of websites.) The databases of these systems have been prepared by
the following procedure. For every compound and solution phase
of a system, an appropriate model is first developed giving the
thermodynamic properties as functions of temperature T, pressure P, and composition. Next, all available thermodynamic and
phase equilibrium data from the literature for the entire system
are simultaneously optimized to obtain one set of critically evaluated, self-consistent parameters of the models for all phases in
two-component; three-component; and, if data are available,
higher-order subsystems. Finally, the models are used to estimate
the properties of multicomponent solutions from the databases of
parameters of the lower-order subsystems. The Gibbs energy minimization software then accesses the databases and, for given sets
of conditions (T, P, composition …), calculates the compositions
and amounts of all phases at equilibrium. By calculating the equilibrium state as T, composition, P, etc. are varied systematically,
the software generates the phase diagram.
xvii
xviii
INTRODUCTION
The theme of Part II is models that are currently used to represent the thermodynamic properties of a solution as functions
of T, P, and composition. These models relate the thermodynamic
properties to the atomic or molecular structure of the solution.
Techniques of data evaluation and optimization are also outlined. However, a thorough discussion of these techniques is
beyond the scope of the present work.
List of Websites
FactSage, Montreal, www.factsage.com.
NPL-MTDATA, www.npl.co.uk.
Pandat, www.computherm.com.
SGTE, Scientific Group Thermodata Europe, www.sgte.org.
Thermocalc, Stockholm, www.thermocalc.com.
1
INTRODUCTION
A phase diagram is a graphic representation of the values of the
thermodynamic variables when equilibrium is established among
the phases of a system. Materials scientists are most familiar with
phase diagrams that involve temperature, T, and composition as
variables. Examples are T-composition phase diagrams for binary
systems such as Fig. 1.1 for the Fe-Mo system, isothermal phase
diagram sections of ternary systems such as Fig. 1.2 for the
Zn-Mg-Al system, and isoplethal (constant composition) sections
of ternary and higher-order systems such as Figs. 1.3A and 1.4.
However, many useful phase diagrams can be drawn that
involve variables other than T and composition. The diagram in
Fig. 1.5 shows the phases present at equilibrium in the Fe-Ni-O2
system at 1200°C as the equilibrium oxygen partial pressure
(i.e., chemical potential) is varied. The x-axis of this diagram is
the overall molar metal ratio in the system. The phase diagram
in Fig. 1.6 shows the equilibrium phases present when an equimolar Fe-Cr alloy is equilibrated at 925°C with a gas phase of varying
O2 and S2 partial pressures. For systems at high pressure, P-T
phase diagrams such as the diagram for the one-component
Al2SiO5 system in Fig. 1.7A show the fields of stability of the various allotropes. When the pressure in Fig. 1.7A is replaced by the
volume of the system as y-axis, the “corresponding” V-T phase
diagram of Fig. 1.7B results. The enthalpy of the system can also
be a variable in a phase diagram. In the phase diagram in Fig. 1.3B,
the y-axis of Fig. 1.3A has been replaced by the molar enthalpy difference (hT h25) between the system at 25°C and a temperature
T. This is the heat that must be supplied or removed to heat or cool
the system adiabatically between 25°C and T.
The phase diagrams shown in Figs. 1.1–1.7 are only a small
sampling of the many possible types of phase diagram sections.
These diagrams and several other useful types of phase diagrams
will be discussed in Part I. Although these diagrams appear to have
very different geometries, it will be shown that actually they all
obey exactly the same geometric rules.
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00001-4
# 2019 Elsevier Inc. All rights reserved.
3
Chapter 1 INTRODUCTION
2600
Liquid
Liquid + bcc
bcc
1233°
1240°
Mu + bcc
bcc + Laves
600
bcc
Sigma + bcc
Sigma
Mu
1100
1498°
1453° R 1368°
1250°
fcc
R
Q 1612°
P
1600
Laves
Temperature (∞C)
2100
308°
bcc + Mu
100
0
Fe
0.2
0.4
0.6
Mole fraction Mo
0.8
1
Mo
Fig. 1.1 Temperature-composition phase diagram at P ¼ 1 bar of the Fe-Mo system
(Pelton, 2014).
Zn
0.7
0.6
3
0.5
3
0.5
0.6
Tau
3
0.4
MgZn
0.4
Mg2Zn3
3
0.3
Laves
0.2
0.8
Mg2Zn11
0.1
0.9
hcp
3
γ+τ
3
Mg
Gamma
0.9
0.8
0.7
0.6
3
3
p
0.5
0.9
hcp
0.8
0.2
0.7
0.3
3
0.1
4
XZn= 0.1
fcc
0.4
Mole fraction
0.3
0.2
β AlMg
0.1
Al
Fig. 1.2 Isothermal phase diagram section at 25°C and P ¼ 1 bar of the Zn-Mg-Al
system (Pelton, 2014).
Chapter 2 provides a review of the fundamentals of thermodynamics as required for the interpretation and calculation of phase
diagrams of all types. In Chapter 3, the Gibbs phase rule is developed in a general form suitable for the understanding of phase
diagrams involving a wide range of variables including chemical
potentials, enthalpy, and volume. Chapter 4 provides a review of
Chapter 1 INTRODUCTION
Liquid
625
Liquid + fcc
Tau
(Gamma + Phi + Tau)
125
hcp + Gamma + MgZn
25
0
(A)
0.2
fcc + Beta
+ Tau
(fcc + Laves
+ Mg2Zn11)
(fcc + Mg 2Zn11 + hcp)
0.4
0.6
0.8
1
Molar ratio Al/(Mg+Al)
30
o
625 C
o
725 C
Liquid
25
Liquid + fcc
20
Liquid + Tau
Liquid
+ hcp
o
525 C
15
Liquid + fcc + Tau
Tau
Enthalpy (h – h25∞C) (kJ mol–1)
fcc + Laves
+ Ta
u
Tau + G
p
mma
hcp +
Gamma + Phi
(Beta + Tau)
+ Ga
225
Liquid
+ Gamma
Beta
325
fcc + Tau
amma
425 Liquid + hcp
hcp + MgZn
hcp + Phi + MgZn
Temperature (∞C)
(Liquid + Tau)
525
10
o
fcc + Tau
425 C
o
325 C
o
5
hcp + Phi
+ MgZn
225 C
o
125 C
0
(B)
Fig. 1.3 Isoplethal phase diagram section at XZn ¼ 0.1 and P ¼ 1 bar of the
Zn-Mg-Al system. (A) Temperature versus composition; (B) enthalpy versus
composition (Pelton, 2014).
the thermodynamics of solutions. Chapter 5 begins with a discussion of the thermodynamic origin of binary T-composition phase
diagrams, presented in the classical manner involving common
tangents to curves of Gibbs energy. Chapter 5 continues with a
thorough discussion of all features of T-composition phase diagrams of binary systems with particular stress on the relationship
between the phase diagram and the thermodynamic properties of
the phases. A similar discussion of T-composition phase diagrams
of ternary systems is given in Chapter 6. Isothermal and isoplethal
sections and polythermal liquidus and solidus projections are
discussed.
5
Chapter 1 INTRODUCTION
850∞C 0.3 wt.% carbon
1
2
3
4
5
6
4
bcc + M7C3
bcc + M7C3 + M23C6
bcc + fcc + M7C3
bcc + fcc + M7C3 + M23C6
fcc + M7C3 + M23C6
bcc + fcc + MC + M7C 3
Zero phase fraction lines
M23C6 (mnprth)
bcc (eactuj)
M7C3 (gvaonqsui)
fcc (dobrsk)
MC (fvcbpql)
m
3
bcc + MC
o
d
bcc + fcc + MC
fcc
0
2
b
g
4
p
q
1 2
3 4r bcs
t
c
u
c
fcc + M7C3
v
0f
6
fcc + MC
+ M7C3
C3
fcc + MC
e
a
7
1
l
bcc + MC
+ M7C3 + M23C6
c M
bc C +
M
2
bcc + MC + M23C6
n
+
Weight percent vanadium
5
bcc + M23C6
+f
cc
5 fcc
h i + M23C6
6
8
10
12
Weight percent chrome
+M
23
C6
j
14 k
16
Fig. 1.4 Phase diagram section of the Fe-Cr-V-C system at 850°C, 0.3 wt% C, and
P ¼ 1 bar (SGTE).
1200∞C
–1
–3
Spinel + Fe2O3
Spinel + Monoxide
log10p(O2) (bar)
6
Spinel
–5
Monoxide
–7 Spinel +
Monoxide
Spinel + fcc
–9
Monoxide
Monoxide + fcc
Monoxide + fcc
–11
fcc
–13
0
0.2
0.4
0.6
0.8
1
Molar ratio Ni/(Fe+Ni)
Fig. 1.5 Phase diagram of the Fe-Ni-O2 system at 1200°C showing equilibrium
oxygen pressure versus overall metal ratio (Pelton, 2014).
In Chapter 7, the geometry of general phase diagram sections
is developed in depth. In this chapter, the following are presented:
the general rules of construction of phase diagram sections, the
proper choice of axis variables and constants required to give a
single-valued phase diagram section with each point of the diagram representing a unique equilibrium state, and a general
925∞C, Molar ratio Cr/(Fe+Cr) = 0.5
0
log10p(S2) (bar)
–2
Pyrrhotite + FeCr2S4
–4
te
oti
rrh
+M
O3
2
Py
–6
rr
Py
Pyrrhotite
–8
tit
ho
S
e+
el
pin
Pyrrhotite + fcc
Pyrrhotite + bcc
–10
Spinel
fcc + Spinel
fcc + M2O3
bcc
–12
bcc + M2O3
–14
–24
–22
–20
–18
log10p(O2) (bar)
–16
–14
Fig. 1.6 Phase diagram of the Fe-Cr-S2-O2 system at 925°C showing equilibrium S2 and O2 partial pressures at constant
molar ratio Cr/(Cr + Fe) ¼ 0.5 (Pelton, 2014).
6
Pressure (kbar)
Kyanite
5
4
Sillimanite
Triple point
3
2
Andalusite
1
0
500
550
(A)
R
650
Kyanite
0.046
0.048
0.050
Andalusite + Kyanite
Molar volume (L mol–1)
0.044
600
Temperature (∞C)
Sillimanite + Kyanite
Sillimanite
Q
Andalusite + Sillimanite
0.052
P
Andalusite
(B)
Fig. 1.7 (A) P-T and (B) V-T phase diagrams of Al2SiO5 (Pelton, 2014).
700
8
Chapter 1 INTRODUCTION
algorithm for calculating all such phase diagram sections
thermodynamically.
Chapter 8 begins with a discussion of the course of equilibrium
solidification of binary, ternary, and multicomponent systems and
includes a classification of invariant reactions. Next, nonequilibrium Scheil-Gulliver solidification is discussed in which solids,
once precipitated, cease to react with the liquid or with each other.
Scheil-Gulliver constituent diagrams are introduced from which
one can visualize the course of Scheil-Gulliver cooling as temperature and composition are varied. The calculation and theory of
these diagrams are discussed.
Chapter 9 treats paraequilibrium in which, during cooling, rapidly diffusing elements reach equilibrium but more slowly diffusing elements remain essentially immobile. Paraequilibrium phase
diagrams are introduced, and the application of the Phase Rule to
these diagrams is developed.
Chapter 10 provides an introduction to phase diagrams with
second-order phase transitions and to ferromagnetic and order/
disorder transitions.
In Chapter 11, phase diagrams for systems involving an aqueous phase including evaporation diagrams, classical Eh-pH diagrams (which are not true phase diagrams), and true aqueous
phase diagrams are discussed.
Finally, Chapter 12 provides a survey of phase diagram compilations and texts on the relationship between thermodynamics
and phase diagrams.
References
Pelton, A.D., 2014. In: Laughlin, D.E., Hono, K. (Eds.), (Chapter 3). Physical Metallurgy. fifth ed. Elsevier, New York.
List of Websites
SGTE, Scientific Group Thermodata Europe, www.sgte.org.
2
THERMODYNAMICS
FUNDAMENTALS
CHAPTER OUTLINE
2.1 The First and Second Laws of Thermodynamics 10
2.1.1 Nomenclature 10
2.1.2 The First Law 10
2.1.3 The Second Law 11
2.1.4 The Fundamental Equation of Thermodynamics 12
2.2 Enthalpy 13
2.2.1 “Absolute” Enthalpy 16
2.2.2 Standard Enthalpy of Formation 16
2.3 Gibbs Energy 16
2.4 Equilibrium and Chemical Reactions 18
2.4.1 Equilibria Involving a Gaseous Phase 18
2.4.2 A Note on Nonideal Gases 20
2.4.3 Predominance Diagrams 22
2.5 Measuring Gibbs Energy, Enthalpy and Entropy 25
2.5.1 Measuring Gibbs Energy Change 25
2.5.2 Measuring Enthalpy Change 26
2.5.3 Measuring Entropy—The Third Law of Thermodynamics
2.6 Gibbs Energy of a Pure Compound as a Function of
Temperature 27
2.7 Auxiliary Functions 28
2.8 The Chemical Potential 29
2.9 Some Other Useful Thermodynamic Equations 30
2.9.1 The Gibbs-Duhem Equation 30
2.9.2 General Auxiliary Functions 31
References 31
List of Websites 31
26
This chapter is intended to provide a review of the fundamentals of thermodynamics as required for the interpretation
and calculation of phase diagrams. It is not intended to be an
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00002-6
# 2019 Elsevier Inc. All rights reserved.
9
10
Chapter 2 THERMODYNAMICS FUNDAMENTALS
introductory treatise on the subject. The development of the
thermodynamics of phase diagrams will be continued in the
succeeding chapters.
2.1
The First and Second Laws of
Thermodynamics
If the thermodynamic system under consideration is permitted
to exchange both energy and mass wth its surroundings, the system is said to be open. If energy but not mass may be exchanged,
the system is said to be closed. The state of a system is defined by
intensive properties, such as temperature and pressure, which are
independent of the mass of the system, and by extensive properties, such as volume and internal energy, which vary directly as the
mass of the system.
2.1.1
Nomenclature
Extensive thermodynamic properties will be represented by
uppercase symbols. For example, G ¼ Gibbs energy in J. Molar
properties will be represented by lowercase symbols. For example,
g ¼ G/n ¼ molar Gibbs energy in J mol1 where n is the total number of moles in the system.
2.1.2
The First Law
The internal energy of a system, U, is the total thermal and
chemical bond energy stored in the system. It is an extensive state
property.
Consider a closed system undergoing a change of state that
involves an exchange of heat, dQ, and work, dW, with its surroundings. Since energy must be conserved,
dUn ¼ dQ + dW
(2.1)
This is the First Law. The convention is adopted whereby energy
passing from the surroundings to the system is positive.
The subscript on dUn indicates that the system is closed (constant
number of moles.)
It must be stressed that dQ and dW are not changes in state
properties. For a system passing from a given initial state to a
given final state, dUn is independent of the process path since it
is the change of a state property; however, dQ and dW are, in general, path-dependent.
Chapter 2 THERMODYNAMICS FUNDAMENTALS
2.1.3
The Second Law
For a rigorous and complete development of the Second Law,
the reader is referred to standard texts on thermodynamics. The
entropy of a system, S, is an extensive state property that is given
by Boltzmann’s equation as
S ¼ kB ln t
(2.2)
where kB is Boltzmann’s constant and t is the multiplicity of the
system. Somewhat loosely, t is the number of possible equivalent
microstates in a macrostate, that is, the number of quantum states
of the system that are accessible under the applicable conditions
of energy, volume, etc. For example, for a system that can be
described by a set of single-particle energy levels, t is the number
of ways of distributing the particles over the energy levels, keeping
the total internal energy constant. At low temperatures, most of
the particles will be in or near the ground state. Hence, t and S will
be small. As the temperature and hence U increase, more energy
levels become occupied. Consequently, t and S increase. For solutions, an additional contribution to t arises from the number of
different possible ways of distributing the atoms or molecules
over the lattice or quasi-lattice sites (see Section 4.6). Again somewhat loosely, S can be said to be a measure of the disorder of a
system.
During any spontaneous process, the total entropy of the universe will increase for the simple reason that disorder is more
probable than order. That is, for any spontaneous process,
dStotal ¼ ðdS + dSsurr Þ 0
(2.3)
where dS and dSsurr are the entropy changes of the system and surroundings, respectively. The existence of a state property S that
satisfies Eq. (2.3) is the essence of the Second Law.
Eq. (2.3) is a necessary condition for a process to occur. However, even if Eq. (2.3) is satisfied, the process may not actually be
observed if there are kinetic barriers to its occurrence. That is, the
Second Law says nothing about the rate of a process that can vary
from extremely rapid to infinitely slow.
It should be noted that the entropy change of the system, dS,
can be negative for a spontaneous process as long as the sum
(dS +dSsurr) is positive. For example, during the solidification of
a liquid, the entropy change of the system is negative in going
from the liquid to the more ordered solid state. Nevertheless, a liquid below its melting point will freeze spontaneously because the
entropy change of the surroundings is sufficiently positive due to
the transfer of heat from the system to the surroundings. It should
11
12
Chapter 2 THERMODYNAMICS FUNDAMENTALS
also be stressed that in passing from a given initial state to a given
final state, the entropy change of the system dS is independent
of the process path since it is the change of a state property.
However, dSsurr is path-dependent.
2.1.4
The Fundamental Equation of
Thermodynamics
Consider an open system at equilibrium with its surroundings
and at internal equilibrium. That is, no spontaneous irreversible
processes are taking place. Suppose that a change of state occurs
in which S, V (volume), and ni (number of moles of component i in
the system) change by dS, dV, and dni. Such a change of state
occurring at equilibrium is called a reversible process, and the corresponding heat and work terms are dQrev and dWrev. We may then
write
X
dU ¼ ðdU=dSÞv,n dS + ðdU=dV Þs:n dV +
μi dni
(2.4)
where
μi ¼ ðdU=dni Þs,v, nj6¼i
(2.5)
μi is the chemical potential of component i which will be discussed
in Section 2.8.
The absolute temperature is given as
T ¼ ðdU=dSÞv,n
(2.6)
We expect that temperature should be defined such that heat
flows spontaneously from high to low T. To show that T as
given by Eq. (2.6) is, in fact, such a thermal potential, consider
two closed systems, isolated from their surroundings but in thermal contact with each other, exchanging only heat at constant
volume. Let the temperatures of the systems be T1 and T2, and
let T1 > T2. Suppose that heat flows from system 1 to system 2,
then dU2 ¼ dU1 > 0. Therefore, from Eq. (2.6),
dS ¼ dS1 + dS2 ¼ dU1 =T1 + dU2 =T2 > 0
(2.7)
That is, the flow of heat from high to low temperature results in an
increase in total entropy, and hence, from the Second Law, it is
spontaneous.
The second term in Eq. (2.4) is ( PdV), the work of reversible
expansion. From Eq. (2.6), the first term in Eq. (2.4) is equal to TdS,
and this is then the reversible heat:
T dS ¼ dQrev
(2.8)
Chapter 2 THERMODYNAMICS FUNDAMENTALS
That is, in the particular case of a process that occurs reversibly,
(dQrev/T) is path-independent since it is equal to the change of
a state property dS. Eq. (2.8) is actually the definition of entropy
change in the classical development of the Second Law.
Eq. (2.4) may now be written as
X
(2.9)
dU ¼ T dS PdV +
μi dni
Eq. (2.9), which results from combining the first and second laws, is
called the fundamental equation of thermodynamics. We have
assumed that the only work term is the reversible work of expansion
(sometimes called “PV work”). In general, in this book, this will be the
case. If other types of work occur, then non-PV terms, dWrev(non-PV),
must be added to Eq. (2.6). For example, if the process is occurring
in a galvanic cell, then dWrev(non-PV) is the reversible electric work
in the external circuit. Eq. (2.9) can thus be written more generally as
X
dU ¼ T dS PdV +
μi dni + dWrevðnonPV Þ
(2.10)
2.2
Enthalpy
Enthalpy, H, is an extensive state property defined as
H ¼ U + PV
(2.11)
Consider a closed system undergoing a change of state that may
involve irreversible processes (such as chemical reactions).
Although the overall process may be irreversible, we shall assume
that any work of expansion is approximately reversible (i.e., the
external and internal pressures are equal) and that there is no
work other than the work of expansion. Then, from Eq. (2.1),
dUn ¼ dQ PdV
(2.12)
From Eqs. (2.11), (2.12), it follows that
dHn ¼ dQ + V dP
(2.13)
Furthermore, if the pressure remains constant throughout the
process, then
dHp ¼ dQp
(2.14)
Integrating both sides of Eq. (2.14) gives
ΔHp ¼ Qp
(2.15)
That is, the enthalpy change of a closed system in passing from an
initial to a final state at constant pressure is equal to the heat
13
14
Chapter 2 THERMODYNAMICS FUNDAMENTALS
exchanged with the surroundings. Hence, for a process occurring
at constant pressure, the heat is path-independent since it is equal
to the change of a state property. This is an important result. As an
example, suppose that the initial state of a system consists of
1.0 mol of C and 1.0 mol of O2 at 298.15 K at 1.0 bar pressure and
that the final state is 1.0 mol of CO2 at the same temperature
and pressure. The enthalpy change of this reaction is.
o
C + O2 ¼ CO2 ΔH298:15
¼ 393:52kJ
ΔHo298.15
(2.16)
(where the superscript on
indicates the standard state
reaction involving pure solid graphite and CO2 and O2 at 1.0 bar
pressure). Hence, an exothermic heat of 393.52 kJ will be
observed, independent of the reaction path, provided only that
the pressure remains constant throughout the process. For
instance, during the combustion reaction, the reactants and products may attain high and unknown temperatures. However, once
the CO2 product has cooled back to 298.15 K, the total heat that
has been exchanged with the surroundings will be 393.52 kJ
independent of the intermediate temperatures.
In Fig. 2.1, the standard molar enthalpy of Fe is shown as a
function of temperature. The y-axis, (hoT ho298.15), is the heat
required to heat 1.0 mol of Fe from 298.15 K to a temperature T
at constant pressure. As usual, the superscripts “o” indicate the
standard state of pure Fe. The slope of the curve is the molar heat
capacity at constant pressure:
Fig. 2.1 Standard molar enthalpy of Fe (FactSage).
Chapter 2 THERMODYNAMICS FUNDAMENTALS
cP ¼ ðdh=dT ÞP
(2.17)
Heat capacities are often represented as functions of temperature
by the semiempirical equation:
cP ¼ a + bT + cT 2
(2.18)
where a, b, and c are constants and T is in kelvins. Empirical terms
with other powers Ti are often also used.
From Eq. (2.17), we obtain the following expression for the heat
required to heat 1.0 mol of a substance from a temperature T1 to a
temperature T2 at constant P (assuming no phase changes in the
interval):
Tð2
ðhT2 hT1 Þ ¼
cP dT
(2.19)
T1
The enthalpy curve can be measured by the technique of drop
calorimetry, or the heat capacity can be measured directly
by adiabatic calorimetry.
The standard equilibrium temperature of fusion (melting)
of Fe is 1811 K as shown in Fig. 2.1. The fusion reaction is a
first-order phase change since it occurs at constant temperature.
The standard molar enthalpy of fusion Δhof is 13.807 kJ mol1
as shown in Fig. 2.1. It can also be seen in Fig. 2.1 that Fe
undergoes two other first-order phase changes, the first from
α (bcc) to γ (fcc) Fe at Toα!γ ¼ 1184.8 K and the second from
γ to δ (bcc) at 1667.5 K. The enthalpy changes are, respectively,
1.013 and 0.826 kJ mol1. The Curie temperature, Tcurie ¼ 1045 K,
which is also shown in Fig. 2.1, will be discussed in Chapter 10.
Hence, for example, the enthalpy change upon heating 1.0 mol
of pure Fe from 298.15 K to a temperature T above the melting
point is given as
hoT
ho298:15
1184:8
ð
¼
cPα dT
1667:5
ð
+ Δhoα!γ
+
298:15
1811
ð
+
1667:5
cPγ dT + Δhoγ!δ
1184:8
cPδ dT + Δhof +
ðT
cPliq dT
(2.20)
1811
where cαP, cγP, cδP, and cliq
P are the molar heat capacities of the
individual phases, each expressed by an equation similar to
Eq. (2.18). In the case of Fe and other magnetic materials, additional terms must be added to cαP as will be discussed in
Chapter 10.
15
16
Chapter 2 THERMODYNAMICS FUNDAMENTALS
2.2.1
“Absolute” Enthalpy
No natural zero exists for the enthalpy because there is no simple natural zero for the internal energy (see Eq. (2.11)). However,
by convention, the “absolute” enthalpy of a pure element in its
stable state at P ¼ 1.0 bar and T ¼ 298.15 K is often set to zero.
Hence, under this convention, the y-axis of Fig. 2.1 could be
labeled simply as hoT.
2.2.2
Standard Enthalpy of Formation
The standard molar enthalpy of formation of a compound is
defined as the enthalpy of formation of 1.0 mol of the pure compound in its stable state from the pure elements in their stable
states at P ¼ 1.0 bar at constant temperature. So, for example,
ΔHo298.15 of the reaction in Eq. (2.16) is the standard enthalpy of
formation of CO2 at 298.15 K.
Under the convention that the standard enthalpies of the
elements are zero at 298.15 K (Section 2.2.1), it then follows that
the “absolute” standard enthalpy of a compound at 298.15 K is
equal to its standard enthalpy of formation from the elements.
o
¼ 393.52 kJ mol1.
That is, hCO
2(298.15)
2.3
Gibbs Energy
The Gibbs energy (also called the Gibbs free energy or simply
the free energy), G, is defined as
G ¼ H TS
(2.21)
As in Section 2.2, we consider a closed system and assume that the
only work term is the work of expansion and that this is reversible.
From Eqs. (2.13), (2.21),
dGn ¼ dQ + V dP T dS SdT
(2.22)
Consequently, for a process occurring at constant T and P in a
closed system,
dGT,P,n ¼ dQP T dS ¼ dHP T dS
(2.23)
Consider the case where the “surroundings” are simply a heat reservoir at a temperature equal to the temperature T of the system.
That is, no irreversible processes occur in the surroundings which
receive only a reversible transfer of heat (dQ) at constant temperature. Therefore, from Eq. (2.8),
dSsurr ¼ dQ=T
(2.24)
Chapter 2 THERMODYNAMICS FUNDAMENTALS
Substituting into Eq. (2.23) and using Eq. (2.3) yields
dGT,P,n ¼ T dSsurr T dS ¼ T dStotal 0
(2.25)
Eq. (2.25) may be considered to be a special form of the Second
Law for a process occurring in a closed system at constant T
and P. From Eq. (2.25), such a process will be spontaneous if dG
is negative.
For our purposes in this book, Eq. (2.25) is the most useful form
of the second law.
An objection might be raised that Eq. (2.25) applies only when
the surroundings are at the same temperature T as the system.
However, if the surroundings are at a different temperature, we
simply postulate a hypothetical heat reservoir that is at the same
temperature as the system and that lies between the system and
surroundings, and we consider the irreversible transfer of heat
between the reservoir and the real surroundings as a second
separate process.
Substituting Eqs. (2.8), (2.24) into Eq. (2.25) gives
dGT, P, n ¼ ðdQ dQrev Þ 0
(2.26)
In Eq. (2.26), dQ is the heat of the actual process, while dQrev is the
heat that would be observed were the process occurring reversibly. If the actual process is reversible, that is, if the system is at
equilibrium, then dQ ¼ dQrev, and dG ¼ 0.
As an example, consider the fusion of Fe in Fig. 2.1. At
Tof ¼ 1811 K, solid and liquid are in equilibrium. That is, at this
temperature, melting is a reversible process. Therefore, at 1811 K,
dgfo ¼ dhof Tdsof ¼ 0
(2.27)
where, again, the superscripts indicate the standard state of pure
Fe at P ¼ 1.0 bar.
Therefore, the molar entropy of fusion at 1811 K is given by
Δsfo ¼ Δhof =T when T ¼ Tfo
(2.28)
Similar equations apply to other first-order phase changes. For
example, for the α ➔ γ transition of Fe in Fig. 2.1,
o
o
¼ Δhoα!γ =T when T ¼ Tα!γ
¼ 1184:8K
(2.29)
Δsα!γ
Eq. (2.28) applies only at the equilibrium melting point. At temperatures below and above this temperature, the liquid and solid
are not in equilibrium. Above the melting point, Δgof < 0, and melting is a spontaneous process (liquid is more stable than solid).
Below the melting point, Δgof > 0, and the reverse process
(solidification) is spontaneous. As the temperature deviates
17
18
Chapter 2 THERMODYNAMICS FUNDAMENTALS
progressively further from the equilibrium melting point, the
magnitude of Δgof , which is the magnitude of the driving force,
increases. We shall return to this subject in Section 5.1.
2.4
Equilibrium and Chemical Reactions
Consider first the question of whether silica fibers in an aluminum matrix at 500°C will react to form mullite, Al6Si2O13:
13
9
SiO2 + 6Al ¼ Si + Al6 Si2 O13
2
2
(2.30)
If the reaction proceeds with the formation of dξ moles of mullite,
then from the stoichiometry of the reaction, dnSi ¼ (9/2) dξ,
dnAl ¼ 6 dξ, and dnSiO2 ¼ 13/2 dξ. Since the four substances
are essentially immiscible at 500°C, their Gibbs energies are equal
to their standard molar Gibbs energies, goi (also known as the standard chemical potentials μoi as defined in Section 2.8), the standard state of a solid or liquid compound being the pure
compound at P ¼ 1.0 bar. The Gibbs energy of the system then varies as
9 o 13 o
o
o
+ gSi
gSiO2 6gAl
¼ ΔG o ¼ 830 kJ
dG=dξ ¼ gAl
6 Si2 O13
2
2
(2.31)
where ΔGo is called the standard Gibbs energy change of the reaction Eq. (2.30) at 500 °C. Since ΔG° < 0, the reaction will proceed
spontaneously so as to minimize G. Equilibrium will never be
attained, and the reaction will proceed to completion.
2.4.1
Equilibria Involving a Gaseous Phase
An ideal gas mixture is one that obeys the ideal gas equation of
state:
PV ¼ nRT
(2.32)
where n is the total number of moles. Further, the partial pressure
of each gaseous species in the ideal mixture is given by
P
P
pi V ¼ ni RT
(2.33)
where n ¼ ni and P ¼ pi (Dalton’s law for ideal gases). R is the
ideal gas constant. The standard state of an ideal gaseous compound is the pure compound at a pressure of 1.0 bar. (See
Section 2.4.2 for the case of nonideal gases.) It can easily be shown
that the partial molar Gibbs energy gi (also called the chemical
potential μi as defined in Section 2.8) of a species in an ideal
Chapter 2 THERMODYNAMICS FUNDAMENTALS
gas mixture is given in terms of its standard molar Gibbs energy goi
(also called the standard chemical potential μoi ) by
gi ¼ gio + RT ln pi
(2.34)
The final term in Eq. (2.34) is purely entropic. As a gas expands at
constant temperature, its entropy increases.
Consider a gaseous mixture of H2, S2, and H2S with partial pressures pH2, pS2, and pH2S. The gases can react according to
2H2 + S2 ¼ 2H2 S
(2.35)
If the reaction (Eq. 2.35) proceeds to the right with the formation
of 2dξ moles of H2S, then the Gibbs energy of the system varies as
dG=dξ ¼ 2gH2 S 2gH2 gS2 ¼
2gHo 2 S 2gHo 2 gSo2 + RT ð2 ln pH2 S 2 ln pH2 ln pS2 Þ ¼
1
¼ ΔG
ΔG o + RT ln p2H2 S p2
H2 pS2
(2.36)
ΔG, which is the Gibbs energy change of the reaction in Eq. (2.35),
is thus a function of the partial pressures. If ΔG < 0, then the reaction will proceed to the right so as to minimize G. In a closed system, as the reaction continues with the production of H2S, pH2S
will increase, while pH2 and pS2 will decrease. As a result, ΔG will
become progressively less negative. Eventually, an equilibrium
state will be reached when dG/dξ ¼ ΔG ¼ 0.
For the equilibrium state, therefore,
1
(2.37)
ΔG o ¼ RT ln K ¼ RT ln p2H2 S p2
H2 pS2
equilibrium
where K, the equilibrium constant of the reaction, is the one
2 1
2
pH
pS2 ) for which the system will
unique value of the ratio ( pH
2S
2
be in equilibrium at the temperature T. If the initial partial pressures are such that ΔG > 0, then the reaction in Eq. (2.35) will proceed to the left in order to minimize G until the equilibrium
condition of Eq. (2.37) is attained.
As a further example, consider the possible precipitation of
graphite from a gaseous mixture of CO and CO2. The reaction is
2CO ¼ C + CO2
(2.38)
Proceeding as above, we can write
dG=dξ ¼ gC + gCO2 2gCO ¼
o
o
gCo + gCO
2gCO
+ RT ln pCO2 p2
CO ¼
2
2
ΔG o + RT ln pCO2 p2
CO ¼ ΔG ¼ RT ln K + RT ln pCO2 pCO
(2.39)
19
20
Chapter 2 THERMODYNAMICS FUNDAMENTALS
If (pCO2 p2
CO) is less than the equilibrium constant K, then precipitation of graphite will occur in order to decrease G.
Real situations are, of course, generally more complex. To treat
the deposition of solid Si from a vapor of SiI4, for example, we
must consider the formation of gaseous I2, I, and SiI2 so that three
independent reaction equations must be written:
SiI4 ðgÞ ¼ SiðsolÞ + 2I2 ðgÞ
(2.40)
SiI4 ðgÞ ¼ SiI2 ðgÞ + I2 ðgÞ
(2.41)
I2 ðgÞ ¼ 2IðgÞ
(2.42)
The equilibrium state, however, is still that that minimizes the
total Gibbs energy of the system. This is equivalent to satisfying
simultaneously the equilibrium constants of the reactions in
Eqs. (2.40)–(2.42).
Equilibria involving reactants and products that are solid or
liquid solutions will be discussed in Section 4.5.
2.4.2
A Note on Nonideal Gases
Ideal gases were discussed in Section 2.4.1. The ideal gas equation of state, Eq. (2.32), can be derived from the kinetic theory of
gases if it is assumed that the gaseous molecules do not interact
with each other except through elastic collisions. Real gases
approach ideal gas behavior as the pressure decreases and as
the temperature increases. At higher pressures and lower temperatures, the molecules are in closer proximity, attractive and repulsive interactions become more important, and the equation of
state of the gas becomes more complex than Eq. (2.32).
As examples of typical deviations from ideal behavior, consider
1.0 mol of a gas at T ¼ 300 K and P ¼ 1.0 bar. From Eq. (2.32), the
volume of an ideal gas is 24.94 L. For the real gases O2, CO2, and
CH3CH2OH (ethanol), the volumes under these conditions deviate from the ideal value by 0.06%, 0.5%, and 5.6%, respectively,
while at 300 K and P ¼ 10.0 bar, the deviations from the ideal volume (2.494 L) are 0.6%, 4.9%, and 56%, respectively. (These values
were calculated from the first virial coefficient as given by the Tsonopoulos equation of state assuming no dissociation.) With larger
molecules, the possibilities for intermolecular interactions are
greater, and so the deviations from ideal behavior become larger.
This is particularly true in the case of polar molecules such as ethanol. From these examples, it can be seen that, except in the case
of the smallest molecules, deviations from ideal gas behavior
must be taken into account in calculations near room temperature, particularly at elevated pressures.
Chapter 2 THERMODYNAMICS FUNDAMENTALS
However, at T ¼ 1000 K, the deviations from ideality are much
smaller. At P ¼ 1.0 bar, the volumes of O2, CO2, and CH3CH2OH all
deviate from the ideal value by <0.03% (again assuming no dissociation), and at P ¼ 10.0 bar, the deviations are still all <0.06%. In
the examples used in this book, we shall deal mainly with gases
with small simple molecules at elevated temperatures and at relatively low pressures. Hence, ideal gas behavior will always be
assumed.
Real gases are described thermodynamically by replacing
Eq. (2.34) with the following equation:
gi ¼ gio + RT ln fi
(2.43)
where fi is called the fugacity. The fugacity is a function of T and P.
For an ideal gas, fi is equal to the partial pressure: fi ¼ pi. For a real
gas, fi approaches pi at high T and low P. A plot of fi versus pi at
constant T for a hypothetical gas is shown in Fig. 2.2. Also shown
in the figure is the ideal gas line (fi ¼ pi) obtained by extrapolating
the limiting slope at pi ¼ 0. The standard state of a gas is then
defined as the point at pi ¼ 1.0 bar on the ideal gas line. Note that
this is a hypothetical standard state. That is, at pi ¼ 1.0 bar, a real
gas is not in its standard state.
2.0
1.8
1.6
al
Re
Fugacity fi (bar)
1.4
1.2
Id
=
l (f i
ea
p )i
1.0
Standard state
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Pressure pi (bar)
Fig. 2.2 Fugacity versus partial pressure for a gas (schematic).
1.2
1.4
21
Chapter 2 THERMODYNAMICS FUNDAMENTALS
For a chemical reaction as in Eq. (2.35), ΔGo is the Gibbs energy
change when all gases are in their standard states. Hence, in the
case of real gases, Eqs. (2.36), (2.37) apply if all the partial pressures pi are replaced by their corresponding fugacities fi.
Tables or equations giving fi as a function of T and P for pure gases
are obtained from their experimental equations of state. However,
care must be exercised in substituting these values into the equilibrium constant equation because the fugacity of a gas depends
not only on T and pi but also upon its interactions with all the
other gases that are present.
2.4.3
Predominance Diagrams
Predominance diagrams are a particularly simple type of phase
diagram that have many applications in the fields of hot corrosion, chemical vapor deposition, etc. Furthermore, their construction clearly illustrates the principles of Gibbs energy
minimization.
A predominance diagram for the Cu-SO2-O2 system at 700°C is
shown in Fig. 2.3. The axes are the logarithms of the partial pressures
of SO2 and O2 in the gas phase. The diagram is divided into areas or
domains of stability of the various solid compounds of Cu, S, and O.
700∞C
3
Cu2SO4
2
1
CuS
CuSO4
Cu2S
0
Cu2O
–1
Ptotal = 1.0 bar
–2
5
SO
u2
C
log10p(SO2) (bar)
22
Point Z
–7
Cu
–3
–4
–20 –18
pO = 10
–2
pSO = 10
2
CuO
2
–16
–14
–12 –10 –8
log10p(O2) (bar)
–6
–4
–2
0
Fig. 2.3 “Predominance” phase diagram of the Cu-SO2-O2 system at 700°C
showing the solid phases at equilibrium as a function of the equilibrium SO2 and O2
partial pressures (FactSage).
Chapter 2 THERMODYNAMICS FUNDAMENTALS
For example, at point Z, where pSO2 ¼ 102 and pO2 ¼ 107 bar, the
stable phase is Cu2O. The conditions for coexistence of two and three
solid phases are indicated, respectively, by the lines and triple points
on the diagram. For example, along the line separating the Cu2O and
2 3/2
p
of the
CuSO4 domains, the equilibrium constant K ¼ pSO
2 O2
following reaction is satisfied:
Cu2 O + 2SO2 + 3=2 O2 ¼ 2CuSO4
(2.44)
Hence, along this line,
log K ¼ ΔG o =RT ¼ 2 log pSO2 3=2 log pO2 ¼ constant (2.45)
This line is thus straight with slope (3/2)/2 ¼ 3/4.
We can construct Fig. 2.3 following the procedure of Bale et al.
(1986). We formulate a reaction for the formation of each solid
phase, always from 1.0 mol of Cu and involving the gaseous species whose pressures are used as the axes (SO2 and O2 in this
example)
1=2
Cu + 0:5O2 ¼ CuO ΔG ¼ ΔG o + RT ln pO2
1=4
Cu + 0:25O2 ¼ 0:5Cu2 O ΔG ¼ ΔG o + RT ln pO2
Cu + SO2 ¼ CuS + O2 ΔG ¼ ΔG o + RT ln pO2 p1
SO2
1
Cu + SO2 + O2 ¼ CuSO4 ΔG ¼ ΔG o + RT ln p1
SO2 pO2
(2.46)
(2.47)
(2.48)
(2.49)
and similarly for the formation of Cu2S, Cu2SO4, and Cu2SO5.
The values of ΔGo are obtained from tables of thermodynamic
properties. For any given values of pSO2 and pO2, ΔG for each formation reaction can then be calculated. The stable compound is
simply the one with the most negative ΔG. If all the ΔG values are
positive, then pure Cu is the stable compound.
By reformulating Eqs. (2.46)–(2.49) in terms of, for example, S2
and O2 rather than SO2 and O2, a predominance diagram with log
pS2 and log pO2 as axes can be constructed. Logarithms of ratios or
products of partial pressures can also be used as axes.
Along the curve shown by the crosses in Fig. 2.3, the total pressure is 1.0 bar. That is, the gas consists not only of SO2 and O2 but
also of other species such as S2 whose equilibrium partial pressures can be calculated. Above this line, the total pressure is
>1.0 bar even though the sum of the partial pressures of SO2
and O2 may be <1.0 bar. Caution must therefore be exercised in
using such diagrams. If the total pressure is >1.0 bar, states above
the Ptotal ¼ 1.0 bar line in Fig. 2.3 are inaccessible. That is, the
23
Chapter 2 THERMODYNAMICS FUNDAMENTALS
0
–50
RT ln p(O2) (kJ)
24
Cu2O(liq)
CuO(sol)
–100
M
Cu2O(sol)
–150
M
–200
Cu(liq)
–250
Cu(sol)
–300
–350
0
200
400
600
800
1000
1200
1400
1600
Temperature (∞C)
Fig. 2.4 Predominance phase diagram (also known as a Gibbs energy versus T
diagram, or an Ellingham diagram) for the Cu-O2 system. Points M and M are the
melting points of Cu and Cu2O, respectively (FactSage).
calculated diagram for a total pressure of 1.0 bar terminates at
this line.
A similar phase diagram of RT ln pO2 versus T for the Cu-O2
system is shown in Fig. 2.4. For the formation reaction
2Cu + ½ O2 ¼ Cu2 O
(2.50)
we can write
0:5
ΔG o ¼ RT ln K ¼ RT ln ðpO2 Þequilibrium
¼ ΔH o T ΔSo
(2.51)
The line between the Cu and Cu2O domains in Fig. 2.4 is thus a
plot of twice the standard Gibbs energy of formation of Cu2O versus T. The temperatures indicated by the symbols M and M are the
melting points of Cu and Cu2O, respectively. This line is thus simply a line taken from the well-known Ellingham diagram, or ΔGo
versus T diagram, for the formation of oxides. However, by drawing vertical lines at the melting points of Cu and Cu2O as shown in
Fig. 2.4, we convert the Ellingham diagram to a phase diagram.
Stability domains for Cu(sol), Cu(liq), Cu2O(sol), Cu2O(liq), and
CuO(sol) are shown as functions of T and of the imposed equilibrium pO2. The lines and triple points indicate conditions for twoand three-phase equilibria.
Chapter 2 THERMODYNAMICS FUNDAMENTALS
2.5
2.5.1
Measuring Gibbs Energy, Enthalpy and
Entropy
Measuring Gibbs Energy Change
In general, the Gibbs energy change of a process can be determined by performing measurements under equilibrium conditions. Consider the reaction in Eq. (2.38), for example.
Experimentally, equilibrium is established between the solid
C and the gas phase, and the equilibrium constant is then found
by measuring the partial pressures of CO and CO2. Eq. (2.39) is
then used to calculate ΔGo.
Another common technique of establishing equilibrium is
by means of an electrochemical cell. A galvanic cell is designed
so that a reaction (the overall cell reaction) can occur only
with the passage of an electric current through an external circuit.
An external voltage is applied that is equal in magnitude
but opposite in sign to the cell potential, E(volts), such that no
current flows. The system is then at equilibrium. With the passage
of dz coulombs through the external circuit, Edz joules of
work are performed. Since the process occurs at equilibrium, this
is the reversible non-PV work, dWrev(non-PV). Substituting
Eq. (2.11) into Eq. (2.21), differentiating and substituting
Eq. (2.10) give
X
dG ¼ V dP SdT +
μi dni + dWrevðnonPV Þ
(2.52)
Hence, for a closed system at constant T and P,
dGT,P,n ¼ dWrevðnonPV Þ
(2.53)
If the cell reaction is formulated such that the advancement of the
reaction by dξ moles is associated with the passage of ndz coulombs through the external circuit, then from Eq. (2.53),
ΔG dξ ¼ Endz
(2.54)
ΔG ¼ nFE
(2.55)
Hence,
where ΔG is the Gibbs energy change of the cell reaction;
F ¼ dz/dξ ¼ 96,500 C mol1 is the Faraday constant, which is the
number of coulombs in a mole of electrons; and the negative sign
follows the usual electrochemical sign convention. Hence, by
measuring the open-circuit voltage E, one can determine ΔG of
the cell reaction.
Eq. (2.55) is called the Nernst equation.
25
26
Chapter 2 THERMODYNAMICS FUNDAMENTALS
2.5.2
Measuring Enthalpy Change
The enthalpy change, ΔH, of a process may be measured
directly by calorimetry. Alternatively, if ΔG has been measured
by an equilibration technique as in Section 2.5.1, then ΔH can
be determined from the measured temperature dependence of
ΔG. From Eq. (2.52), for a process in a closed system with no
non-PV work at constant pressure,
ðdG=dT ÞP, n ¼ S
(2.56)
Substitution of Eq. (2.21) into Eq. (2.56) gives
ðdðG=T Þ=dð1=T ÞÞP,n ¼ H
(2.57)
ðdH=dT ÞP,n ¼ T ðdS=dT ÞP, n
(2.58)
Eqs. (2.56)–(2.58) apply to both the initial and final states of a process. Hence,
ðdðΔGÞ=dT ÞP, n ¼ ΔS
(2.59)
ðdðΔG=T Þ=dð1=T ÞÞP, n ¼ ΔH
(2.60)
dððΔH Þ=dT ÞP,n ¼ T ðdðΔSÞ=dT ÞP,n
(2.61)
Eqs. (2.56)–(2.61) are all forms of the Gibbs-Helmholtz equation.
2.5.3
Measuring Entropy—The Third Law of
Thermodynamics
As is the case with the enthalpy, ΔS of a process can be determined from the measured temperature dependence of ΔG by
means of Eq. (2.59). This is known as the second law method of
measuring entropy.
Another method of measuring entropy involves the third law
of thermodynamics that states that the entropy of a perfect crystal
of a pure substance at internal equilibrium at a temperature of 0 K
is zero. Note that the third law is not a convention (like the convention regarding “absolute” enthalpies in Section 2.2.1). Rather,
it follows from Eq. (2.2) and the concept of entropy as disorder. For
a perfectly ordered system at absolute zero, t ¼ 1, and S ¼ 0.
We shall first derive an expression for the change in entropy as
a substance is heated. Substituting Eq. (2.19) into Eq. (2.58) and
integrating gives the following expression for the entropy change
as 1.0 mol of a substance is heated at constant P from T1 to T2,
assuming no phase changes in the interval:
Chapter 2 THERMODYNAMICS FUNDAMENTALS
Tð2
ðsT2 sT1 Þ ¼
ðcP =T ÞdT
(2.62)
T1
The similarity to Eq. (2.19) is evident, and a plot of (sT2 sT1) versus T is similar in appearance to Fig. 2.1 with discontinuities at
each first-order transition point equal to ΔsTotr ¼ ΔhTotr/TTotr where
ΔhTotr, ΔsTotr, and TTotr are the standard enthalpy, entropy, and temperature of transition, respectively (see Eqs. 2.28, 2.29). So, for
example, the entropy change upon heating 1.0 mol of Fe from
298.15 K to a temperature T above the melting point is given as
(cf. Fig. 2.1 and Eq. 2.20)
sTo
o
s298:15
1184:8
ð
¼
cPα =T dT + Δhoα!γ =1184:8
298:15
1667:5
ð
γ cP =T dT + Δhoγ!δ =1667:5
+
1184:8
1811
ð
+
cPδ =T dT + Δhof =1811 +
1667:5
ðT cPliq =T dT
1811
(2.63)
Setting T1 ¼ 0 in Eq. (2.62) and using the third law gives
Tð2
sT 2 ¼
ðcP =T ÞdT
(2.64)
0
Hence, if cP has been measured down to temperatures sufficiently
close to 0 K, Eq. (2.64) can be used to calculate the absolute entropy.
This is known as the third law method of measuring entropy.
Note that the third law method permits the measurement of
the absolute entropy of a substance, whereas the second law
method permits the measurement only of entropy changes ΔS.
2.6
Gibbs Energy of a Pure Compound as a
Function of Temperature
From Eq. (2.21), the molar Gibbs energy upon heating a pure
substance at constant P from 298 K to a temperature T is
o
o
o
o
¼ hT TsoT ho298 298s298
gT g298
o
o
(2.65)
ðT 298Þs298
¼ hoT ho298 T sTo s298
27
28
Chapter 2 THERMODYNAMICS FUNDAMENTALS
where (hoT ho298) and (soT so298) are given by equations like
Eqs. (2.20), (2.63).
2.7
Auxiliary Functions
It was shown in Section 2.3 that Eq. (2.25) is a special form of
the Second Law for a process occurring at constant T and P. Other
special forms of the Second Law can be derived in a
similar manner.
It follows directly from Eqs. (2.3), (2.12), (2.24), for a process at
constant V and S, that
dUV, S,n ¼ T dStotal 0
(2.66)
Similarly, from Eqs. (2.3), (2.13), (2.24),
dHP,S,n ¼ T dStotal 0
(2.67)
Eqs. (2.66), (2.67) are not of direct practical use since there is no
simple means of maintaining the entropy of a system constant
during a change of state. A more useful function is the Helmholtz
energy, A, defined as
A ¼ U TS
(2.68)
dAn ¼ dQ P dV T dS SdT
(2.69)
From Eqs. (2.12), (2.68),
Then, from Eqs. 2.3 and 2.24, for processes at constants T and V,
dAT,V,n ¼ T dStotal 0
(2.70)
Eqs. (2.25), (2.66), (2.67), (2.70) are special forms of the second law
for closed systems. In this regard, H, A, and G are sometimes called
auxiliary functions. If we also consider open systems, then many
other auxiliary functions can be defined as will be shown in
Section 2.9.2.
In Section 2.3, the very important result was derived that the
equilibrium state of a system corresponds to a minimum of the
Gibbs energy G. This is used as the basis of most algorithms for
calculating chemical equilibria and phase diagrams. An objection
might be raised that Eq. (2.25) only applies at constant T and P.
What if equilibrium were approached at constant T and V, for
instance? Should we not then minimize A as in Eq. (2.70)? The
response is that the restrictions on Eqs. (2.25), (2.70) only apply
to the paths followed during the approach to equilibrium. However,
for a system at an equilibrium temperature, pressure, and volume, it
clearly does not matter along which hypothetical path the approach
to equilibrium occurred. The reason that most algorithms minimize G rather than other auxiliary functions is simply that it is
Chapter 2 THERMODYNAMICS FUNDAMENTALS
generally easiest and most convenient to express thermodynamic
properties in terms of the independent variables T, P, and ni.
2.8
The Chemical Potential
The chemical potential was defined in Eq. (2.5). However, this
equation is not particularly useful directly since it is difficult to see
how S can be held constant.
Substituting Eq. (2.11) into Eq. (2.21), differentiating and
substituting Eq. (2.9) gives
X
(2.71)
dG ¼ V dP SdT +
μi dni
from which it can be seen that μi is also given by
μi ¼ ð∂G=∂ni ÞT, P, nj6¼i
(2.72)
Differentiating Eq. (2.11) with substitution of Eq. (2.9) gives
X
dH ¼ T dS + V dP +
μi dni
(2.73)
whence
μi ¼ ð∂H=∂ni ÞT, V, nj6¼i
(2.74)
Similarly, it can be shown that
μi ¼ ð∂A=∂ni ÞT, V, nj6¼i
(2.75)
Among these equations for μi, the most useful is Eq. (2.72). The
chemical potential of a component i of a system is given by the
increase in Gibbs energy as the component is added to the system
at constant T and P while keeping the number of moles of every
other component of the system constant.
The chemical potentials of its components are intensive properties of a system; for a system at a given T, P, and composition,
the chemical potentials are independent of the mass of the system.
That is, with reference to Eq. (2.72), adding dni moles of component
i to a solution of a fixed composition will result in a change in Gibbs
energy dG that is independent of the total mass of the solution.
The reason that this property is called a chemical potential is
illustrated by the following thought experiment. Imagine two systems, I and II, at the same temperature and pressure and separated by a membrane that permits the passage of only one
component, say component 1. The chemical potentials of component 1 in systems I and II are μl1 ¼ ∂ Gl/∂ nl1 and μll1 ¼ ∂ Gll/∂ nll1. Component 1 is transferred across the membrane with dnI1 ¼ dnII
1.
The change in the total Gibbs energy accompanying this transfer
is then
(2.76)
dG ¼ d GI + G II ¼ μI1 μII1 dnII1
29
30
Chapter 2 THERMODYNAMICS FUNDAMENTALS
If μl1 > μll1, then d(GI +GII) is negative when dnll1 is positive. That is,
the total Gibbs energy is decreased by a transfer of component 1
from system I to system II. Hence, component 1 is transferred
spontaneously from a system of higher μ1 to a system of lower μ1.
For this reason, μ1 is called the chemical potential of component 1.
Similarly, by using the appropriate auxiliary functions, it can be
shown that μi is the potential for the transfer of component i at
constant T and V, at constant S and V, etc.
An important principle of phase equilibrium can now be
stated. When two or more phases are in equilibrium, the chemical
potential of any component is the same in all phases. This criterion
for phase equilibrium is thus equivalent to the criterion of minimizing G. This will be discussed further in Section 5.1.
2.9
Some Other Useful Thermodynamic
Equations
In this section, some other useful thermodynamic relations are
derived. Consider an open system that is initially empty. That is,
initially, S, V, and n are all equal to zero. We now “fill up” the system
by adding a previously equilibrated mixture. The addition is made
at constant T, P, and composition. Hence, it is also made at constant μi (of every component), since μi is an intensive property.
The addition is continued until the entropy, volume, and number
of moles of each of the components in the system are S, V, and ni.
Integrating Eq. (2.9) from the initial (empty) to the final state then
gives.
X
(2.77)
U ¼ TS PV +
μi ni
It then follows that
H ¼ TS +
X
ni μi
X
A ¼ PV +
ni μi
and most importantly,
G¼
2.9.1
(2.78)
(2.79)
X
n i μi
(2.80)
The Gibbs-Duhem Equation
By differentiation of Eq. (2.77),
dU ¼ T dS + SdT PdV V dP +
X
ni dμi +
X
μi dni
(2.81)
Chapter 2 THERMODYNAMICS FUNDAMENTALS
By comparison of Eqs. (2.9), (2.81), it follows that
X
SdT V dP +
ni dμi ¼ 0
(2.82)
This is the general Gibbs-Duhem equation.
2.9.2
General Auxiliary Functions
It was stated in Section 2.7 that many auxiliary functions can
be defined for open systems. Consider, for example, the auxiliary
function:
G 0 ¼ U ðPV Þ TS k
X
ni μi ¼ G 1
k
X
ni μ i
(2.83)
1
Differentiating Eq. (2.83) with substitution of Eq. (2.71) gives
dG 0 ¼ V dP SdT k
X
1
ni dμi +
C
X
μi dni
(2.84)
k+1
where C is the number of components in the system. It then follows that
0
dGT,P,μ
<0
i ð1ik Þ, ni ðk + 1iC Þ
(2.85)
is the criterion for a spontaneous process at constant T, P, μi (for
1 i k), and ni (for k + 1 i C) and that equilibrium is
achieved when d G0 ¼ 0. The chemical potential of a component
may be held constant, for example, by equilibration with a gas
phase as, for example, by fixing the partial pressure of O2, SO2, etc.
Finally, it also follows that the chemical potential may be
defined very generally as
μi ¼ ð∂G 0 =∂ni ÞT, P, μi ð1ikÞ,ni ðk + 1iC Þ
(2.86)
The discussion of generalized auxiliary functions will be continued in Section 7.7.
Reference
Bale, C.W., Pelton, A.D., Thompson, W.T., 1986. Can. Metall. Q. 25, 107–112.
List of Websites
FactSage, Montreal, www.factsage.com.
31
3
THE GIBBS PHASE RULE
CHAPTER OUTLINE
3.1 The Phase Rule and Binary Temperature-Composition Phase
Diagrams 35
3.1.1 Three-Phase Invariants in Binary Temperature-Composition
Phase Diagrams 37
3.2 Other Examples of Applications of the Phase Rule 38
Reference 40
The geometry of all types of phase diagrams is governed by the
Gibbs Phase Rule that applies to any system at equilibrium:
F ¼C P +2
(3.1)
The phase rule will be derived in this section in a general form
along with several examples. Further examples will be presented
in later sections.
In Eq. (3.1), C is the number of components in the system. This
is the minimum number of substances required to describe the
elemental composition of every phase present. As an example,
for pure H2O at 25°C and 1.0 bar pressure, C ¼ 1. For a gaseous
mixture of H2, O2, and H2O at high temperature, C ¼ 2.
(Although there are many species present in the gas including
O2, H2, H2O, and O3, these species are all at equilibrium with each
other. At a given T and P, it is sufficient to know the overall elemental composition in order to be able to calculate the concentration
of every species. Hence, C ¼ 2. C can never exceed the number of
elements in the system.)
For the systems shown in Figs. 1.1–1.7 respectively, C ¼ 2, 3, 3,
4, 3, 4, and 1.
There is generally more than one way to specify the components of a system. For example, the Fe-Si-O system could also
be described as the FeO-Fe2O3-SiO2 system, the FeO-SiO2-O2 system, etc. The formulation of a phase diagram can often be
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00003-8
# 2019 Elsevier Inc. All rights reserved.
33
34
Chapter 3 THE GIBBS PHASE RULE
simplified by a judicious definition of the components as will be
discussed in Section 7.9.
In Eq. (3.1), P is the number of phases present at equilibrium at
a given point on the phase diagram. For example, in Fig. 1.1, at any
point within the region labeled “liquid + bcc,” P ¼ 2. At any point
in the region labeled “liquid,” P ¼ 1.
In Eq. (3.1), F is the number of degrees of freedom or the variance of the system at a given point on the phase diagram. It is the
number of variables that must be specified in order to completely
specify the state of the system. Let us number the components as
1, 2, 3, …, C and designate the phases as α, β, γ, …. In most derivations of the phase rule, the only variables considered are T, P,
and the compositions of the phases. However, since we also want
to treat more general phase diagrams in which chemical potentials, volumes, enthalpies, etc. can also be variables, we shall
extend the list of variables as follows:
T ,P
β β
β
α
, X1 , X2 , …, XC1
,…
X1α , X2α , …, XC1
μ1 ,μ2 ,μ3 , …
V α , V β ,V γ ,…
Sα , Sβ ,Sγ ,…
where Xαi is the mole fraction of component i in phase α, μi is the
chemical potential of component i, and Vα and Sα are the volume
and entropy of phase α. Alternatively, we may substitute mass
fractions for mole fractions and enthalpies Hα, Hβ, … for Sα, Sβ, ….
It must be stressed that the overall composition of the system is
not a variable in the sense of the phase rule. The composition variables are the compositions of the individual phases.
The total number of variables is thus equal to (2 +P(C 1) +
C + 2P).
(Only (C 1) independent composition variables Xi are
required for each phase since the sum of the mole fractions is
unity.)
At equilibrium, the chemical potential μi of each component i
is the same in all phases (see Section 2.8) as are T and P. μi is an
intensive variable (see Section 2.8) that is a function of T, P, and
composition. Hence,
(3.2)
μαi T , P, X1α , X2α , …, ¼ μβi T , P, X1β , X2β , …, ¼ ⋯ ¼ μi
This yields PC equations relating the variables. Furthermore,
the volume of each phase is a function of T, P, and composition, Vα ¼ (T, P, Xα1, Xα2, …, ), as is the entropy of the phase
Chapter 3 THE GIBBS PHASE RULE
(or, alternatively, the enthalpy). This yields 2P additional equations relating the variables.
F is then equal to the number of variables minus the number of
equations relating the variables at equilibrium:
F ¼ ð2 + P ðC 1Þ + C + 2P Þ PC 2P ¼ C P + 2
(3.3)
Eq. (3.1) is thereby derived.
3.1
The Phase Rule and Binary
Temperature-Composition Phase
Diagrams
As a first example of the application of the phase rule, consider
the temperature-composition (T-X) phase diagram of the Bi-Sb
system at a constant applied pressure of 1.0 bar in Fig. 3.1. The
abscissa of Fig. 3.1 is the composition, expressed as mole fraction
of Sb, XSb. Note that XBi ¼ 1 XSb. Phase diagrams are also often
drawn with the composition axis expressed as weight percent.
At all compositions and temperatures in the area above the line
labeled “liquidus,” a single-phase liquid solution will be observed,
while at all compositions and temperatures below the line labeled
“solidus,” there will be a single-phase solid solution. A sample at
equilibrium at a temperature and overall composition between
700
o
630.6
Temperature (∞C)
600
Liquid
526o
0.60
Liquidus line
B
A
500
0.37
o
450
0.89
(Liquid + Solid)
0.80
P
R
Q
349o
D
Solidus line
400
0.16
C
0.60
300
Solid
o
271.4
200
0
Bi
0.2
0.4
0.6
Mole fraction XSb
0.8
1
Sb
Fig. 3.1 Temperature-composition phase diagram at P ¼ 1 bar of the Bi-Sb system
(Pelton, 2014).
35
36
Chapter 3 THE GIBBS PHASE RULE
these two curves will consist of a mixture of solid and liquid
phases, the compositions of which are given by the liquidus
and solidus compositions at that temperature. For example, a
sample of overall composition XSb ¼ 0.60 at T ¼ 450°C (at point
R in Fig. 3.1) will consist, at equilibrium, of a mixture of a liquid
of composition XSb ¼ 0.37 (point P) and solid of composition
XSb ¼ 0.80 (point Q). The line PQ is called a tie-line or conode.
Binary temperature-composition phase diagrams are generally
plotted at a fixed pressure, usually 1.0 bar. This eliminates one
degree of freedom. In a binary system, C ¼ 2. Hence, for binary
isobaric T-X diagrams, the phase rule reduces to
F ¼3P
(3.4)
Binary T-X diagrams contain single-phase areas and two-phase
areas. In the single-phase areas, F ¼ 3 1 ¼ 2. That is, temperature
and composition can be specified independently. These regions
are thus called bivariant. In two-phase regions, F ¼ 3 2 ¼ 1. If,
say, T is specified, then the compositions of both phases are determined by the ends of the tie-lines. Two-phase regions are thus
termed univariant. Note that although the overall composition
can vary over a range within a two-phase region at constant T,
the overall composition is not a parameter in the sense of the
phase rule. Rather, it is the compositions of the individual phases
at equilibrium that are the parameters to be considered in counting the number of degrees of freedom.
As the overall composition is varied at 450°C between points
P and Q, the compositions of the liquid and solid phases remain
fixed at P and Q, and only the relative proportions of the two
phases change. From a simple mass balance, we can derive the
lever rule for binary systems: (moles of liquid)/(moles of solid) ¼
QR/PR where QR and PR are the lengths of the line segments.
Hence, at 450°C, a sample with overall composition XSb ¼ 0.60
consists of liquid and solid phases in the molar ratio
(0.80–0.60)/(0.60–0.37) ¼ 0.87. Were the composition axis
expressed as weight percent, then the lever rule would give the
weight ratio of the two phases.
Suppose that a liquid Bi-Sb solution with composition
XSb ¼ 0.60 is cooled very slowly from an initial temperature above
526°C. When the temperature has decreased to the liquidus temperature 526°C (point A), the first solid appears, with a composition at point B (XSb ¼ 0.89). As the temperature is decreased
further, solid continues to precipitate with the compositions of
the two phases at any temperature being given by the liquidus
and solidus compositions at that temperature and with their relative proportions being given by the lever rule. Solidification is
Chapter 3 THE GIBBS PHASE RULE
complete at 349 °C, the last liquid to solidify having composition
XSb ¼ 0.16 (point C).
The process just described is known as equilibrium cooling. At
any temperature during equilibrium cooling, the solid phase has a
uniform (homogeneous) composition. In the preceding example,
the composition of the solid phase during cooling varies along the
line BQD. Hence, in order for the solid grains to have a uniform
composition at any temperature, diffusion of Sb from the center
to the surface of the growing grains must occur. Since solid-state
diffusion is a relatively slow process, equilibrium cooling conditions are only approached if the temperature is decreased very
slowly. If a sample of composition XSb ¼ 0.60 is cooled rapidly
from the liquid, concentration gradients will be observed in the
solid grains, with the concentration of Sb decreasing toward the
surface from a maximum of XSb ¼ 0.89 (point B). Furthermore,
in this case, solidification will not be complete at 349°C since at
349°C, the average concentration of Sb in the solid grains will
be greater than XSb ¼ 0.60. These considerations are discussed
more fully in Chapter 8.
At XSb ¼ 0 and XSb ¼ 1 in Fig. 3.1, the liquidus and solidus curves
meet at the equilibrium melting points or temperatures of fusion
of Bi and Sb, which are Tof(Bi) ¼ 271.4°C and Tof(Sb) ¼ 630.6°C. At
these two points, the number of components is reduced to 1.
Hence, from the phase rule, the two-phase region becomes invariant (F ¼ 0), and solid and liquid coexist at the same temperature.
The phase diagram is influenced by the total pressure, P. Unless
otherwise stated, T-X diagrams are usually presented for
P ¼ constant ¼ 1.0 bar. For equilibriums involving only solid and
liquid phases, the phase boundaries are typically shifted only by
the order of a few hundredths of a degree per bar change in P.
Hence, the effect of pressure upon the phase diagram is generally
negligible unless the pressure is of the order of hundreds of bars.
On the other hand, if gaseous phases are involved, the effect of
pressure is very important. The effect of pressure will be discussed
in Section 5.2.
3.1.1
Three-Phase Invariants in Binary
Temperature-Composition Phase Diagrams
When three phases are at equilibrium in a binary system at
constant pressure, then from Eq. (3.4), F ¼ 0. Hence, the compositions of all three phases and T are fixed, and the system is said to
be invariant. An example is the line PQR in Fig. 1.1 in the T-X
37
38
Chapter 3 THE GIBBS PHASE RULE
diagram of the Fe-Mo system. At any overall composition of the
system lying between points P and R at 1612°C, three phases will
be in equilibrium: liquid, sigma, and bcc, with compositions at
points P, Q, and R, respectively. Several other invariants can also
be seen in Fig. 1.1. Binary T-X diagrams and invariant reactions
will be discussed in detail in Chapter 5.
3.2
Other Examples of Applications
of the Phase Rule
Despite its apparent simplicity, the phase rule is frequently
misunderstood and incorrectly applied. In this section, a few
additional illustrative examples of the application of the phase
rule will be briefly presented. These examples will be elaborated
upon in subsequent sections.
In the single-component Al2SiO5 system (C ¼ 1) shown in
Fig. 1.7, the phase rule reduces to
F ¼3P
(3.5)
Three allotropes of Al2SiO5 are shown in the figure. In the singlephase regions, F ¼ 2. In these bivariant regions, both P and T
(Fig. 1.7A) or both V and T (Fig. 1.7B) can be specified independently. When two phases are in equilibrium, F ¼ 1. Two-phase
equilibrium is therefore represented by the univariant lines in
the P–T phase diagram in Fig. 1.7A; when two phases are in equilibrium, T and P cannot be specified independently. When all
three allotropes are in equilibrium, F ¼ 0. Three-phase equilibrium can thus only occur at the invariant triple point (530°C,
3.859 kbar) in Fig. 1.7A.
The invariant triple point in Fig. 1.7A becomes the invariant
line PQR at T ¼ 530°C in the V-T phase diagram in Fig. 1.7B. When
the three phases are in equilibrium, the total molar volume of the
system can lie anywhere on this line depending upon the relative
amounts of the three phases present. However, the molar volumes
of the individual allotropes are fixed at points P, Q, and R. It is the
molar volumes of the individual phases that are the system variables in the sense of the phase rule. Similarly, the two-phase univariant lines in Fig. 1.7A become the two-phase univariant areas in
Fig. 1.7B. The lever rule can be applied in these regions to give the
ratio of the volumes of the two phases at equilibrium.
As another example, consider the phase diagram section for
the ternary (C ¼ 3) Zn-Mg-Al system shown in Fig. 1.3A. Since
the total pressure is constant, one degree of freedom is eliminated.
Consequently,
Chapter 3 THE GIBBS PHASE RULE
F ¼4P
(3.6)
Although the diagram is plotted at constant 10 mol% Zn, this is a
composition variable of the entire system, not of any individual
phase. That is, it is not a variable in the sense of the phase rule,
and setting it constant does not reduce the number of degrees
of freedom. Hence, for example, the three-phase regions in
Fig. 1.3A are univariant (F ¼ 1); at any given T, the compositions
of all three phases are fixed, but the three-phase region extends
over a range of temperature. Ternary temperature-composition
phase diagrams will be discussed in detail in Chapter 6.
Finally, we shall examine some diagrams with chemical potentials as variables. As a first example, the phase diagram of the
Fe-Ni-O2 system in Fig. 1.5 is a plot of the equilibrium oxygen
pressure (fixed, e.g., by equilibrating the solid phases with a gas
phase of fixed pO2) versus the molar ratio Ni/(Fe + Ni) at constant
T and constant total hydrostatic pressure. For an ideal gas from
Eq. (2.34),
μi ¼ μoi + RT ln pi
(3.7)
μoi
where
is the standard-state (at pi ¼ 1.0 bar) chemical potential
of the pure gas at temperature T. Hence, if T is constant, μi varies
directly as ln pi. The y-axis in Fig. 1.5 thus varies directly as μO2.
Fixing T and total hydrostatic pressure eliminates two degrees
of freedom. Hence, F ¼ 3 P. Therefore, Fig. 1.5 has the same
topology as a binary T-X diagram like Fig. 1.1, with single-phase
bivariant areas, two-phase univariant areas with tie-lines, and
three-phase invariant lines at fixed log pO2 and fixed compositions
of the three phases.
As a second example, in the isothermal predominance diagram
of the Cu-SO2-O2 system in Fig. 2.3, the x- and y-axis variables vary
as μO2 and μSO2. Since C ¼ 3 and T is constant,
F ¼4P
(3.8)
The diagram has a topology similar to the P-T phase diagram of a
one-component system as in Fig. 1.7A. Along the lines of the diagram, two solid phases are in equilibrium, and at the triple points,
three solid phases coexist. However, a gas phase is also present
everywhere. Hence, at the triple points, there are really four
phases present at equilibrium, while the lines represent threephase equilibriums. Hence, the diagram is consistent with
Eq. (3.8).
It is instructive to look at Fig. 2.3 in another way. Suppose that
we fix a constant hydrostatic pressure of 1.0 bar (e.g., by placing
the system in a cylinder fitted with a piston). This removes one
39
40
Chapter 3 THE GIBBS PHASE RULE
further degree of freedom, so that now F ¼ 3 P. However, as long
as the total equilibrium gas pressure is less than 1.0 bar, there will
be no gas phase present, and so, the diagram is consistent with the
phase rule. On the other hand, above the 1.0 bar curve shown by
the line of crosses on Fig. 2.3, the total gas pressure exceeds
1.0 bar. Hence, the phase diagram calculated at a total hydrostatic
pressure of 1.0 bar must terminate at this curve.
The phase rule may be applied to the four-component isothermal Fe-Cr-S2-O2 phase diagram in Fig. 1.6 in a similar way.
Remember that fixing the molar ratio Cr/(Fe + Cr) equal to 0.5
does not eliminate a degree of freedom since this composition
variable applies to the entire system, not to any individual phase.
Reference
Pelton, A.D., 2014. In: Laughlin, D.E., Hono, K. (Eds.), (Chapter 3). Physical Metallurgy. fifth ed. Elsevier, New York.
4
FUNDAMENTALS OF THE
THERMODYNAMICS OF
SOLUTIONS
CHAPTER OUTLINE
4.1 Gibbs Energy of Mixing 41
4.2 Tangent Construction 43
4.3 Partial Molar Properties 43
4.4 Relative Partial Molar Properties 44
4.4.1 A Note on Standard States 45
4.5 Activity 46
4.6 Ideal Raoultian Solutions 46
4.7 Excess Properties 48
4.8 Activity Coefficients 49
4.9 Regular Solution Theory 50
4.10 Multicomponent Solutions 52
References 52
List of Websites 52
4.1
Gibbs Energy of Mixing
Liquid gold and copper are completely miscible at all compositions. The Gibbs energy of 1 mol of liquid solution, g, at 1100°C is
drawn in Fig. 4.1 as a function of composition expressed as mole
fraction XCu of copper. (Note that XAu ¼ 1 XCu.) The curve of g
varies between the standard molar Gibbs energies of pure liquid
Au and Cu, goAu and goCu. From Eq. (2.72), it follows that, for a pure
component, goi and μoi are identical.
The function Δgm shown in Fig. 4.1 is called the molar Gibbs
energy of mixing of the liquid solution. It is defined as
(4.1)
Δgm ¼ g XAu μoAu + XCu μoCu
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00004-X
# 2019 Elsevier Inc. All rights reserved.
41
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
ΔmCu = RT ln aCu
Gibbs energy
(Joules mol−1)
° (= g°Cu)
mCu
Δgm
ΔmAu = RT ln aAu
° )
m°Au(= gAu
g
mCu
mAu
0
Au
0.2
0.4
0.6
0.8
Mole fraction XCu
1
Cu
Fig. 4.1 Molar Gibbs energy of liquid Au-Cu alloys at 1100°C illustrating the tangent
construction (Pelton, 2014).
It can be seen that Δgm is the Gibbs energy change associated with
the isothermal mixing of XAu moles of pure liquid Au and XCu moles
of pure liquid Cu to form 1 mol of solution according to the reaction:
XAu AuðliqÞ + XCu Cu ðliqÞ ¼ 1 mole liquid solution
(4.2)
Note that for the solution to be stable, it is necessary that Δgm be
negative. A plot of Δgm versus composition for the Au-Cu system
at 1100°C is shown in Fig. 4.2.
0
–2000
–4000
ΔhAu
E
Δhm = hE g
–6000
Joules
42
–8000
–TΔsAu
ΔgAu
Δgid
–TΔsm
–10,000
ΔhCu
–TΔsCu
–12,000
Δgm
–14,000
–16,000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction Cu
0.7
0.8
0.9
1.0
Fig. 4.2 Integral and partial molar properties in liquid Au-Cu solutions at 1100°C.
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
4.2
Tangent Construction
An important construction is illustrated in Fig. 4.1. If a tangent
is drawn to the curve of g at a certain composition (XCu ¼ 0.4 in
Fig. 4.1), then the intercepts of this tangent on the axes at XAu ¼ 1
and XCu ¼ 1 are equal to μAu and μCu, respectively, at this
composition.
To prove this, take Eq. (2.80) for a binary system A-B and divide
by (nA + nB):
g ¼ XA μA + XB μB
(4.3)
dg ¼ ðXA dμA + XB dμB Þ + ðμA dXA + μB dXB Þ
(4.4)
Differentiation gives
Writing the Gibbs-Duhem equation, Eq. (2.82), at constant
T and P for a binary system and dividing by (nA + nB) gives
(XAdμA + XBdμB) ¼ 0.
Comparison with Eq. (4.4) then gives
dg ¼ μA dXA + μB dXB
(4.5)
Since dXA ¼ dXB, it can be seen that Eqs. (4.3), (4.5) are equivalent to the tangent construction shown in Fig. 4.1.
These equations may also be rearranged to give the following
useful expression for a binary system:
μi ¼ g + ð1 Xi Þdg=dXi
4.3
(4.6)
Partial Molar Properties
The Gibbs energy of a solution is the sum of enthalpic and entropic terms, (G ¼ H TS). Differentiating with respect to ni as in
Eq. (2.72) yields
ð∂G=∂ni ÞT,P, nj6¼i ¼ ð∂H=∂ni ÞT,P, nj6¼i T ð∂S=∂ni ÞT,P, nj6¼i
(4.7)
which may be written as
μi ¼ hi Tsi
(4.8)
where hi and si are called the partial molar enthalpy and entropy of
component i in the solution. (In this terminology, μi is sometimes
called the partial molar Gibbs energy, gi, of component i.) (Note that
the terms in Eq. (4.7) are derivatives of the extensive properties G,
H, and S, not of the molar properties g, h, and s.) As discussed in
Section 2.8, μi is an intensive property. So too are hi and si.
43
44
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
A plot of h versus XCu in a liquid Au-Cu solution will closely
resemble Fig. 4.1 with the molar enthalpy of mixing of the liquid
solution defined, similarly to Eq. (4.1), as
(4.9)
Δhm ¼ h XAu hoAu + XCu hoCu
where hoAu and hoCu are the molar enthalpies of pure liquid Au and
Cu. They are the values of hAu and hCu when XAu ¼ 1 and XCu ¼ 1,
respectively. Δhm is the enthalpy associated with the mixing reaction in Eq. (4.2). A plot of Δhm versus composition for the Au-Cu
liquid solution at 1100°C is shown in Fig. 4.2.
An equation similar to Eq. (4.3) may be written:
h ¼ XAu hAu + XCu hCu
(4.10)
and the tangent construction applies. That is, if a tangent is drawn
to the curve of h at a given composition, then the intercepts of this
tangent on the axes at XAu ¼ 1 and XCu ¼ 1 are equal to hAu and hCu,
respectively, at this composition. An equation analogous to
Eq. (4.6) may also be written relating hi to h.
Similar equations apply for the entropy functions. The molar
entropy change of the mixing reaction in Eq. (4.2) is
o
o
(4.11)
+ XCu sCu
Δsm ¼ s XAu sAu
and on a plot of s versus composition, the tangent construction
applies. A plot of ( TΔsm) versus composition is shown in Fig. 4.2.
Other partial molar properties may be defined similarly to
Eqs. (4.7), (4.8) by differentiation of any extensive property of a
solution. For example, the partial molar volume of component i
is defined as
vi ¼ ð∂V =∂ni ÞT,P, nj6¼i
(4.12)
Equations similar to Eq. (4.10) can be written for all solution properties (h, s, v, …). That is, the molar property of a solution is equal
to the weighted sum of the corresponding partial properties. However, one must not take the word “partial” too literally. An integral
solution property is the result of interactions among the components. It cannot be divided into “parts” due to each component.
4.4
Relative Partial Molar Properties
The difference between the chemical potential μi (also called
the partial molar Gibbs energy gi) of a component in solution
and the chemical potential μoi (or goi ) of the same component in
a standard state is called the relative chemical potential (or relative
partial Gibbs energy), Δμi (or Δgi). It is usual to choose, as standard
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
state, the pure component in the same phase at the same temperature. The activity ai of the component relative to the chosen standard state is then defined in terms of Δμi by the following
equation, as illustrated in Fig. 4.1:
Δμi ¼ μi μoi ¼ RT ln ai
(4.13)
From Fig. 4.1, it can be seen that
Δgm ¼ XAu ΔμAu + XCu ΔμCu ¼ RT ðXAu ln aAu + XCu ln aCu Þ
(4.14)
The isothermal Gibbs energy of mixing can be divided into enthalpic and entropic terms, as can the relative chemical potentials:
Δgm ¼ Δhm T Δsm
(4.15)
Δμi ¼ Δhi T Δsi
(4.16)
where Δhi and Δsi are the relative partial enthalpy and entropy of
component i defined as
Δhi ¼ hi hoi
(4.17)
Δsi ¼ si sio
(4.18)
It follows from Eqs. (4.14)–(4.16) that
Δhm ¼ XAu ΔhAu + XCu ΔhCu
(4.19)
Δsm ¼ XAu ΔsAu + XCu ΔsCu
(4.20)
Hence, if Δgm, Δhm, or Δsm is plotted versus composition, then the
tangent construction gives the corresponding relative partial
properties as illustrated in Fig. 4.2.
4.4.1
A Note on Standard States
So far, we have defined the standard state of a solid or liquid
element or compound as the pure (i.e., not dissolved in solution)
solid or liquid as the case may be. (To be rigorous, we must also
specify a total hydrostatic pressure of 1.0 bar.) Unless otherwise
specified, this is usually the default standard state. However, other
standard states may sometimes be introduced for convenience.
For example, dilute solution standard states will be introduced
in Sections 5.9 and 5.10. In such cases, μoi in Eq. (4.13) is the chemical potential in the chosen standard state. Hence, the numerical
value of the activity ai depends not only on μi but also on the
choice of standard state. When an activity is quoted, the standard
state must always be specified.
See Sections 5.9 and 5.10 for a further discussion of standard
states.
45
46
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
4.5
Activity
The activity of a component in a solution was defined by
Eq. (4.13). Since ai varies monotonically with μi, it follows that
when two or more phases are in equilibrium, the activity of any
component is the same in all phases, provided that the activity
in every phase is expressed with respect to the same standard state.
For more on this subject, see Sections 5.9–5.11.
The use of activities in calculations of chemical equilibrium
conditions is illustrated by the following example. A liquid solution of Au and Cu at 1100 °C with XCu ¼ 0.6 is exposed to an atmosphere in which the oxygen partial pressure is pO2 ¼ 103 bar. Will
Cu2O be formed? The reaction is
2CuðliqÞ + 0:5O2 ðgÞ ¼ Cu2 OðsolÞ
(4.21)
where the Cu (liq) is in solution. If the reaction proceeds with
the formation of dn moles of Cu2O, then 2dn moles of Cu are
consumed, and the Gibbs energy of the Au-Cu solution changes
by 2(dGl/dnCu)dn. The total Gibbs energy of the system then
varies as
dG=dn ¼ μoCu2 O 0:5μO2 2 dG l =dnCu
¼ μoCu2 O 0:5μO2 2μCu
¼ μoCu2 O 0:5μoO2 2μoCu 0:5RT ln pO2 2RT ln aCu
0:5 2
¼ ΔG o + RT ln pO
a
Cu ¼ ΔG
2
(4.22)
where Eqs. (2.72), (3.7), and (4.13) have been used.
For the reaction in Eq. (4.21) at 1100°C, ΔG° ¼ 68.73 kJ (from
FactSage). The activity of Cu in the liquid alloy at XCu ¼ 0.6 is
aCu ¼ 0.385 (from FactSage). Substitution into Eq. (4.22) with
pO2 ¼ 103 bar gives
dG=dn ¼ ΔG ¼ 7:50 kJ
(4.23)
Hence, under these conditions, the reaction entails a decrease in
the total Gibbs energy, and so the copper will be oxidized.
4.6
Ideal Raoultian Solutions
An ideal Raoultian solution is usually defined as one in which
the activity of a component with respect to the pure component
standard state is equal to its mole fraction:
¼ Xi
aideal
i
(4.24)
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
With a judicious choice of standard state, this definition can also
encompass ideal Henrian solutions, as will be discussed in
Sections 5.9–5.11.
This Raoultian definition of ideality is generally only applicable
to simple substitutional solutions on a single lattice. There are
more useful definitions for other types of solutions such as interstitial solutions, ionic solutions, solutions of defects, and polymer
solutions. That is, the most convenient definition of ideality
depends upon the solution model. Solution models are discussed
in Part II. In the present section, Eq. (4.24) for an ideal substitutional solution will be developed with the liquid Au-Cu solution
as example.
In the ideal substitutional solution model, it is assumed that Au
and Cu atoms are nearly alike, with nearly identical radii and electronic structures. This being the case, there will be no change in
bonding energy or volume upon mixing, so that the enthalpy of
mixing is zero:
¼0
Δhideal
m
(4.25)
Furthermore, and for the same reason, the Au and Cu atoms will
be randomly distributed over the lattice sites. (In the case of a liquid solution, we can think of the lattice sites as the instantaneous
atomic positions.) For a random distribution of NAu gold atoms
and NCu copper atoms over (NAu + NCu) sites, Boltzmann’s equation (Eq. 2.2) can be used to calculate the configurational entropy
of the solution. This is the entropy associated with the spatial distribution of the particles over the lattice sites:
Sconfig ¼ kB ln ðNAu + NCu Þ!=NAu !NCu !
(4.26)
The configurational entropies of pure Au and Cu are zero since all
lattice sites are occupied by the same element. Hence, the configconfig
, will be equal to Sconfig. The
urational entropy of mixing, ΔSm
solution also possesses a nonconfigurational contribution to the
entropy related to the distribution of phonons and electrons over
vibrational and electronic energy levels. However, because of the
assumed close similarity of Au and Cu, these distributions do not
change when the pure components are mixed, and so, there is no
nonconfigurational contribution to the entropy of mixing of an
ideal solution. Hence, the entropy of mixing is equal to Sconfig.
Application of Stirling’s approximation, which states that ln N!
[(N ln N) N] when N is very large, yields
NAu
NCu
ideal
¼ Sconfig ¼ kB NAu ln
+ NCu ln
ΔSm
NAu + NCu
NAu + NCu
(4.27)
47
48
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
For 1 mol of solution, (NAu + NCu) ¼ N° where N° is Avogadro’s
number. We also note that (kB N°) is equal to the ideal gas constant
R. Hence,
ideal
Δsm
¼ RðXAu ln XAu + XCu ln XCu Þ
(4.28)
Therefore, since the ideal enthalpy of mixing is zero,
ideal
¼ RT ðXAu ln XAu + XCu ln XCu Þ
Δgm
(4.29)
By comparing Eqs. (4.14), (4.29), we obtain.
¼ RT ln aideal
¼ RT ln Xi
Δμideal
i
i
(4.30)
Hence, Eq. (4.24) has been demonstrated for an ideal substitutional solution.
Ideal solutions will be discussed further in Part II.
4.7
Excess Properties
In reality, Au and Cu atoms are not identical, and so Au-Cu
solutions are not perfectly ideal. The difference between a solution property and its value in an ideal solution is called an excess
property. The excess Gibbs energy and excess entropy, for example,
are defined as
ideal
g E ¼ Δgm Δgm
(4.31)
ideal
sE ¼ Δsm Δsm
(4.32)
and
Since the ideal enthalpy of mixing is zero, the excess enthalpy is
equal to the enthalpy of mixing:
¼ Δhm
hE ¼ Δhm Δhideal
m
(4.33)
g E ¼ hE TsE ¼ Δhm TsE
(4.34)
Hence,
The excess molar heat capacity of a solution is given as
cPE ¼ dhE =dT ¼ cP X1 cPð1Þ + X2 cPð2Þ
(4.35)
where cP is the heat capacity of the solution and cP(1) and cP(2) are
the heat capacities of the components. Note that the excess heat
capacity of an ideal solution is zero.
Excess partial properties are defined similarly:
¼ RT ln ai RT ln Xi
μEi ¼ Δμi Δμideal
i
(4.36)
siE ¼ Δsi Δsiideal ¼ Δsi ðR ln Xi Þ
(4.37)
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
(where μEi is equivalent to gEi ). Also,
μEi ¼ hEi TsEi ¼ Δhi TsEi
(4.38)
Equations analogous to Eqs. (4.14), (4.19), (4.20) relate the integral
and partial excess properties. For example, in Au-Cu solutions,
E
E
g E ¼ XAu μEAu + XCu μECu and sE ¼ XAu sAu
+ XCu sCu
(4.39)
Tangent constructions similar to that of Fig. 4.1 can thus also be
employed for excess properties, and an equation analogous to
Eq. (4.6) can be written:
μEi ¼ g E + ð1 Xi Þdg E =dXi
(4.40)
Similar equations relate hEi to hE and sEi to sE.
The Gibbs-Duhem equation, Eq. (2.82), also applies to excess
properties. At constant T and P,
XAu dμEAu + XCu dμECu ¼ 0
(4.41)
and similar Gibbs-Duhem equations may be written for the partial
excess enthalpies and entropies.
A plot of gE versus composition in liquid Au-Cu alloys at 1100°C
is shown in Fig. 4.2. If a tangent is drawn to this curve at a given
composition, the intercepts of the tangent on the axes at XAu ¼ 1
and XCu ¼ 1 are equal to μEAu and μECu, respectively.
In Au-Cu alloys, gE is negative. That is, Δgm is more negative
than Δgideal
m , and so the solution is thermodynamically more stable
than an ideal solution. We thus say that Au-Cu solutions exhibit
negative deviations from ideality. If gE > 0, then a solution is less
stable than an ideal solution and is said to exhibit positive
deviations.
It can be seen in Fig. 4.2 that hE and gE are very nearly equal.
That is, in liquid Au-Cu solutions, sE is very small (see Eq. 4.34).
4.8
Activity Coefficients
The activity coefficient of a component in a solution is
defined as
γ i ¼ ai =Xi
(4.42)
μEi ¼ RT ln γ i
(4.43)
Hence, from Eq. (4.36),
In an ideal solution, γ i ¼ 1, and μEi ¼ 0 for all components. If γ i < 1,
. That is, the compothen μEi < 0, and from Eq. (4.36), Δμi < Δμideal
i
nent i is more stable in the solution than it would be in an ideal
49
50
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
solution of the same composition. If γ i > 1, then μEi > 0, and the
driving force for the component to enter into solution is less than
in the case of an ideal solution.
Activity coefficients with respect to different standard states
are discussed in Sections 5.9–5.11.
4.9
Regular Solution Theory
Many years ago, Van Laar (1908) showed that many observed
features of simple binary solutions can be illustrated, at least qualitatively, by simple regular solution theory. A simple binary regular
solution of components A and B is one for which
g E ¼ XA XB ðω ηT Þ
(4.44)
where ω and η are parameters independent of temperature and
composition. Substituting Eq. (4.44) into Eq. (4.40) yields, for
the partial properties, the following:
μEA ¼ XB2 ðω ηT Þ, μEB ¼ XA2 ðω ηT Þ
(4.45)
Several liquid and solid solutions conform approximately to regular solution behavior, particularly if gE is small. Examples may be
found for alloys, molecular solutions, and ionic solutions such as
molten salts and oxides, among others. (The very low values of gE
observed for gaseous solutions generally conform very closely to
Eq. 4.44.)
To understand why this should be so, we require only a very
simple model. Suppose that the atoms or molecules of the components A and B mix substitutionally on a single lattice. If the
atomic (or molecular) sizes and electronic structures of A and
B are similar, then the distribution will be nearly random, and
the configurational entropy will be nearly ideal as in Eq. (4.28).
Hence,
g E ¼ Δhm TsEðnonconfigÞ
E(non -config)
(4.46)
where s
is the nonconfigurational part of the entropy of
mixing (which is equal to zero in an ideal solution).
Assuming that the bond energies εAA, εBB, and εAB of firstnearest-neighbor pairs are independent of temperature and composition and that the average nearest-neighbor coordination
number, Z, is also constant, we calculate the enthalpy of mixing
resulting from the change in the total energy of first-nearestneighbor pair bonds as follows.
In 1 mol of solution, there are (N°Z/2) nearest-neighbor pair
bonds, where N° is Avogadro’s number. Since the distribution is
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
assumed to be random, the probability that a given bond is an
(A-A) bond is equal to X2A. The probabilities of (B-B) and (A-B)
bonds are X2B and 2XAXB. The contribution to the molar enthalpy
of mixing resulting from the change in total energy of firstnearest-neighbor bonds is then equal to the sum of the energies
of the first-nearest-neighbor bonds in 1 mol of solution, minus
the energy of the (A-A) bonds in XA moles of pure A and the energy
of the (B-B) bonds in XB moles of pure B:
ΔhFNN ¼ ðN o Z=2Þ XA2 εAA + XB2 εBB + 2XA XB εAB
ðN o Z=2ÞðXA εAA + XB εBB Þ
¼ ðN o Z Þ½εAB ðεAA + εBB Þ=2XA XB
(4.47)
¼ C1 XA XB
where C1 is a constant.
With exactly the same assumptions and arguments, we can
derive equations for the enthalpy of mixing contributions resulting from the change in total energy of second-nearest-neighbor,
ΔhSNN; third-nearest-neighbor, ΔhTNN; and other pair bonds.
These are clearly all of the same form, CiXAXB. If we now assume
that the actual observed enthalpy of mixing, Δhm, results mainly
from these changes in the total energy of all pair bonds, then,
Δhm ¼ ωX A XB
(4.48)
where ω ¼ C1 + C2 + C3 + ….
If we now define σ AA, σ BB, and σ AB as the vibrational entropies
of pair bonds and follow an identical argument to that just presented for the pair bond energies, we obtain
sEðnonconfigÞ ¼ ðN o Z=2Þ½σ AB ðσ AA + σ BB =2Þ ¼ ηX A XB
(4.49)
Eq. (4.44) has thus been derived.
If (A-B) bonds are stronger than (A-A) and (B-B) bonds, then
(εAA σ ABT) < [(εAA σ AAT)/2 + (εB σ BBT)/2] so that (ω ηT) < 0
and gE < 0. That is, the solution is rendered more stable. If the
(A-B) bonds are relatively weaker, then the solution is rendered
less stable, (ω ηT) > 0 and gE > 0.
Simple nonpolar molecular solutions and simple ionic solutions such as molten salts often exhibit approximately regular
behavior. The assumption of additivity of the energy of pair bonds
is probably reasonably realistic for van der Waals or coulomb
forces. For metallic alloys, the concept of a pair bond is, at best,
vague, and metallic solutions tend to exhibit larger deviations
from regular behavior.
51
52
Chapter 4 FUNDAMENTALS OF THE THERMODYNAMICS OF SOLUTIONS
In many solutions, it is found that | ηT | | ω | in Eq. (4.44). That
is, gE Δhm ¼ ωXAXB, and so, to a first approximation, gE is independent of T. This is more often the case in nonmetallic solutions
than in metallic solutions.
The regular solution model will be developed further in
Chapter 14. The simple regular solution model developed here
will be used to illustrate several observed features of binary phase
diagrams in Section 5.6.
4.10 Multicomponent Solutions
The equations of this chapter have been derived with a binary
solution as example. However, the equations apply equally to systems of any number of components, either with no changes or by
including additional terms. For instance, in a solution of components A-B-C-D…, Eqs. (4.1), (4.14) when combined become
g ¼ XA μoA + XB μoB + XC μoC + ⋯ + RT ðXA ln aA + XB ln aB + XC ln aC + ⋯Þ
(4.50)
References
Pelton, A.D., 2014. In: Laughlin, D.E., Hono, K. (Eds.), (Chapter 3). Physical Metallurgy. fifth ed. Elsevier, New York.
Van Laar, J.J., 1908. Z. Phys. Chem. 63, 216–253.
List of Websites
FactSage, Montreal, www.factsage.com.
5
THERMODYNAMIC ORIGIN
OF PHASE DIAGRAMS
CHAPTER OUTLINE
5.1 Temperature-Composition Phase Diagrams in Systems with
Complete Solid and Liquid Miscibility 53
5.2 Binary Pressure-Composition Phase Diagrams 58
5.3 Minima and Maxima in Two-Phase Regions 59
5.4 Miscibility Gaps 61
5.5 Simple Eutectic Systems 63
5.5.1 Eutectic Microstructure and the Lever Rule 65
5.6 Thermodynamic Origin of Simple Binary Phase Diagrams
Illustrated by Regular Solution Theory 65
5.7 Immiscibility—Montectics 67
5.8 Intermediate Phases 70
5.9 Limited Mutual Solubility—Ideal Henrian Solutions 73
5.10 Henry’s Law, Raoult’s Law and Standard States 76
5.11 Single Ion Activities 78
5.12 The “Activity” of a Solution 79
5.13 Geometry of Binary Temperature-Composition Phase
Diagrams 80
5.14 Effects of Grain Size, Coherency, and Strain Energy 83
References 84
List of Websites 84
5.1
Temperature-Composition Phase
Diagrams in Systems with Complete
Solid and Liquid Miscibility
In this section, we first consider the thermodynamic origin of
simple “lens-shaped” phase diagrams in binary systems with
complete liquid and solid miscibility. An example of such a diagram was given in Fig. 3.1. Another example is the Ge-Si phase
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00005-1
# 2019 Elsevier Inc. All rights reserved.
53
54
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.1 Ge-Si phase diagram at P ¼ 1 bar (based on data from Hansen, 1958) and Gibbs energy curves at three
temperatures, illustrating the common tangent construction (Pelton, 2014).
diagram in the lowest panel of Fig. 5.1. In the upper three panels of
Fig. 5.1, the molar Gibbs energies of the solid and liquid phases, gs
and gl at three temperatures, are shown to scale. As illustrated in
the top panel, gs varies with composition between the standard
chemical potentials of pure solid Ge and of pure solid Si, μo(s)
Ge
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
l
and μo(s)
Si , while g varies between the standard chemical potentials
o(l)
of the pure liquid components, μo(l)
Ge and μSi . The difference
o(l)
o(s)
between μGe and μGe is equal to the standard molar Gibbs energy
o(s)
of fusion (melting) of pure Ge, Δgof(Ge) ¼ (μo(l)
Ge μGe ). Similarly for
o
o(l)
o(s)
Si, Δgf(Si) ¼ (μSi μSi ). The Gibbs energy of fusion of a pure component may be written as
Δgfo ¼ Δhof T Δsfo
(5.1)
where Δhof and Δsof are the standard molar enthalpy and entropy of
fusion.
Since, to a first approximation, Δhof and Δsof are independent of T
(i.e., clp csp), Δgof is approximately a linear function of T. If T > Tf°,
then Δgof is negative; if T < Tf°, then Δgof is positive. Hence, as seen
in Fig. 5.1, as T decreases, the gs curve descends relative to gl. At
1500°C, gl < gs at all compositions. Therefore, by the principle that
a system always seeks the state of minimum Gibbs energy at constant
T and P, the liquid phase is stable at all compositions at 1500°C.
At 1300°C, the curves of gs and gl cross. The line P1Q1, which is
the common tangent to the two curves, divides the composition
range into three sections. For compositions between pure Ge
and P1, a single-phase liquid is the state of minimum Gibbs
energy. For compositions between Q1 and pure Si, a single-phase
solid solution is the stable state. Between P1 and Q1, a total Gibbs
energy lying on the tangent line P1Q1 may be realized if the system
adopts a state consisting of two phases with compositions at P1
and Q1 with relative proportions given by the lever rule (see
Section 3.1). Since the tangent line P1Q1 lies below both gs and
gl, this two-phase state is more stable than either phase alone.
Furthermore, no other line joining any point on the gl curve to
any point on the gs curve lies below the line P1Q1. Hence, this line
represents the true equilibrium (i.e., minimum Gibbs energy)
state of the system, and the compositions P1 and Q1 are the liquidus and solidus compositions at 1300°C.
As T is decreased to 1100°C, the points of common tangency
are displaced to higher concentrations of Ge. For T < 938°C, gs < gl
at all compositions.
It was shown in Fig. 4.1 that if a tangent is drawn to a Gibbs
energy curve, then the intercept of this tangent on the axis at
Xi ¼ 1 is equal to the chemical potential μi of component i. The
common tangent construction of Fig. 5.1 thus ensures that the
chemical potentials of Ge and Si are equal in the solid and liquid
phases at equilibrium. That is,
μlGe ¼ μsGe
(5.2)
μlSi ¼ μsSi
(5.3)
55
56
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
This equality of chemical potentials was shown in Section 2.8 to
be a criterion for phase equilibrium. That is, the common tangent
construction simultaneously minimizes the total Gibbs energy
and ensures the equality of the chemical potentials, thereby showing that these are equivalent criteria for equilibrium between
phases as was discussed in Section 2.8.
If we rearrange Eq. (5.2), subtracting the Gibbs energy of fusion
o(s)
of pure Ge, Δgof(Ge) ¼ (μo(l)
Ge μGe ), from each side, we obtain
oðlÞ
oðsÞ
oðlÞ
oðsÞ
μlGe μGe μsGe μGe ¼ μGe μGe
(5.4)
Using Eq. (4.13), we can write (Eq. 5.4) as
ΔμlGe ΔμsGe ¼ ΔgfoðGeÞ
(5.5)
RT ln alGe RT ln asGe ¼ ΔgfoðGeÞ
(5.6)
or
where alGe is the activity of Ge (with respect to pure liquid Ge as
standard state) in the liquid solution on the liquidus and asGe is
the activity of Ge (with respect to pure solid Ge as standard state)
in the solid solution on the solidus. Starting with Eq. (5.3), we can
derive a similar expression for the other component:
RT ln alSi RT ln asSi ¼ ΔgfoðSiÞ
(5.7)
Eqs. (5.6), (5.7) are equivalent to the common tangent construction.
It should be noted that absolute values of chemical potentials
o(l)
cannot be defined. Hence, the relative positions of μo(l)
Ge and μSi in
Fig. 5.1 are arbitrary. However, this is immaterial for the preceding
o(s)
discussion, since displacing both μo(l)
Si and μSi by the same arbio(l)
o(s)
trary amount relative to μGe and μGe will not alter the compositions of the points of common tangency.
The shape of the two-phase (solid + liquid) “lens” on the phase
diagram is determined by the Gibbs energies of fusion, Δgof of the
components and by the mixing terms, Δgs and Δgl. In order to
observe how the shape is influenced by varying Δgof , let us consider a hypothetical system A-B in which Δgs and Δgl are ideal
Raoultian (Eq. 4.29). Let Tof(A) ¼ 800 K and Tof(B) ¼ 1200 K. Furthermore, assume that the entropies of fusion of A and B are equal and
temperature-independent. The enthalpies of fusion are then
given from Eq. (5.1) by the expression Δhof ¼ Tof Δsof since Δgof ¼ 0
when T ¼ Tof (see Eq. 2.28). Calculated phase diagrams for Δsof ¼ 3,
10, and 30 J mol1 K1 are shown in Fig. 5.2. A value of Δsof 10
J mol1 K1 is typical of most metals. When the components are
ionic compounds such as ionic oxides and halides, Δsof can be significantly larger since there are several ions per formula unit.
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.2 Phase diagram of a system A-B with ideal solid and liquid solutions.
Melting points of A and B are 800 and 1200 K, respectively. Diagrams calculated for
entropies of fusion Dsof(A) ¼ Dsof(B) ¼ 3, 10, and 30 J mol1 K1 (Pelton, 2014).
Hence, two-phase “lenses” in binary ionic salt or oxide phase diagrams tend to be wider than those encountered in alloy systems. If
we are considering vapor-liquid equilibria rather than solid-liquid
equilibria, then the shape is determined by the entropy of vaporization, Δsov. Since Δsov is usually an order of magnitude larger than
Δsof , two-phase (liquid + vapor) lenses tend to be very wide. For
equilibria between two solid solutions of different crystal structures, the shape is determined by the entropy of solid-solid transformation, which is usually smaller than the entropy of fusion by
approximately an order of magnitude. Therefore two-phase (solid
+ solid) lenses tend to be narrow.
57
58
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
5.2
Binary Pressure-Composition Phase
Diagrams
Let us consider liquid-vapor equilibrium in a system with
complete liquid miscibility, using as example the Zn-Mg
system. Curves of gv and gl can be drawn at any given T and P,
as in the upper panel of Fig. 5.3, and the common tangent
construction then gives the equilibrium vapor and liquid
Fig. 5.3 Pressure-composition phase diagram of the Zn-Mg system at 977°C
calculated for ideal vapor and liquid solutions. Upper panel illustrates common
tangent construction at a constant pressure (Pelton, 2014).
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
compositions. The phase diagram depends upon the Gibbs energies of vaporization of the pure components, Δ gov(Zn) ¼ (μvZn μo(l)
Zn )
)
as
shown
in
Fig.
5.3.
and Δgov(Mg) ¼ (μvMg μo(l)
Mg
To generate the isothermal pressure-composition (P-X) phase
diagram in the lower panel of Fig. 5.3, we require the Gibbs energies
of vaporization of the components as functions of P. Assuming monatomic ideal vapors and assuming that pressure has negligible effect
upon the Gibbs energy of the liquid, we can write, using Eq. (3.7),
ΔgvðiÞ ¼ ΔgvoðiÞ + RT ln P
(5.8)
where Δgov(i) is the standard Gibbs energy of vaporization (when
P ¼ l.0 bar), which is given by
ΔgvoðiÞ ¼ ΔhovðiÞ T ΔsvoðiÞ
(5.9)
For example, the standard enthalpy of vaporization of Zn is
115,310 J mol1 at its normal boiling point of 1180 K (FactSage).
Assuming that Δhov(Zn) is independent of T, we calculate from
Eq. (5.9) that Δsov(Zn) ¼ 115,310/1180 ¼ 97.71 J mol1 K1. From
Eq. (5.8), Δgv(Zn) at any T and P is thus given by
ΔgvðZnÞ ¼ ð115310 97:71T Þ + RT ln P
(5.10)
A similar expression can be derived for the other component, Mg.
At constant temperature, then, the curve of gv in Fig. 5.3
descends relative to gl as the pressure is lowered, and the P-X
phase diagram is generated by the common tangent construction.
The diagram at 977°C in Fig. 5.3 was calculated under the assumption of ideal liquid and vapor mixing (gE(l) ¼ 0, gE(v) ¼ 0).
P-X phase diagrams involving liquid-solid and solid-solid
equilibria can be calculated in a similar fashion through the following general equation that follows from Eq. (2.71) and that gives
the effect of pressure upon the Gibbs energy change for the transformation of one mole of pure component i from an α-phase to a
β-phase at constant T:
ΔgðiÞα!β ¼ ΔgðoiÞα!β
ðP +
viβ viα dP
(5.11)
P¼1
o
Δg(i)α!β
where
is the standard (P ¼ l.0 bar) Gibbs energy of transformation and vβi and vαi are the molar volumes.
5.3
Minima and Maxima in Two-Phase
Regions
As discussed in Section 4.7, the Gibbs energy of mixing Δgm
may be expressed as the sum of an ideal term Δgideal
and an
m
59
60
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
excess term gE. As was shown in Section 5.1, if the solid and liquid phases are both ideal and close to ideal, a “lens-shaped”
two-phase region results. However, in most systems, even
approximately ideal behavior is the exception rather than
the rule.
Curves of gs and gl for a hypothetical system A-B are shown
schematically in Fig. 5.4 at a constant temperature (below the
melting points of pure A and B) such that the solid state is the
stable state for both pure components. However, in this system,
gE(l) < gE(s) so that gs presents a flatter curve than does gl, and there
exists a central composition region in which gl < gs. Hence, there
are two common tangent lines, P1Q1 and P2Q2. Such a situation
gives rise to a phase diagram with a minimum in the two-phase
region, as observed in the Na2CO3-K2CO3 system shown in
Fig. 5.5. At a composition and temperature corresponding to
the minimum point, liquid and solid of the same composition
exist in equilibrium.
A two-phase region with a minimum point as in Fig. 5.5 may be
thought of as a two-phase “lens” that has been “pushed down” by
virtue of the fact that the liquid is relatively more stable than
the solid. Thermodynamically, this relative stability is expressed
as gE(l) < gE(s).
Conversely, if gE(l) > gE(s) to a sufficient extent, then a two-phase
region with a maximum will result. Such maxima in (liquid +
solid) or (solid + solid) two-phase regions are nearly always associated with the existence of an intermediate phase, as will be discussed in Section 5.9.
Fig. 5.4 Isothermal Gibbs energy curves for solid and liquid phases in a system A-B
in which gE(l) < gE(s). A phase diagram of the type of Fig. 5.5 results (Pelton, 2014).
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.5 Temperature-composition phase diagram of the K2CO3-Na2CO3 system at
P ¼ 1 bar (see FactSage).
5.4
Miscibility Gaps
If g > 0, then the solution is thermodynamically less stable
than an ideal solution. This can result from a large difference in
size of the component atoms, ions, or molecules; from differences
in bond types, valencies, and electronic structure; or from many
other factors.
In the Au-Ni system, gE is positive in the solid phase. In the top
panel of Fig. 5.6, gE(s) is plotted at 1200 K (Hultgren et al., 1973),
is also plotted at
and the ideal Gibbs energy of mixing Δgideal
m
1200 K. The sum of these two terms is the Gibbs energy of mixing
of the solid solution, Δgsm, which is plotted at 1200 K and at other
temperatures in the central panel of Fig. 5.6. Now, from Eq. (4.29),
is always negative and varies directly with T, whereas gE
Δgideal
m
generally varies less rapidly with temperature. As a result, the sum
+ gE becomes less negative as T decreases. However,
Δgsm ¼ Δgideal
m
curve at XAu ¼ 1
from Eq. (4.29), the limiting slopes to the Δgideal
m
and XNi ¼ 1 are both infinite, whereas the limiting slopes of gE are
always finite (Henry’s law; see Section 5.10). Hence, Δgsm will always
be negative as XAu ! 1 and XNi ! 1 no matter how low the temperature. As a result, below a certain temperature, the curve of Δgsm will
exhibit two negative “humps.” Common tangent lines P1Q1, P2Q2,
and P3Q3 to the two humps at different temperatures define the ends
of tie-lines of a two-phase solid-solid miscibility gap in the Au-Ni
phase diagram, which is shown in the lower panel in Fig. 5.6
(Hultgren et al., 1973). The peak of the gap occurs at the critical or
consolute temperature and composition, Tc and Xc.
E
61
62
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.6 Phase diagram (based on data from Hultgren et al., 1973) and Gibbs energy
curves of solid solutions for the Au-Ni system at P ¼ 1 bar. Letter s indicates
spinodal points (Pelton, 2014).
An example of a miscibility gap in a liquid phase will be given
in Fig. 5.10.
Below the critical temperature, the curve of Δgsm exhibits two
inflection points, indicated by the letter s in Fig. 5.6. These are
known as the spinodal points. On the phase diagram, their locus
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
traces out the spinodal curve. The spinodal curve is not part of the
equilibrium phase diagram, but it is important in the kinetics of
phase separation.
When gE(s) is positive for the solid phase in a system, it is usually also the case that gE(l) < g(s) since the unfavorable factors (such
as a difference in atomic dimensions) that cause gE(s) to be positive will have less of an effect upon gE(l) in the liquid phase owing
to the greater flexibility of the liquid structure to accommodate
different atomic sizes, valencies, etc. Hence, a solid-solid miscibility
gap is often associated with a minimum in the two-phase (solid +
liquid) region, as is the case in the Au-Ni system.
5.5
Simple Eutectic Systems
The more positive gE is in a system, the higher is Tc, and the
wider is the miscibility gap at any temperature. Suppose that
gE(s) is so positive that Tc is higher than the minimum in the (solid
+ liquid) region. The result will be a phase diagram such as that of
the Ag-Cu system shown in Fig. 5.7. The upper panel of Fig. 5.7
shows the Gibbs energy curves at 850°C. The two common tangents define two two-phase regions. As the temperature is
decreased below 850°C, the gs curve descends relative to gl, and
the two points of tangency P1 and P2 approach each other until,
at T ¼ 780°C, P1 and P2 become coincident at the composition
E. That is, at T ¼ 780°C, there is just one common tangent line contacting the two portions of the gs curve at compositions A and
B and contacting the gl curve at E. This temperature is known
as the eutectic temperature, TE, and the composition E is the eutectic composition. For temperatures below TE, gl lies completely
above the common tangent to the two portions of the gs curve,
and so for T < TE, a solid-solid miscibility gap is observed. The
phase boundaries of this two-phase region are called the solvus
lines. The word eutectic is from the Greek for “to melt well,” since
the eutectic composition E is a minimum melting temperature.
This description of the thermodynamic origin of simple eutectic phase diagrams is strictly correct only if the pure solid components A and B have the same crystal structure. Otherwise, a curve
for gs that is continuous at all compositions cannot be drawn. In
this case, each terminal solid solution (α and β) must have its own
Gibbs energy curve (gα and gβ), and these curves do not join up
with each other in the central composition region. However, a
simple eutectic phase diagram as in Fig. 5.7 still results.
Suppose a liquid Ag-Cu solution of composition XCu ¼ 0.22
(composition P1) is cooled from the liquid state very slowly under
63
64
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.7 Temperature-composition phase diagram at P ¼ 1 bar and Gibbs energy
curves at 850°C for the Ag-Cu system. Solid Ag and Cu have the same (fcc) crystal
structure (see SGTE).
equilibrium conditions. At 850°C, the first solid appears with composition Q1. As T decreases further, solidification continues with
the liquid composition following the liquidus curve from P1 to
E and the composition of the solid phase following the solidus
curve from Q1 to A. The relative proportions of the two phases
at any T are given by the lever rule (Section 3.1). At a temperature
just above TE, two phases are observed: a solid α of composition
A called the proeutectic or primary constituent and a liquid of
composition E. At a temperature just below TE, two solids with
compositions A and B are observed. Therefore, at TE, during cooling, the following binary eutectic reaction occurs:
liquid ! α ðsolidÞ + β ðsolidÞ
(5.12)
Under equilibrium conditions, the temperature will remain constant at T ¼ TE until all the liquid has solidified, and during the
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
reaction, the compositions of the three phases will remain fixed at
A, B, and E. For this reason, the eutectic reaction is called an
invariant reaction as was discussed in Section 3.1.1. This eutectic
solidification results in a eutectic constituent of overall composition E consisting of an intimate mixture of α and β.
5.5.1
Eutectic Microstructure and the Lever Rule
A binary eutectic grain is a two-phase microstructural constituent consisting of an intimate mixture of two solid phases. Common morphologies include alternating lamellae of the two phases,
or globules or needles of one phase in a matrix of the other phase.
0
After equilibrium cooling of an alloy of composition X in
Fig. 5.7 from above the liquidus to just below TE, the microstructure will consist of proeutectic α grains of composition A and
eutectic grains of composition E. The relative amounts of these
two constituents are related by the lever rule:
moles eutectic consituent
AX 0
¼ 0
moles pro eutectic constituent X E
(5.13)
The relative amounts of the α and β phases within a eutectic grain
are also related by the lever rule:
moles α in eutectic EB
¼
moles β in eutectic AE
(5.14)
If the alloy is subsequently cooled to 500°C and annealed at this
temperature until equilibrium is attained, β-phase precipitates
of composition W will be formed within the proeutectic α matrix
that will have the residual composition V. By the lever rule,
moles β precipitated in pro eutectic grains AV
¼
moles residual α in pro eutectic grains
AW
(5.15)
The same ratio AV/AW relates the relative amounts of β-phase precipitates and residual α-phase in the α-phase lamellae of the
eutectic grains.
5.6
Thermodynamic Origin of Simple Binary
Phase Diagrams Illustrated by Regular
Solution Theory
Many years ago, Van Laar (1908) showed that the thermodynamic origin of a great many of the observed features of binary
phase diagrams can be illustrated, at least qualitatively, by simple
regular solution theory as developed in Section 4.9.
65
66
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.8 shows several phase diagrams calculated for a hypothetical system A-B containing a solid and a liquid phase with
melting points of Tof(A) ¼ 800 K and Tof(B) ¼ 1200 K and with entropies of fusion of both A and B set to 10 J mol1 K1 which is a
Fig. 5.8 Topological changes in the phase diagram for a system A-B with regular solid and liquid phases brought
about by systematic changes in the regular solution parameters os and ol. Melting points of pure A and B are 800
and 1200 K. Entropies of fusion of both A and B are 10.0 J mol1 K1 (Pelton and Thompson, 1975). Dashed curve in panel
(D) is a metastable liquid miscibility gap.
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
typical value for metals. The solid and liquid phases are both
regular with temperature-independent excess Gibbs energies:
g EðsÞ ¼ ωs XA XB and g EðlÞ ¼ ωl XA XB
(5.16)
The parameters ωs and ωl have been varied systematically to
generate the various panels of Fig. 5.8.
In panel (N), both phases are ideal. Panels (L) to (R) exhibit
minima or maxima depending upon the sign and magnitude of
(gE(l) gE(s)), as was discussed in Section 5.3. In panel (H), the liquid is ideal, but positive deviations in the solid give rise to a solidsolid miscibility gap as discussed in Section 5.4. On passing from
panel (H) to panel (C), an increase in gE(s) results in a widening of
the miscibility gap so that the solubilities of A in solid B and of B in
solid A decrease. Panels (A) to (C) illustrate that negative deviations in the liquid cause a relative stabilization of the liquid with
resultant lowering of the eutectic temperature.
In Fig. 5.8E, positive deviations in the liquid have given rise to a
liquid-liquid miscibility gap.
It is of interest to derive an equation for the consolute temperature of the miscibility gap of a regular solution. The Gibbs energy
of mixing of a regular solution (assuming sE ¼ 0) is
Δgm ¼ RT ðXA ln XA + XB ln XB Þ + ωXA XB
(5.17)
For a value of ωl ¼ 20 kJ mol1 as in Fig. 5.8E, plots of Δgm versus
XB are shown in Fig. 5.9. Below 1203 K, the curves exhibit two negative humps resulting in a miscibility gap as discussed in
Section 5.4. Because of the symmetry of Eq. (5.17), the gap is symmetrical about XA ¼ XB ¼ 0.5. The consolute temperature TC can be
seen to be given by substituting Eq. (5.17) into the following equation and solving at XB ¼ 0.5:
TC ¼ ∂2 Δgm =∂XB2 ¼ 0
(5.18)
This gives
TC ¼ ω=2R
When ω ¼ 20 kJ mol
5.7
1
(5.19)
as in Fig. 5.8E, TC ¼ 1203 K.
Immiscibility—Montectics
The CaO-SiO2 system shown in Fig. 5.10 exhibits a liquidliquid miscibility gap.
Suppose that a liquid of composition XSiO2 ¼ 0.8 is cooled
slowly from high temperatures. At T ¼ 1830°C the miscibility gap
boundary is crossed, and a second liquid layer appears with a
67
68
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
1000
500
800 K
0
Δgm (J mol–1)
–500
1000 K
–1000
–1500
–2000
1203 K
–2500
–3000
–3500
A
1400 K
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
B
Mole fraction B
Fig. 5.9 Gibbs energy of mixing for a regular solution A-B with o ¼ 20 kJ mol1.
composition XSiO2 ¼ 0.96. As the temperature is lowered further,
the composition of each liquid phase follows its respective phase
boundary until, at 1689°C, the SiO2-rich liquid has a composition
of XSiO2 ¼ 0.988 (point B), and in the CaO-rich liquid, XSiO2 ¼ 0.72
(point A). At any temperature, the relative amounts of the two
phases are given by the lever rule.
At 1689°C, the following invariant binary monotectic reaction
occurs upon equilibrium cooling:
Liquid B ! Liquid A + SiO2 ðsolidÞ
(5.20)
The temperature remains constant at 1689°C, and the compositions of the phases remain constant until all of liquid B is consumed. Cooling then continues with precipitation of solid SiO2,
with the equilibrium liquid composition following the liquidus
from point A to the eutectic E.
Returning to Fig. 5.8, we see in panel (D) that the positive deviations in the liquid in this case are not large enough to produce
immiscibility, but they do result in a flattening of the liquidus,
which indicates a “tendency to immiscibility.” If the nucleation
of the solid phases can be suppressed by sufficiently rapid cooling,
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
69
2400
2200
Liquid
o
2154
Liquid + CaO
o
2017
2000
o
Tc = 1895
CaO + Ca2SiO4
o
Ca2SiO4(β)
1799
1400
1437
o
1300
1200
CaO + Ca2SiO4
0
CaO
0.2
o
β
α
A
o
o
1464
P1
Q1
o
1463
0.4
1689
o
B
Liquid + cristobalite
1540
P2
Q2
CaSiO3
1600
2 Liquids
(Liq + CaSiO3)
Ca3Si2O7
CaO + Ca3SiO5
Ca2SiO4(α)
1800
Ca3SiO5
Temperature (°C)
(Liq + Ca2SiO4)
Liquid + cristobalite
2600
o
1465
o
E 1437
Liquid + tridymite
CaSiO3 + tridymite
0.6
0.8
Mole fraction SiO2
1
SiO2
Fig. 5.10 CaO-SiO2 phase diagram at P ¼ 1 bar and Gibbs energy curves at 1500°C illustrating Gibbs energies of fusion
and the formation of the compound CaSiO3 (see FactSage; Pelton, 2001). The temperature difference between the
peritectic at 1464°C and the eutectic at 1463°C has been exaggerated for the sake of clarity.
then a metastable liquid-liquid miscibility gap will be observed as
shown in Fig. 5.8D. For example, in the Na2O-SiO2 system, a flattened (or “S-shaped”) SiO2-liquidus heralds the existence of a
metastable miscibility gap of importance in glass technology.
70
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
5.8
Intermediate Phases
The phase diagram of the Ag-Mg system (Hultgren et al., 1973)
is shown in Fig. 5.11D. An intermetallic phase, β0 , is seen centered
approximately about the composition XMg ¼ 0.5. The Gibbs energy
curve at 1050 K for such an intermetallic phase has the form
’
shown schematically in Fig. 5.11A. The curve gβ rises quite rapidly
on either side of its minimum, which occurs near XMg ¼ 0.5.
As a result, the single-phase-β0 region appears on the phase diagram only over a limited composition range. This form of the Gibbs
energy curve results from the fact that when XAg XMg, a particularly stable crystal structure exists in which Ag and Mg atoms preferentially occupy different sublattices. The two common tangents
0
P1Q1 and P2Q2 give rise to a maximum in the two-phase (β + liquid)
region of the phase diagram. (Although the maximum is observed
very near XMg ¼ 0.5, there is no thermodynamic reason for the maximum to occur exactly at this composition.)
Another intermetallic phase, the ε-phase, is also observed
in the Ag-Mg system (Fig. 5.11D). The phase is associated with
a peritectic invariant ABC at 744 K. The Gibbs energy curves
are shown schematically at the peritectic temperature in
’
Fig. 5.11C. One common tangent line can be drawn to gl, gβ,
ε
and g at 744 K.
Suppose that a liquid alloy of composition XMg ¼ 0.7 is cooled
very slowly from the liquid state. At a temperature just above
0
744 K, a liquid phase of composition C and a β -phase of composition A are observed at equilibrium. At a temperature just below
0
744 K, the two phases at equilibrium are β of composition A and ε
of composition B. The following invariant binary peritectic reaction thus occurs upon cooling:
Liquid + β0 ðsolidÞ ! ε ðsolidÞ
(5.21)
This reaction occurs isothermally at 744 K with all three phases at
fixed compositions (at points A, B, and C). For an alloy with overall
composition between points A and B, the reaction proceeds until
all the liquid has been consumed. In the case of an alloy with over0
all composition between B and C, the β -phase will be the first to
be completely consumed.
Peritectic reactions occur upon cooling generally with the formation of the product solid (ε in this example) on the surface of
0
the reactant solid (β ), thereby forming a coating that can reduce
or prevent further contact between the reactant solid and liquid.
Further reaction may thus be greatly retarded so that equilibrium
conditions can only be achieved by extremely slow cooling.
The Gibbs energy curve for the ε-phase, gε, in Fig. 5.11C rises
more rapidly on either side of its minimum than does the Gibbs
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
71
Fig. 5.11 Ag-Mg phase diagram at P ¼ 1 bar (after Hultgren et al., 1973) and Gibbs energy curves at three temperatures
(Pelton, 2014).
72
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
0
energy gβ of the β -phase. Consequently, the width of the singlephase region over which the ε-phase exists (sometimes called
its range of stoichiometry or homogeneity range) is narrower than
0
that of the β -phase.
In the upper panel of Fig. 5.10 for the CaO-SiO2 system, Gibbs
energy curves at 1500°C for the liquid and CaSiO3 phases are
shown schematically. g0.5(CaSiO3) rises extremely rapidly on either
side of its minimum. (We write g0.5(CaSiO3) for 0.5 mol of the compound in order to normalize to a basis of 1 mol of components
CaO and SiO2.) As a result, the points of tangency Q1 and Q2 of
the common tangents P1Q1 and P2Q2 nearly (although not exactly)
coincide. Hence, the range of stoichiometry of the CaSiO3 phase is
very narrow (although theoretically never zero). The two-phase
regions labeled (liquid + CaSiO3) in Fig. 5.10 are the two sides
of a two-phase region that passes through a maximum at 1540°
C just as the (β0 + liquid) region passes through a maximum in
Fig. 5.11D. Because the CaSiO3 single-phase region is extremely
narrow, we refer to CaSiO3 as a stoichiometric compound. Any
deviation in composition from the stoichiometric 1:1 ratio of
CaO to SiO2 results in a very large increase in its Gibbs energy.
The ε-phase in Fig. 5.11 is based on the stoichiometry AgMg3.
The Gibbs energy curve, Fig. 5.11C, rises extremely rapidly on the
Ag side of the minimum but somewhat less steeply on the Mg side.
As a result, Ag is virtually insoluble in AgMg3, while Mg is sparingly
soluble. Such a phase with a narrow range of homogeneity is often
called a nonstoichiometric compound. At low temperatures, the β0 phase exhibits a relatively narrow range of stoichiometry about
the 1:1 AgMg composition and can properly be called a compound. However, at higher temperatures, it is debatable whether
a phase with such a wide range of composition should be called a
“compound.”
From Fig. 5.10, it can be seen that if stoichiometric CaSiO3 is
heated, it will melt isothermally at 1540°C to form a liquid of
the same composition. Such a compound is called congruently
melting or simply a congruent compound. The compound Ca2SiO4
in Fig. 5.9 is also congruently melting. The β0 -phase in Fig. 5.11 is
also congruently melting at the composition of the liquidus/solidus maximum.
It should be noted with regard to congruent melting that the
limiting slopes dT/dX of both branches of the liquidus at the congruent melting point (i.e., at the maximum of the two-phase
region) are zero since we are dealing with a maximum in a twophase region (similar to the minimum in the two-phase region
in Fig. 5.5).
The AgMg3 (ε) compound in Fig. 5.11D is said to melt incongruently. If solid AgMg3 is heated, it will melt isothermally at 744 K by
’
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
the reverse of the peritectic reaction (Eq. 5.21), to form a liquid of
composition C and another solid phase, β0 , of composition A.
Another example of an incongruently melting compound is
Ca3Si2O7 in Fig. 5.10, which melts incongruently (or peritectically)
to form liquid and Ca2SiO4 at the peritectic temperature of 1464°C.
An incongruently melting compound is always associated
with a peritectic. However, the converse is not necessarily true.
A peritectic is not always associated with an intermediate phase.
See, for example, Fig. 5.8I.
For purposes of phase diagram calculations involving stoichiometric compounds such as CaSiO3, we may, to a good approximation, consider the Gibbs energy curve, g0.5(CaSiO3), in the upper
panel of Fig. 5.10 to have zero width. All that is then required is
the value of g0.5(CaSiO3) at the minimum. This value is usually
expressed in terms of the Gibbs energy of fusion of the compound
Δg f (0.5CaSiO3) or of the Gibbs energy of formation Δgform(0.5CaSiO3) of
the compound from the pure solid components CaO and SiO2
according to the reaction: 0.5 CaO (sol) + 0.5 SiO2(sol) ¼ 0.5
CaSiO3(sol). Both of these quantities are interpreted graphically
in Fig. 5.10.
5.9
Limited Mutual Solubility—Ideal
Henrian Solutions
In Section 5.5, the region of two solids in the Ag-Cu phase diagram of Fig. 5.7 was described as a miscibility gap. That is, only
one continuous gs curve was assumed. If, somehow, the appearance of the liquid phase could be suppressed, the two solvus lines
in Fig. 5.7, when projected upward, would meet at a critical point,
above which one continuous solid solution would exist at all
compositions.
Such a description is justifiable only if the pure solid components have the same crystal structure, as is the case for Ag and
Cu. However, consider the Ag-Mg system, Fig. 5.11, in which
the terminal (Ag) solid solution is face-centered-cubic and the terminal (Mg) solid solution is hexagonal-close-packed. In this case,
one continuous curve for gs cannot be drawn. Each solid phase
must have its own separate Gibbs energy curve, as shown scheand μo(fcc)
matically in Fig. 5.11B at 800 K. In this figure, μo(hcp)
Mg
Ag
are the standard molar Gibbs energies of pure hcp Mg and pure
is the standard molar Gibbs energy of pure
fcc Ag, while μo(hcp-Mg)
Ag
(hypothetical) hcp Ag in the hcp-Mg phase.
Since the solubility of Ag in the hcp-Mg phase is limited, we
can, to a good approximation, describe it as a Henrian ideal
73
74
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
solution. That is, when a solution is sufficiently dilute in one component, we can approximate μEsolute ¼ RT ln γ solute by its value in an
infinitely dilute solution. That is, if Xsolute is small, we set
γ solute ¼ γ osolute where γ osolute , the Henrian activity coefficient, is
the value at Xsolute ¼ 0. That is, for sufficiently dilute solutions,
we assume that γ solute is independent of composition. Physically,
this means that in a very dilute solution, there is negligible interaction among solute particles because they are so far apart.
Hence, each additional solute particle added to the solution
makes the same contribution to the excess Gibbs energy of the
solution and so μEsolute ¼ dGE/dnsolute ¼ constant.
From the Gibbs-Duhem equation (Eq. 4.41), if dμEsolute ¼ 0, then
E
dμsolvent ¼ 0. Hence, in a Henrian solution, γ solvent is also constant
and equal to its value in an infinitely dilute solution. That is,
γ solvent ¼ 1, and the solvent behaves ideally. In summary, then,
for dilute solutions (Xso!vent 1.0) Henry’s law applies. Henry’s
law is usually defined as
γ solvent 1
γ solute γ osolute ¼ constant
(5.22)
but see Section 5.11 for a more rigorous definition.
Care must be exercised for solutions other than simple substitutional solutions. Henry’s law applies only if the ideal activity is
defined correctly for the applicable solution model. Solution
models are discussed in Part II.
Henrian activity coefficients can usually be expressed as
approximately linear functions of temperature:
RT ln γ oi ¼ a bT
(5.23)
where a and b are constants. If data are limited, it can further be
assumed that b 0 so that RT ln γ oi constant.
Treating the hcp-Mg phase in the Ag-Mg system (Fig. 5.11B) as
a Henrian solution, we write
oð fccÞ
oðhcpÞ
+ RT XAg ln aAg + XMg ln aMg
g hcp ¼ XAg μAg + XMg μMg
oð fccÞ
oðhcpÞ
¼ XAg μAg + XMg μMg
+ RT XAg ln γ oAg XAg + XMg ln XMg
(5.24)
γ oAg
where aAg and
are the activity and activity coefficient of silver
with respect to pure fcc silver as standard state. Let us now combine terms as follows:
h
i
oð fccÞ
oðhcpÞ
g hcp ¼ XAg μAg + RT ln γ oAg + XMg μMg
(5.25)
+ RT XAg ln XAg + XMg ln XMg
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Since γ oAg is independent of composition, let us define
oðhcpMgÞ
μAg
oð fccÞ
¼ μAg
+ RT ln γ oAg
(5.26)
and
From Eqs. (5.25), (5.26), it can be seen that, relative to μo(hcp)
Mg
defined in this way,
to the hypothetical standard state μo(hcp-Mg)
Ag
the hcp solution is ideal. Eqs. (5.25), (5.26) are illustrated in
Fig. 5.11B. It can be seen that as γ oAg becomes larger, the point
of tangency N moves to higher Mg concentrations. That is, as
μo(fcc)
) becomes more positive, the solubility of Ag
(μo(hcp-Mg)
Ag
Ag
in hcp-Mg decreases.
The chemical activity of a component was defined in Eq. (4.13).
As discussed in Section 4.4.1, the activity depends not only on the
chemical potential μi but also on the choice of the standard state
chemical potential μoi . If we define the activity of Ag in the hcp
solution as
oð fccÞ
hcp
RT ln aAg ¼ μAg μAg
(5.27)
then this is the activity of Ag with respect to pure fcc Ag as standard state and, as just shown in the dilute Henrian region where
XAg is small, aAg XAg γ oAg. On the other hand, if we define the activity of Ag as
oðhcpMgÞ
RT ln a0Ag ¼ μAg μAg
hcp
(5.28)
then this is the activity relative to the hypothetical (hcp-Mg) standard state and, as just shown, in the dilute Henrian region,
a0Ag XAg.
That is, a0Ag as defined by Eq. (5.28) is not the same as aAg
defined by Eq. (5.27) even though μhcp
Ag is the same in both equations. For more on this subject, see Section 5.11.
as defined by Eq. (5.26) is
It must be stressed that μo(hcp-Mg)
Ag
is not the same as, for examsolvent-dependent. That is, μo(hcp-Mg)
Ag
for Ag in dilute hcp-Cd solutions. However, for purple, μo(hcp-Cd)
Ag
poses of developing consistent thermodynamic databases for
multicomponent solution phases, it is advantageous to select
one single set of values for the chemical potentials of pure elements and compounds in metastable or unstable crystal struc, which are independent of the solvent.
tures, such as μo(hcp)
Ag
Such values, sometimes called lattice stabilities, may be obtained
(i) from measurements at high pressures if the structure is stable
at high pressure, (ii) as averages of the values for several solvents,
or (iii) from first-principle quantum mechanical or molecular
dynamic calculations. Tables (Dinsdale, 1991) of such values for
many elements in metastable or unstable states have been
75
76
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
prepared and agreed upon by the international community working in the field of phase diagram evaluation and thermodynamic
database optimization. The use of such solvent-independent
values, of course, means that even in very dilute solutions, nonzero excess Gibbs energy terms are usually required.
5.10 Henry’s Law, Raoult’s Law and
Standard States
Henry’s law is most often defined as in Eq. (5.22). However, a
more correct and less restrictive definition is
lim daB =dXB ¼ γ oB 6¼ ∞
(5.29)
XB ¼0
The more restrictive form, Eq. (5.22), simply says that, over a composition range sufficiently close to XB ¼ 0, the solute activity aB
may be approximated as lying on the line labeled “ideal Henrian”
in Fig. 5.12.
The Gibbs-Duhem equation, Eq. (2.82), combined with
Eq. (4.13) may be written as follows
XA d ln aA + XB d ln aB ¼ 0
1
mB = moB
Ide
al
Activity aA and aB
aB = 1
Ra
ou
ltia
0.8
n g
A
0.6
=
1,
aA
=X
A
aA
aB
0.4
,a
ian
ult
0.2
al
o
Ra
Ide
0
0
(5.30)
0.2
gB
=X
B
mB = moB'
B
a'B =1
=1
= 1,
nrian f B
Ideal He
0.4
o
= XB
,
g B = g B a'B
0.6
Mole fraction XB
Fig. 5.12 Activities of A and B in a binary system A-B.
0.8
1
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Hence,
XA daA XB dγ B
1 dXA
lim ðXA d ln aA + XB d ln ðγ B XB ÞÞ ¼ lim
X B ¼0
X B ¼0 aA dXA
γ B dXB
¼0
(5.31)
Hence,
lim daA =dXA ¼ 1:0
(5.32)
XA ¼1
Eq. (5.32) is Raoult’s law which is illustrated in Fig. 5.12. Consequently, over a composition range sufficiently close to XA ¼ 1,
the solvent activity aA may be approximated as lying on the line
labeled “ideal Raoultian γ A ¼ 1” in Fig. 5.12. That is, Henry’s law
and Raoult’s law are one and the same.
The Raoultian activity aB and Raoultian activity coefficient γ B
are defined by
RT ln aB ¼ RT lnðγ B XB Þ ¼ μB μoB
(5.33)
where the standard state chemical potential μoB lies on the ideal
Raoultian line at XB ¼ 1 as shown in Fig. 5.12.
We now define a Henrian activity a0B and a Henrian activity
coefficient fB by
0
RT ln a0B ¼ RT lnðfB XB Þ ¼ μB μoB
(5.34)
0
where the hypothetical standard state chemical potential μoB lies
on the ideal Henrian line at XB ¼ 1 as illustrated in Fig. 5.12. We
say that the standard state is pure B relative to a reference state
of B at infinite dilution in A. The activity coefficient fB ¼ 1 along
the Ideal Henrian line. In the actual solution in Fig. 5.12,
γ B < 1 and fB > 1.
By combining Eqs. (5.33), (5.34), we see that
0
(5.35)
RT ln aB =a0B ¼ μoB μoB ¼ constant
Hence, the ratios aB/a0B and γ B/fB are constant, independent of the
composition. When XB ¼ 0, γ B/fB ¼ γ oB/1. Hence, γ B/fB ¼ aB/a0B ¼ γ oB
at all compositions. That is, aB and a0B are represented by the same
curve in Fig. 5.12; only the vertical scale is multiplied by a
constant factor.
Sometimes in dilute metallic solutions, particularly in Fe as
solvent, activities are defined as
a00B ¼ fB00 ðw=oÞB
(5.36)
77
78
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
where (w/o)B is the weight percentage of solute B and the activity
coefficient f B00 ¼ 1 when (w/o)B ¼ 0 (i.e., the reference state is B at
infinite dilution in A):
00
(5.37)
RT ln a00B ¼ RT ln fB00 ðw=oÞB ¼ μB μoB
00
where μoB is the chemical potential of B in a standard state lying on
an ideal Henrian line at (w/o)B ¼ 1 when the composition axis is
00
(w/o)B rather than XB. Note that μoB is a hypothetical standard state
lying on the ideal Henrian line. It is not the same as μB in the actual
solution at (w/o)B ¼ 1.
To convert from aB00 to aB, we combine Eqs. (5.33), (5.37):
f 00 ðw=oÞB o00
RT ln a00B =aB ¼ RT ln B
¼ μB μoB ¼ constant
γ B XB
(5.38)
The constant is evaluated at a composition where fB00 and γ B are
both known, namely, at XB ¼ 0 where fB00 ¼ 1 and γ B ¼ γ oB. When
XB ¼ 0, (w/o)B/XB ¼ 100 MB/MA.
where MA and MB are the molecular weights. Hence, at all
compositions
a00B =aB ¼
100MB
¼ constant
γ oB MA
(5.39)
In aqueous solutions, activities of solutes are generally
expressed as
RT ln ai ¼ RT lnðfi mi Þ ¼ μi μo∗
(5.40)
i
where μo∗
i is the chemical potential of solute i in a hypothetical
standard state lying on an ideal Henrian line at the composition
mi ¼ 1 where mi is the molality of i defined as mi ¼ moles of i
per kilogram of solvent (H2O) and where fi ¼ 1 when mi ¼ 0. (See
also Section 14.10.2 regarding Wagner’s interaction parameter formalism when compositions are expressed per unit mass of
solvent.)
5.11 Single Ion Activities
In a dilute solution of NaCl in H2O,
μNaCl ¼ ∂G=∂nNaCl ¼ ∂G=∂nNa + + ∂G=∂nCl
(5.41)
∂nNa + ¼ ∂nCl ¼ ∂nNaCl
(5.42)
where
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
The activity of NaCl relative to a one-molal standard state is
aNaCl ¼ mNa + mCl fNaCl
(5.43)
where fNaCl ¼ 1 at infinite dilution.
If we write
μNa + ¼ ∂G=∂nNa + and μCl ¼ ∂G=∂nCl
(5.44)
aNaCl ¼ aNa + aCl ¼ ðmNa + fNa + ÞðmCl fCl Þ
(5.45)
then
where fNaCl ¼ fNa+ fCl. However, these definitions of μNa+ and μCl
do not correspond to measureable quantities since one cannot
experimentally transfer a measureable amount of a singlecharged species into a solution.
It is nevertheless useful in certain models such as the Debye€ ckel equations to formally calculate theoretical single-ion
Hu
activities and to calculate the activities of neutral components
therefrom. The single-ion activities, however, are constructs of
the particular solution model.
5.12
The “Activity” of a Solution
Consider a system with three solution phases (α, β, and γ) and a
stoichiometric compound δ, and suppose that their molar Gibbs
energies at a given temperature are as shown in Fig. 5.13. At this
temperature and for any overall composition between points
P and Q, a two-phase mixture of the phases α and β is stable, with
phase compositions P and Q, and the phases γ and δ are not present at equilibrium. At any overall system composition between
points P and Q, the activity, aδ, of the stoichiometric δ-phase is
given by the construction shown in the figure where aδ < 1.0.
The further is μoδ above the equilibrium tangent line PQ, the smaller is the value of aδ. The activity of the compound is thus a convenient indicator of how close the compound is to being stable. If
aδ is close to 1.0, then the compound might actually be observed
in practice if conditions vary slightly from equilibrium or are
changed slightly.
It is useful to define a similar property indicating how close a
solution phase such as the γ-phase is to being stable. A suggestion
(Hack, 2008) is shown in the figure. A line is drawn tangent to the
Gibbs energy curve of the γ-phase and parallel to the equilibrium
tangent line PQ. As shown in the figure, the vertical distance RV
between the two lines is equal to RT ln aγ where aγ may be called
79
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
γ
Molar Gibbs energy
80
mδ
o
R
δ - phase
RTlnaδ< 0
RTlnaγ< 0
a
P
0
mδ
Equilibriu
0.20
m tangen
0.40
V
t line
0.60
β
Q
0.80
1.00
Mole fraction B
Fig. 5.13 Gibbs energy curves at constant temperature in a system A-B illustrating the definition of the “activity” of a
solution phase.
the “activity” of the γ-phase. This activity is constant at all overall
system compositions between points P and Q.
Note that this definition of the “activity” of a solution is not
standard thermodynamic terminology, but is nevertheless a useful concept. When a solution phase is stable, its “activity” is equal
to 1.0. The further it is from being stable, the smaller is its
“activity.”
The definition may be extended to ternary and higher-order
systems by taking the vertical distance between the equilibrium
tangent plane and a parallel plane tangent to the solution in
question.
5.13 Geometry of Binary TemperatureComposition Phase Diagrams
In Section 3.1.1, it was shown that when three phases are at
equilibrium in a binary system at constant pressure, the system
is invariant (F ¼ 0). There are two general types of three-phase
invariants in binary phase diagrams. These are the eutectic-type
and peritectic-type invariants as illustrated in Fig. 5.14. Let
the three phases concerned be called α, β, and γ, with β as the
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Fig. 5.14 Geometric units of binary phase diagrams, illustrating rules of
construction (Pelton, 2014).
central phase as shown in the figure. The phases α, β, and γ
can be solid, liquid, or gaseous. At a eutectic-type invariant, the
following invariant reaction occurs isothermally as the system is
cooled:
β!α+γ
(5.46)
while at a peritectic-type invariant, the invariant reaction upon
cooling is
α+γ!β
(5.47)
Some examples of eutectic-type invariants are as follows:
(i) Eutectics in which α ¼ solid1, β ¼ liquid, and γ ¼ solid2. The eutectic reaction is liq ! s, + s2. (ii) Monotectics in which α ¼ liquid1, β ¼
liquid2, and γ ¼ solid. The monotectic reaction is liq2 ! liq1, + s.
(iii) Eutectoids in which α ¼ solid1, β ¼ solid2, and γ ¼ solid3. The
eutectoid reaction is s2 ! s1 + s3. (iv) Catatectics in which α ¼ liquid,
β ¼ solid1, and γ ¼ solid2. The catatectic reaction is s1 ! liq + s2.
81
82
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
Some examples of peritectic-type invariants are as follows:
(i) Peritectics in which α ¼ liquid, β ¼ solid1, and γ ¼ solid2. The
peritectic reaction is liq + s2 ! sI. (ii) Syntectics (Fig. 5.8K) in which
α ¼ liquid1, β ¼ solid, and γ ¼ liquid2. The syntectic reaction is
liq1 + liq2 ! s. (iii) Peritectoids in which α ¼ solid1, β ¼ solid2,
and γ ¼ solid3. The peritectoid reaction is s1 + s3 ! s2.
An important rule of construction that applies to invariants in
binary phase diagrams is illustrated in Fig. 5.14. This extension
rule states that at an invariant, the extension of a boundary of a
two-phase region must pass into the adjacent two-phase region
and not into a single-phase region. Examples of both correct
and incorrect constructions are given in Fig. 5.14. To understand
why the “incorrect extension” shown is not correct, consider that
the (α + γ)/γ-phase boundary line indicates the composition of the
γ-phase in equilibrium with the α-phase, as determined by the
common tangent to the Gibbs energy curves. Since there is no reason for the Gibbs energy curves or their derivatives to change
slope discontinuously at the invariant temperature, the extension
of the (α + γ)/γ-phase boundary also represents the stable phase
boundary under equilibrium conditions. Hence, for this line to
extend into a region labeled as single-phase-γ is incorrect. This
extension rule is a particular case of the general Schreinemakers’
rule (see Section 7.7.1).
Two-phase regions in binary phase diagrams can terminate
(i) on the pure component axes (at XA ¼ 1 or XB ¼ 1) at a transformation point of pure A or B, (ii) at a critical point of a miscibility
gap, or (iii) at an invariant. Two-phase regions can also exhibit
maxima or minima. In this case, both phase boundaries must pass
through their maximum or minimum at the same point as shown
in Fig. 5.14.
All the geometric units of construction of binary phase diagrams have now been discussed. The phase diagram of a binary
alloy system will usually exhibit several of these units. As an example, the Fe-Mo phase diagram was shown in Fig. 1.1. The invariants in this system are peritectics at 1612, 1498, and 1453°C;
eutectoids at 1240, 1233, and 308°C; and peritectoids at 1368
and 1250°C. The two-phase (liquid + bcc) region passes through
a minimum at XMo ¼ 0.2. Between 910 and 1390°C is a two-phase
(α + γ) “γ-loop.” Pure Fe adopts the fcc γ structure between these
two temperatures but exists as the bcc α-phase at higher and lower
temperatures. Mo, on the other hand, is more soluble in the bcc
< μo(fcc-Fe)
as discussed
than in the fcc structure. That is, μo(bcc-Fe)
Mo
Mo
in Section 5.10. Therefore, small additions of Mo stabilize the bcc
structure.
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
In the CaO-SiO2 phase diagram, Fig. 5.10, we observe eutectics
at 1437, 1463, and 2017°C; a monotectic at 1689°C; and a peritectic
at 1464°C. The compound Ca3SiO5 dissociates upon heating to
CaO and Ca2SiO4 by a peritectoid reaction at 1799°C and dissociates upon cooling to CaO and Ca2SiO4 by a eutectoid reaction at
1300°C. Maxima are observed at 2154 and 1540°C. At 1465°C, there
is an invariant associated with the tridymite ! cristobalite transition of SiO2. This is either a peritectic or a catatectic depending
upon the relative solubility of CaO in tridymite and cristobalite.
However, these solubilities are very small and unknown.
At 1437°C, Ca2SiO4 undergoes an allotropic transformation. This
gives rise to two invariants, one involving Ca2SiO4(α), Ca2SiO4(β),
and Ca3SiO5 and the other Ca2SiO4(α), Ca2SiO4(β), and Ca3Si2O7.
Since the compounds are all essentially stoichiometric, these
two invariants are observed at almost exactly the same temperature of 1437°C, and it is not possible to distinguish whether they
are of the eutectoid or peritectoid type.
5.14
Effects of Grain Size, Coherency, and
Strain Energy
The foregoing phase diagram calculations have assumed that
the grain size of the solid phases is large enough that surface
(interfacial) energy contributions to the Gibbs energy can be
neglected. For very fine grain sizes in the submicron range, however, the interfacial energy can be appreciable and can have an
influence on the phase diagram. This effect can be important in
the development of materials with grain sizes in the nanometer
range. Interfacial energy increases the Gibbs energy of a phase
and hence decreases its thermodynamic stability. In particular,
the liquidus temperature of a fine-grained solid will be lowered.
If a suitable estimate can be made for the grain boundary energy,
then it can be added to the Gibbs energy expression for purposes
of calculating phase equilibriums.
If the crystal lattice is continuous across the boundary between
two solid phases, the boundary is said to be coherent. The mismatch in lattice parameters then creates a positive strain energy
that increases the Gibbs energy of the system. For example, in
Fig. 5.6, if the two-phase solid/solid boundary is coherent, then
it is rendered less stable relative to the single-phase solid region
with the result that the consolute temperature TC is decreased.
If a suitable estimate of the strain energy can be made, then it
can be added to the Gibbs energy expression for the purpose of
calculating the phase equilibria.
83
84
Chapter 5 THERMODYNAMIC ORIGIN OF PHASE DIAGRAMS
References
Dinsdale, A.T., 1991. Calphad J. 15, 317–425.
Hack, K., 2008. SGTE Casebook: Thermodynamics at Work, second ed. CRC Press,
Boca Raton, FL.
Hansen, M., 1958. Constitution of Binary Alloys, second ed. McGraw-Hill Book
Company Inc., New York
Hultgren, R.P.D., Desai, D., Hawkins, T., Gleiser, M., Kelley, K.K., Wagman, D.D.,
1973. Selected Values of Thermodynamic Properties of Binary Alloys. American
Society of Metals, Metals Park, OH.
Pelton, A.D., 2001. Kostorz, G. (Ed.), Phase Transformations in Materials.
(Chapter 1). Wiley, Weinheim.
Pelton, A.D., 2014. Laughlin, D.E., Hono, K. (Eds.), Physical Metallurgy. fifth ed.
(Chapter 3). Elsevier, New York.
Pelton, A.D., Thompson, W.T., 1975. Prog. Solid State Chem. 10, 119–155.
Van Laar, J.J., 1908. Z. Phys. Chem. 63, 216–253.
List of Websites
FactSage, Montreal, www.factsage.com.
SGTE, Scientific Group Thermodata Europe, www.sgte.org.
6
TERNARY TEMPERATURECOMPOSITION PHASE
DIAGRAMS
CHAPTER OUTLINE
6.1 The Ternary Composition Triangle 85
6.2 Ternary Space Model 86
6.3 Polythermal Projections of Liquidus Surfaces 88
6.4 Ternary Isothermal Sections 91
6.4.1 Topology of Ternary Isothermal Sections 93
6.5 Ternary Isopleths (Constant Composition Sections) 95
6.5.1 Quasibinary Phase Diagrams 96
6.6 First-Melting (Solidus) Projections of Ternary Systems 97
6.7 Phase Diagram Projections in Quaternary and Higher-Order
Systems 99
6.7.1 Liquidus Projections 99
6.7.2 Solidus Projections 100
References 102
List of Websites 102
6.1
The Ternary Composition Triangle
In a ternary system with components A-B-C, the sum of the
mole fractions is unity, (XA +XB +XC) ¼ 1. Hence, there are two
independent composition variables. A representation of composition, symmetrical with respect to all three components, may be
obtained with the equilateral composition triangle as shown in
Fig. 6.1 for the Bi-Sn-Cd system. Compositions at the corners of
the triangle correspond to the pure components. Compositions
along the edges of the triangle correspond to the three binary subsystems Bi-Sn, Sn-Cd, and Cd-Bi. Lines of constant mole fraction
XBi are parallel to the Sn-Cd edge, while lines of constant XSn and
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00006-3
# 2019 Elsevier Inc. All rights reserved.
85
86
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
Fig. 6.1 Projection of the liquidus surface of the Bi-Sn-Cd system onto the ternary
composition triangle. Small arrows show the crystallization path of an alloy of
overall composition at point a (P ¼ 1 bar) (Pelton, 2014). Data from Bray, H.F., Bell, F.D.,
Harris, S.J., 1961–1962. J. Inst. Met. 90, 24.
XCd are parallel to the Cd-Bi and Bi-Sn edges, respectively. For
example, at point a in Fig. 6.1, XBi ¼ 0.05, XSn ¼ 0.45, and
XCd ¼ 0.50.
Similar equilateral composition triangles can be drawn with
coordinates in terms of weight % of the three components.
6.2
Ternary Space Model
A ternary temperature-composition phase diagram at constant
total pressure may be plotted as a three-dimensional “space
model” within a right triangular prism with the equilateral composition triangle as base and temperature as vertical axis. Such a
space model for a simple eutectic ternary system A-B-C is illustrated in Fig. 6.2. The three vertical faces of the prism are found
on the phase diagrams of the three binary subsystems, A-B,
B-C, and C-A that, in this example, are all simple eutectic binary
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
Fig. 6.2 Perspective view of ternary space model of a simple eutectic ternary
system. e1, e2, and e3 are the binary eutectics, and E is the ternary eutectic.
The base of the prism is the equilateral composition triangle (pressure ¼ constant)
(Pelton, 2014).
systems. The binary eutectic points are e1, e2, and e3. Within the
prism, we see three liquidus surfaces descending from the melting
points of pure A, B, and C. Compositions on these surfaces correspond to compositions of liquid in equilibrium with A-, B-, and
C-rich solid phases.
In a ternary system at constant pressure, the Gibbs Phase Rule,
Eq. (3.1), becomes
F ¼4P
(6.1)
When the liquid and one solid phase are in equilibrium, P ¼ 2.
Hence, F ¼ 2, and the system is bivariant. A ternary liquidus is thus
a two-dimensional surface. We may fix two variables, say T and one
composition coordinate of the liquid, but then, the other liquid
composition coordinate and the composition of the solid are fixed.
The A- and B-liquidus surfaces in Fig. 6.2 intersect along the
line e1E. Liquids with compositions along this line are therefore
in equilibrium with A-rich and B-rich solid phases simultaneously. That is, P ¼ 3, and so, F ¼ 1. Such “valleys” are thus called
univariant lines. The three univariant lines meet at the ternary
eutectic point E at which P ¼ 4 and F ¼ 0. This is an invariant point
since the temperature and the compositions of all four phases in
equilibrium are fixed.
87
88
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
6.3
Polythermal Projections of Liquidus
Surfaces
A two-dimensional representation of the ternary liquidus surface may be obtained as an orthogonal projection upon the base
composition triangle. Such a polythermal projection of the liquidus of the Bi-Sn-Cd system (Bray et al., 1961-1962) is shown in
Fig. 6.1. This is a simple eutectic ternary system with a space
model like that shown in Fig. 6.2. The constant temperature lines
in Fig. 6.1 are called liquidus isotherms. The univariant valleys are
shown as heavier lines. By convention, the large arrows indicate
the directions of decreasing temperature along these lines.
Let us consider the sequence of events occurring during the
equilibrium cooling from the liquid of an alloy of overall composition a in Fig. 6.1. Point a lies within the field of primary crystallization of Cd. That is, it lies within the composition region in
Fig. 6.1 in which Cd-rich solid will be the first solid to precipitate
upon cooling. As the liquid alloy is cooled, the Cd-liquidus surface
is reached at T 465 K (slightly below the 473 K isotherm). A solid
Cd-rich phase begins to precipitate at this temperature. Now, in
this particular system, Bi and Sn are nearly insoluble in solid
Cd, so that the solid phase is virtually pure Cd. (Note that this fact
cannot be deduced from Fig. 6.1 alone.) Therefore, as solidification proceeds, the liquid becomes depleted in Cd, but the ratio
XSn/XBi in the liquid remains constant. Hence, the composition
path followed by the liquid (its crystallization path) is a straight
line passing through point a and projecting to the Cd-corner of
the triangle. This crystallization path is shown in Fig. 6.1 as the
line ab.
In the general case in which a solid solution rather than a pure
component or stoichiometric compound is precipitating, the
crystallization path will not be a straight line. However, for equilibrium cooling, a straight line joining a point on the crystallization path at any T to the overall composition point a will extend
through the composition, on the solidus surface, of the solid
phase in equilibrium with the liquid at that temperature. That
is, the compositions of the two equilibrium phases and the overall
system composition always lie on the same straight tie-line as is
required by mass balance considerations.
When the composition of the liquid has reached point b in
Fig. 6.1 at T 435 K, the relative proportions of the solid Cd and
liquid phases at equilibrium are given by the lever rule applied
to the tie-line dab: (moles of liquid)/(moles of Cd) ¼ da/ab where
da and ab are the lengths of the line segments. Upon further cooling, the liquid composition follows the univariant valley from b to
E, while Cd and Sn-rich solids coprecipitate as a binary eutectic
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
mixture. When the liquidus composition attains the ternary
eutectic composition E at T 380 K, an invariant ternary eutectic
reaction occurs:
liquid ! s1 + s2 + s3
(6.2)
where s1, s2, and s3 are the three solid phases and where the compositions of all four phases (as well as T) remain fixed until all liquid has solidified.
In order to illustrate several of the features of polythermal projections of liquidus surfaces, a projection of the liquidus surface of
a hypothetical system A-B-C is shown in Fig. 6.3. For the sake of
simplicity, isotherms are not shown; only the univariant lines
are drawn, with arrows to show the directions of decreasing temperature. The binary subsystems A-B and C-A are simple eutectic
systems, while the binary subsystem B-C contains one congruently melting binary phase, ε, and one incongruently melting
binary phase, δ, as shown in the insert in Fig. 6.3. The letters e
Fig. 6.3 Projection of the liquidus surface of a system A-B-C. The binary subsystems
A-B and C-A are simple eutectic systems. The binary phase diagram B-C is shown in the
insert. All solid phases are assumed to be pure stoichiometric components or
compounds. Small arrows show crystallization paths of alloys of compositions at points
a and b (Pelton, 2014).
89
90
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
and p indicate binary eutectic and peritectic points. The δ and ε
phases are called binary compounds since they have compositions
within a binary subsystem. Two ternary compounds, η and ζ, with
compositions within the ternary triangle, as indicated in Fig. 6.3,
are also found in this system. All compounds and pure solid A, B,
and C (the α-, β-, and γ-phases) are assumed to be stoichiometric
(i.e., there is no solid solubility). The fields of primary crystallization of all the solids are indicated in parentheses in Fig. 6.3. The
composition of the ε phase lies within its field, since it is a congruently melting compound, while the composition of the δ phase
lies outside of its field since it is an incongruently melting compound. Similarly for the ternary compounds, η is a congruently
melting compound, while ζ is incongruently melting. For the congruent compound η, the highest temperature on the η-liquidus
occurs at the composition of η.
The univariant lines meet at a number of ternary eutectic points
E (three arrows converging on the point), a ternary peritectic point
P (one arrow entering, two arrows leaving the point), and several
0
ternary quasi-peritectic points P (two arrows entering, one arrow
leaving). Two saddle points s are also shown. These are points of
maximum T along the univariant line but of minimum T on the
liquidus surface along a section joining the compositions of the
two solids. For example, s1 is a maximum point along the univar0
iant E1P3 but is a minimum point on the liquidus along the
straight line ζs1η.
Let us consider the events occurring during equilibrium cooling from the liquid of an alloy of overall composition a in Fig. 6.3.
The primary crystallization product will be the ε phase. Since this
is a pure stoichiometric solid, the crystallization path of the liquid
will be along a straight line passing through point a and extending
to the composition of ε as shown in the figure.
Solidification of ε continues until the liquid attains a composition on the univariant valley. Thereafter, the liquid composition
descends along the valley toward the point P’1 in coexistence with
0
ε and ζ. At point P1, the invariant ternary quasi-peritectic reaction
occurs isothermally:
liquid + ε ! δ + ζ
(6.3)
Since there are two reactants, there are two possible outcomes:
(i) The liquid is completely consumed before the ε phase, and
0
solidification will be complete at the point P1; (ii) ε is completely
consumed before the liquid, and solidification will continue with
0
decreasing T along the univariant line P1E1 with coprecipitation
of δ and ζ until, at E1, the liquid will solidify eutectically
(liquid ! δ + ζ + η). TO determine whether outcome (i) or (ii)
occurs, we use the mass balance criterion that, for three-phase
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
equilibrium, the overall composition a must always lie within the
tie-triangle formed by the compositions of the three phases. Now,
the triangle joining the compositions of δ, ε, and ζ does not contain the point a, but the triangle joining the compositions of δ, ζ,
0
and liquid at P1 does contain the point a. Hence, outcome
(ii) occurs.
An alloy of overall composition b in Fig. 6.3 solidifies with ε as
primary crystallization product until the liquid composition contacts the univariant line. Thereafter, coprecipitation of ε and β
occurs with the liquid composition following the univariant valley
until the liquid reaches the peritectic composition P. An invariant
ternary peritectic reaction then occurs isothermally:
liquid + ε + β ! ζ
(6.4)
Since there are three reactants, there are three possible outcomes:
(i) The liquid is consumed before either ε or β and solidification
terminates at P, (ii) ε is consumed first and solidification then con0
tinues along the path PP3, or (iii) β is consumed first and solidifi0
cation continues along the path PP1. Which outcome actually
occurs depends on whether the overall composition b lies within
the tie-triangle (i) εβζ, (ii) βζP, or (iii) εζP. In the example shown,
outcome (i) will occur.
A polythermal projection of the liquidus surface of the Zn-MgAl system is shown in Fig. 6.4. There are 11 primary crystallization
fields of nine binary solid solutions (with limited solubility of the
third component) and of two ternary solid solutions. There are
three ternary eutectic points, three saddle points, three ternary
peritectic points, and five ternary quasi-peritectic points.
6.4
Ternary Isothermal Sections
Polythermal projections of the liquidus surface do not provide
information on the compositions of the solid phases at equilibrium. However, this information can be presented at any one
temperature on an isothermal section such as that shown for
the Bi-Sn-Cd system at 423 K in Fig. 6.5. This phase diagram is a
constant temperature slice through the space model of Fig. 6.2.
The liquidus lines bordering the one-phase liquid region of
Fig. 6.5 are identical to the 423 K isotherms of the liquidus projection in Fig. 6.1. Point c in Fig. 6.5 is point c on the univariant line in
Fig. 6.1. An alloy with overall composition in the one-phase liquid
region of Fig. 6.5 at 423 K will consist of a single liquid phase. If the
overall composition lies within one of the two-phase regions, then
the compositions of the two phases are given by the ends of the
tie-line that passes through the overall composition. For example,
91
92
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
Zn
>
0. 9
0. 8
>
P'
0. 2
400
450
0. 4
0. 6
0
50
>
0. 3
0. 7
0. 1
Mg2Zn11
0
> > 40 50
4
hcp
e 400
p
e
E
0. 5
Laves
0. 4
500
s
P' < P
XZn = 0.1
550
>
0
45
0
40
4
Tau
0. 8
Phi
<
< 0
0
0. 9
>
<
s
>>
> >>
0
45
Gamma
0.8
e
0.7
0.6
600
P'
>E
>
>
0. 2
>
0.9
fcc
>
A
0. 7
0. 3
P'
<
>
h
cp
0
50 0
55
0
60
0. 1
>
<
P
p
e< P
E P'
B
Mg
s
0. 6
p
0. 5
Mg2Zn3
MgZn
550
0.5
e e
0.4
Mole fraction
650
0.3
0.2
βAlMg
0.1
Al
Fig. 6.4 Polythermal projection of the liquidus surface of the Zn-Mg-Al system at P ¼ 1 bar (T ¼ °C) (see FactSage).
Fig. 6.5 Isothermal section of the Bi-Sn-Cd system at 423 K at P ¼ 1 bar. Extents of solid solubility in Bi and Sn have been
exaggerated for clarity of presentation (Pelton, 2014). Data from Bray, H.F., Bell, F.D., Harris, S.J., 1961–1962. J. Inst. Met. 90, 24.
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
a sample with overall composition p in Fig. 6.5 will consist of a liquid of composition q on the liquidus and a solid Bi-rich alloy of
composition r on the solidus. The relative proportions of the
two phases are given by the Lever Rule:
ðmoles of liquidÞ=ðmoles of solidÞ ¼ pr=pq
where pr and pq are the lengths of the tie-line segments.
In the case of solid Cd, the solid phase is nearly pure Cd, so all tielines of the (Cd + liquid) region converge nearly to the corner of
the triangle. In the case of the Bi- and Sn-rich solids, some solid solubility is observed. (The actual extent of this solubility is exaggerated
in Fig. 6.5 for the sake of clarity of presentation.) Alloys with overall
compositions rich enough in Bi or Sn to lie within the single-phase
(Sn) or (Bi) regions of Fig. 6.5 will consist at 423 K of single-phase
solid solutions. Alloys with overall compositions at 423 K in the
two-phase (Cd + Sn) region will consist of two solid phases.
Alloys with overall compositions within the three-phase triangle dcf will, at 423 K, consist of three phases: Cd- and Sn-rich solids
with compositions at d and f and liquid of composition c. To
understand this better, consider an alloy of composition a in
Fig. 6.5, which is the same composition as the point a in
Fig. 6.1. In Section 6.3, we saw that when an alloy of this composition is cooled, the liquid follows the path ab in Fig. 6.1 with primary precipitation of Cd and then follows the univariant line with
coprecipitation of Cd and Sn so that at 423 K, the liquid is at the
composition point c and two solid phases are in equilibrium with
the liquid.
6.4.1
Topology of Ternary Isothermal Sections
At constant temperature, the Gibbs energy of each phase in a
ternary system is represented as a function of composition by a
surface plotted in a right triangular prism (as in Fig. 6.2) with
Gibbs energy as vertical axis and the composition triangle as base.
Just as the compositions of phases at equilibrium in binary systems are determined by the points of contact of a common tangent line to their isothermal Gibbs energy curves (Fig. 5.1), so
the compositions of phases at equilibrium in a ternary system
are given by the points of contact of a common tangent plane
to their isothermal Gibbs energy surfaces (or, equivalently as discussed in Section 5.1, by minimizing the total Gibbs energy of the
system.) A common tangent plane can contact two Gibbs energy
surfaces at an infinite number of pairs of points, thereby generating an infinite number of tie-lines within a two-phase region on an
isothermal section. A common tangent plane to three Gibbs
93
94
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
Fig. 6.6 A tie-triangle in a ternary isothermal (and isobaric) section illustrating the
lever rule and the extension rule (Schreinemakers’ rule—Sections 7.3 and 7.7.1).
energy surfaces contacts each surface at a unique point, thereby
generating a three-phase tie-triangle.
Hence, the principal topological units of construction of an
isothermal ternary phase diagram are three-phase (α + β + γ)
tie-triangles as in Fig. 6.6 with their accompanying two-phase
and single-phase areas. Each corner of the tie-triangle contacts
a single-phase region, and from each edge of the triangle, there
extends a two-phase region. The edge of the triangle is a limiting
tie-line of the two-phase region.
For overall compositions within the tie-triangle, the compositions of the three phases at equilibrium are fixed at the corners of
the triangle. The relative proportions of the three phases are given
by the lever rule of tie-triangles, which can be derived from mass
balance considerations. At an overall composition q in Fig. 6.6, for
example, the relative proportion of the γ-phase is given by projecting a straight line from the γ-corner of the triangle (point c)
through the overall composition q to the opposite side of the triangle, point p. Then, (moles of γ)/(total moles) ¼ qplcp if compositions are expressed as mole fractions, or (weight of γ)/(total
weight) ¼qp/cp if compositions are in weight percent.
Isothermal ternary phase diagrams are generally composed of
a number of these topological units. An example for the Zn-Mg-Al
system at 25°C is shown in Fig. 1.2. At 25°C, hcp-Mg, hcp-Zn, and
Mg2Zn11 are nearly stoichiometric compounds, while Mg and Zn
have very limited solubility in fcc-Al. The gamma and βAlMg
phases are binary compounds that exhibit very narrow ranges
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
of stoichiometry in the Mg-Al binary system and in which Zn is
very sparingly soluble. The Laves, Mg2Zn3, and MgZn phases
are virtually stoichiometric binary compounds in which Al is soluble to a few percent. Their ranges of stoichiometry are thus
shown in Fig. 1.2 as very narrow lines. The tau phase is a ternary
phase with a small single-phase region of stoichiometry as shown.
An examination of Fig. 1.2 will show that it is composed of the
topological units of Fig. 6.6.
An extension rule, a particular case of Schreinemakers’ rule (see
Sections 7.3 and 7.7.1), for ternary tie-triangles is illustrated in
Fig. 6.6 at points a and b. At each corner of the tie-triangle, the
extension of the boundaries of the single-phase regions, indicated
by the broken lines, must either both project into the triangle as at
point a, or must both project outside the triangle as at point b. Furthermore, the angle between the boundaries must not be greater
than 180°.
Another important rule of construction, whose proof is evident, is that within any two-phase region, tie-lines must never
cross one another.
It should also be noted that equations analogous to Eqs. (5.6),
(5.7) may be written relating the activities of a component at the
two ends of a two-phase tie-line.
6.5
Ternary Isopleths (Constant
Composition Sections)
A vertical isopleth, or constant composition section, through
the space model of the Bi-Sn-Cd system, is shown in Fig. 6.7.
The section follows the line AB in Fig. 6.1.
The phase fields in Fig. 6.7 indicate the phases that are present
when an alloy with an overall composition on the line AB is equilibrated at any temperature. For example, consider the cooling,
from the liquid state, of an alloy of composition a that is on the
line AB (see Fig. 6.1). At T 465 K, precipitation of the solid
(Cd) phase begins at point a in Fig. 6.7. At T 435 K (point b in
Figs. 6.1 and 6.7), the solid (Sn) phase begins to appear. Finally,
at the eutectic temperature TE, the ternary eutectic reaction
(Eq. 6.2) occurs, leaving solid (Cd) + (Bi) + (Sn) at lower temperatures. The intersection of the isopleth with the univariant lines
in Fig. 6.1 occurs at points f and g that are also shown in
Fig. 6.7. The intersection of this isopleth with the isothermal section at 423 K is shown in Fig. 6.5. The points s, t, u, and v of Fig. 6.5
are also shown in Fig. 6.7.
95
96
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
Fig. 6.7 Isopleth (constant composition section) of the Bi-Sn-Cd system at P ¼ 1 bar
following the line AB of Fig. 6.1. (Extents of solid solubility in the terminal solid
solutions have been exaggerated for clarity of presentation.) (Pelton, 2014).
It is important to note that on an isopleth, the tie-lines do not,
in general, lie in the plane of the diagram. Therefore, the diagram
provides information only on which phases are present, not on
their compositions or relative proportions. The phase boundary
lines on an isopleth do not in general indicate the compositions
or relative proportions of the phases but only the temperature
at which a phase appears or disappears for a given overall composition. The lever rule cannot be applied on an isoplethal section.
An isoplethal section of the Zn-Mg-Al system at 10 mol% Zn is
shown in Fig. 1.3A. The liquidus surface of the diagram may be
compared with the liquidus surface along the line where XZn ¼ 0.1
in Fig. 6.4 and to the section at XZn ¼ 0.1 in the isothermal section
at 25°C in Fig. 1.2. Point p on the two-phase (tau + gamma) field in
Fig. 1.3A at 25°C corresponds to point p in Fig. 1.2 where the equilibrium compositions of the gamma and tau phases are given by
the ends of the tie-line. These individual phase compositions cannot be read from Fig. 1.3A.
The geometric rules that apply to isoplethal sections will be
discussed in Chapter 7.
6.5.1
Quasibinary Phase Diagrams
The binary CaO-SiO2 phase diagram in Fig. 5.10 is actually an
isopleth of the ternary Ca-Si-O system along the line nO ¼
(nCa +2nSi). However, all tie-lines in Fig. 5.10 lie within (or virtually
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
97
within) the plane of the diagram because nO ¼ (nCa +2nSi) in every
phase. Therefore, the diagram is called a quasi-binary phase diagram. Similarly, the K2CO3-Na2CO3 phase diagram in Fig. 5.5 is a
quasi-binary isoplethal section of the quaternary K-Na-C-O system.
6.6
First-Melting (Solidus) Projections
of Ternary Systems
Liquidus projections of phase diagrams of ternary and higherorder systems are commonly found in the literature. Solidus projections, on the other hand, are virtually never seen, and their
interpretation has been systematically discussed only recently
(Eriksson et al., 2013).
The liquidus projection of the Zn-Mg-Al system was shown in
Fig. 6.4. The solidus projection for this system is shown in Fig. 6.8.
It can be seen that for each ternary eutectic point (E), ternary peritectic point (P), and ternary quasi-peritectic point (P0 ) on the
liquidus projection in Fig. 6.4, there is a corresponding isothermal
tie-triangle on the solidus projection, each labeled with the temperature of the invariant reaction. For example, at a temperature
just below 471°C, a solid alloy with a composition within the
Zn
o
T C
hcp
0.9
0.1
Mg2 Zn11
630
0.8
0.2
351
0.7
0.3
Laves
580
0.6
0.4
449
33
7
c+
34
8
0.8
0.7
47
0
460
GAMMA
0.6
330
fcc
48
0
392
0.2
0.1
0.9
380
71
0.9
Mg
hcp
Ta
u4
0.8
H I MA
+ PGAM
p
c
h +
430
38
0
0.7
i
Ph
371
+
49
0
0.3
34
5
La
ve
s
37
0
0.6
u
Ta
0.4
480
fc
0.5
0.5
Mg2 Zn3
Mg12 Zn13
530
366
451
0.5 4510.4
Beta0.3
Mole fraction
0.2
0.1
fcc
Al
Fig. 6.8 First-melting projection of the phase diagram of the Zn-Mg-Al system (temperature in °C) (Eriksson et al., 2013).
98
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
triangle labeled “fcc + Laves + Tau 471” will, at thermodynamic
equilibrium, consist of these three phases. This alloy will first melt
isothermally (peritectically in this case) at 471°C to form a liquid
phase with a composition at the peritectic point labeled A in
Fig. 6.4. (To prevent the diagram from becoming too crowded,
only one other tie-triangle in Fig. 6.8 has been fully labeled.)
Each isothermal tie-triangle in Fig. 6.8 is surrounded on every
side by nonisothermal two-solid-phase regions containing
isothermal tie-lines and is in contact at every corner with nonisothermal single-solid-phase regions similar to Fig. 6.6. In each
two-solid-phase region of Fig. 6.8, an equilibrated solid alloy with
overall composition on a tie-line will first start to melt at the temperature of the tie-line, at which temperature the two solid phases
will have compositions at the ends of the tie-line. In the singlesolid-phase regions, the isotherms indicate the solidus temperatures. (Again, only a few of the tie-line temperatures have been
labeled in Fig. 6.8 in order to prevent the diagram from becoming
too crowded.) Note that although the single-solid-phase regions
labeled MgZn, Mg2Zn3, Mg2Zn11, beta, and phi appear in Fig. 6.8
as lines, they are actually areas of very small but finite width.
It can be seen that the solidus projection in Fig. 6.8 has a topology similar to that of an isothermal section of a ternary phase diagram as discussed in Section 6.4.1. The only difference is that the
two-solid-phase and single-solid-phase regions in a solidus projection are not isothermal. An isothermal section of the
Zn-Mg-Al system at 340°C (just below the lowest ternary eutectic
temperature) is shown in Fig. 6.9. The topological similarity to
Fig. 6.8 is evident, the only difference being the appearance in
Fig. 6.9 of the binary miscibility gap in the fcc phase of the
Al-Zn system whose consolute temperature of 350°C lies below
the solidus. The fact that solidus projections and isothermal sections have similar topologies clearly results from the fact that a tietriangle in a solidus projection at an invariant temperature T0 is
identical to the corresponding tie-triangle in an isothermal section at a temperature just below T0.
In the Zn-Mg-Al system, as in the large majority of systems, the
solidus projection shows the temperature at which a liquid phase
first appears. That is, the solidus projection is a first-melting projection. However, in a few uncommon systems that contain a catatectic invariant or retrograde solid solubility, the solidus
temperature at certain compositions may not be single-valued.
Over limited composition ranges in such systems, the solidus projection and the first-melting projection will not be identical, and
over these ranges, the topological rules just discussed do not
apply. In these exceptional cases, it is preferable to plot the
first-melting projection because it is always single-valued.
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
hcp
+ Mg2Zn11
+ fcc
M
g2
Zn
11
+
La
ve
s
+
fcc
La fcc
v
es
La
+f
ve
cc
s
+
+f
Tau
Ta
cc
u
+
fc
c
0.7
0.4
0.6
0.3
Laves
Mg 2 Zn 3
0.2
0.8
Mg 2 Zn11
hcp
0.1
0.9
Zn
0.8
i
Ph
0.2
fcc + fcc
0.7
0.3
0.6
0.4
0.5
0.5
Mg12 Zn13
Mg
b
a 0.9
0.8
0.7
0.9
0.1
fcc
ma
am
G
i + + hcp
Ph
Tau
+ fcc
Gamma
0.6
0.5
0.4
+ Be
ta
Beta0.3
Mole fraction
0.2
0.1
Al
fcc
Fig. 6.9 Isothermal section at 340°C of the Zn-Mg-Al phase diagram (Eriksson
et al., 2013).
These exceptions, as well as first-melting projections for quaternary and higher-order systems, are discussed by Eriksson et al.
(2013).
6.7
6.7.1
Phase Diagram Projections in
Quaternary and Higher-Order Systems
Liquidus Projections
A projection of the liquidus surface of the quaternary Mg-SrBa-Ca system at a constant Ca mole fraction XCa ¼ 0.25 is shown
in Fig. 6.10. The axes of the triangle show the relative amounts of
Mg, Sr, and Ba.
Although this diagram resembles a liquidus projection of a ternary system, it is important to note that crystallization paths like
those shown in Figs. 6.1 and 6.4 cannot be drawn on this diagram.
That is, at any given composition, the diagram shows only the temperature at which the first solid phase precipitates. As solidification
progresses, the liquidus composition no longer lies on the surface
of Fig. 6.10 since the Ca content of the liquid is no longer XCa ¼ 0.25.
99
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
0.4
fcc
0.6
0.4
0.5
0.5
0.6
0.3
0.7
0.2
0.8
0.1
0.9
Sr/(Mg + Sr + Ba)
bcc
0.3
0.7
0
64
0.8
0
0.2
56
0
0
40
0.9
0
40
0
0.7
48
0.8
560
640
0.9
Mg/(Mg + Sr + Ba)
48
Laves C14
0.1
100
0.6
0.5
0.4
Mole fraction
0.3
0.2
0.1
Ba/(Mg + Sr + Ba)
Fig. 6.10 Projection of the liquidus surface of the Mg-Sr-Ba-Ca phase diagram at
constant XCa ¼ 0.25 (temperature in °C) (FactSage).
The case is different for the liquidus projection of the quaternary MgO-Cr2O3-Al2O3-O2 system shown in Fig. 6.11 since, in this
diagram, it is the partial pressure of the fourth component, rather
than its composition, that is held constant. Hence, crystallization
paths can be drawn on the diagram. Note that compositions read
from the composition triangle give only the molar ratios MgO/
(MgO + Cr2O3 + Al2O3), Cr2O3/(MgO + Cr2O3 + Al2O3), and Al2O3/
(MgO + Cr2O3 + Al2O3) on the liquidus in the four-component system. However, the liquid may be deficient in oxygen. That is, some
CrO may be present. From an experimental standpoint, the composition triangle gives the relative amounts of MgO, Cr2O3, and
Al2O3 in an initial mixture of the three oxides that was subsequently equilibrated with air at higher temperatures (cf. Fig. 7.15).
6.7.2
Solidus Projections
The first-melting (solidus) projection of the quaternary ZnMg-Al-Y system at a constant yttrium mole fraction XY ¼ 0.05 is
shown in Fig. 6.12. This diagram shows the temperatures of first
melting at thermodynamic equilibrium for alloys with an overall
yttrium mole fraction XY ¼ 0.05. The axes of the diagram give the
molar ratios Zn/(Zn + Mg + Al), Mg/(Zn + Mg + Al), and Al/(Zn + Mg
+ Al). Note that the compositions of the individual solid phases
cannot be read directly from the figure. The regions labeled with
temperatures in Fig. 6.12 are regions in which melting occurs
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
101
0.3
0.7
0.2
0.8
0.1
0.9
Cr2O3
0.6
o
0.5
0.5
00
0.4
22
M2O3(corundum)
0.4
22
00
2700
o
Monoxide
210
0
o
o
o
0.9
o
23
00
o
2400
2500
2600
0.1
0.8
0.2
0.7
0.3
0.6
Spinel
o
2000
MgO
0.9
0.7
0.8
0.6
0.5
0.4
0.3
o
0.1
0.2
Mole fraction
Al2O3
Fig. 6.11 Projection of the liquidus surface of the MgO-Cr2O3-Al2O3-O2 system at constant pO2 ¼ 0.21 bar (temperature
in °C) (FactSage).
Zn
0.9
0.7
0.6
440
34
46
9
3
0.5
0.5
1
34
0.4
467
452
0.4
355
0.6
0.3
467
379
a
b
d
46
6
364
392
468
385
0.8
0.7
0.6
h
f g
e
0.5
0.4
447
0.3
0.9
531
0.2
0.8
363
476
468
c
0.7
0.9
0.3
429
353
3
48
0.1
0.2
0.8
360
8
48
1
37
Mg
348
0.1
a = Tau + Al3Y
b = Tau + Al3Y + Al4MgY
c = Tau + Al3Y + fcc
d = Tau + Al3Y + Al4MgY + fcc
e = Gamma + Al4MgY
f = Gamma + Al4MgY + Phi
g = Gamma + Al4MgY + Tau
h = Gamma + Al4MgY
+ Phi + Tau
0.2
0.1
Al
Moles/(Zn+Mg+Al)
Fig. 6.12 First-melting projection of the phase diagram of the Zn-Mg-Al-Y system at constant mole fraction of yttrium,
XY ¼ 0.05 (temperature in °C) (Eriksson et al., 2013).
102
Chapter 6 TERNARY TEMPERATURE-COMPOSITION PHASE DIAGRAMS
isothermally by an invariant reaction involving four solid phases.
All other regions of the diagram are nonisothermal. The isotherms
within these regions indicate the temperatures of first melting of
alloys with one, two, or three solid phases at equilibrium. (Note
that these are not tie-lines.) In order to prevent the diagram from
becoming too crowded, only a few regions have been labeled.
Just as ternary first-melting projections consist of the same
structural units as ternary isothermal sections (see Section 6.6),
quaternary and higher-order first-melting projections consist of
the same structural units as quaternary and higher-order isothermal sections. See Eriksson et al. (2013) for a discussion.
References
Bray, H.F., Bell, F.D., Harris, S.J., 1961-1962. J. Inst. Met. 90, 24.
Eriksson, G., Bale, C.W., Pelton, A.D., 2013. J. Chem. Thermodyn. 67, 63–73.
Pelton, A.D., 2014. Laughlin, D.E., Hono, K. (Eds.), Physical Metallurgy. fifth ed.
ch. 3. Elsevier, NY.
List of Websites
FactSage, Montreal, www.factsage.com.
7
GENERAL PHASE DIAGRAM
SECTIONS
CHAPTER OUTLINE
7.1 Corresponding Potentials and Extensive Variables 104
7.2 The Law of Adjoining Phase Regions 105
7.3 Nodes in Phase Diagram Sections 107
7.4 Zero Phase Fraction Lines 108
7.4.1 General Algorithm to Calculate Any Phase Diagram Section 109
7.5 Choice of Variables to Ensure That Phase Diagram Sections are
Single-Valued 110
7.6 Corresponding Phase Diagrams 114
7.6.1 Enthalpy-Composition Phase Diagrams 117
7.7 The Thermodynamics of General Phase Diagram Sections 118
7.7.1 Schreinemakers’ Rule 122
7.8 Interpreting Phase Diagrams of Oxide Systems and Other Systems
Involving Two or More Oxidation States of a Metal 122
7.9 Choice of Components and Choice of Variables 127
7.10 Phase Diagrams of Reciprocal Systems 128
7.11 Choice of Variables to Ensure Straight Tie-Lines 129
7.12 Other Sets of Corresponding Variable Pairs 130
7.13 Extension Rules for Polythermal Liquidus Projections 131
7.14 Phase Fraction Lines 131
References 131
List of Websites 131
Phase diagrams involving temperature and composition as
variables were discussed in Chapters 5 and 6 for binary and ternary systems. In the present chapter, we shall discuss the geometry of general phase diagram sections involving these and other
variables such as total pressure, chemical potentials, volume, and
enthalpy for systems of any number of components. A few examples of such diagrams have already been shown in Figs. 1.1–1.7,
2.3, and 2.4. It will now be shown that all these phase diagrams,
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00007-5
# 2019 Elsevier Inc. All rights reserved.
103
104
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
although seemingly quite different geometrically, actually all obey
one simple set of geometric rules.
The rules that will be developed apply to two-dimensional
phase diagram sections in which two thermodynamic variables
are plotted as axes, while other variables are held constant. We
shall develop a set of sufficient conditions for the choice of axis
variables and constants that ensure that the phase diagram section is single-valued, with each point on the diagram representing
a unique equilibrium state. Although these rules do not apply
directly to phase diagram projections such as in Figs. 6.1, 6.3,
and 6.4, such diagrams can be considered to consist of portions
of several phase diagram sections projected onto a common
plane. A geometric rule specific to projections will be given in
Section 7.13. Finally, since all single-valued phase diagram sections obey the same set of geometric rules, one single algorithm
can be used for their calculation as will be demonstrated in
Section 7.4.1.
In Sections 7.1–7.6, we shall first present the geometric rules
governing general phase diagram sections without giving detailed
proofs. Rigorous proofs will follow in Sections 7.7 and 7.9.
7.1
Corresponding Potentials and Extensive
Variables
In a system with C components, we select the following (C + 2)
thermodynamic potentials (intensive variables): T, P, μ1, μ2, … μC.
For each potential, ϕi there is a corresponding extensive variable qi
related by the equation:
ϕi ¼ ð∂U=∂qi Þqj ðj6¼iÞ
(7.1)
The corresponding pairs of potentials and extensive variables are
listed in Table 7.1. The corresponding pairs are found together in
the terms of the fundamental equation, Eq. (2.9), and in the GibbsDuhem equation (Eq. 2.82).
When a system is at equilibrium, the potential variables are the
same for all phases (all phases are at the same T, P, μ1, μ2, …), but
the extensive variables can differ from one phase to the next (the
phases can all have different volumes, entropies, and compositions). In this regard, it is important to note the following. If the
only variables that are held constant are potential variables, then
the compositions of all phases will lie in the plane of the phase
diagram section, and the lever rule may be applied. Some examples are found in Figs. 1.1, 1.2, and 1.5. However, if an extensive
variable such as composition is held constant, then this is no
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
105
Table 7.1 Corresponding Pairs of Potentials fi and
Extensive Variables qi
fi
qi
T
S
P
V
m1
n1
m2
n2
longer necessarily true as was discussed, for example, in
Section 6.5 regarding isoplethal sections such as Figs. 1.3, 1.4,
and 6.7.
7.2
The Law of Adjoining Phase Regions
The following law of adjoining phase regions (Palatnik and
Landau, 1956) applies to all single-valued phase diagram sections:
as a phase boundary line is crossed, one and only one phase either
appears or disappears.
This law is clearly obeyed in any phase diagram section in
which both axis variables are extensive variables such as Figs. 1.2,
1.3B, 1.4, and 6.5. When one axis variable is a potential, however, it
may at first glance appear that the law is not obeyed for invariant
lines. For instance, as the eutectic line is crossed in the binary Tcomposition phase diagram in Fig. 5.7, one phase disappears, and
another appears. However, the eutectic line in this figure is not a
simple phase boundary, but rather an infinitely narrow threephase region. This is illustrated schematically in Fig. 7.1 where
L
α+L
α+β+L
T
α
β+L
β
α+β
A
XB
B
Fig. 7.1 An isobaric temperature-composition phase diagram with the eutectic
line “opened up” to show that it is an infinitely narrow three-phase region
(Pelton, 2001).
…
…
mC
nC
106
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
the three-phase region has been “opened up” to show that the
region is enclosed by upper and lower phase boundaries that
are coincident. In passing from the (β + L) to the (α + β) region,
for example, we first pass through the (α + β + L) region. Hence,
the law of adjoining phase regions is obeyed. The three-phase
region is infinitely narrow because all three phases at equilibrium must be at the same temperature. Another example is
seen in the isoplethal section in Fig. 6.7. At the ternary eutectic
temperature, TE, there is an infinitely narrow horizontal fourphase invariant region (L + (Bi) + (Cd) + (Sn)). As yet another
example, the y-axis of Fig. 1.5 is the chemical potential of oxygen which is the same in all phases at equilibrium. This diagram exhibits two infinitely narrow horizontal three-phase
invariant regions.
Similarly, if both axis variables are potentials, there may be
several infinitely narrow univariant phase regions. For example,
in the P-T diagram of Fig. 1.7A, all lines are infinitely narrow
two-phase regions bounded on either side by coincident phase
boundaries as shown schematically in Fig. 7.2 where all twophase regions have been “opened up.” The three-phase invariant triple point is then seen to be a degenerate three-phase
triangle. Other examples of diagrams in which both axis variables are potentials are seen in Figs. 1.6, 2.3, and 2.4. All lines
in Figs. 2.3 and 2.4 are infinitely narrow two-phase univariant
regions. In Fig. 1.6, some lines are infinitely narrow threephase univariant regions, while others are simple phase
boundaries.
β
α+β
P
β+γ
α + β+ γ
α
α+β
γ
T
Fig. 7.2 A P-T diagram in a one-component system “opened up” to show that the
two-phase lines are infinitely narrow two-phase regions (Pelton, 2001).
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
7.3
Nodes in Phase Diagram Sections
All phase boundary lines in any single-valued phase diagram
section meet at nodes where exactly four lines converge as illustrated in Fig. 7.3. N phases (α1, α2, … αN), where N 1, are common
to all four regions. Two additional phases, β and γ, are disposed as
shown in Fig. 7.3, giving rise to two (N + 1)-phase regions and one
(N + 2)-phase region. At every node, the generalized Schreinemakers’ rule requires that the extensions of the boundaries of
the N-phase region must either both lie within the (N + 1)-phase
regions as in Fig. 7.3 or both lie within the (N + 2)-phase region.
The generalized Schreinemakers’ rule will be proved in
Section 7.7.1.
An examination of Fig. 1.4, for example, reveals that all nodes
involve exactly four boundary lines. Schreinemakers’ rule is illustrated on this figure by the dashed extensions of the phase boundaries at the nodes labeled a, b, and c and in Fig. 6.7 by the dashed
extension lines at nodes f and g.
For phase diagrams in which one or both axis variables are
potentials, it might at first glance seem that fewer than four
boundaries converge at some nodes. However, all nodes can still
be seen to involve exactly four-phase boundary lines when all infinitely narrow phase regions are “opened up” as in Figs. 7.1 and 7.2.
The extension rule illustrated in Fig. 6.6 for tie-triangles in ternary isothermal sections can be seen to be a particular case of
Schreinemakers’ rule, as can the extension rules show in
Fig. 5.14 for binary T-composition phase diagrams. Finally, an
extension rule for triple points on P-T phase diagrams is illustrated in Fig. 1.7A; the extension of each univariant line must pass
into the opposite bivariant region. This rule, which can be seen to
apply to Figs. 2.3 and 2.4 as well, is also a particular case of
Schreinemakers’ rule.
(α1 + α2 + … + αn)
(α1 + α2 + … + αn)
+β
p
(α1 + α2 + … + αn)
+γ
(α1 + α2 + … + αn)
+β+γ
Fig. 7.3 A node in a general phase diagram section showing the zero-phasefraction lines of the b and g phases. Schreinemakers’ rule is obeyed (Pelton, 2001).
107
108
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
An objection might be raised that a minimum or a maximum in
a two-phase region in a binary temperature-composition phase
diagram, as in Figs. 5.5, 5.10 and 5.11, represents an exception
to Schreinemakers’ rule. However, the extremum in such cases
is not actually a node where four-phase boundaries converge,
but rather a point where two boundaries touch. Such extrema
in which two-phase boundaries touch with zero slope may occur
for a C-phase region in a phase diagram of a C-component system
when one axis is a potential. For example, in an isobaric
temperature-composition phase diagram of a four-component
system, we may observe a maximum or a minimum in a fourphase region separating two three-phase regions. A similar maximum or minimum in a (C-n)-phase region, where n > 0, may also
occur along particular composition paths. This will be discussed
in more detail in Section 8.3.
Other apparent exceptions to Schreinemakers’ rule
(such as five boundaries meeting at a node) can occur in
special limiting cases, for example if an isoplethal section of a ternary system passes exactly through a ternary eutectic
composition.
7.4
Zero Phase Fraction Lines
All phase boundaries on any single-valued phase diagram section are zero-phase-fraction (ZPF) lines, an extremely useful concept introduced by Morral and coworkers (Bramblett and Morral,
1984; Morral, 1984; Gupta et al., 1986) which is a direct consequence of the law of adjoining phase regions. There is one ZPF line
associated with each phase. On one side of its ZPF line, the phase
occurs, while on the other side, it does not, as illustrated by the
colored lines in Fig. 1.4. There is a ZPF line for each of the five
phases in Fig. 1.4. The five ZPF lines account for all the phase
boundaries on the two-dimensional section. Phase diagram sections plotted on triangular coordinates as in Figs. 1.2 and 6.5 similarly consist of one ZPF line for each phase.
With reference to Fig. 7.3, it can be seen that at every node, two
ZPF lines cross each other.
In the case of phase diagrams in which one or both axis variables are potentials, certain phase boundaries may be coincident
as was discussed in Section 7.2. Hence, ZPF lines for two different
phases may be coincident over part of their lengths as is illustrated
in Fig. 7.4. In Figs. 2.3 and 2.4, every line on the diagrams is actually two coincident ZPF lines.
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
a
2800
Zero phase fraction lines
α (abcd)
Liquid (aecf)
β (fbeg)
Liquid
Temperature (°C)
2600
Liquid + α
e
2400
b
L+β
c
Solid α
2200
f
Solid β
2 Solids (α + β)
2000
1800
1600 g
0
MgO
0.2
0.4
0.6
Mole fraction XCaO
0.8
d 1
CaO
Fig. 7.4 Binary T-composition phase diagram at P ¼ 1 bar of the MgO-CaO system
showing zero-phase-fraction (ZPF) lines of the phases (Pelton, 2014).
7.4.1
General Algorithm to Calculate Any Phase
Diagram Section
Since all single-valued phase diagram sections obey the same
geometric rules described in Sections 7.1–7.3, one single general
algorithm can be used to calculate any phase diagram section
thermodynamically. As discussed in Section 2.3, the equilibrium
state of a system (i.e., the amount and composition of every
phase) can be calculated for a given set of conditions (T, P, overall
composition, chemical potentials, etc.) by software that minimizes the total Gibbs energy of the system, retrieving the thermodynamic data for each phase from databases. To calculate any
phase diagram section, one first specifies the desired axis variables and their limits, as well as the constants of the diagram.
The software then scans the four edges of the diagram, using
Gibbs energy minimization to find the ends of each ZPF line. Next,
each ZPF line is calculated and tracked from one end to the other
using Gibbs energy minimization to determine the locus of points
at which the phase is just on the verge of appearing. Should a ZPF
line not intersect any edge of the diagram, it will be discovered by
the program while it is tracing one of the other ZPF lines. When
ZPF lines for all phases have been drawn, the diagram is complete.
This is the algorithm used by the FactSage system software which
109
110
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
has been used to calculate most of the phase diagrams shown in
this book.
In a ternary liquidus projection as in Fig. 6.4, the univariant
lines are “double ZPF lines” along which the liquid is stable and
two solid phases are also stable but present in zero amounts.
The univariant lines can be calculated by a procedure similar to
that used for calculating phase diagram sections except that temperature and composition must be varied simultaneously as the
double ZPF lines are tracked across the diagram. A similar procedure is used to calculate first-melting projections as in Fig. 6.8.
7.5
Choice of Variables to Ensure That
Phase Diagram Sections are
Single-Valued
In a system of C components, a two-dimensional phase diagram section is obtained by choosing two axis variables and holding (C 1) other variables constant. However, not every choice of
variables will result in a single-valued phase diagram with a
unique equilibrium state at every point. For example, on the PV diagram for H2O shown schematically in Fig. 7.5, at any point
in the area where the (S + L) and (L + G) regions overlap, there
are two possible equilibrium states of the system. Similarly, the
diagram of S2 pressure versus the mole fraction of Ni at constant
T and P in the Cu-Ni-S2 system in Fig. 7.6 exhibits a region in
which there are two possible equilibrium states.
P
S+L
L+G
S+G
V
Fig. 7.5 Schematic P-V diagram for H2O. This is not a single-valued phase diagram
section (Pelton, 2001).
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
111
−15
log10p(S2)(bar)
−15.5
Cu2S + Ni3S2
−16
fcc + Cu2S
−16.5
fcc + Ni3S2
−17
−17.5
−18
0
0.2
0.4
0.6
0.8
1
Mole fraction nNi/(nCu+nNi+nS2)
Fig. 7.6 Equilibrium S2 pressure versus mole fraction of Ni in the Cu-Ni-S2 system at 427°C. This is not a single-valued
phase diagram (Pelton, 2014).
In order to ensure that a phase diagram section is single-valued
at every point, the following procedure for the choice of variables
is sufficient:
(1) From each corresponding pair (ϕi and qi) in Table 7.1, select
either ϕi or qi, but not both. At least one of the selected variables must be an extensive variable qi.
(2) Define the size of the system as some function of the
selected extensive variables. (Examples: (n1 + n2) ¼ constant,
n2 ¼ constant, or V ¼ constant.) Then, normalize the chosen
extensive variables.
(3) The chosen potentials and normalized extensive variables
are then the (C + 1) independent phase diagram variables.
Two of these are selected as axes, and the others are held
constant.
Note that this procedure constitutes a set of sufficient (but not
necessary) conditions for the diagram to be single-valued. That is,
a diagram that violates these conditions may still be single-valued,
but there is no assurance that this will be the case.
The procedure is illustrated by the following examples. In each
example, the selected variables, one from each conjugate pair, are
underlined.
112
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
System Fe-Mo (Fig. 1.1):
P μFe μMo
ϕi : T
qi : S V nFe nMo
Size: (nFe + nMo).
Phase diagram variables: T, P, nMo/(nFe + nMo) ¼ XMo.
System Fe-Cr-S2-O2 (Fig. 1.6):
P μFe μCu μS2 μO2
ϕi : T
qi : S V nFe nCu nS2 nO2
Size: (nFe + nCu).
Phase diagram variables: T, P, μO2, μS2, nCr/(nFe + nCr).
System Zn-Mg-Al isopleth (Fig. 1.3A):
ϕi : T
P μZn μMg μAl
qi : S V nZn nMg nAl
Size: (nMg + nAl).
Phase diagram variables: T, P, nAl/(nMg + nAl), nZn/(nMg + nAl) ¼ XZn/
(1 XZn).
System Fe-Cr-V-C (Fig. 1.4):
ϕi : T
P μFe μCr μV μC
qi : S V mFe mCr mV mC
Size: (mFe + mCr + mV + mC) ¼ m.
Phase diagram variables: T, P, mC/m, mV/m, mCr/m.
Note that it is permissible to replace the masses of the constituents in moles ni by the masses in other mass units mi.
System Mg-Al-Ce-Mn-Y-Zn (Fig. 7.7):
ϕi : T P μMg μAl μCe μMn μY μZn
qi : S V mMg mAl mCe mMn mY mZn
Size: (mMg + mAl + mCe + mMn + mY + mZn) ¼ m.
Phase diagram variables: T, P, mMg/m, mAl/m, mCe/m, mMn/m,
mY/m, mZn/m.
System Al2SiO5 (Fig. 1.7B):
P μAl2 SiO5
ϕi : T
qi : S V nAl2 SiO5
Size: nAl2SiO5.
Phase diagram variables: T, V/nAl2SiO5.
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
System Cu-Ni-S2 (Fig. 7.8):
P μCu μNi μS2
ϕi : T
qi : S V nCu nNi nS2
Size: (nCu + nNi).
Phase diagram variables: T, P, μS2, nNi/(nCu + nNi).
This choice of variables for the Cu-Ni-S2 system gives the
single-valued phase diagram in Fig. 7.8. For the same system in
Fig. 7.6, an incorrect choice of variables was made, leading to a
diagram that is not single-valued. The x-axis variable of Fig. 7.6
contains the term nS2. That is, both variables (μS2 and nS2) of the
same corresponding pair were selected, thereby violating the first
selection rule. Similarly, the selection of both variables (P and V)
from the same corresponding pair results in the diagram of Fig. 7.5
that is not single-valued. Similar examples of phase diagrams that
are not single-valued have been presented by Hillert (1997).
Other examples of applying the procedure for choosing variables will be given in Sections 7.8–7.10.
In several of the phase diagrams in this chapter, log pi or RT In
pi is substituted for μi as an axis variable or constant. From
Eq. (3.7), it can be seen that this substitution can clearly be made
if T is constant. However, even when T is an axis variable of the
phase diagram as in Fig. 2.4, this substitution is still permissible
since μoi is a monotonic function of T. The substitution of logpi
(Mg) + (Mn) + Al2Y
Liquid + (Mg)
Liquid + (Mn,Al)
600
Temperature ( °C)
Liquid
Liquid + (Mn)
Liquid + Al2Y + (Mg) + (Mn,Al)
Liquid + (Mn,Al) + (Mg) + Al3Y
Liquid + (Mn,Al) + (Mg) + Al4MgY
400
(Mg) + Al4MgY + (Mn,Al) + Al11Ce3
(Mg) + Al4MgY + (Mn,Al) + Gamma + Al11Ce3
200
(Mg) + Al3Y + (Mn,Al) + Gamma + Al11Ce3
0
0
4
8
12
Weight percent Al
16
20
Fig. 7.7 Isoplethal section of the Mg-Al-Ce-Mn-Y-Zn system at 0.05% Ce, 0.5% Mn,
0.1% Y, and 1.0% Zn (weight %) at P ¼ 1 bar (FactSage).
113
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
427°C
−15
Cu2S + Ni3S2
log10p(S2) (bar)
114
fcc + Ni3S2
−16
fcc + Cu2S
−17
fcc
−18
0
0.2
0.4
0.6
Molar ratio nNi/(nCu+nNi)
0.8
1
Fig. 7.8 Equilibrium S2 pressure versus molar ratio nNi/(nCu +nNi) in the Cu-Ni-S2
system at 427°C. This is a single-valued phase diagram (Pelton, 2014).
or RT ln pi for μi results in a progressive expansion and displacement of the axis with increasing T that preserves the overall geometry of the diagram.
7.6
Corresponding Phase Diagrams
If a potential axis, ϕi, of a phase diagram section is replaced by
its normalized conjugate variable qi, then the new diagram and the
original diagram may be said to be corresponding phase diagrams.
An example is seen in Fig. 1.7, where the P-axis of Fig. 1.7A is
replaced by the V-axis of Fig. 1.7B. Another example is shown in
Fig. 7.9 for the Fe-O system. Placing corresponding diagrams
beside each other as in Figs. 1.7 and 7.9 is useful because the information contained in the two diagrams is complementary. Note
how the invariant triple points of one diagram of each pair correspond to the invariant lines of the other. Note also that as ϕi,
increases, qi increases. That is, in passing from left to right at constant T across the two diagrams in Fig. 7.9, the same sequence of
phase regions is encountered. This is a general result of the thermodynamic stability criterion that will be discussed in Section 7.7.
A set of corresponding phase diagrams for a multicomponent
system with components 1, 2, 3, …,C are shown in Fig. 7.10. The
potentials ϕ4, ϕ5, … ϕC+2 are held constant. The size of the system
(in the sense of Section 7.5) is q3. The invariant triple points of
Liquid iron + liquid oxide
Liquid iron
1900
Liquid
iron
Liquid oxide
δ-Iron
1700
Temperature (K)
Liquid oxide
Wustite
Wustite
1500
γ-Iron
+
Wustite
1300
1100
Wustite
+
Magnetite
α-Iron
+
Wustite
900
α-Iron + Magnetite
700
0
γ-Iron
Hematite
+
Magnetite
0.50
0.54
Magnetite
α-Iron
Hematite
+
Oxygen
0.58
Hematite
0.62 -500
-400
-300
-200
-100
0
−1
RT ln pO2 (kJ mol )
Mole fraction XO
Fig. 7.9 Corresponding T-XO and T (mO2 mOo2 ) phase diagrams of the Fe-O system. Data from Muan, A., Osborn, E.F., 1965.
q1/q3
Phase Equilibria Among Oxides in Steelmaking. Addison Wesley, Reading, MA.
(C)
(D)
β
F1
α
α
(A)
β
q2/q3
(B)
F2
Fig. 7.10 A corresponding set of phase diagrams for a C-component system when the potentials f4, f5, …, fC+2 are
constant. From Pelton, A.D., Schmalzried, H., 1973. Metall. Trans. 4, 1395–1404.
116
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
Fig. 7.10B correspond to the invariant lines of Fig. 7.10A and D and
to the invariant tie-triangles of Fig. 7.10C. The critical points of
Fig. 7.10B where the two-phase regions terminate correspond to
the critical points of the “miscibility gaps” in Figs 7.10A, C, and D.
A set of corresponding phase diagrams for the Fe-Cr-O2 system
at T ¼ 1300°C are shown in Fig. 7.11. Fig. 7.11B is a plot of μCr
versus μO2. Fig. 7.11A is similar to Fig. 1.5. Fig. 7.11C is a plot of
nO/(nFe + nCr) versus nCr/(nFe + nCr) where the size of the system
Fig. 7.11 Corresponding phase diagrams of the Fe-Cr-O2 system at 1300°C. From
Pelton, A.D., Schmalzried, H., 1973. Metall. Trans. 4, 1395–1404.
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
is defined by (nFe + nCr). The coordinates of Fig. 7.11C are known as
Ja€necke coordinates. Fig. 7.11C may be “folded up” to give the
more familiar isothermal ternary section of Fig. 7.11D plotted
on the equilateral composition triangle.
7.6.1
Enthalpy-Composition Phase Diagrams
The T-composition phase diagram of the Mg-Si system is shown
in Fig. 7.12A. A corresponding phase diagram could be constructed
by replacing the T-axis by the molar entropy because (T,S) is a corresponding pair in Table 7.1, although such a diagram would not
be of much practical interest. However, just as (dS/dT) > 0 at
1425
Liquid
Temperature (°C)
1225
Liquid + Si
1025
Liquid + Mg2Si
825
Liquid + Mg2Si
625
Mg2Si + Si
425
Mg + Mg2Si
225
25
0
(A)
0.4
0.6
0.8
1
Mole fraction XSi
2425°C
2025°C
5°C
222
Si
1625
1825°C
°C
C
°
1425
80
Liquid
70
60
50
Liquid + Mg
40
Liquid + Mg2Si
30
20
10
Liquid + Si
Liquid
+ Mg2Si
°C
25
12
Enthalpy (h - h25°C) (kJ mol-1)
90
(B)
0.2
Mg
10
25
°
C
Liquid + Mg2Si + Si
Liquid
+ Mg + Mg2Si
Mg + Mg2Si
825°C
625°C
Mg2Si + Si
425°C
225°C
0
Fig. 7.12 Corresponding T-composition and enthalpy-composition phase diagrams
of the Mg-Si system at P ¼ 1 bar. The enthalpy axis is referred to the enthalpy of the
system at equilibrium at 25°C (Pelton, 2014).
117
118
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
constant P and composition, so also is (dH/dT) > 0. Hence, an
H-composition phase diagram will have the same geometry as
an S-composition diagram. (This will be demonstrated more rigorously in Section 7.12.)
Fig. 7.12B is the enthalpy-composition phase diagram corresponding to the T-composition diagram of Fig. 7.12A. The y-axis
is (h h25) where h25 is the molar enthalpy of the system in equilibrium at 25°C. Referencing the enthalpy to h25 slightly distorts
the geometry of the diagram because h25 is a function of composition. In particular, the tie-lines are no longer necessarily perfectly straight lines. However, this choice of reference renders
the diagram more practically useful since the y-axis variable then
directly gives the heat that must be added or removed to heat or
cool the system between 25°C and any temperature T. The isothermal lines shown in Fig. 7.12B are not part of the phase diagram but
have been calculated and plotted to make the diagram more useful. It should be stressed that Fig. 7.12B is a phase diagram section
obeying all the geometric rules of phase diagram sections discussed in Sections 7.1–7.4. It consists of ZPF lines and can be calculated using the same algorithm as for any other phase diagram
section as discussed in Section 7.4.1. Note that the three-phase
tie-triangles are isothermal.
Another H-composition phase diagram is shown in Fig. 1.3B
for the Zn-Mg-Al system along the isoplethal section where
XZn ¼ 0.1.
7.7
The Thermodynamics of General Phase
Diagram Sections
This section provides a more rigorous derivation of the geometry of general phase diagram sections that was discussed in
Sections 7.2–7.6.
In Section 2.9.2, a general auxiliary function G0 was introduced.
Let us now define an even more general auxiliary function, B, as
follows:
B¼U m
X
ϕi qi 0 m ðC + 2Þ
(7.2)
i¼1
where the ϕi and qi include all corresponding pairs in Table 7.1 (if
m ¼ 0, then B ¼U). Following the same reasoning as in
Section 2.9.2, we obtain for a system at equilibrium
dB ¼ m
X
1
qi dϕi +
C
+2
X
m+1
ϕi dqi
(7.3)
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
α
β
Coexistence
f3
line
β
α
f2
β
α
f1
Fig. 7.13 f1-f2-f3 surfaces of a and b phases in a C-component system when f4,
f5, …, fC+2 are constant. Reprinted with permission from Hillert, M., 2008. Phase
Equilibria, Phase Diagrams and Phase Transformations—Their Thermodynamic Basis,
second ed. Cambridge Univ. Press, Cambridge.
and the equilibrium state at constant ϕi (0 i m) and qi (m +
1 i C + 2) is approached by minimizing B:
dBϕi ð0imÞ,qi ðm + 1iC + 2Þ 0
(7.4)
Fig. 7.13 is a plot of the ϕ1-ϕ2-ϕ3 surfaces of two phases, α and β, in
a C-component system when ϕ4, ϕ5, …, ϕC+2 are all held constant.
For example, in a one-component system, Fig. 7.13 would be a
plot of the P-T-μ surfaces. Let us now choose a pair of values ϕ1
and ϕ2 and derive the condition for equilibrium. Since all potentials except ϕ3 are now fixed, the appropriate auxiliary function,
from Eqs. (7.2), (2.77) (see also Eqs. 2.80, 2.83), is B ¼ q3ϕ3. Furthermore, let us fix the size of the system by setting q3 constant. Then,
from Eq. (7.4), the equilibrium state for any given pair of values of
ϕ1 and ϕ2 is given by setting dB ¼ q3dϕ3, ¼ 0 that is by minimizing
ϕ3. This is illustrated in Fig. 7.13 where the base plane of the figure
is the ϕ1-ϕ2 phase diagram of the system at constant ϕ4, ϕ5 …, ϕC +2
(Hillert, 2008). For example, in a one-component system, if ϕ1, ϕ2,
and ϕ3 are P, T, and μ, respectively, then the P-T phase diagram is
given by minimizing μ, which is the molar Gibbs energy of the
one-component system. Clearly therefore, a ϕ1-ϕ2 diagram at constant ϕ4, ϕ5, …, ϕC+2 is single-valued.
As another example, consider the T μO2 phase diagram of the
Fe-O system at constant hydrostatic pressure in Fig. 7.9. The
potentials of the system are T, P, μO2, and μFe. At constant P and
for any given values of T and μO2, the equilibrium state of the system is given by minimizing μFe. Similarly, the predominance
119
120
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
diagram of the Cu-SO2-O2 system in Fig. 2.3 can be calculated by
minimizing μCu. This can be shown to be equivalent to the procedure described in Section 2.4.3.
A general ϕ1-ϕ2 phase diagram at constant ϕ4, ϕ5, …, ϕC+2 was
shown in Fig. 7.10B. The size of the system is defined by fixing q3.
We wish now to show that, when Fig. 7.10B is “opened up” by
replacing ϕ2 by q2 to give the corresponding phase diagram in
Fig. 7.10A, the sequence of phase regions encountered in moving
horizontally across Fig. 7.10A is the same as in Fig. 7.10B. For
example, when the two-phase (α + β) line in Fig. 7.10B is
“opened” up to form the two-phase (α + β) region in Fig. 7.10A,
the α and β phases remain on the left and right sides, respectively,
of the two-phase region.
We first derive the general thermodynamic stability criterion
for a single-phase region. Imagine two bordering microscopic
regions, I and II, within a homogeneous phase and suppose that
a local fluctuation occurs in which an amount dqn > 0 of the extensive property qn is transferred from region I to region II at constant
ϕi (0 i m) and constant qi (m + 1 i C + 2) where i 6¼ n. Suppose that, as a result of the transfer, ϕn increases in region I and
decreases in region II, that is, that (dϕn/dqn) < 0. A potential gradient is thus set up with ϕIn > ϕnII. From Eq. (7.3), further transfer of
qn from region I to II then results in a decrease in B: dB ¼ (ϕII ϕI)
dqn < 0. From Eq. (7.4), equilibrium is approached by minimizing
B. Hence, the fluctuation grows, with continual transfer from
region I to region II. That is, the phase is unstable. Therefore,
the criterion for a phase to be thermodynamically stable is
dϕn =dqn > 0
(7.5)
for any choice of constant ϕi or qi constraints (i 6¼ n). Since
ϕn ¼ (∂ B/∂ qn), the stability criterion can also be written as
2
∂ B=∂q2n ϕi ð0imÞ ,qi ðm + 1iC + 2Þ;i6¼n > 0
(7.6)
An important example is illustrated in Fig. 5.6 for the solid phase
of the Au-Ni system. In this case, the appropriate auxiliary function is the Gibbs energy, G. In the central composition regions
of the Gibbs energy curves in Fig. 5.6, between the spinodal points
indicated by the letter s, (∂2G/∂ n2Ni)T, P, nFe < 0. Hence, at compositions between the spinodal points, microscopic composition fluctuations will grow since this leads to a decrease in the total Gibbs
energy of the system. Hence, the phase is unstable and phase separation can occur by spinodal decomposition. Other well-known
examples of the stability criterion are (dT/dS) > 0 (entropy
increases with temperature) and (dP/d( V)) > 0 (pressure
increases with decreasing volume).
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
Returning now to Fig. 7.10, it can be seen from Eq. (7.5) that
moving horizontally across a single-phase region in Fig. 7.10B corresponds to moving in the same direction across the same region
in Fig. 7.10A. Furthermore, in the two-phase (α + β) region, suppose that ϕ2 is increased by an amount dϕ2 > 0 keeping all other
potentials except ϕ3 constant, thereby displacing the system into
the single-phase β region. From the Gibbs-Duhem equation,
Eq. (2.82),
dϕα3 ¼ ðq2 =q3 Þα dϕ2 and dϕβ3 ¼ ðq2 =q3 Þβ dϕ2
(7.7)
where (q2/q3)α and (q2/q3)β are the values at the boundaries of the
two-phase region.
Since, as has just been shown, the equilibrium state is given by
minimizing ϕ3, it follows that
(7.8)
d ϕβ3 ϕα3 =dϕ2 ¼ ðq2 =q3 Þα ðq2 =q3 Þβ < 0
Hence, the α-phase and β-phase regions lie to the left and right,
respectively, of the (α + β) region in Fig. 7.10A, and so, Fig. 7.10A
is single-valued.
Since the stability criterion also applies when extensive variables
are held constant, it can similarly be shown that when Fig. 7.10A is
“opened up” by replacing the ϕ1 axis by q1/q3 to give Fig. 7.10A, the
sequence of the phase regions encountered as the diagram is traversed vertically is the same as in Fig. 7.10A and that Fig. 7.10C is
single-valued. Finally, it can be shown that the phase diagrams will
remain single-valued if a constant ϕ4 is replaced by q4/q3. Fig. 7.10A,
for example, is a section at constant ϕ4 through a three-dimensional
ϕ1-ϕ4-(q2/q3) diagram that may be “opened up” by replacing ϕ4 by
(q4/q3) to give a ϕ1-(q4/q3)-(q2/q3) diagram that is necessarily
single-valued because of the stability criterion. Hence, sections
through this diagram at constant (q4/q3) will also be single-valued.
Consider now the node in a general phase diagram section
shown in Fig. 7.3. At any point on either phase boundary of the
(α1 + α2 + … + αN) region, one additional phase, β or γ, becomes stable. Since there is no thermodynamic reason why more than one
additional phase should become stable at exactly the same point,
the law of adjoining phase regions (Section 7.2) is evident. These
phase boundaries intersect at the node, below which there must
clearly be a region in which β and γ are simultaneously stable. If
the x-axis of the diagram is an extensive variable, then this region
will have a finite width since there is no thermodynamic reason
for the β-phase to become unstable at exactly the same point that
the γ-phase becomes stable. It then follows that exactly four
121
122
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
boundaries meet at a node. Of course, if the x-axis is a potential
variable, then the two-phase boundaries of the (α1 + α2 + … + αN +
β + γ) region will necessarily be coincident.
7.7.1
Schreinemakers’ Rule
Schreinemakers’ rule (Schreinemakers, 1915; Pelton 1995),
which was discussed in Section 7.3, will now be proved for
the general case. Consider the point p in Fig. 7.3 that lies in the
(α1 + α2 + … + αN + γ) region arbitrarily close to the node. If precipitation of γ was prohibited by kinetic constraints, point p would lie
in the (α1 + α2 + … + αN + β) region since it lies below the dashed
extension of the phase boundary line as shown on the figure.
Let B be the appropriate auxiliary function. If precipitation of
γ is prohibited, then dB/dnβ < 0, where dnβ is the amount of
β-phase (of fixed composition) that precipitates. That is, precipitation of β decreases B. If γ is permitted to precipitate, then precipitation of β is prevented. Therefore,
d
dB
>0
(7.9)
dnγ dnβ
Since the order of differentiation does not matter,
d
dB
>0
dnB dnγ
(7.10)
Hence, the extension of the other boundary of the (α1 + α2 + … +
αN) region must pass into the (α1 + α2 + … + αN + β) region as shown
in Fig. 7.3. Similarly, it can be shown that if the extension of one
boundary of the (α1 + α2 + … + αN) region passes into the (α1 +
α2 + … + αN + β + γ) region, then so must the extension of the other
(Pelton, 1995).
It was shown in Section 6.6 that solidus projections have the
same topology as isothermal sections. Hence, Schreinemakers’
rule also applies to the tie-triangles of solidus projections.
7.8
Interpreting Phase Diagrams of Oxide
Systems and Other Systems Involving
Two or More Oxidation States of a Metal
Three-phase diagrams for the Fe-Si-O2 system are shown in
Figs. 7.14–7.16. Diagrams of these types are often a source of
confusion.
Fig. 7.14 is often called the “phase diagram of the Fe3O4-SiO2
system.” However, it is not a binary phase diagram nor is it a
quasi-binary phase diagram. Rather, it is an isoplethal section
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
2000
Temperature (°C)
1800
2 Liquids
Liquid
1600
Liquid + SiO2 (cristobalite)
Liquid + Spinel
1400
Liquid + SiO2 (tridymite)
Liquid + Spinel + SiO2
1200
Spinel + SiO2 + Fe2SiO4
1000
0
0.2
0.4
0.6
0.8
1
Molar ratio SiO2/(Fe3O4+SiO2)
Fig. 7.14 Isoplethal section across the Fe3O4-SiO2 join of the Fe-Si-O2 system
(P ¼ 1 bar) (Pelton, 2014).
2000
Temperature (°C)
1800
Liquid
2 Liquids
Liquid + SiO2 (cristobalite)
1600
Liquid + SiO2 (tridymite)
Spinel + Liquid
Spinel + SiO2
1400
1200
SiO2 + Fe2O3
1000
0
0.2
0.4
0.6
0.8
1
Molar ratio SiO2/(Fe2O3+SiO2)
Fig. 7.15 “Fe2O3-SiO2 phase diagram in air” (pO2 ¼ 0.21 bar) (Pelton, 2014).
across the Fe3O4-SiO2 join of the Fe-Si-O2 composition triangle. As
with any isoplethal section, tie-lines in Fig. 7.14 do not necessarily
lie in the plane of the diagram. That is, the x-axis gives only the
overall composition of the system, not the compositions of individual phases. In general, the lever rule does not apply. In the
scheme of Section 7.5, Fig. 7.14 is described as follows:
P μFe3 O4 μSiO2 μO2
ϕi : T
qi : S V nFe3 O4 nSiO2 nO2
123
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
Size: (nFe3O4 + nSiO2)
Phase diagram variables: T, P, nSiO2/(nFe3O4 + nSiO2), nO2/(nFe3O4 +
nSiO2), with nO2/(nFe3O4 + nSiO2) ¼ 0. Note that the components
have been defined judiciously as Fe3O4-SiO2-O2 rather than
Fe-Si-O2. While this is not necessary, it clearly simplifies the
description.
Fig. 7.15 is often called the “Fe2O3-SiO2 phase diagram at constant pO2.” This diagram is actually a ternary section along a constant pO2 line as shown schematically in Fig. 7.17. Note that the
position of this line for any give pO2 will vary with temperature.
At lower temperatures, the line approaches the Fe2O3-SiO2 join.
Since pO2 is the same in all phases at equilibrium, the compositions of all phases and the overall composition lie along the
constant pO2 line. Hence, Fig. 7.15 has the same topology as binary
T-composition phase diagrams. All tie-lines lie in the plane of the
diagram as illustrated in Fig. 7.17, and the lever rule applies. In
the scheme of Section 7.5, Fig. 7.15 is described as
P μFe2 O3 μSiO2 μO2
ϕi : T
qi : S V nFe2 O3 nSiO2 nO2
Size: (nFe2O3 + nSiO2)
Phase diagram variables: T, P, nSiO2/(nFe2O3 + nSiO2), μO2.
2000
Liquid oxide + Fe(liq)
1800
2 Liquid oxides
1600
Temperature (°C)
124
Liquid oxide + Fe(liq) + SiO2 (cristobalite)
Liquid oxide + Fe(δ) + SiO2 (cristobalite)
Liquid oxide + Fe(δ)
1400
1200
Liquid oxide + Fe(δ) + SiO2 (tridymite)
Liquid oxide + Fe(γ)
Monoxide +
Liquid oxide + Fe(γ)
Liquid oxide + Fe(γ) + SiO2 (tridymite)
Fe2SiO4 + Fe(γ) + SiO2 (tridymite)
Monoxide + Fe2SiO4 + Fe(γ)
1000
Fe2SiO4 + Fe(α) + SiO2 (tridymite)
800
Monoxide + Fe2SiO4 + Fe(α)
Fe(α) + Fe2SiO4 + SiO2 (high quartz)
600
Spinel + Fe2SiO4 + Fe(α)
400
0
0.2
Fe(α) + Fe2SiO4 + SiO2 (low quartz)
0.4
0.6
0.8
Molar ratio SiO2/(FeO+SiO2)
Fig. 7.16 “FeO-SiO2 phase diagram at Fe saturation” (Pelton, 2014).
1
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
O2
Fe2O3
SiO2
Fe3O4
PO2 = 0.21 bar
FeO
Tie-line
Fe
Si
Fig. 7.17 Constant pO2 line in the Fe-Si-O2 system illustrating the construction and
interpretation of Fig. 7.15 (Pelton, 2014).
The phase boundary lines in Fig. 7.15 give the ratio nSiO2/
(nFe2O3 + nSiO2) in the individual phases of the Fe2O3-SiO2-O2 system at equilibrium. However, they do not give the total composition of the phases that may be deficient in oxygen. That is, some of
the Fe may be present in a phase as FeO.
Fig. 7.15 may also be visualized as a projection of the equilibrium phase boundaries from the O2-corner of the composition triangle onto the Fe2O3-SiO2 join as illustrated in Fig. 7.17. From an
experimental standpoint, the x-axis of Fig. 7.15 represents the relative amounts of an initial mixture of Fe2O3 and SiO2 that was
charged into an apparatus at room temperature and that was subsequently equilibrated with air at higher temperatures. Fig. 7.16 is
often called the “FeO-SiO2 phase diagram at Fe saturation.” Usually, this diagram is drawn without indicating the presence of Fe in
each phase field, although metallic Fe is always present as a separate phase. The horizontal lines in Fig. 7.16 that indicate the allotropic transformation temperatures and melting point of Fe are
also generally not drawn. Diagrams such as Fig. 7.16 are best
regarded and calculated as isoplethal sections with excess Fe.
Fig. 7.16 was calculated by the following scheme:
P μFeO μSiO2 μFe
ϕi : T
qi : S V nFeO nSiO2 nFe
125
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
Size: (nFeO + nSiO2)
Phase diagram variables: T, P, nSiO2/(nFeO + nSiO2), nFe/(nFeO +
nSiO2) with nFe/(nFeO + nSiO2) ¼ 0.0001. That is, the components of
the system have been defined as FeO, SiO2, and Fe with a sufficient
excess of Fe so as to always give a metallic phase at equilibrium. Note
that the phase boundary compositions in Fig. 7.16 give only the ratio
nSiO2/(nFeO + nSiO2) in each phase but not the total compositions of
the phases which may contain an excess or a deficit of iron relative
to the FeO-SiO2 join. In this system, the solubility of Si in Fe is very
small and so has a negligible effect on the phase diagram. However,
in other systems such as Fe-Zn-O2, the metallic phase at equilibrium
could even contain more Zn than Fe, depending upon the conditions.
As a final example, consider the isothermal section of the
Pb-PbO-ZnO-Fe3O4-SiO2-CaO system at Pb-saturation shown in
Fig. 7.18. The scheme for this diagram is
ϕi : T P μPb μPbO μZnO μFe3 O4 μSiO2 μCaO
qi : S V mPb mPbO mZnO mFe3 O4 mSiO2 mCaO
Size: (mPbO + mZnO + mFe3O4 + mSiO2 + mCaO) ¼ m
Phase diagram variables: T, P, μPb, mCaO/m, mSiO2/m, mPbO/
mZnO, mPbO/mFe3O4.
PbO/ZnO=6.52
PbO+ZnO+Fe3O4
PbO/Fe3O4=3.60
10
Lead saturation
L+S+Z
5
3O
4
D=α'Ca2SiO4
Zn
O+
Fe
90
C=lime
L+D+Z
85
+Z
L+C+D
O+
Pb
%
M=melilite
L+S
20
L
L+M
80
SiO
L+C
Z=zincite
10
95
%
Wt
80
L=liquid
S=spinel
L+Z
L+C+Z
90
Wt
95
D
L+
L+C+D
75
L+D+M
126
90
10
20
Wt % CaO
Fig. 7.18 Isothermal section of the Pb-PbO-ZnO-Fe3O4-SiO2-CaO system at
Pb-saturation at 1150°C. Curves are phase fraction lines (in weight %) of the liquid
phase ( Jak et al., 2000).
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
Since the saturating metallic phase is nearly pure liquid Pb, it is
a sufficiently good approximation to plot the diagram at a constant activity of Pb of 1.0.
The “phase fraction lines” in Fig. 7.18 are discussed in
Section 7.14.
7.9
Choice of Components and Choice of
Variables
As was discussed in Chapter 3, the components of a given system may be defined in different ways. A judicious selection can
simplify the choice of variables for a phase diagram as has just
been illustrated in Section 7.8. As another example, the selection
of variables for the isothermal log pSO2-log pO2 phase diagram of
the Cu-SO2-O2 system in Fig. 2.3 according to the scheme of
Section 7.5 is clearly simplified if the components are formally
selected as Cu-SO2-O2 rather than, for example, as Cu-S-O.
In certain cases, different choices of variables can result in different phase diagrams. For the same system as in Fig. 2.3, suppose
that we wish to plot an isothermal predominance diagram with
log pSO2 as y-axis but with log aCu as the x-axis. The gas phase will
consist principally of O2, S2, and SO2 species. As the composition
of the gas phase varies at constant total pressure from pure O2 to
pure S2, the partial pressure of SO2 passes through a maximum
where the gas phase consists mainly of SO2. Therefore, for the
phase diagram to be single-valued, the log pSO2 axis must encompass gaseous mixtures that consist mainly either of (O2 + SO2) or of
(SO2 + S2), but not both. Let us define the components as Cu-SO2O2 and choose the phase diagram variables as follows:
P μCu μSO2 μO2
ϕi : T
qi : S V nCu nSO2 nO2
Size: nO2
Phase diagram variables: T, P, μCu, μSO2, nO2.
As pSO2 increases at constant T, P, and nO2, pO2 decreases, while
pS2 always remains small. The phase diagram then essentially
encompasses only the Cu-SO2-O2 subsystem of the Cu-S2-O2 system. A phase field for Cu2O will appear on the diagram, for example, but there will be no field for Cu2S. If, on the other hand, we
choose the components as Cu-SO2-S2 with the size of the system
defined as nS2 ¼ constant, then pS2 varies along the pSO2 axis, while
pO2 always remains small. The phase diagram then encompasses
only the Cu-SO2-S2 subsystem. A phase field for Cu2S appears, but
there will be no field for Cu2O.
127
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
This example serves to illustrate again how the procedure for
choosing variables given in Section 7.5 provides sufficient conditions for a phase diagram to be single-valued.
7.10 Phase Diagrams of Reciprocal Systems
A phase diagram section at T ¼ 750°C and P ¼ 1.0 bar of the
reciprocal salt system NaCl-CaCl2-NaF-CaF2 is shown in
Fig. 7.19. A reciprocal salt system is one consisting or two or more
cations and two or more anions. Since excess Na, Ca, F, and Cl are
virtually insoluble in the ionic salts, this system is, to a very close
approximation, a quasi-ternary isoplethal section of the Na-CaCl-F system in which
ðnF + nCl Þ ¼ ð2nCa + nNa Þ
(7.11)
That is, the total cationic charge is equal to the total anionic
charge. The compositions of all phases lie in the plane of the diagram and so the lever rule applies.
The corners of the composition square in Fig. 7.19 represent
the pure salts, while the edges of the square represent compositions in common-ion quasi-binary systems. The choice of phase
CaF2
NaF
1
Liquid + NaF + CaF2
Liquid + NaF
2
Ca
2
Liquid + CaF2
aC
ls
.s
)+
Li
qu
id
qu
0.4 (NaCl s.s)
Li
)+
+ Liquid
.s
ls
aC
(N
0.2
C
+
Liq
u
aF
+
F
id
0.6
Liquid + CaF2
(NaCl s.s) +
CaF2
id
0.8
Molar ratio F/(F+Cl)
Liquid
(N
128
(NaCl s.s) + Liquid
0
0
NaCl
0.2
0.4
0.6
0.8
1
CaCl2
Molar ratio 2Ca/(F+Cl) = 2Ca/(2Ca + Na)
Fig. 7.19 Phase diagram at T ¼ 750°C and P ¼ 1 bar of the reciprocal system
NaCl-CaCl2-NaF-CaF2 (FactSage).
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
diagram variables is most simply formulated using the procedure
of Section 7.5, as follows:
P μNa μCa μF μCl
ϕi : T
qi : S V nNa nCa nF nCl
Size: (nF + nCl) ¼ (2nCa + nNa)
Phase diagram variables:
T , P,nF =ðnF + nCl Þ,2nCa =ð2nCa + nNa Þ, ð2nCa + nNa Þ=ðnF + nCl Þ ¼ 1
The axis variables of Fig. 7.19 are sometimes called “charge equivalent ionic fractions” in which the number of moles of each ion is
weighted by its absolute charge.
Note that if Na, Ca, F, or Cl were actually significantly soluble in
the salts, the diagram could still be plotted as in Fig. 7.19, but it
would then no longer be a quasi-ternary section, and tie-lines
would not necessarily lie in the plane of the diagram nor would
the lever rule apply.
7.11
Choice of Variables to Ensure Straight
Tie-Lines
Fig. 7.19 could have been drawn with the axis variables chosen
instead as nF/(nF +nCl) and nCa/(nCa +nNa), that is as molar ionic
fractions rather than charge equivalent ionic fractions. In this
case, the diagram would be a distorted version of Fig. 7.19 but
would still be a valid phase diagram obeying all the geometric
rules. However, tie-lines in two-phase regions would no longer
be straight lines. That is, a straight line joining the compositions
of two phases at equilibrium would not pass through the overall
composition of the system. It is clearly desirable from a practical
standpoint that tie-lines be straight.
It can be shown, through simple mass balance considerations
(Pelton and Thompson, 1975), that the tie-lines of a phase diagram in which both axes are composition variables will only be
straight lines if the denominators of the two composition variable
ratios are the same. This is the case in Fig. 7.19 where the denominators are the same because of the condition of Eq. (7.11). For
another example, see Fig. 7.10C. It can thus be appreciated that
the procedure of Section 7.5 provides a sufficient condition for
tie-lines to be straight by the stratagem of normalizing all composition variables with respect to the same system “size.”
It can also be shown (Pelton and Thompson, 1975) that ternary
phase diagram sections plotted on a composition triangle always
have straight tie-lines.
129
130
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
7.12 Other Sets of Corresponding
Variable Pairs
The set of corresponding variable pairs in Table 7.1 is only one
of several possible sets.
Substituting S from Eq. (2.78) into the Gibbs-Duhem equation,
Eq. (2.82), and rearranging terms yield
X
(7.12)
Hdð1=T Þ ðV =T ÞdP +
ni d ðμi =T Þ ¼ 0
Hence, another set of corresponding variable pairs is that given in
Table 7.2.
With this set of conjugate variable pairs replacing those in
Table 7.1, the same procedure as described in Section 7.5 can
be used to choose variables to construct single-valued phase diagrams. For example, the procedure for the H-composition phase
diagram in Fig. 7.12B is as follows:
P
μMg =T μSi =T
ϕi : 1=T
nSi
qi : H V =T nMg
Size: (nMg + nSi)
Phase diagram variables: H, P, nMg/(nMg + nSi).
This demonstrates that Fig. 7.12B is, in fact, a single-valued
phase diagram. Although in this particular case, an S-composition
phase diagram would also be single-valued, and very similar in
form to Fig. 7.12B, H cannot simply be substituted for S in
Table 7.1 in all cases.
Many other sets of conjugate variable pairs can be obtained
through appropriate thermodynamic manipulation. The set
shown in Table 7.2 and many other such sets have been discussed
by Hillert (1997). For example, (A/T,T/P), (TU/P,1/T), and (μi/T,
ni) are one such set. In conjunction with the selection rules of
Section 7.5, all these sets of conjugate pairs can be used to construct single-valued phase diagram sections. However, such diagrams would probably be of very limited practical utility.
Table 7.2 Other Corresponding Pairs of Potentials fi
and Extensive Variables qi
fi
qi
1/T
H
P
V/T
m1/T
n1
m2/T
n2
…
…
mC/T
nC
Chapter 7 GENERAL PHASE DIAGRAM SECTIONS
7.13
Extension Rules for Polythermal
Liquidus Projections
An extension rule for the univariant lines on polythermal projections is illustrated in Fig. 6.4 at point A. At any intersection
point of three univariant lines, the extension of each univariant
line passes between the other two. The proof, although straightforward, is rather lengthy and so will not be reproduced here.
7.14
Phase Fraction Lines
In Section 7.4, the concept of zero-phase-fraction lines was
introduced. It is often of practical interest to plot phase fraction
lines for fractions other than zero. As an example, in Fig. 7.18,
95%, 90%, etc. phase fraction lines for the liquid phase are shown.
For instance, along the 80% phase fraction line, 20% (by weight) of
the system is in the solid state. For a discussion of phase fraction
lines, see Bramblett and Morral (1984).
References
Bramblett, T.R., Morral, J.E., 1984. Bull. Alloy Phase Diagrams 5, 433.
Gupta, H., Morral, J.E., Nowotny, H., 1986. Scr. Metall. 20, 889–894.
Hillert, M., 1997. J. Phase Equilib. 18, 249–263.
Hillert, M., 2008. Phase Equilibria, Phase Diagrams and Phase Transformations—
Their Thermodynamic Basis, second ed. Cambridge Univ. Press, Cambridge.
Jak, E., Decterov, S.A., Zhao, B., Pelton, A.D., Hayes, P.C., 2000. Metall. Mater. Trans.
31B, 621–630.
Morral, J.E., 1984. Scripta Metall. 18, 407–410.
Muan, A., Osborn, E.F., 1965. Phase Equilibria Among Oxides in Steelmaking. Addison Wesley, Reading, MA.
Palatnik, L.S., Landau, A.I., 1956. Zh. Fiz. Khim. 30, 2399.
Pelton, A.D., Schmalzried, H., 1973. Metall. Trans. 4, 1395–1404.
Pelton, A.D., 1995. J. Phase Equilib. 16, 501–503.
Pelton, A.D., 2001. In: Kostorz, G. (Ed.), Phase Transformations in Materials. Wiley,
Weinheim (Chapter 1).
Pelton, A.D., 2014. Laughlin, D.E., Hono, K. (Eds.), Physical Metallurgy. fifth ed.
(Chapter 3). Elsevier, New York.
Pelton, A.D., Thompson, W.T., 1975. Prog. Solid State Chem. 10, 119–155.
Schreinemakers, F.A.H., 1915. Proc. K. Akad. Wet. 18, 116. Amsterdam (Section of
Sciences).
List of Websites
FactSage, Montreal, www.factsage.com
131
8
EQUILIBRIUM AND
SCHEIL-GULLIVER
SOLIDIFICATION
CHAPTER OUTLINE
8.1 Equilibrium Solidification 133
8.2 General Nomenclature for Invariant and Other Reactions
8.3 Quasi-Invariant Reactions 135
8.4 Nonequilibrium Scheil-Gulliver Solidification 136
8.5 Scheil-Gulliver Constituent Diagrams 137
8.5.1 Binary Systems 137
8.5.2 Ternary Systems 140
8.5.3 Topology and Calculation of Scheil-Gulliver
Constituent Diagrams 147
References 148
List of Websites 148
8.1
134
Equilibrium Solidification
The course of equilibrium solidification can be followed through
the use of isoplethal sections, such as that for the six-component
Mg alloy shown in Fig. 7.7. Polythermal liquidus projections are also
useful in this respect. Consider an alloy with composition 80 mol%
Mg, 15% Al, and 5% Zn at point B in Fig. 6.4. As this alloy is cooled at
equilibrium, solid hcp Mg begins to precipitate at the liquidus temperature that is just below 500°C. The liquid composition then follows the crystallization path shown in Fig. 6.4 until it attains a
composition on the univariant line. The liquid composition then
follows the univariant line toward the point P0 with coprecipitation
of hcp and the gamma phase. (For a discussion of equilibrium crystallization paths, see Section 6.3.) At point P0, a quasi-peritectic reaction occurs. All the foregoing information can be deduced from
Fig. 6.4. However, from this diagram alone, it is not possible to
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00008-7
# 2019 Elsevier Inc. All rights reserved.
133
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
100
80
Mass percent
134
uid
Liq
60
40
hc
Gamma
20
0
p
Phi
300
350
400
450
500
550
Temperature (°C)
Fig. 8.1 Calculated amounts of phases present during equilibrium cooling of an
alloy of composition 85 mol% Mg, 15% Al, and 5% Zn (point B in Fig. 6.4) (see
FactSage).
determine the amounts and compositions of the various phases at
equilibrium because the extent of the solid solutions is not shown
nor, for the same reason, is it possible to determine whether solidification is complete at the point P0 or whether the liquid persists to
lower temperatures.
However, with thermodynamic calculations, this information is
readily obtained. Software is available, with tabular or graphic
display, to calculate the amounts and compositions of every phase
during equilibrium solidification and to display the sequence of
reactions during solidification. In the current example, the reaction
sequence, calculated using the FactSage software and databases, is
Liq: ! hcp 498 405° C, bivariant
Liq: ! hcp + γ 405 364° C, univariant
γ + Liq: ! ϕ + hcp 364° C, invariant
The solidification is calculated to terminate at 364 °C with complete consumption of the liquid by the quasi-peritectic reaction.
A graphic display of the calculated amounts of all phases during
equilibrium cooling is shown in Fig. 8.1.
8.2
General Nomenclature for Invariant
and Other Reactions
As discussed in Section 5.13, in a binary isobaric temperaturecomposition phase diagram, there are two possible types of
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
invariant reactions: “eutectic type” (β ! α + γ) and “peritectic type”
(α + γ ! β). In a ternary system, there are “eutectic-type” (α ! β +
γ + δ), “peritectic-type” (α + β + γ ! δ), and “quasi-peritectic-type”
(α + β ! γ + δ) invariants (Section 6.3). In a system of C components, the number of types of isobaric invariant reactions is equal
to C. A reaction with one reactant, such as (α ! β + γ + δ), is clearly
a “eutectic-type” invariant reaction, but in general, there is no
standard terminology. These reactions may be conveniently
described according to the numbers of reactants and products
(in the direction that occurs upon cooling). Hence, the reaction
(α + β ! γ + δ + ε) is a 2 ! 3 reaction, the reaction (α ! β + γ + δ)
is a 1 ! 3 reaction, and so on.
A similar terminology can be used for univariant, bivariant,
and other isobaric reactions. For example, a 1 ! 2 reaction
(α ! β + γ) is invariant in a binary system but univariant in a
ternary system. The ternary peritectic-type 3 ! 1 reaction (α + β
+ γ ! δ) is invariant in a ternary system, univariant in a quaternary
system, bivariant in a quinary system, etc.
8.3
Quasi-Invariant Reactions
According to the Phase Rule, an invariant reaction occurs in a
C-component system when (C + 1) phases are at equilibrium at
constant pressure. Apparent exceptions are observed where an
isothermal reaction occurs even though fewer than (C + 1) phases
are present. However, these are just limiting cases.
An example is shown in Fig. 5.5. A liquid with a composition
exactly at the liquidus/solidus minimum in a binary system will
solidify isothermally to give a single solid phase of the same composition. Similarly, at a maximum in a two-phase region in a
binary system, the liquid solidifies isothermally to give a single
solid phase, for example, at the congruent melting points of the
β0 phase in Fig. 5.11 and of the Ca2SiO4 and CaSiO3 compounds
in Fig. 5.10. Such maxima are also associated with congruently
melting compounds in ternary and higher-order systems, for
example, the compound η in Fig. 6.3.
In a ternary system, a minimum point may be observed on a
univariant line as shown in Fig. 8.2. In this hypothetical system,
there are two solid phases: pure stoichiometric A and a solid solution of B and C in which A is only slightly soluble. The A-B and C-A
binary subsystems thus have simple eutectic phase diagrams,
while the B-C system resembles Fig. 3.1. A liquid with a composition at the minimum point m on the univariant line joining the
two binary eutectic points e1 and e2 will solidify isothermally to
give solid A and a solid solution of composition at point p. Point
135
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
A
Tie-line
136
e1
B
m
p
e2
C
Solid solution
Fig. 8.2 Liquidus projection of a system A-B-C with a minimum on the univariant
line joining the binary eutectic points e1 and e2. Solid phases are stoichiometric
A and a binary solid solution B-C (Pelton, 2014).
m is a local liquidus minimum, but the isothermal solidification at
point m is not a ternary eutectic reaction since the latter involves
three solid phases. Similar quasi-invariant reactions can also
occur at liquidus minima in higher-order systems.
At point m in Fig. 8.2, the eutectic quasi-invariant reaction
Liq ! α + β occurs isothermally. A line joining two binary peritectic points p1 and p2 in a ternary system can also pass through a
minimum at which an isothermal peritectic quasi-invariant reaction occurs: Liq + α ! β.
Another example of a quasi-invariant reaction occurs at saddle
points such as the points s in Figs. 6.3 and 6.4. Saddle points occur
at maxima on univariant lines. A ternary liquid with a composition
exactly at a saddle point will solidify isothermally, at constant
composition, to give the two solid phases with which it is in equilibrium. Higher-order saddle points can be found in systems
where C 4.
8.4
Nonequilibrium Scheil-Gulliver
Solidification
During cooling of an alloy from the liquid state at complete
thermodynamic equilibrium, all solid solution phases remain
homogeneous through internal diffusion, and all peritectic reactions proceed to completion. Such conditions are generally
approached only when the cooling rate is extremely slow. In
practice, conditions often approach more closely to those of
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
Scheil-Gulliver cooling (Scheil, 1942) in which solids, once precipitated, cease to react with the liquid or with each other; there is no
diffusion in the solids; the liquid phase remains homogeneous; and
the liquid is in equilibrium with the surfaces of the solid phases.
The course of Scheil-Gulliver solidification can be calculated
by thermodynamic database computing systems (FactSage,
Thermo-Calc, Pandat). To calculate the course of Scheil-Gulliver
cooling at a fixed overall composition, the following simulation
strategy is adopted. At a given overall composition, the liquidus
temperature is first calculated by decreasing the temperature
incrementally and performing equilibrium calculations with each
decrease until a solid phase is observed to precipitate. Then, starting at the liquidus, the temperature is decreased by an increment
ΔT, and the compositions and amounts of precipitated solids at
equilibrium with the liquid are calculated. These solids are then
“removed” from further equilibrium calculations, but their compositions and amounts are retained in memory. The remaining
liquid is then cooled by successive increments ΔT, and the process
is repeated until all the liquid has solidified. Generally, a value of
ΔT ¼ 5 K is sufficient to give a good approximation to ScheilGulliver cooling conditions, but occasionally, a smaller value
may be required.
In this way, Scheil-Gulliver cooling can be simulated, and plots
of the amounts of the phases versus temperature during cooling
similar to Fig. 8.1 can be drawn but only at one composition at
a time. By analogy with equilibrium phase diagrams that apply
to equilibrium cooling, it would be very useful to calculate and
plot Scheil-Gulliver constituent diagrams, which would permit
one to better visualize the course of Scheil-Gulliver solidification
as temperature and composition are varied. The theory and calculation of such diagrams have been presented in a recent article
(Pelton et al., 2017).
8.5
8.5.1
Scheil-Gulliver Constituent Diagrams
Binary Systems
The equilibrium phase diagram and Scheil-Gulliver constituent diagram of the Al-Li system are shown in Figs. 8.3 and 8.4,
respectively. The composition paths followed by the liquid phase
during solidification of alloys of overall compositions XLi ¼ 0.10
and XLi ¼ 0.65 are shown.
In the case of equilibrium cooling of an alloy of composition
XLi ¼ 0.10 in Fig. 8.3, solidification begins on the liquidus at
918 K with precipitation of a primary fcc phase, and solidification
137
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
1000
918
XLi = 0.65
XLi = 0.10
901
904
900
Liquid + AlLi
fcc
800
793
AlLi
700
Liquid + Al2Li3
600
604
fcc + AlLi
Al2Li3
500
400
Liquid + Al4Li9
452
bcc
Al4Li9
T (K)
Liquid
874
Al2Li3 + Al4Li9
138
bcc + Al
300
0
0.2
0.4
0.6
4Li9
0.8
1
Mole fraction Li
Fig. 8.3 Calculated equilibrium phase diagram of the Al-Li system illustrating course of equilibrium solidification
of alloys of compositions XLi ¼ 0.10 and XLi ¼ 0.65 (Pelton et al., 2017).
1000
XLi = 0.10
918
Liq + AlLi
Liq + fcc
874
800
Liq + AlLi
700
AlLi + eut (fcc + AlLi)
793
fcc + eut (fcc + AlLi)
T (K)
901
600
500
400
Liq + AlLi + Al2Li3
604
Liq + bcc
Liq + Al2Li3
Liq + Al4Li9
900
XLi = 0.65
Liquid
Liq + AlLi + Al2Li3 + Al4Li9
452
AlLi + Al2Li3 + Al4Li9
+ eut ( Al4Li9 + bcc)
bcc + eut
300
0
0.2
0.4
0.6
Mole fraction Li
0.8
1
Fig. 8.4 Calculated Scheil-Gulliver constituent diagram of the Al-Li system illustrating course of Scheil-Gulliver
solidification of alloys of compositions XLi ¼ 0.10 and XLi ¼ 0.65 (Pelton et al., 2017).
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
is complete at 904 K when the alloy consists of a single homogeneous solid phase with composition XLi ¼ 0.10.
In the case of Scheil-Gulliver cooling of the same alloy in
Fig. 8.4, precipitation of the primary fcc constituent begins on
the liquidus at 918 K and continues down to the eutectic temperature of 874 K where the following eutectic reaction occurs
isothermally until all the liquid has been consumed:
Liquid ! fcc + AlLi
(8.1)
Hence, at all temperatures below 874 K at XLi ¼ 0.10, the
microstructural constituents are an inhomogeneous fcc primary
constituent and a eutectic constituent (fcc + AlLi) as shown
in Fig. 8.4.
In the case of equilibrium cooling of an alloy of composition
XLi ¼ 0.65 in Fig. 8.3, solidification begins on the liquidus at
901 K. The primary AlLi solution precipitates between 901 and
793 K. At 793 K, the peritectic reaction
Liquid + AlLi ! Al2 Li3
(8.2)
occurs isothermally until all the AlLi phase has been consumed.
Cooling then continues between 793 and 604 K with precipitation
of Al2Li3. At 604 K, the peritectic reaction
Liquid + Al2 Li3 ! Al4 Li9
(8.3)
occurs isothermally until all the liquid has been consumed, and
solidification is complete at 604 K. In the case of Scheil-Gulliver
cooling in Fig. 8.4, primary precipitation of AlLi occurs between
901 and 793 K. Since peritectic reactions cannot occur during
Scheil-Gulliver cooling, the reaction in Eq. (8.2) does not take place,
and the primary AlLi constituent remains unchanged at all temperatures below 793 K. Between 793 and 604 K, Al2Li3 precipitates. At
604 K, the peritectic reaction (Eq. 8.3) does not occur, and all the
Al2Li3 that precipitated remains unchanged at all temperatures
below 604 K. Between 604 and 452 K, precipitation of Al4Li9 occurs.
At the eutectic temperature of 452 K, the eutectic reaction
Liquid ! Al4 Li9 + bcc
(8.4)
occurs isothermally, and solidification is complete at this temperature. As a result, on the Scheil-Gulliver constituent diagram in
Fig. 8.4 at XLi ¼ 0.65 at all temperatures below 452 K, there is a
region showing four microstructural constituents: primary AlLi
(inhomogeneous), Al2Li3, Al4Li9, and (Al4Li9 + bcc) eutectic.
The fields on a Scheil-Gulliver constituent diagram are labeled
not with the names of phases but with the names of the microstructural constituents that may be stoichiometric compounds (Al2Li3
139
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
and Al4Li9), inhomogeneous solutions (AlLi), or eutectic constituents. Hence, Scheil-Gulliver diagrams are more properly called
“constituent diagrams” rather than phase diagrams.
In general, unlike equilibrium phase diagrams, Scheil-Gulliver
constituent diagrams do not indicate the compositions or relative
amounts of the solid solution constituents. For example, at a given
temperature above 793 K in Fig. 8.4, the average composition of
the AlLi constituent cannot be read directly from the diagram.
Furthermore, the relative amounts of the constituents cannot be
read directly from the diagrams (i.e., the lever rule does not apply).
However, this information can be readily calculated and displayed.
8.5.2
Ternary Systems
It can be appreciated from Figs. 8.3 and 8.4 that, with some
experience, one could construct a Scheil-Gulliver constituent diagram for a binary system “by inspection” from the corresponding
equilibrium phase diagram. However, this is no longer true in ternary or higher-order systems. As an example, the Au-Bi-Sb system
is chosen because it permits several features of ternary ScheilGulliver constituent diagrams to be illustrated in a single system.
The projection of the liquidus surface is shown in Fig. 8.5. In this
0.8
0.7
800
0. 3
A
0.2
AuSb2
Equilibrium crystallization path
Scheil-Gulliver crystallization path
0.1
0.9
Sb
D
0.4
0.6
p
0.5
0.4
0. 6
X (651)
0.3
dra
0.2
600
569.2
P
70
0
0.9
fcc
0.8
10
00
l
AuSb2
80
0
Au
2B
E
i
0.8
0.7
0.6
0.5
0.7
he
bo
Au
0.9
om
Rh
12
00
0.5
700
e
0.1
140
0.4
Au2Bi Mole fraction
p 0.3
512.3
0.2 e 0.1
Bi
Fig. 8.5 Calculated liquidus projection of the Au-Bi-Sb system (temperatures in
kelvin). Crystallization paths during equilibrium cooling (black circles) and ScheilGulliver cooling (blue diamonds) are shown for two alloys (A and D) whose
compositions are given in Table 8.1 (Pelton et al., 2017).
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
system, the following four solid phases are observed: a Au-rich fcc
phase in which Sb is very slightly soluble; a binary Bi-Sb rhombohedral phase, miscible above 443 K at all Bi/Sb ratios, in which the
solubility of Au is negligible; a stoichiometric Au2Bi phase in
which the solubility of Sb is negligible; and an AuSb2 phase exhibiting a solubility of up to 12.5 mol% Au2Bi. In Fig. 8.5, lowercase e
and p indicate binary eutectic and peritectic compositions.
Uppercase E and P indicate ternary eutectic and peritectic points
at 512.3 and 569.2 K, respectively.
Consider the point A in Fig. 8.5 whose coordinates (Sb 70-Au
28.5-Bi 1.5) are shown in Table 8.1. The calculated crystallization
path during equilibrium cooling is shown in the diagram by the
black circles. A calculated summary of the reaction steps during
equilibrium cooling is shown in Table 8.1. The liquidus is reached
at 765.4 K. Next, the rhombohedral phase precipitates until the
liquid composition reaches the univariant line at 729.7 K. The liquid composition then follows the univariant line as the following
peritectic reaction occurs:
Liquid + Rhombohedral ! AuSb2
(8.5)
Solidification ceases at 704.8 K when the liquid has been
completely consumed.
The course of Scheil-Gulliver solidification starting at point A is
very different. The calculated crystallization path is shown in
Fig. 8.5 by the blue diamonds, and a calculated summary of the
reaction steps is shown in Table 8.1. Precipitation of the rhombohedral phase follows virtually the same crystallization path as in
equilibrium cooling because, at this composition, the rhombohedral phase is nearly pure Sb. However, when the liquid composition reaches the univariant line at 729.7 K, the peritectic reaction
(Eq. 8.5) is prohibited. Hence, the crystallization path continues
across the AuSb2 phase field with precipitation of the AuSb2 solution down to 578.3 K. It then follows the univariant line from 578.3
to 569.2 K with precipitation of a binary (AuSb2 + fcc) eutectic. The
ternary peritectic reaction at 569.2 K is prohibited; hence, the crystallization path continues along the univariant line with precipitation of a binary (AuSb2 + Au2Bi) eutectic down to the ternary
eutectic point at 512.3 K where the ternary eutectic reaction
occurs isothermally until all the liquid is consumed.
As a second example, consider point D in Fig. 8.5 whose composition is given in Table 8.1. The calculated crystallization paths
for equilibrium solidification (black circles) and Scheil-Gulliver
solidification (blue diamonds) are shown in Fig. 8.5, and the reaction steps are summarized in Table 8.1. As can be seen in the figure, the equilibrium and Scheil-Gulliver paths diverge during the
141
142
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
Table 8.1 Calculated Steps During Scheil-Gulliver
and Equilibrium Cooling From the Liquid State to
Final Disappearance of the Liquid at Several
Compositions Shown in Figs. 8.5 and 8.6
Scheil-Gulliver Cooling
Point A: Sb70-Au28.5-Bi1.5
Liquid cooling
LIQ ! RHOMBO
LIQ ! AuSb2
LIQ ! AuSb2 + FCC (e1)
LIQ ! AuSb2 + Au2Bi (e3)
LIQ ! AuSb2 + Au2Bi + RHOMBO (E)
Point B: Sb70-Au26-Bi4
Liquid cooling
LIQ ! RHOMBO
LIQ ! AuSb2
LIQ ! AuSb2 + Au2Bi (e3)
LIQ ! AuSb2 + Au2Bi + RHOMBO (E)
Point C: Sb70-Au20-Bi10
Liquid cooling
LIQ ! RHOMBO
LIQ ! AuSb2
LIQ ! AuSb2 + RHOMBO (e4)
LIQ ! AuSb2 + Au2Bi + RHOMBO (E)
Point D: Sb70-Au5-Bi25
Liquid cooling
LIQ ! RHOMBO
LIQ ! RHOMBO + AuSb2 (e4)
LIQ ! AuSb2 + Au2Bi + RHOMBO (E)
Point E: Sb70-Au0.2-Bi29.8
Liquid cooling
LIQ ! RHOMBO
LIQ ! RHOMBO + Au2Bi (e4)
LIQ ! AuSb2 + Au2Bi + RHOMBO (E)
Equilibrium Cooling
(1000–765.4 K)
(765.4–729.7 K)
(729.7–578.3 K)
(578.3–569.2 K)
(569.2–512.3 K)
(512.3 K isothermal)
Point A: Sb70-Au28.5-Bi1.5
Liquid cooling
LIQ ! RHOMBO
LIQ + RHOMBO ! AuSb2
(1000–765.4 K)
(765.4–729.7 K)
(729.7–704. 8 K)
Point D: Sb70-Au5-Bi25
Liquid cooling
LIQ ! RHOMBO
LIQ ! RHOMBO + AuSb2 (e4)
(1000–816.4 K)
(816.4–648.7 K)
(648.7–616.1 K)
(1000–770.9 K)
(770.9–723.5 K)
(723.5–538.5 K)
(538.5–512.3 K)
(512.3 K isothermal)
(1000–783 K)
(783.8–706.2 K)
(706.2–547.9 K)
(547.9–512.3 K)
(512.3 K isothermal)
(1000–816.4 K)
(816.4–630.9 K)
(630.9–512.3 K)
(512.3 K isothermal)
(1000–826.7 K)
(826.7–513.5 K)
(513.5–512.3 K)
(512.3 K isothermal)
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
precipitation of the rhombohedral solution because the solid
solution remains homogeneous during equilibrium cooling,
whereas during Scheil-Gulliver cooling, only the surfaces of the
solid grains are in equilibrium with the liquid. After reaching
the univariant line, the crystallization paths in both the equilibrium and Scheil-Gulliver cases follow the univariant line as the
following binary eutectic reaction occurs:
Liquid ! Rhombohedral + AuSb2
(8.6)
(Note that at temperatures above point X on the univariant line in
Fig. 8.5, the univariant reaction is peritectic (Eq. 8.5), whereas at
temperatures below 651 K, the reaction is eutectic (Eq. 8.6). In the
case of equilibrium cooling, the reaction in Eq. (8.6) proceeds
until all the liquid is consumed at 616.1 K. In the case of ScheilGulliver cooling, the reaction (Eq. 8.6) continues down to the ternary eutectic temperature.
On the liquidus surface in Fig. 8.6, the Scheil-Gulliver crystallization paths at compositions A and D from Fig. 8.5 are repeated along
with Scheil-Gulliver paths at three additional compositions (B, C,
and E) whose compositions and reaction paths are shown in
0.7
0.6
0.5
0.5
l
dra
0.4
he
750
0.3
D E
bo
p
C
om
A B
Rh
AuSb2
0.2
0.8
0.1
0.9
Sb
0.4
0.6
AuSb2
e
0.3
0.2
0.1
60
0
(569.2)
0.9
fcc
0
0
0.8
70
65
0.7
X
Au
2 Bi
Au
0.9
0.8
0.7
Au2Bi
0.6
0.5
0.4
p 0.3
(512.3)
0.2
e 0.1
Bi
Mole fraction
Fig. 8.6 Calculated liquidus projection of the Au-Bi-Sb system (temperatures in
kelvin). Crystallization paths during Scheil-Gulliver cooling are shown for five alloys
(A, B, C, D, and E) whose compositions are given in Table 8.1 (Pelton et al., 2017).
143
144
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
Table 8.2 Microstructural Constituents in the
Au-Bi-Sb System after Scheil-Gulliver
Cooling is Complete
Fcc (solution)
AuSb2 (solution)
Au2Bi
Rhombohedral (solution)
e1 ¼ (fcc + AuSb2) eutectic
e2 ¼ (Au2Bi + rhombo) eutectic
e3 ¼ (Au2Bi + AuSb2) eutectic
e4 ¼ (AuSb2 + rhombo) eutectic
E ¼ (Au2Bi + AuSb2 + rhombo) eutectic
Table 8.1. All the crystallization paths continue to the ternary eutectic point at 512.3 K. Necessarily, in any Scheil-Gulliver solidification,
the final disappearance of the liquid will always occur at a local
minimum on the liquidus surface. (In this regard, see also Fig. 8.4.)
The possible microstructural constituents that can result from
Scheil-Gulliver cooling of alloys in the Au-Bi-Sb system are listed
in Table 8.2. These include four single-phase constituents, four
binary eutectic constituents (denoted as e1, e2, e3, and e4), and
one ternary eutectic constituent (denoted as E).
The calculated Scheil-Gulliver constituent diagram of the
Au-Bi-Sb system at 450 K is shown in Fig. 8.7. This diagram shows
the constituents present after Scheil-Gulliver cooling from above
the liquidus to a temperature below the final disappearance of the
liquid phase. Fig. 8.8 is a superposition of Figs. 8.6 and 8.7. For
example, point A lies in the region labeled (Rh + AuSb2 + e1 + e3 + E), and as can be seen from Table 8.1, these are the
five constituents present after complete solidification of this alloy.
Similarly, points B, C, D, and E can be seen to lie in the regions
consistent with the precipitation sequences shown in Table 8.1.
For comparison with Fig. 8.7, the equilibrium isothermal phase
diagram for the system at 450 K is shown in Fig. 8.9.
In Fig. 8.10 is a calculated isoplethal Scheil-Gulliver constituent diagram along a join at constant mole fraction XSb ¼ 0.70.
The compositions of points A, B, C, D, and E, which all lie at
XSb ¼ 0.70, are shown along the x-axis in Fig. 8.10. Scheil-Gulliver
cooling proceeds down vertical (constant composition) lines in
this diagram. The transition temperatures at the compositions
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
145
0.7
0.6
0.5
0.4
0.3
0.7
e4
E
0.8
+
0.2
0.6
+
2 Bi
0.7
+e
3
2Bi + e3 + E
0.6
0.5
0.9
0.1
0.5
+E
2
+E
0.8
e4
Sb
+ e3
E
0.9
Rh + e2 + E
Au
f
fcc + Au
0.4
Au
Sb 2
e3 +
+E
fcc + Au
Au
+
Rh
e4 + E
Au
e1 +
+ e3
0.3
uSb 2 +
b2 +
AuS
e1
cc +
0.2
Rh + A
Rh + AuSb2 + e3 + E
0.8
Rh + AuSb2 + e1 + e3 + E
0.1
0.9
Sb
+E
0.4
0.3
2Bi + e2 + E Mole fraction
0.2
Bi
0.1
Au2Bi + e2 + E
Fig. 8.7 Calculated isothermal Scheil-Gulliver constituent diagram of the Au-Bi-Sb system at 450 K (Pelton et al., 2017).
D E
C
0.5
0.5
0.4
0.6
AB
0.3
0.7
0.2
0.8
0.1
0.9
Sb
0.4
Au
0.9
0.1
0.8
0.2
0.7
0.3
0.6
X
0.9
0.8
0.7
0.6
0.5
0.4
Mole fraction
Fig. 8.8 Superposition of Figs. 8.6 and 8.7 (Pelton et al., 2017).
0.3
0.2
0.1
Bi
146
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
0.2
o
Rh
0.8
0.1
0.9
Sb
0.7
0.3
o
mb
he
0.6
l+A
0.5
b2
0.4
b2 +
Au
2 Bi +
fcc
0.3
0.7
0.8
0.1
0.9
AuS
0.8
0.2
0.6
0.9
0.5
uS
AuSb2 + fcc
Au
0.4
dra
AuSb2
AuSb2 + Au2Bi + Rhombo
0.7
0.6
Au2Bi
0.5
0.4
Mole fraction
0.3
0.2
0.1
Bi
Au2Bi + Rhombo
Fig. 8.9 Calculated isothermal equilibrium phase diagram of the Au-Bi-Sb system at 450 K (Pelton et al., 2017).
900
L
800
L + Rh
L + Rh + AuSb2 + e1
600
L + Rh + AuSb2
L + Rh + e2
L + Rh + AuSb2 + e1 + e3
Rh + AuSb2 + e1 + e3 + E
500
400
A
300
0
Rh + AuSb2 + e3 + E
L + Rh + AuSb2 + e3
L + Rh + AuSb2 + e4
Rh + e4 + E
Rh + AuSb2 + e4 + E
B
0.05
D
C
0.1
L + Rh + e4
0.15
Mole fraction Bi
0.2
0.25
Rh + e2 + E
T (K)
700
E
0.3
Fig. 8.10 Calculated isoplethal Scheil-Gulliver constituent diagram of the Au-Bi-Sb system at XSb ¼ 0.70. Compositions
of points A, B, C, D, and E as in Table 8.1 and Fig. 8.8 (Pelton et al., 2017).
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
147
900
Liquid
800
Liquid + Rhombohedral
T (K)
700
Liquid + Rhom
bohedral + Au
Sb2
600
Rhombohedral + AuSb2
500
400
Rhombohedral + Rhombohedral
300
D
A
0
0.05
0.1
0.15
0.2
0.25
0.3
Mole fraction Bi
Fig. 8.11 Calculated isoplethal equilibrium phase diagram of the Au-Bi-Sb system at XSb ¼ 0.70. Compositions of points
A and D as in Table 8.1 and Fig. 8.5 (Pelton et al., 2017).
A, B, C, D, and E can be seen to correspond to the temperatures
shown in Table 8.1. For comparison, the equilibrium isoplethal
phase diagram along the same join is shown in Fig. 8.11 along with
the compositions of points A and D.
8.5.3
Topology and Calculation of Scheil-Gulliver
Constituent Diagrams
For equilibrium phase diagrams, the law of adjoining phase
regions was discussed in Section 7.2.
For Scheil-Gulliver constituent diagrams, an analogous law of
adjoining constituent regions may be postulated: as a constituent
boundary is crossed, one and only one constituent appears or disappears, again with the proviso that some lines on a diagram can
actually be two coincident or nearly coincident boundaries. For
example, the vertical line descending from the eutectic point at
XLi ¼ 0.23 in Fig. 8.4 is actually two coincident boundaries on
either side of an infinitely narrow single-constituent ((fcc + AlLi)
eutectic) region, and the line between the regions labeled (Rh+
AuSb2 + e3 + E) and (Rh + AuSb2 + e4 + E) in Fig. 8.7 is actually two
boundaries on either side of an infinitely narrow (Rh+ AuSb2 + E)
region.
As a consequence of the law of adjoining phase regions, as discussed in Section 7.4, an equilibrium phase diagram consists of a
148
Chapter 8 EQUILIBRIUM AND SCHEIL-GULLIVER SOLIDIFICATION
collection of zero-phase-fraction (ZPF) lines, one for each phase,
and a general algorithm for calculating phase boundaries, based
on this concept, was discussed in Section 7.4.1.
Similarly, as a consequence of the law of adjoining constituent
regions, a Scheil-Gulliver constituent diagram consists of a collection of zero constituent fraction (ZCF) lines, one for each constituent. Hence, a strategy analogous to that employed for
constructing equilibrium phase diagrams can be used to calculate
Scheil-Gulliver constituent diagrams. The software first scans
around the frame of the diagram to find the ends of the ZCF lines.
It then tracks and plots each ZCF line from one end to the other,
keeping the constituent just on the verge of appearing (i.e., thermodynamically stable but at zero amount). Once all the ZCF lines
have been calculated, the diagram is complete.
It may be noted that some ZCF lines are also ZPF lines. However, this is not necessarily the case; that is, the regions on both
side of a ZCF line may contain the same phases but not the same
constituents. For example, the line descending vertically from the
eutectic composition at XLi ¼ 0.23 in Fig. 8.4 is not a ZPF line.
Clearly, in an isoplethal Scheil-Gulliver constituent diagram in
which the vertical axis is temperature, as the temperature
decreases along a vertical path at constant composition, one additional solid constituent appears as each boundary is crossed, and
solid constituents that are already present at higher temperatures
never disappear.
Also, clearly, below the temperature of final disappearance of
the liquid, Scheil-Gulliver constituent diagrams are independent
of temperature since all solid-state reactions such as eutectoid
and peritectoid reactions and phase separation are prohibited.
However, caution should be exercised in interpreting ScheilGulliver constituent diagrams at room temperature since massive
transformations, martensitic transformations, and spinodal
decomposition can occur even during Scheil-Gulliver cooling
but are not shown in the diagrams.
References
Pelton, A.D., 2014. In: Laughlin, D.E., Hono, K. (Eds.), Physical Metallurgy. fifth ed.
(Chapter 3). Elsevier, New York.
Pelton, A.D., Eriksson, G., Bale, C.W., 2017. Metall. Mater. Trans. 48, 3113–3129.
Scheil, E., 1942. J. Inst. Met. 34, 70–72.
List of Websites
FactSage, Montreal, www.factsage.com
Pandat, www.computherm.com.
Thermo-Calc, Stockholm, www.thermocalc.com
9
PARAEQUILIBRIUM PHASE
DIAGRAMS AND MINIMUM
GIBBS ENERGY DIAGRAMS
CHAPTER OUTLINE
9.1 The Geometry of Paraequilibrium Phase Diagram Sections
9.2 Minimum Gibbs Energy Phase Diagrams 155
References 158
150
In certain solid systems, some elements diffuse much faster
than others. Hence, if an initially homogeneous single-phase system at high temperature is rapidly cooled and then held at a lower
temperature, a temporary paraequilibrium state may result in
which the rapidly diffusing elements have reached equilibrium
but the more slowly diffusing elements have remained essentially
immobile (Hultgren, 1947; Hillert, 2008; Kozeschnik and Vitek,
2000). The best known and most industrially important example
occurs when homogeneous austenite is quenched and annealed;
interstitial elements such as C and N are much more mobile than
the metallic elements. Of course, in reality, some diffusion of the
metallic elements will always occur, so that paraequilibrium is a
limiting state that is never fully realized but may, nevertheless,
be reasonably closely approached in many cases.
At paraequilibrium, the ratios of the slowly diffusing elements
are the same in all phases and are equal to their ratios in the initial
single-phase high-temperature alloy. Hence, the calculation of
paraequilibrium only involves modifying the Gibbs energy minimization algorithm by the addition of this constraint. For a complete discussion, see Pelton et al. (2014). Paraequilibrium phase
diagram sections can subsequently be calculated by exactly the
same procedure as is used to calculate full equilibrium phase diagrams as was discussed in Section 7.4.1.
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00009-9
# 2019 Elsevier Inc. All rights reserved.
149
150
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
9.1
The Geometry of Paraequilibrium
Phase Diagram Sections
In this section, the application of the Phase Rule to paraequilibrium phase diagrams is discussed as described in the recent article by Pelton et al. (2014).
Since a paraequilibrium calculation only involves adding an
additional constraint, the law of adjoining phase regions (LAPR)
for equilibrium phase diagram sections (see Section 7.2) also
applies to paraequilibrium phase diagram sections.
In the Fe-Cr-C system at elevated temperatures, the range of
homogeneous austenite (fcc) extends from pure Fe to approximately 8 mol% C and 15 mol% Cr. Alloys with compositions in this
range, when cooled rapidly and then held at a lower temperature,
may exhibit a temporary paraequilibrium state. A vertical section
of the full equilibrium Fe-Cr-C phase diagram section is shown in
Fig. 9.1 where the molar ratio C/(Fe + Cr) is plotted versus temperature, T, at a constant molar metal ratio Cr/(Fe + Cr) ¼ 0.04. In
Fig. 9.1 and other figures in this section, M23C6, M7C3, and
cementite are solutions of Fe and Cr carbides. On all these calculated diagrams, the formation of graphite has been suppressed.
1100
1000
fcc
fcc + Cementite + M7C3
900
fcc + bcc
M7C3 + fcc
T (°C)
800
fcc + bcc + M
bcc
a
700
c
7C3
b
751°
M7C3 + bcc
bcc + Cementite + M7C3
600
M7C3 + bcc + Cr3C2
500
489°
f
e
C6
3
M2
d
C3
M7
bcc + Cementite + Cr3C2
bc
M23C6 + bcc
+
c
300
bcc + Cr3C2
+
400
0
0.01
0.02
0.03
0.04
0.05
0.06
Molar ratio C/(Fe+Cr)
Fig. 9.1 Full equilibrium phase diagram section of the Fe-Cr-C system. Molar ratio C/(Fe + Cr) versus T at constant molar
ratio Cr/(Fe + Cr) ¼ 0.04 (formation of graphite suppressed) (Pelton et al., 2014).
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
151
1100
1000
900
fcc
a
T (°C)
800
b
700
Cementite + fcc
fcc + bcc
d
c
706.3°
bcc
600
Cementite + bcc
500
400
300
0
0.01
0.02
0.03
0.04
0.05
0.06
Molar ratio C/(Fe+Cr)
Fig. 9.2 Paraequilibrium phase diagram section of the Fe-Cr-C system when C is the only diffusing element. Molar ratio
C/(Fe + Cr) versus T at constant molar ratio Cr/(Fe + Cr) ¼ 0.04 (formation of graphite suppressed) (Pelton et al., 2014).
In Fig. 9.2 is shown the paraequilibrium phase diagram for
exactly the same section as in Fig. 9.1, calculated for the case
where C is the only diffusing element. Since the molar ratio
Cr/(Fe + Cr) ¼ 0.04 is now constant and the same in every phase,
the diagram in this particular example resembles a full equilibrium T-composition phase diagram of a two-component
system, the “components” being Fe0.96Cr0.04 and C. The three-phase
(fcc+bcc+cementite) region bcd now appears as an isothermal
invariant, similar to a binary eutectoid.
The x-axis in Fig. 9.2 is the molar ratio C/(Fe + Cr), not the carbon mole fraction XC ¼ C/(Fe + Cr + C). Similarly, the diagram is
calculated at constant ratio Cr/(Fe + Cr), not at constant Cr mole
fraction XCr ¼ Cr/(Fe + Cr + C). If the diagram is recalculated at
constant XCr ¼ 0.04 with the x-axis as XC, it will be nearly identical
to Fig. 9.2 since the C content is very small. However, the threephase region will no longer be isothermal because now the ratio
of the nondiffusing elements Cr/(Fe + Cr) varies with XC and so
the diagram can no longer be considered to be that of a pseudobinary system. This T-XC paraequilibrium diagram at constant XCr
is plotted in Fig. 9.3 where the temperature axis has been greatly
expanded to show clearly that the three-phase field is not isothermal. Note, however, that the three-phase field is still infinitely
152
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
707
706.8
fcc
706.6
fcc + bcc
706.4
Cementite + fcc
T (°C)
706.2
bcc
706
705.8
705.6
Cementite + bcc
705.4
705.2
705
0
0.01
0.02
0.04
0.03
XC = C/(C+Fe+Cr) molar ratio
0.05
0.06
Fig. 9.3 Paraequilibrium phase diagram section of the Fe-Cr-C system when C is the only diffusing element. Mole
fraction XC versus T at constant mole fraction XCr ¼ 0.04 (formation of graphite suppressed) (Pelton et al., 2014).
narrow. At any value of XC, the molar ratio Cr/(Fe + Cr) has a fixed
value that must be the same in every phase. This additional constraint removes a degree of freedom so that from the Phase Rule
(Eq. 3.1), for the three-phase paraequilibrium, F ¼ 0. In other
words, at any given XC, there is only one temperature where the
three phases can coexist, but this temperature varies with XC.
The LAPR clearly applies to both Figs. 9.2 and 9.3 as, indeed, it
does to all paraequilibrium phase diagram sections.
Another isothermal paraequilibrium phase diagram section
of the Fe-Cr-C system is shown in Fig. 9.4 where the molar ratio
Cr/(Fe + Cr) is plotted versus the molar ratio C/(Fe + Cr) at constant T ¼ 775°C. Tie-lines are horizontal in this diagram since
all phases at paraequilibrium have the same Cr/(Fe + Cr) ratio.
Since this ratio is the same in all phases, it acts, as far as the
geometry of the diagrams is concerned, like a “potential variable”
(such as T, pressure, or chemical potential (see Section 7.1)),
which is the same in all phases at equilibrium, rather than like
a normal composition variable, which, in general, is not the same
in phases at equilibrium. Indeed, Fig. 9.4 clearly has the same
topology as a full equilibrium binary T-composition phase diagram. When the ratios C/(Fe + Cr) and Cr/(Fe + Cr) are replaced
by XC ¼ C/(Fe + Cr + C) and XCr ¼ Cr/(Fe + Cr + C), respectively, a
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
153
0.25
bcc
M23C6 + bcc
Molar ratio Cr/(Fe+Cr)
0.2
fcc + bcc
M23C6 + fcc
0.15
fcc
0.1
Cementite + fcc
0.05
0
0
0.02
0.04
0.06
0.08
0.1
Molar ratio C/(Fe+Cr)
Fig. 9.4 Paraequilibrium phase diagram section of the Fe-Cr-C system when C is the only diffusing element. Molar
ratio C/(Fe + Cr) versus molar ratio Cr/(Fe + Cr) at constant T ¼ 775°C. Tie-lines are shown (formation of graphite
suppressed) (Pelton et al., 2014).
topologically similar diagram results; however, the tie-lines and
the invariant lines are no longer horizontal, but lie along loci of
constant Cr/(Fe + Cr) ratio. Nevertheless, the three-phase fields
are still infinitely narrow, and of course, the LAPR applies.
Yet another section of the paraequilibrium phase diagram of
the Fe-Cr-C system is shown in Fig. 9.5. At a constant molar ratio
C/(Fe + Cr) ¼ 0.05, temperature is plotted versus the molar ratio
Cr/(Fe + Cr). Both axis variables, T and Cr/(Fe + Cr), are the same
in all phases at paraequilibrium. That is, as just stated, the ratio
Cr/(Fe + Cr) acts like a “potential variable.” From the Phase Rule,
Eq. (3.1), when three phases are at equilibrium at constant pressure, F ¼ 3–3 + 1 ¼ 1. Hence, the three-phase fields that separate
the two-phase fields in Fig. 9.5 are infinitely narrow univariant
lines. The molar ratio C/(Fe + Cr) is a “normal” composition variable that need not be the same in all phases at equilibrium. Hence,
keeping this ratio constant does not decrease the number of
degrees of freedom. Note that the lines in Fig. 9.5 separating the
fcc field from the (fcc + cementite) and (fcc + M23C6) fields are
not infinitely narrow three-phase univariant lines but, rather,
are simple phase boundaries. The point in Fig. 9.5 where the four
154
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
1000
950
fcc
900
T (°C)
850
800
fcc + Cementite
fcc + M23C6
750
700
650
bcc + M23C6
bcc + Cementite
600
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Molar ratio Cr/(Fe+Cr)
Fig. 9.5 Paraequilibrium phase diagram section of the Fe-Cr-C system when C is the only diffusing element. Molar ratio
Cr/(Fe + Cr) versus T at constant molar ratio C/(Fe + Cr) ¼ 0.05 (formation of graphite suppressed) (Pelton et al., 2014).
univariant lines converge is an infinitely small four-phase invariant field. The LAPR applies.
Figure 9.6 is a paraequilibrium phase diagram section for the
four-component Fe-Cr-C-N system for the case when both
C and N, but not Fe or Cr, diffuse. Temperature is plotted versus
the molar ratio Cr/(Fe + Cr) with the two composition variable C/
(Fe + Cr) and N/(Fe + Cr) constant. From the Phase Rule, when four
phases are at equilibrium at constant pressure, F ¼ 4 4 + 1 ¼ 1.
Hence, four-phase fields appear as infinitely narrow univariant
lines. In Fig. 9.6, the four-phase univariant regions are the (fcc
+ bcc + cementite + M23C6) and (fcc + bcc + cementite + M2(C,N))
lines. The other lines in Fig. 9.6 are simple phase boundaries.
From the preceding examples, it can be seen that paraequilibrium phase diagram sections obey exactly the same rules as
equilibrium phase diagrams with the following proviso. Since
the molar ratios of nondiffusing elements are the same in all
phases at paraequilibrium, these ratios act, as far as the geometry
of the diagram is concerned, like “potential” variables (such as T,
pressure, or chemical potentials) rather than like “normal” composition variables that need not be the same in all phases. Furthermore and for the same reason, holding a ratio of nondiffusing
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
155
900
800
fcc
700
T (°C)
fcc + bcc
600
fcc + bcc + M23C6
fcc + bcc + Cementite
500
400
bcc + Cementite + M2(C, N)
300
0
0.05
0.1
0.15
0.2
0.25
Molar ratio Cr/(Fe+Cr)
Fig. 9.6 Paraequilibrium phase diagram section of the Fe-Cr-C-N system when C and N are the only diffusing elements.
Molar ratio Cr/(Fe + Cr) versus T at constant molar ratios C/(Fe + Cr) ¼ N/(Fe + Cr) ¼ 0.02 (formation of graphite
suppressed) (Pelton et al., 2014).
elements constant decreases the number of degrees of freedom of
the system by 1, whereas holding a “normal” composition variable
constant does not.
In the preceding examples, all compositions are expressed as
molar ratios. However, replacing molar ratios by mass ratios will,
of course, have no effect on the topology of the diagrams.
9.2
Minimum Gibbs Energy Phase Diagrams
If a paraequilibrium calculation is performed under the constraint that no elements diffuse, then the ratios of all elements
remain the same as in the initial homogeneous high-temperature
state. Hence, such a calculation will simply yield the single homogeneous phase with the minimum Gibbs energy at the temperature
and overall composition of the calculation. Such calculations are of
practical interest in physical vapor deposition (PVD) when deposition from the vapor phase is so rapid that phase separation does not
occur, resulting in a single-phase solid deposit.
The full equilibrium phase diagram of the Ni-Cr system is
shown in Fig. 9.7. The corresponding minimum Gibbs energy diagram is shown in Fig. 9.8. Clearly, two-phase fields appear as
156
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
1900
1700
Liquid
T (°C)
1500
1300
bcc
1345°
fcc
1100
fcc + bcc
900
700
500
0
0.2
0.4
0.6
XCr = Cr/(Ni+Cr) molar ratio
0.8
1
Fig. 9.7 Full equilibrium phase diagram of the Ni-Cr system. Mole fraction Cr versus T (Pelton et al., 2014).
1900
1700
Liquid
1500
T (°C)
1331°
1300
bcc
fcc
1100
900
700
500
Sigma
0
0.2
0.4
0.6
0.8
1
XCr = Cr/(Ni+Cr) molar ratio
Fig. 9.8 Minimum Gibbs energy phase diagram of the Ni-Cr system. Mole fraction Cr versus T (Pelton et al., 2014).
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
0.3
0.7
0.2
0.8
0.1
0.9
Fe
0.5
0.5
0.4
0.6
bcc
0.6
0.4
fcc
Ni
bcc
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.9
0.1
0.8
0.2
0.7
0.3
Sigma
Cr
Mole fraction
Fig. 9.9 Minimum Gibbs energy phase diagram section of the Fe-Ni-Cr system at
600°C (Pelton et al., 2014).
univariant lines. These lines are the loci of the intersections of the
curves of Gibbs energy versus composition and are sometimes
called To lines. Each To line in Fig. 9.8 lies within the corresponding two-phase region in Fig. 9.7.
As can be seen in Fig. 9.8, at lower temperatures, the sigma
phase is the phase with the lowest Gibbs energy in alloys containing approximately 63%–73% Cr, even though this phase does not
appear in the equilibrium phase diagram. This was pointed out
previously by Saunders and Miodownik (1986) and Spencer
(2001) who noted that sigma deposits have been observed in
vapor-deposited Ni-Cr samples over this approximate composition range at 25°C.
Another example of a calculated minimum Gibbs energy phase
diagram is shown in Fig. 9.9.
Clearly, minimum Gibbs energy phase diagrams obey the geometric rules summarized at the end of Section 9.1.
157
158
Chapter 9 PARAEQUILIBRIUM PHASE DIAGRAMS AND MINIMUM GIBBS ENERGY DIAGRAMS
References
Hillert, M., 2008. Phase Equilibria, Phase Diagrams and Phase Transformations—
Their Thermodynamic Basis, second ed. Cambridge Univ. Press, Cambridge.
Hultgren, A., 1947. Trans. ASM 39, 915.
Kozeschnik, E., Vitek, J.M., 2000. Calphad J. 24, 495–502.
Pelton, A.D., Koukkari, P., Pajarre, R., Eriksson, G., 2014. J. Chem. Thermodyn.
72, 16–22.
Saunders, N., Miodownik, A.P., 1986. J. Mater. Res. 1, 38–46.
Spencer, P., 2001. Z. Metallkd. 92, 1145–1150.
10
SECOND-ORDER AND
HIGHER-ORDER TRANSITIONS
CHAPTER OUTLINE
10.1 Equations for Thermodynamic Properties due to Magnetic
Ordering 163
References 164
List of Websites 164
All the transitions discussed until now have been first-order
transitions. When a first-order phase boundary is crossed, a
new phase appears with extensive properties that are discontinuous with those of the other phases.
An example of a transition that is not first order is the ferromagnetic to paramagnetic transition of Fe, Co, Ni, and other ferromagnetic materials that contain unpaired electron spins. Below
its Curie temperature, Tcurie ¼ 1043 K (770°C) (see Fig. 2.1), Fe is
ferromagnetic since its spins tend to be aligned parallel to one
another. This alignment occurs because it lowers the internal
energy of the system. At 0 K at equilibrium, the spins are fully
aligned. As the temperature increases, the spins become progressively less aligned, since this disordering increases the entropy of
the system, until at Tcurie the disordering is complete and the
material becomes paramagnetic. There is no two-phase region,
and no abrupt change occurs at Tcurie which is simply the temperature at which the disordering becomes complete.
The following model is vastly oversimplified and is presented
only as the simplest possible model illustrating the essential features of a second-order phase transition.
Suppose that we may speak of localized spins and, furthermore, that a spin is oriented either “up” or “down.” Let the fraction
of “up” and “down” spins be ξ and (1 ξ). Suppose further that
when two neighboring spins are aligned, a stabilizing internal
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00010-5
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159
160
Chapter 10 SECOND-ORDER AND HIGHER-ORDER TRANSITIONS
energy contribution ε < 0 results. Assuming that the spins are randomly distributed, the probability of two “up” spins being aligned
is ξ2 and of two “down” spins being aligned is (1 ξ)2. The resultant contribution to the molar Gibbs energy is
h
i
(10.1)
g ¼ RT ðξ ln ξ + ð1 ξÞ ln ð1 ξÞÞ + ξ2 + ð1 ξÞ2 ε
The equilibrium value of ξ is that that minimizes g. Setting
dg/dξ ¼ 0 gives
ξ=ð1 ξÞ ¼ exp ð2ð1 2ξÞε=RT Þ
(10.2)
At T ¼ 0 K, ξ ¼ 0 (or, equivalently, ξ ¼ 1). That is, the spins are
completely aligned. A plot of ξ versus T from Eq. (10.2) shows that
ξ increases with T, attaining the value ξ ¼ ½ when T ¼ ε/R. At all
higher temperatures, ξ ¼ ½, which is the only solution of Eq. (10.2)
above Tcurie ¼ ε/R. As the spins become progressively more disordered with increasing temperature below Tcurie, energy is
absorbed, thereby increasing the heat capacity. Most of the disordering occurs over a fairly small range of temperature below Tcurie.
This can be seen in Fig. 2.1 where the slope of the enthalpy curve
increases just below Tcurie and then quite abruptly decreases again
above this temperature.
In this extremely simple model, there is a discontinuity in the
slope cP ¼ dh/dT at Tcurie. That is, there is a discontinuity in the
second derivative of the Gibbs energy. Such a transition is called
a second-order transition.
In Fig. 10.1, the heat capacity of α-Fe (FactSage), which is the
slope of Fig. 2.1, is shown. It can be seen that, although cP
decreases rapidly above the Curie temperature, the decrease is
not discontinuous. This is due to the fact that some short-range
ordering persists above Tcurie so that the transformation is not
exactly second order. Transitions involving discontinuities in
the third, fourth, and other derivatives of G are called third-order,
fourth-order, etc. transitions.
For more details on magnetism, see Section 10.1.
The T-composition phase diagram of the Fe-Ni system is shown
in Fig. 10.2. The Curie temperature varies with composition as it traverses the bcc phase field as a line. Nickel also undergoes a magnetic
transition at its Curie temperature of 360°C. The approximately
second-order transition line crosses the fcc phase field as shown.
At point P in Fig. 10.2, the second-order transition line widens
into a two-phase miscibility gap, and the magnetic transition
becomes first order. Point P is called a tricritical point. (It may
be noted that some other versions of the Fe-Ni phase diagram
do not show a tricritical point (Massalski et al., 2001).)
Chapter 10 SECOND-ORDER AND HIGHER-ORDER TRANSITIONS
62.5
60
Tcurie
57.5
55
52.5
cp (J mol-1 K-1)
50
Actual cP
47.5
45
42.5
40
37.5
35
32.5
cP without magnetic contribution
30
27.5
25
22.5
20
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800
T (K)
Fig. 10.1 Heat capacity of a-Fe (FactSage).
1000
Tcurie(bcc)
900
Tcurie(fcc)
600
bcc
Temperature (∞C)
C)
800
770°
700
500
P
fcc
400
504°
L12
(ordered)
360°
300
200
100
0
0
Fe
0.2
0.4
0.6
Mole fraction Ni
0.8
1
Ni
Fig. 10.2 Temperature-composition phase diagram of the Fe-Ni system at P ¼ 1 bar showing the magnetic
transformations (see SGTE; Massalski et al., 2001) (FactSage).
161
Chapter 10 SECOND-ORDER AND HIGHER-ORDER TRANSITIONS
1600
Liquid
1400
bcc - A2
(disordered)
1200
fcc
Al5Fe4
1000
Al5Fe2
Temperature (° C)
800
Al61Fe31
162
600
B2
(ordered
body-centered)
Al13Fe4
0
Al
0.2
0.4
0.6
Mole fraction Fe
0.8
1
Fe
Fig. 10.3 Temperature-composition phase diagram of the Al-Fe system at P ¼ 1 bar
showing the B2 ! A2 order–disorder transition (see SGTE) (FactSage).
The magnetic transition is an example of an order-disorder transition. Another important type of order-disorder transition involves
an ordering of the crystal structure. An example is seen in the Al-Fe
system in Fig. 10.3. At low temperatures, the body-centered phase
B2 exhibits long-range ordering (LRO). Two sublattices can be distinguished, one consisting of the corner sites of the unit cells and
the other of the body-centered sites. The Al atoms preferentially
occupy one sublattice, while the Fe atoms preferentially occupy
the other. This is the ordered B2 (CsCl) structure. As the temperature increases, the phase becomes progressively disordered, with
more and more Al atoms moving to the Fe sublattice and vice versa
until, at the transition line, the disordering becomes complete, with
the Al and Fe atoms distributed equally over both sublattices. At
this point, the two sublattices become indistinguishable, the LRO
disappears, and the structure becomes the A2 (bcc) structure. As
with the magnetic transition, there is no two-phase region and
no abrupt change at the transition line which is simply the temperature at which the disordering becomes complete. As with the magnetic transition, most of the disordering takes place over a fairly
small range of temperature just below the transition line. The heat
capacity thus increases as the transition line is approached from
below and then decreases sharply as the line is crossed. The transition is thus second-order or approximately second-order.
Chapter 10 SECOND-ORDER AND HIGHER-ORDER TRANSITIONS
The B2/A2 transition line may terminate at a tricritical point
like point P of Fig. 10.2. Although it is not shown in Fig. 10.3, there
is most likely a tricritical point near 660°C.
Order–disorder transitions need not necessarily be of second
or higher order. In Fig. 10.2, the L12 phase has an ordered facecentered structure with the Ni atoms preferentially occupying
the face-centered sites of the unit cells and the Fe atoms the corner sites. This structure transforms to the disordered fcc
(A1) structure by a first-order transition with a discontinuous
increase in enthalpy at the congruent point (504°C). There is some
controversy as to whether or not reported higher-order transition
lines in many systems may actually be very narrow two-phase
regions.
Although there is no LRO in the high-temperature disordered
structures, appreciable short-range ordering remains. For example, in the disordered A2 phase in the Al-Fe system, nearestneighbor (Fe-Al) pairs are still preferred. Modeling of ordered
phases and of order-disorder transitions is discussed in
Section 18.2.
10.1
Equations for Thermodynamic
Properties due to Magnetic Ordering
Inden (2001) and Hillert and Jarl (1978) have proposed the following empirical equation for the contribution of ferromagnetic
ordering to the Gibbs energy of magnetic phases:
Gmag ¼ RT ln ðβ + 1Þg ðτÞ where τ ¼
T
Tcurie
(10.3)
and β is the magnetic moment
1
τ3
1
79τ
474 1
τ9
τ15
+
+
+
1
p 1
where g ðτÞ ¼
140p 497
6 135 600
D
when τ 1
1 τ5 τ15 τ25
g ðτ Þ ¼
+
+
when τ > 1
D 10 315 1500
and where D ¼
518 11692 1
+
p 1
1125 15975
(10.4)
(10.5)
(10.6)
where p ¼ 0.28 for fcc phases and p ¼ 0.40 for bcc phases. The
magnetic moment, β, is an adjustable parameter that depends
upon the particular compound.
163
164
Chapter 10 SECOND-ORDER AND HIGHER-ORDER TRANSITIONS
Temperature derivatives of Eq. (10.3) then give the magnetic
contributions to the entropy, enthalpy, and heat capacity. These
contributions are then added to expressions for the properties
of the hypothetical magnetically disordered (paramagnetic)
phase. In Fig. 10.1, a plot of cP of paramagnetic α-Fe is shown,
which is given by an equation of the form of Eq. (2.18), along with
a plot of the actual heat capacity that includes the magnetic
contribution.
The semiempirical expression of Eq. (10.3) has been shown to
provide a reasonably good representation of the magnetic contribution to the thermodynamic properties of many magnetic
phases with only the two adjustable parameters Tcurie and β.
Essentially, the same equation can be used for the contribution
of antiferromagnetic ordering to the thermodynamic properties,
the adjustable parameters being the magnetic moment and the
el transition temperature.
antiferromagnetic/paramagnetic Ne
References
Hillert, M., Jarl, M., 1978. Calphad J. 2, 227–238.
Inden, G., 2001. Atomic ordering. In: Kostorz, G. (Ed.), Phase Transformations in
Materials. (Chapter 8). Wiley VCH, Weinheim.
Massalski, T.B., Okamoto, H., Subramanian, P.R., Kacprzak, L., 2001. Binary Alloy
Phase Diagrams, In: second ed. American Society of Metals, Metals Park, OH.
List of Websites
FactSage, Montreal. www.factsage.com.
SGTE, Scientific Group Thermodata Europe, www.sgte.org.
11
PHASE DIAGRAMS OF SYSTEMS
WITH AN AQUEOUS PHASE
CHAPTER OUTLINE
11.1 Evaporation Paths 165
11.2 Eh-pH Diagrams 167
11.3 True Aqueous Phase Diagrams 172
11.3.1 Other Examples of True Aqueous Phase Diagram Sections
11.3.2 Re-plotting the Diagrams in Eh-pH Coordinates 180
11.3.3 Summary 181
References 182
List of Websites 182
11.1
176
Evaporation Paths
The isothermal phase diagram section at 125°C of the H2OKNO3-NaNO3 system is shown in Fig. 11.1. The gas phase is not
shown. There are three solid phases: NaNO3 and two KNO3-rich
solid solutions with different crystal structures called K1 and K2.
Suppose that we begin with an unsaturated liquid solution of
composition A and that H2O is subsequently gradually removed
by evaporation. During evaporation, the overall composition of
the system follows the line AB. When the overall composition
reaches point C, precipitation of essentially pure NaNO3 commences. During this initial precipitation, the liquid composition
follows the liquidus (also known in aqueous chemistry as the saturation limit) from point C to point D as shown by the arrows. The
overall system composition is now at point E. Thereafter, coprecipitation of NaNO3 and phase K1 occurs:
Liquid ðat point DÞ➔NaNO3 + K1 ðat point FÞ + H2 O ðgasÞ
(11.1)
During this reaction, the compositions of the liquid and K1 remain
constant at points D and F, respectively. The reaction (Eq.11.1)
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00011-7
# 2019 Elsevier Inc. All rights reserved.
165
166
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
0.5
0.5
0.4
0.6
0.3
0.7
0.2
0.8
0.1
0.9
H2O
0.6
0.4
Liquid
A
0.2
Liquid + K2
Liquid + NaNO3
N
D
Liquid + K1
0.9
0.1
K
0.8
C
I
Liquid + K1 + K2
0.7
0.3
G
E
Liquid + K1 + NaNO3
KNO3
J L M0.9
H
0.8
F
0.7
0.6
0.5
Mass fraction
0.4
B 0.3
0.2
0.1
NaNO3
Fig. 11.1 Isothermal section at 125°C of the H2O-NaNO3-KNO3 system (FactSage) (gas phase suppressed) showing
equilibrium evaporation paths.
continues until the overall composition is at point B and all the
H2O has been removed.
Suppose now that the initial composition is at point G. During
evaporation, the overall system composition follows the line GH.
We shall first consider the case of equilibrium evaporation in
which the system remains at complete thermodynamic equilibrium. When the overall composition reaches point I, precipitation
of the K2 phase begins. The first precipitate has a composition at
point J at the end of the tie-line IJ. As the liquid follows the liquidus
(saturation limit) from point I to point K, the composition of the
K2 phase passes from point J to point L, remaining homogeneous
via solid-state diffusion. When the liquid reaches point K, the following reaction occurs:
Liquid ðat point KÞ + K2 ðat point LÞ➔K1ðat point MÞ + H2 O ðgasÞ
(11.2)
until all the K2 phase has been consumed. Subsequently, the liquid follows the liquidus from point K to point N at the end of the
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
tie-line HN. At this point, all the H2O has been removed, and there
is a single homogeneous solid K1 solution of composition H.
In practice, equilibrium conditions will rarely be attained. Let
us now consider the limiting nonequilibrium case in which no
solid-state diffusion occurs in the solid precipitates; that is, once
precipitated, a solid does not change its composition. Precipitation of K2 commences when the liquid composition reaches point
I and subsequently the liquid composition follows the saturation
limit from point I to point K. When the liquid is at point K, the solid
phase is now inhomogeneous with a composition varying from
point J to point L. The reaction in Eq. (11.2) now does not occur.
The liquid continues along the liquidus from point K to point
D with precipitation of a K1 phase whose composition varies from
point M to point F. With the liquid at point D, coprecipitation of
NaNO3 and of K1 of composition F then occurs by the reaction
in Eq. (11.1) until all the H2O has been removed. The final solids
then consist of an inhomogeneous K2 phase, an inhomogeneous
K1 phase, and a coprecipitated mixture of K1 and NaNO3.
This limiting nonequilibrium evaporation is clearly analogous
to Scheil-Gulliver cooling as discussed in Section 8.4, with reaction in Eqs. (11.1), (11.2) being analogous to eutectic and peritectic reactions, respectively.
11.2
Eh-pH Diagrams
Phase equilibria in systems containing an aqueous phase are
frequently depicted in classical Eh-pH diagrams, many of which
have been collected for practical applications in Pourbaix’s Atlas
(Pourbaix, 1974). The construction and uses of classical Eh-pH
diagrams have been discussed by Huang (2016) and by Woods
et al. (1987). A classical Eh-pH diagram (also called a Pourbaix diagram) for the H2O-Fe system at 298.15 K is shown in Fig. 11.2 with
the molalities (moles per kilogram of H2O) of all Fe-containing
aqueous species set to m ¼ 106. The x-axis is the pH of the aqueous solution, and the y-axis is the redox potential Eh relative to a
standard hydrogen electrode. This is not a true phase diagram section in the sense of the rules enunciated in Chapter 7. Rather, it is a
“predominance diagram” inasmuch as the fields labeled “aqueous
with Fe3+, aqueous with Fe2+, and aqueous with FeOH+” are actually all part of just one single-phase aqueous region. The three
areas shown in the figure are the areas in which the predominant
(highest concentration) Fe-containing solutes are Fe3+, Fe2+, and
FeOH+. In a true phase diagram, the lines CA and BD would not
appear. The diagram in Fig. 11.2 was calculated by fictitiously
167
168
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
1.6
1.4
1.2
Aqueous with
Fe[3+]
Aqueous +
Fe2O3(s)
1.0
0.8 C
A
Eh (volts)
0.6
O2 (1
0.4
0.2
Aqueous with
Fe[2+]
0
B
−0.2
−0.4
Aqueous + Fe3O4(s)
Aqueous with
FeOH[+]
−0.6
E
D
−0.8
H2 (1
Aqueous + Fe(OH)2(s)
−1.0
−1.2
−2
bar)
bar)
Aqueous + Fe(s)
0
2
4
6
pH
8
10
12
14
Fig. 11.2 Classical Eh-pH diagram of the H2O-Fe system at 298.15 K with molalities of all Fe-containing aqueous species
set to m ¼ 106 (FactSage).
treating the “aqueous with Fe3 +” region as a separate “phase” in
which Fe3+ is the only Fe-containing ion (at m ¼ 106) and similarly for the other two regions. An Eh-pH diagram is thus similar
to the predominance diagram of the Cu-SO2-O2 system in Fig. 2.3.
It should also be pointed out that in a two-phase region such as
the “aqueous + Fe2O3” field, the concentration of Fe-containing
species in the aqueous phase is less than m ¼ 106 since most
of the Fe is precipitated in the solid phase. That is, m ¼ 106
applies only when no solid phases are present. (Usually, on EhpH diagrams, two-phase (aqueous + solid) regions are labeled
with just the name of the solid phase, e.g., simply as “Fe2O3,”
the presence of the aqueous phase being understood.)
Classical Eh-pH diagrams are calculated by assuming that the
aqueous phase is an ideal Henrian solution (see Section 5.10) in
which there are no interactions among solute species. Nonideality
cannot be taken into account because it would then be necessary
to specify the concentrations of all solute species. In Fig. 11.2, for
+
example, ions such as Cl, NO
3 , and Na that may be present in
the aqueous solution are not specified. That is, electric neutrality
of the aqueous phase is generally not a condition of the
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
calculations. However, this is permissible as long as the solution is
an ideal Henrian solution since then the diagram is independent
of the concentrations of solute species other than those
containing Fe.
The standard state for a solute species in an aqueous solution is
generally taken by convention to be an ideal Henrian solution in
which the molality of the species is m ¼ 1.0 (see Sections 5.10
and 5.11). Hence, in an ideal aqueous solution, the activity of a
solute species is equal to its molality.
The x-axis of Fig. 11.2 is the pH defined as
pH ¼ log 10 aH +
(11.3)
+
where aH+ is the activity of the H ion (equal to mH+ in Eh-pH
diagrams).
The y-axis in Fig. 11.2 is the redox potential Eh relative to a
standard hydrogen (H2/H+) electrode with pH2 ¼ 1.0 bar and
aH+ ¼ 1.0. Suppose, for example, that an electrochemical cell is
constructed with one electrode being a standard hydrogen electrode, and the other electrode placed in the aqueous solution. If
a potential Eh < 0.7097 V (point C in Fig. 11.2) is applied between
the electrodes, then Fe2+ ions will be predominant. That is, the
equilibrium
Fe2 + + H + ¼ Fe3 + + 0:5 H2 ðgasÞ
(11.4)
is shifted to the left. If Eh > 0.7097 V, then the reaction is shifted to
the right, and Fe3+ ions predominate.
Consider now an electrochemical cell with one electrode being
a standard hydrogen electrode (pH2 ¼ 1.0 and aH+ ¼ 1.0), and the
other a hydrogen electrode with a hydrogen pressure pH2 in a solution where the hydrogen ion activity is aH+. The cell reaction is
for which
H + ðat activity aH + Þ + 0:5 H2 ðat pH2 ¼ 1Þ
¼ 0:5 H2 ðat pH2 Þ + H + ðat activity aH + ¼ 1Þ
(11.5)
0:5
=aH +
ΔG ¼ ΔGo + RT ln pH
2
(11.6)
Since ΔGo ¼ 0, using the Nernst equation (Eq. 2.55), we may write
Eh ¼ ðRT =F Þð0:5 ln pH2 ln aH + Þ
¼ ð2:303RT =F Þð0:5 log 10 pH2 pHÞ
(11.7)
Since H2O is always present, the effective hydrogen and oxygen
partial pressures are coupled through the equilibrium:
H2 OðaqÞ ¼ H2 ðgÞ + 0:5 O2 ðgÞ
(11.8)
169
170
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
for which the equilibrium constant is
0:5
K ¼ pH2 pO
=aH2 O
2
(11.9)
where aH2O is the activity of H2O (approximately equal to 1.0 in
dilute solutions) and where (aq) and (g) indicate species in the
aqueous phase and gaseous species, respectively. Substituting
Eq. (11.9) into Eq. (11.7) gives
Eh ¼ ð2:303RT =F Þð0:25log 10 pO2 pH 0:5 log K 0:25 log aH2 O Þ
(11.10)
Hence, imposing an effective hydrogen pressure Eq. (11.7) or
effective O2 pressure (Eq. 11.10) by equilibration with a gaseous
atmosphere or by other means is equivalent to imposing a voltage
Eh relative to a standard hydrogen electrode.
Eqs. (11.10), (11.7) are illustrated in Figs. 11.3 and 11.4,
respectively.
Along the dashed lines in Fig. 11.2, pH2 and pO2 calculated from
Eqs. (11.7), (11.10) are equal to 1.0 bar. If the total pressure is
1.0 bar, then at applied potentials Eh above the O2(1 bar) line,
Eh = 1
Eh = 0.8
Eh = 1
−10
Eh = 0.6
Eh = 0.8
log10 p(O2) (bar)
−20
Eh = 0.4
Eh = 0.6
−30
Eh = 0.2
pH = 1
−40
pH = 2
−50
pH = 3
Eh = 0.4
pH = 4
pH = 5
pH = 6
pH = 7
pH = 8
pH = 9
Eh = 0 pH = 10 pH = 11 pH = 12
Eh = 0.2
Eh = −0.2
−60
Eh = 0
Eh = −0.4
Eh = −0.2
−70
Eh = −0.4
Eh = −0.6
−80
0
2
4
6
8
10
12
pH
Fig. 11.3 The relationships among Eh, pH, and the oxygen potential logpO2 at T ¼ 298.15 K from Eq. (11.10) when the
activity of H2O ¼ 1.0 (Pelton et al., 2018).
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
Eh = −0.4
−5
171
Eh = −0.6
Eh = −0.2
log10 p(H2) (bar)
Eh = −0.4
Eh = 0
−10
Eh = −0.2
−15
pH = 1
Eh = 0.2
pH = 2
pH = 3
pH = 4
pH = 5
pH = 6
pH = 7
pH = 8
Eh = 0.4
−20
pH = 9 pH = 10
pH = 11 pH = 12
Eh = 0
Eh = 0.2
Eh = 0.6
Eh = 0.4
−25
Eh = 0.6
Eh = 0.8
−30
0
2
4
8
6
10
12
pH
Fig. 11.4 The relationships among Eh, pH, and the hydrogen potential log log pH2 at T ¼ 298.15 K from Eq. (11.7) (Pelton
et al., 2018).
O2 gas will be evolved, and below the H2(1 bar) line, H2 gas will be
evolved. So, for instance, it is not possible to electrochemically
reduce Fe2+ in aqueous solution to Fe metal since H2 will be
evolved instead. That is, H2O is only thermodynamically stable
between the dashed lines.
As a sample calculation, we calculate the slope of the line BE in
Fig. 11.2 along which FeOH+ as the predominant Fe-containing
species at m ¼ 106 is in equilibrium with Fe3O4. The equilibrium
may be written as.
3FeOH + + 2H + + H2 O ¼ Fe3 O4 + 5H + + H2
(11.11)
with
ΔG o ¼ RT ln K
¼ RT ½ð ln pH2 2 ln aH + Þ + 5ð2:303Þ log 10 aH + 3 ln mFeOH + (11.12)
From Eqs. (11.3), (11.7), the slope of line BE is thus 5(2.303)RT/
2F ¼ 6.76.
172
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
11.3 True Aqueous Phase Diagrams
In order to address the limitations of classical Eh-pH diagrams
as discussed in the preceding section, we may consider the thermodynamic criteria for true phase diagram sections as enunciated in Chapter 7 and apply these criteria to systems with an
aqueous phase, as has been discussed in a recent article (Pelton
et al., 2018).
After defining electrically neutral components of the system,
one defines the “size” of the system as the mass (or number of
moles) of one or more components. The axis variables and constants of the diagram are then temperature, T, mass (or molar)
ratios and relative chemical potentials of electrically neutral components. (As discussed in Chapter 7, in the general case, other
properties such as pressure, volume, and enthalpy can be axis variables or constants, but these will not be considered here.) If the
criteria and rules described in Section 7.5 are followed, then not
only will the diagram be a true phase diagram section but also
it will consist of ZPF lines as discussed in Section 7.4 and can
therefore be calculated using the same general algorithm that is
used for all true phase diagram sections as described in
Section 7.4.1. The algorithm is general and versatile, permitting
the calculation of diagrams with a variety of axis variables under
a wide variety of constraints for real (i.e., nonideal) solutions and
for any number of components.
Although the redox potentials Eh and pH are not axis variables
of true phase diagram sections, software can calculate and plot
iso-Eh and isopH lines on the diagrams.
A classical Eh-pH diagram for the Cu-H2O system with
mCu ¼ 107 is shown in Fig. 11.5. We shall now calculate a similar
true phase diagram section.
Fig. 11.6 is a calculated phase diagram section for the H2O-O2HCl-NaOH-Cu system. Data were retrieved automatically from
the FactSage databases for aqueous solutions and for pure compounds. The “size” of the system (see Section 7.5) is 1.0 mass unit
of H2O, as initially defined before the system reaches equilibrium.
The diagram is calculated at a constant total overall Cu molality
mCu ¼ 107 mol/kg H2O and at a constant total overall molality
(mHCl + mNaOH) ¼ 0.1 mol/kg H2O. The x-axis is the total overall
molar ratio NaOH/(HCl + NaOH). The y-axis is the oxygen potential, logpO2, which is related to Eh by Eq. (11.10).
The aqueous phase diagram in Fig. 11.6 and the classic Pourbaix diagram in Fig. 11.5 can be seen to have corresponding
domains and topologies. However, the phase boundaries (black
lines) in Fig. 11.6 are true phase boundaries and are all ZPF lines.
1.2
1.0
0.8
Point A
Eh (volts)
0.6
Cu[2+]
CuO(s)
0.4
O
2
0.2
Cu[+]
0
(1 b
ar)
Point C
Cu2O(s)
Point B
−0.2
−0.4
Cu(s)
−0.6
−0.8
−1.0
H2 (
1 ba
r)
−2
2
0
4
6
8
10
12
14
16
pH
Fig. 11.5 Classical Eh-pH (Pourbaix) diagram of the Cu-H2O system at T ¼ 298.15 K at a Cu molality ¼ 107.
Dashed lines show limits at which O2 or H2 pressures exceed 1.0 bar. The aqueous phase is only
thermodynamically stable between these lines (points A, B, and C correspond to the points in Fig. 11.6) (FactSage).
Eh = 0.6
−10
−20
Eh = 0.8
pH = 4
Eh = 0.8
pH = 6
pH = 5
Point A
log p(O2) (bar)
Eh = 0.6
pH = 8
pH = 9
−30
aqueous
CuO + aqueous
Eh = 0.2
Point C
Eh = 0.4
−40
Eh = 0.4
pH = 7
pH = 10
Eh = 0
−50
Eh = 0.2
pH = 4
−60
pH = 7
pH = 6
pH = 5
Point B
Cu2O + aqueous
Eh = 0
Eh = −0.2
Cu + aqueous
−70
Eh = −0.4
Eh = −0.4
Eh = −0.2
−80
0.4994
0.4996
0.4998
0.5
0.5002
Molar ratio NaOH/(HCl+NaOH)
0.5004
0.5006
Fig. 11.6 True phase diagram section of the aqueous system H2O-O2-HCl-NaOH-Cu at T ¼ 298.15 K at a constant
total overall Cu molality ¼ 107 mol/kg H2O and a constant total overall molality of
(HCl +NaOH) ¼ 0.1 mol/kg H2O. x-axis is the molar ratio NaOH/(HCl + NaOH). y-axis is the oxygen potential. Iso-Eh and
isopH lines are calculated (Pelton et al., 2018).
174
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
The diagram in Fig. 11.6 can be considered to be that of a system consisting of the following electroneutral components: 1.0 Kg
H2O, 107 mol Cu, 0.1 mol (HCl + NaOH), and a variable amount
of oxygen as determined by pO2. The region labeled simply as
Cu2O on the Eh-pH predominance diagram in Fig. 11.5 is, on
the true phase diagram of Fig. 11.6, labeled as Cu2O + aqueous
because the aqueous phase is still present. The total amount of
Cu in the aqueous solution and solid Cu2O taken together is
107 mol. The molality of Cu in the aqueous phase is thus equal
to 107 only in the single-phase aqueous field. The boundary of
the single-phase aqueous field is the solubility limit of Cu when
mCu in the aqueous phase is 107. The line separating the Cu+
and Cu2+ fields in Fig. 11.5 does not appear in Fig. 11.6 because
this is actually all one single-phase aqueous region. (However, it
will be shown later how the position of this line can be calculated.)
Eh and pH are not proper axis variables or constants of a true
phase diagram since they are not relative chemical potentials of
neutral components of the system. However, at any point on
the diagram, the software can calculate the equilibrium composition of the aqueous phase and the H+ ion activity. Since Eh and pH
are related to the H+ ion activity and pO2 by Eqs. (11.3), (11.10), the
automatic calculation and plotting of iso-Eh and isopH lines on a
calculated diagram are straightforward. These lines have been
plotted in Fig. 11.6.
After a diagram has been calculated and displayed, the equilibrium amounts and compositions of all phases can be rapidly and
easily calculated at any point on the diagram. For example, using
the FactSage software with Fig. 11.6 on the screen, by placing the
cursor at point A (coordinates x ¼ 0.4995 and y ¼ 20.00) and
“clicking” or alternatively by entering the coordinates of the point
on a drop-down menu, one generates the output shown in
Table 11.1 that gives the equilibrium composition of the aqueous
solution at this point in the single-phase region. It can be seen that
at this oxidation potential, the Cu is almost entirely in the 2 + oxidation state as Cu2+. The values of Eh and pH at the point are also
calculated and displayed.
The output generated by repeating the calculation at point
B (x ¼ 0.4995 and y ¼ 60.00) is also shown in Table 11.1. At this
lower oxidation potential, it can be seen that the dissolved Cu is
mainly in the 1 + oxidation state as CuCl1
2 complex ions. These
results may be compared with the classical Eh-pH diagram in
Fig. 11.5 where the regions of predominance of the 1+ and 2+
states and the positions of the points A, B, and C are shown.
Point C (coordinates x ¼ 0.5001 and y ¼ 35.00) in Fig. 11.6 lies
in the two-phase (aqueous + CuO) region. The total amount of Cu
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
Table 11.1 Calculated Equilibrium States at
Points in Fig. 11.6
Point A: log p(O2)/bar ¼ 20, NaOH/(HCl + NaOH) (mol/mol) ¼ 0.4995
1.0000 mol (18.036 g) aqueous solution
Molalities
5.0005E 02
4.9905E 02
9.9711E 05
9.9911E 08
9.8934E 11
3.7501E 14
9.1944E 17
+…………………
(Eh ¼ 0.696 V, pH 4.001)
Point B: log p(O2)/bar ¼ 60, NaOH/(HCl + NaOH) (mol/mol) ¼ 0.4995
1.0000 mol (18.036 g) aqueous solution
Molalities
5.0005E 02
4.9905E 02
9.9811E 05
9.9640E 08
2.4430E 10
9.8835E 11
2.6573E 11
+…………………
(Eh ¼ 0.105 V, pH 4.001)
Point C: log p(O2)/bar ¼ 35, NaOH/(HCl + NaOH) (mol/mol) ¼ 0.5001
1.0000 mol (18.036 g) aqueous solution
Molalities
4.9965E 02
4.9945E 02
1.9987E 05
4.9357E 10
4.1338E 09
1.0160E 11
9.7177E 12
+…………………
(Eh ¼ 0.161 V, pH 9.307)
+1.7220E-09 mol (1.3698E-07 g) CuO solid
Cl
Na+
H+
Cu2+
OH
CuCl
2
Cu+
Cl
Na+
H+
CuCl
2
Cu+
OH
Cu2+
Na+
Cl
OH
H+
CuCl
2
Cu+
Cu2+
Species lists have been truncated to show only those species with the highest concentrations.
175
176
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
in the aqueous and CuO phases taken together is 107 mol/kg of
H2O. The calculated equilibrium phase assemblage at point C is
shown in Table 11.1. It can be seen that the aqueous phase contains very little Cu since nearly all the Cu has precipitated to
form CuO.
Note that the equilibrium system composition in Table 11.1 for
point C is given per mole of aqueous solution at equilibrium.
A small amount of H2O (approximately 0.1 mol per initial kilogram of H2O) has been formed by the reaction of the NaOH and
HCl. The total amount of Cu in the aqueous plus solid CuO phases
taken together is 107 mol/kg of H2O initially present before the
reaction.
On the classical Eh-pH diagram in Fig. 11.5, there are dashed
lines showing the limits at which pO2 and pH2 exceed 1.0 bar. The
aqueous phase is only thermodynamically stable between these
lines. In Fig. 11.6, the limit of pO2 ¼ 1.0 bar is simply the horizontal
x-axis at logpO2 ¼ 0. The limit of pH2 ¼ 1.0 bar is also a line at constant pO2 in Fig. 11.6 that can be calculated from Eq. (11.9) as
logpO2 ¼ 2 log K (setting aH2O 1.0). Alternatively, one could recalculate the diagram with logpH2 as the vertical axis as will be shown
later in Fig. 11.8.
11.3.1
Other Examples of True Aqueous Phase
Diagram Sections
The diagram of Fig. 11.7 is similar to Fig. 11.6 except that the
total Cu content is higher. As a result, fields with solid CuCl
appear. As well, this figure was calculated with the GEOPIGSUPCRT (2008) aqueous database, and the activities of ions at
finite concentrations (i.e., not at infinite dilution) were calculated
by means of the Debye-Davies equation.
Fig. 11.8 is similar to Fig. 11.7 except that the y-axis is now the
hydrogen potential logpH2. Fig. 11.8 is essentially Fig. 11.7 with the
y-axis inverted as can be seen from Eq. (11.9).
Classical Eh-pH diagrams for systems with two or more metallic elements can generally only be approximated by superimposing the diagrams for the individual elements. This restriction no
longer holds in the case of true aqueous phase diagrams. In
Fig. 11.9 is a calculated diagram for the H2O-O2-HCl-Cu-Au system with total overall molalities of Cu and Au, each equal to
0.01 mol/kg H2O. The nonideal GEOPIG-SUPCRT (2008) database
was used for the aqueous phase, and aqueous ion activities were
calculated with the Debye-Davies equation. Data for the solid fcc
solution between Au and Cu were taken from the SGTE solution
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
177
Eh = 1
−10
Aqueous
Eh = 0.8
log p(O2) (bar)
−20
−30
Eh = 0.6
pH = 3
pH = 4
pH = 11
pH = 10
pH = 5
Eh = 0.4
pH = 12
Eh = 0.2
pH = 8
Eh = 0
Eh = 0.4
−40
CuCl + Cu2O
+ aqueous
Eh = 0.2
−50
pH = 3
pH = 4
CuCl + aqueous
−60
CuO + aqueous
CuCl + Cu
+ aqueous
Cu2O + aqueous
Eh = −0.2
pH = 5
pH = 10
pH = 11
pH = 2
Eh = −0.4 pH = 12
Cu + aqueous
−70
Eh = 0
−80
0.44
0.46
Eh = −0.6
0.48
0.5
0.52
Molar ratio NaOH/(HCl+NaOH)
0.54
0.56
Fig. 11.7 Same as Fig. 11.6 except at a Cu molality ¼ 0.005, with the nonideal GEOPIG-SUPCRT (2008) database used for
the aqueous phase and with activities calculated from the Debye-Davies equation (Pelton et al., 2018).
Eh = −0.6
Eh = 0
−5
Aqueous + Cu
pH = 11
Eh = −0.4
−15
Aqueous + CuCl
−20
−25
Aqueous
log p(H2) (bar)
−10
pH = 3
CuCl + Cu
pH = 4
pH = 9
+ aqueous
Eh = 0.2
CuCl + Cu2O
+ aqueous
pH = 4
Eh = −0.2
Aqueous + Cu2O
Eh = 0
Eh = 0.4
pH = 6
Eh = 0.6
−30
pH = 5 pH = 10
pH = 11
Eh = 0.2
Aqueous + CuO
−35
−40Eh = 1
0.44
pH = 3
Eh = 0.8
0.46
0.48
Eh = 0.4
0.5
0.52
Molar ratio NaOH/(HCl+NaOH)
Fig. 11.8 Same as Fig. 11.7 except the y-axis is the H2 potential (Pelton et al., 2018).
0.54
0.56
178
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
0
Aqueous
−10
−20
Eh = 0.6
Eh = 0.8
pH = 1
log p(O2) (bar)
Eh = 0.8
Eh = 1
Eh = 1
pH = 2
−30
pH = 3
pH = 4
Aqueous + fcc + CuO
Eh = 0.4
Eh = 0.6
Aqueous + fcc
−40
Aqueous + Au3Cu + CuO
pH = 5 Aqueous + Au3Cu + Cu2O
pH = 5
Aqueous + Au3Cu + fcc
Eh = 0.4
−50
pH = 1
−60
pH = 3
pH = 2
Eh = 0.2
Aqueous + Au3Cu
Eh = 0.2
Aqueous + AuCu + Au3Cu
−70
Eh = 0
Eh = 0
−80
0
0.5
1
1.5
2
2.5
-log (HCl/H2O) (mol/kg)
3
3.5
4
Fig. 11.9 True phase diagram section of the H2O-O2-HCl-Cu-Au system at T ¼ 298.15 K at total overall molalities
of Cu and Au ¼ 0.01 mol/kg. fcc phase is a nonideal Cu-Au alloy with data taken from the SGTE solution database
(see SGTE). y-axis is the oxygen potential. x-axis is log10 mHCl where mHCl ¼ total overall molality of HCl (mol/kg H2O)
(Pelton et al., 2018).
database (see SGTE). At low temperatures, the fcc phase shows
disordered but nonideal mixing at high Au or high Cu concentrations. In the intermediate range of compositions, three ordered
fcc phases occur: Au3Cu, AuCu, and AuCu3. These have been
approximated as stoichiometric compounds in the calculation.
The single-phase aqueous region only appears at very low pH
and high oxygen potential (high Eh) where Au enters the solution
mainly as AuCl2 and AuCl3 2 ions.
The y-axis of a calculated diagram is not limited to logpO2 or
logpH2. For example, Fig. 11.10 shows the solubility of Cu in an
aqueous solution as a function of HCl content at a constant oxygen potential. Temperature could also be selected as an axis
variable.
Aqueous solutions can become acidified by exposure to CO2
through the formation of HCO3 ions. Fig. 11.11 shows the phase
equilibria in the H2O-O2-CO2-Cu system at a total Cu molality of
107 as a function of logpO2 and logpCO2. By performing an equilibrium calculation at any point on the diagram, one can calculate
−1.2
Aqueous + CuCl + Cu2O
log10(Cu/H2O) (mol/kg)
−1.4
Eh = 0.34 Eh = 0.3
pH = 3
Eh = 0.38
pH = 2
Aq
Eh = 0.4
ue
−1.6
−1.8
Aqueous + Cu2O
ou
s
+
−2
Eh = 0.4
−2.2
−2.4
Cu
Eh = 0.26
Cl
Eh = 0.38
pH = 2
Eh = 0.36
pH = 4
Eh = 0.34
Eh = 0.28
Eh = 0.32
pH = 3
Aqueous
Eh = 0.42
−2.6
−2.8
−3
1
1.2
1.4
1.6
2.4
1.8
2
2.2
-log10(HCl/H2O) (mol/kg)
2.6
2.8
3
Fig. 11.10 True phase diagram section of the H2O-O2-HCl-Cu system at T ¼ 298.15 K at a constant oxygen
potential pO2 ¼ 1050 bar. x-axis is log10 mHCl where mHCl ¼ total overall molality of HCl (mol/kg H2O). y-axis is
log10 mCu where mCu is the total overall molality of Cu (Pelton et al., 2018).
−20
Eh = 0.6
−25
Eh = 0.4
−30
pH = 7
CuO+ aqueous
pH = 6
−35
log p(O2) (bar)
Eh = 0.4
−40
Eh = 0.2
pH = 5
aqueous
−45
Eh = 0.2
Cu2O + aqueous
−50
pH = 7
−55
Eh = 0
Cu + aqueous
−60
−65
−10
−9
−8
−7
pH = 5
pH = 6
−5
−6
log p(CO2) (bar)
Eh = 0
−4
−3
−2
−1
Fig. 11.11 True phase diagram section of the H2O-O2-CO2-Cu system at T ¼ 298.15 K and a constant total overall
molality of Cu ¼ 107 mol/kg H2O. x- and y-axes give the equilibrium O2 and CO2 pressures (Pelton et al., 2018).
180
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
−20
Eh = 0.8
−25
Eh = 0.8
Eh = 0.6
Aqueous + CuO
log10 p(O2) (bar)
−30
Eh = 0.4
−35
Eh = 0.6
−40
pH = 5
Aqueous
−45
pH = 2
pH = 1
Aqueous + Cu2O
pH = 3
pH = 4
Eh = 0.4
−50
Cu2O + CuCl + Aqueous
Aqueous + Cu
−55
Aqueous + CuCl
−60
0
0.5
1
CuCl + Cu + Aqueous
Eh = 0.2
1.5
2
2.5
-log10(HCl/H2O) (mol/kg)
3
3.5
4
Fig. 11.12 True phase diagram section of the H2O-O2-CO2-Cu-HCl system at T ¼ 298.15 K, a total overall Cu
molality ¼ 0.003 mol/kg H2O, and a constant CO2 pressure ¼ 0.01 bar. y-axis is the oxygen potential. x-axis is log10 mHCl
where mHCl ¼ total overall molality of HCl (mol/kg H2O) (Pelton et al., 2018).
the equilibrium HCO3 ion concentration and the concentration
of dissolved neutral CO2 species and CO3 2 ions in the aqueous
solution.
Fig. 11.12 illustrates phase equilibria that could be associated
with the corrosion of Cu in acidic chloride solutions at a CO2 partial pressure (0.01 bar) and redox potentials typical of the conditions encountered in the corrosion of some ancient bronzes
(Chase et al., 2007). At a higher pCO2 ¼ 0.1 bar, the corrosion products are Cu2CO3(OH)2 (malachite) or Cu2O, depending upon
logpO2 (or Eh).
11.3.2
Re-plotting the Diagrams in Eh-pH
Coordinates
Every phase boundary of a true aqueous phase diagram consists of a series of (pH, Eh) coordinates. Hence, after a true phase
diagram has been calculated, the phase boundaries can be
replotted in Eh-pH coordinates. As an example, Fig. 11.6 is
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
181
0.6
0.5
Aqueous + CuO
Eh (volts)
0.4
Aqueous
0.3
0.2
0.1
Aqueous + Cu2O
0
−0.1
Aqueous + Cu
−0.2
4
5
6
7
pH
8
9
10
Fig. 11.13 Fig. 11.6 replotted in Eh-pH coordinates. System H2O-O2-HCl-NaOH-Cu at T ¼ 298.15 K at a constant total
overall Cu molality ¼ 107 mol/kg H2O and a constant total overall molality of (HCl + NaOH) ¼ 0.1 mol/kg H2O. pH
determined by varying the ratio NaOH/(HCl + NaOH). Note that this is not a classical Eh-pH diagram (Pelton et al., 2018).
replotted in Eh-pH coordinates in Fig. 11.13. It must be stressed
that such a diagram cannot be called “the Eh-pH diagram of
the H2O-Cu system” because it is dependent upon the fact that
the pH is determined by the presence of NaOH and HCl and by
their molalities.
As an additional example, Fig. 11.11 is replotted in Eh-pH coordinates in Fig. 11.14.
11.3.3
Summary
In summary, it can be seen that true aqueous phase diagram
sections containing calculated isopH and iso-Eh lines show the
phase equilibria as a function of Eh and pH, as do classical EhpH (Pourbaix) diagrams, while providing much more information
and flexibility inasmuch as a very wide choice of axis variables and
constants is available, the number of components is not limited,
and the diagrams can be calculated for real nonideal concentrated
aqueous solutions. As well, software permits the easy calculation
and display of the equilibrium amounts and compositions of
182
Chapter 11 PHASE DIAGRAMS OF SYSTEMS WITH AN AQUEOUS PHASE
CuO + aqueous
0.25
Eh (volts)
Aqueous
0.2
0.15
Cu2O + aqueous
0.1
Cu + aqueous
0.05
5
5.5
6
6.5
7
pH
Fig. 11.14 Fig. 11.11 replotted in Eh-pH coordinates. System H2O-O2-CO2-Cu at T ¼ 298.15 K and a constant total overall
molality of Cu ¼ 107 mol/kg H2O. Eh and pH determined by varying O2 and CO2 partial pressures. Note that this is not a
classical Eh-pH diagram (Pelton et al., 2018).
every phase at any point on a calculated diagram. Finally, once
calculated, the diagrams can be replotted in Eh-pH coordinates.
References
Chase, W.T., Notis, M., Pelton, A.D., 2007. Proc. ICOM Conversation Committee
Working Group on Metals, Amsterdam.
Huang, H.-H., 2016. Metals (Basel, Switzerland). 6, 23/21–23/30.
Pelton, A.D., Eriksson, G., Hack, K., Bale, C.W., 2018. Monatsh. Chem. 149, 395–409.
Pourbaix, M.J.N., 1974. Atlas of Electrochemical Equilibrium in Aqueous Solutions.
NACE, Houston TX. CEBELCOR, Brussels.
Woods, R., Yoon, R.H., Young, C.A., 1987. Int. J. Miner. Process. 20, 109–120.
List of Websites
FactSage, Montreal, www.factsage.com.
GEOPIG-SUPCRT Database, 2008. http://geopig.asu.edu/sites/default/files/
slop07.dat.
SGTE, Scientific Group Thermodata Europe, www.sgte.org.
SGTE Solution database, 2017. http://www.sgte.net/en/thermochemical-databases.
12
BIBLIOGRAPHY ON PHASE
DIAGRAMS
CHAPTER OUTLINE
12.1 Phase Diagram Compilations
12.2 Further Reading 184
References 184
List of Websites 185
12.1
183
Phase Diagram Compilations
From 1979 to the early 1990s, the American Society for Metals
undertook a project to evaluate critically all binary and ternary
alloy phase diagrams. All available literature on phase equilibria,
crystal structures and, often, thermodynamic properties were critically evaluated in detail by international experts. Many evaluations have appeared in the Journal of Phase Equilibria (formerly
Bulletin of Alloy Phase Diagrams), (ASM Intl., Materials Park,
OH), which continues to publish phase diagram evaluations. Condensed critical evaluations of 4700 binary alloy phase diagrams
have been published in three volumes (Massalski et al., 2001).
The ternary phase diagrams of 7380 alloy systems have also been
published in a 10-volume compilation (Villars et al., 1995). Both
binary and ternary phase diagrams from these compilations are
available on the website of the ASM alloy phase diagram center
(see American Society for Metals). Many of the evaluations have
also been published by ASM as monographs on phase diagrams
involving a particular metal as a component.
The Scientific Group Thermodata Europe (SGTE) group has
produced several subvolumes within volume 19 of the Landolt-B€ rnstein series (SGTE, 1999–2009) of compilations of thermodyo
namic data for inorganic materials and of phase diagrams and
integral and partial thermodynamic quantities of binary alloys.
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00012-9
# 2019 Elsevier Inc. All rights reserved.
183
184
Chapter 12 BIBLIOGRAPHY ON PHASE DIAGRAMS
Phase diagrams for over 9000 binary, ternary, and multicomponent ceramic systems (including oxides, halides, carbonates, and
sulfates) have been compiled in the multivolume series, Phase
Diagrams for Ceramists (Levin et al., 1964). Earlier volumes were
noncritical compilations. However, more recent volumes have
included critical commentaries. See also the Am. Ceram. Soc./
NIST website for an online version.
Hundreds of critically evaluated and optimized phase diagrams of metallic, oxide, salt, and sulfide systems may be downloaded from the FactSage website.
12.2 Further Reading
A text by Hillert (2008) provides an in-depth thermodynamic
treatment of phase equilibriums and solution modeling.
A classical discussion of phase diagrams in metallurgy was
given by Rhines (1956). Prince (1966) presents a detailed treatment of the geometry of multicomponent phase diagrams.
A series of five volumes edited by Alper (1970–1978) discusses
many aspects of phase diagrams in materials science. Bergeron
and Risbud (1984) give an introduction to phase diagrams, with
particular attention to applications in ceramic systems.
In the Calphad Journal (Pergamon) and in the Journal of Phase
Equilibria (ASM Intl., Materials Park, OH), many articles on the
relationships between thermodynamics and phase diagrams are
to be found.
The SGTE Casebook (Hack, 2008) illustrates, through many
examples, how thermodynamic calculations can be used as a
basic tool in the development and optimization of materials
and processes of many different types.
References
Alper, A.M., 1970–1978. Phase Diagrams – Materials Science and Technology.
vol. 1–5. Academic, New York.
Bergeron, C.G., Risbud, S.H., 1984. Introduction to Phase Equilibria in Ceramics.
American Ceramic Society, Columbus, Ohio.
Hack, K., 2008. SGTE Casebook: Thermodynamics at Work, second ed. CRC Press,
Boca Raton, FL.
Hillert, M., 2008. Phase Equilibria, Phase Diagrams and Phase Transformations –
Their Thermodynamic Basis, second ed. Cambridge University Press,
Cambridge.
Levin, E.M., Robbins, C.R., McMurdie, H.F., 1964. Phase Diagrams for Ceramists.
Am. Ceramic Soc., Columbus, OH (Supplements to 2018).
Massalski, T.B., Okamoto, H., Subramanian, P.R., Kacprzak, L., 2001. Binary Alloy
Phase Diagrams, second ed. American Society of Metals, Metals Park, OH.
Chapter 12 BIBLIOGRAPHY ON PHASE DIAGRAMS
Prince, A., 1966. Alloy Phase Equilibria. Elsevier, Amsterdam.
Rhines, F.N., 1956. Phase Diagrams in Metallurgy. McGraw-Hill, New York.
€ r Werkstoffchemie, RWTH, Aachen, ThermoSGTE group, 1999–2009. Lehrstuhl fu
€rnstein Group IV. vol. 19.
dynamic Properties of Inorganic Materials, Landolt-Bo
Springer, Berlin.
Villars, P., Prince, A., Okamoto, H., 1995. Handbook of Ternary Alloy Phase Diagrams. American Society of Metals, Metals Park, OH, pp. 887–891.
List of Websites
American Ceramic Society/NIST, http://phase.ceramics.org.
American Society for Metals, alloy phase diagram center, www1.asminternational.
org/asmenterprise/apd.
FactSage, Montreal, www.factsage.com.
185
INTRODUCTION
13
As discussed in the general introduction, the last 40 years has
witnessed a rapid development of large evaluated optimized thermodynamic databases and of software that accesses these databases
to calculate phase diagrams by Gibbs energy minimization.
The databases for multicomponent solution phases are prepared by first developing an appropriate mathematical model,
based upon the structure of the solution, that gives the thermodynamic properties as functions of temperature and composition
for every phase of a system. Next, all available thermodynamic
data (calorimetric data, activity measurements, etc.) and phase
equilibrium data from the literature for the entire system are evaluated and simultaneously “optimized” to obtain one set of critically evaluated self-consistent parameters of the models for all
phases in two-component and, if data are available, in threecomponent and higher-order systems. Finally, the models are
used to estimate the properties of multicomponent solutions
from the databases of parameters of lower-order subsystems.
More recently, experimental data are being supplemented by
“virtual data” from first principles and molecular dynamics
calculations.
The optimized model equations are consistent with thermodynamic principles and with theories of solutions. The phase diagrams
can be calculated from these thermodynamic equations. Hence,
one set of self-consistent equations describes simultaneously all
the thermodynamic properties and the phase diagrams. This technique of analysis greatly reduces the amount of experimental data
needed to fully characterize a system. All data can be tested for internal consistency. The data can be interpolated and extrapolated more
accurately. All the thermodynamic properties and the phase
diagram can be represented and stored by means of a relatively
small set of coefficients.
This technique permits the estimation of phase diagrams of
multicomponent systems based upon data from lower-order subsystems and allows any desired phase diagram section or
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00013-0
# 2019 Elsevier Inc. All rights reserved.
189
190
Chapter 13 INTRODUCTION
projection to be calculated and displayed rapidly. As an example,
in Fig. 7.7 is a calculated phase diagram section for a sixcomponent system that might be of interest in the design of
new Mg alloys. This diagram can be calculated on a laptop computer in about 1 min. Hence, many sections of a multicomponent
system can be quickly generated for study.
The computer calculation of phase diagrams presents many
additional advantages. For example, metastable phase boundaries
can be readily calculated by simply removing one or more stable
phases from the calculation. Another advantage is the possibility
of calculating phase compositions at points on an isoplethal section. For instance, compositions of individual phases at equilibrium cannot be read directly from Fig. 7.7. However, with the
calculated diagram on the computer screen and with appropriate
software (see FactSage), one can simply place the cursor at any
desired point on the diagram and “click” in order to calculate
the amounts and compositions of all phases in equilibrium at that
point. Other software can follow the course of equilibrium cooling
or of nonequilibrium Scheil-Gulliver cooling as was discussed in
Chapter 8. And, of course, the thermodynamic databases permit
the calculation of chemical potentials that are essential for calculating kinetic phenomena such as diffusion. Coupling thermodynamic databases and software with databases of diffusion
coefficients and software for calculating diffusion (as with the
Dictra program of Thermo-Calc (see Thermo-Calc)) provides a
powerful tool for the study of the heat treatment of alloys.
Among the largest integrated database computing systems
with applications in metallurgy and materials science are
Thermo-Calc, FactSage, MTDATA, and Pandat. (See list of websites.) These systems all combine large evaluated and optimized
databases with advanced Gibbs energy minimization software.
As well, the SGTE group (SGTE, 2017) has developed a large database for metallic systems.
All these databases are extensive. For example, the SGTE alloy
database contains optimized data for 79 components involving
hundreds of solution phases and stoichiometric compounds,
603 completely assessed and optimized binary systems, and 156
optimized ternary and higher-order systems involving 319 solution phases and 1166 stoichiometric compounds. Equally extensive optimized databases have been developed for systems of
oxides, salts, etc. (See Thermo-Calc, FactSage, MTDATA, and
Pandat).
The theme of Part II is models that are currently used to represent the thermodynamic properties of solutions as functions
Chapter 13 INTRODUCTION
of T, P, and composition. These models relate the thermodynamic
properties to the atomic or molecular structure of the solution.
Techniques of data evaluation and optimization are also outlined. However, a thorough discussion of these techniques is
beyond the scope of the present work.
List of Websites
FactSage, Montreal, www.factsage.com.
NPL-MTDATA, www.npl.co.uk.
Pandat, www.computherm.com.
SGTE, Scientific Group Thermodata Europe, www.sgte.org.
SGTE Solution Database, 2017. http://www.sgte.net/en/thermochemical-databases.
Thermo-Calc, Stockholm, www.thermocalc.com.
191
14
SINGLE-LATTICE
RANDOM-MIXING (BRAGGWILLIAMS—BW) MODELS
CHAPTER OUTLINE
14.1 Ideal Raoultian Solutions 194
14.2 Regular Solution Theory: Binary Systems 195
14.3 Polynomial Expansion of the Excess Gibbs Energy: Binary
Systems 196
14.3.1 Respecting the Gibbs-Helmholz Equation 197
14.3.2 Redlich-Kister Polynomials 198
14.3.3 A Shortcoming of Polynomial Expansions 198
14.3.4 A Caveat for Optimizations When Data Are Available Only Over a
Limited Composition Range 199
14.4 Solutions With Two or More Sublattices But With Only One
Sublattice of Variable Composition 200
14.4.1 Asymmetric Common-Ion Molten Salt Solutions:
Temkin Model 200
14.5 Solutions With Limited Solubility: Lattice Stabilities 201
14.6 Darken’s Quadratic Formalism 202
14.7 Introduction to Coupled Thermodynamic/Phase Diagram
Optimization: Binary Systems 203
14.7.1 Optimization Algorithms 206
14.7.2 Use of “Virtual Data” 206
14.8 Multicomponent Systems 207
14.8.1 Kohler Model vs Muggianu Model 212
14.8.2 Sample Calculation: The KF-LiF-NaF System 213
14.8.3 Ternary Polynomial Terms 214
14.8.4 Extension to Systems of Four or More Components 216
14.8.5 Corresponding Expressions for Partial Properties 219
14.9 Liquid Solutions: Coordination Equivalent Fractions 219
14.9.1 Extension to Multicomponent Solutions 222
14.10 Wagner’s Interaction Parameter Formalism and the Unified
Interaction Parameter Formalism 222
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00014-2
# 2019 Elsevier Inc. All rights reserved.
193
194
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
14.10.1 Approximation for ε23 224
14.10.2 Interaction Parameter Formalism With Molar Ratios
14.11 Thermal Vacancies 226
References 226
225
We begin with models based on a single lattice. We further begin
with random-mixing Bragg-Williams (BW) or “point approximation” models in which species are distributed randomly over the
lattice sites with no correlation among their positions. This subject
was introduced in Section 4.6 where it was shown that the ideal
configurational entropy of mixing, that is, the entropy of mixing
associated with the random spatial distribution of the species on
the lattice, is given by Eq. (4.28) which can clearly be generalized
to systems with more than two components as
Δsideal ¼ RðX1 ln X1 + X2 ln X2 + X3 ln X3 + ::…Þ
(14.1)
where the site fraction Xi is the fraction of the lattice sites occupied
by species i.
(Note: In this chapter, it will often be more convenient to
designate the components of a solution as 1-2-3- … rather than
A-B-C- … as was done in Chapter 4.)
14.1 Ideal Raoultian Solutions
As discussed in Section 4.6, for a substitutional solution in
which the mixing species are nearly alike with nearly identical
radii and electronic structures, there will be little change in bonding energy or volume upon mixing. As a result, the enthalpy of
mixing will be small. Furthermore, there will be little change in
the vibrational and electronic energy levels upon mixing. Hence,
the nonconfigurational entropy of mixing resulting from the distribution of phonons and electrons over these levels will be small.
Also, and for the same reasons, the species will be nearly randomly
distributed over the lattice sites. That is, the excess configurational entropy will be small. A solution in which the enthalpy of
mixing is zero and the entropy of mixing is equal to the
random-mixing configurational entropy, Eq. (14.1), is called an
ideal Raoultian solution. Its Gibbs energy of mixing thus results
entirely from the ideal random configurational entropy:
ideal
¼ RT ðX1 ln X1 + X2 ln X2 + X3 ln X3 + ::…Þ
Δgm
(14.2)
As a general rule, one expects solutions to approach more closely
to ideal behavior (i) at higher temperatures since high
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
temperature favors the entropy term in Eq. (4.15) and random
mixing maximizes the configurational entropy, (ii) in liquid more
than in solid solutions because atoms of different sizes can mix
with less strain when not constrained to a rigid solid lattice,
and (iii) if the species that are mixing are chemically similar
and (iv) of similar size.
Generally speaking, the assumption of ideal solution behavior
for a solid or liquid solution is rarely satisfactory as a quantitative
model unless all these criteria are very closely met. For example,
although liquid Au-Cu solutions at 1100°C satisfy the first three of
these criteria, it can be seen from Fig. 4.2 that the Gibbs energy of
mixing is approximately twice as large as Δgideal.
14.2
Regular Solution Theory: Binary
Systems
Regular solution theory was introduced in Section 4.9 as a first
approximation to deviations from ideal behavior. The criteria for
approximately regular solution behavior are the same as criteria
(i)–(iv) as just presented in Section 14.1. If these criteria are reasonably closely met, the distribution might intuitively be expected
to be nearly random such that the configurational entropy of mixing is given by Eq. (14.1). A quantitative confirmation of this intuitive approximation will be given in Section 16.2.
Based on a pair-bond concept of lattice energy and vibrational
entropy, it was shown in Section 4.9 that the enthalpy of mixing,
nonconfigurational entropy, and excess Gibbs energy of mixing in
a binary system should, to a first approximation, be approximately parabolic functions of composition as in Eqs. (4.48),
(4.49), with maxima or minima at the equimolar composition.
As an example, plots of Δhm and gE for liquid Au-Cu solutions were
shown in Fig. 4.2. It can be seen that the curves are quite close to
parabolic with minima near XAu ¼ XCu ¼ 0.5. If gE for this solution is
approximated by the regular solution expression gE ¼ 27,000
XAuXCu J mol1, the deviation from the curve in Fig. 4.2 is never
>690 J mol1.
As discussed in Section 4.9, the excess entropy is the sum of a
configurational term due to deviations from the random spatial
distribution of the species and a nonconfigurational term:
sE ¼ sEðconfigÞ + sEðnonconfigÞ
(14.3)
However, it must be stressed that it is not possible to separate
these two terms by any measurement of a macroscopic property.
In regular solution theory, sE(config) is assumed to be zero.
195
196
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
However, experimentally, only the total excess entropy, sE, can be
measured. The separation into configurational and nonconfigurational terms is an artifact of the model being used. Furthermore,
even though sE might be reasonably closely fitted to a model equation of the parabolic form of Eq. (4.49), this does not necessarily
mean that the excess entropy is all, or even mainly,
nonconfigurational.
14.3 Polynomial Expansion of the Excess
Gibbs Energy: Binary Systems
The excess Gibbs energy of a binary solution of components
1–2 may be expressed as a polynomial expansion in the site fractions as follows:
g E ¼ α12 X1 X2
where the “α-function” α12 is given by
X
i
L 12 ðX1 X2 Þi
α12 ¼
(14.4)
(14.5)
i0
with the ideal entropy and ideal Gibbs energy of mixing being
given by Eqs. (14.1), (14.2). The coefficients iL12 are independent
of composition but may be functions of temperature. If we write
i
L 12 ¼ i h 12 T i s 12
(14.6)
where ih12 and is12 are coefficients (which are not necessarily independent of temperature), then
Xi
h 12 ðX1 X2 Þi
(14.7)
Δhm ¼ X1 X2
i0
and
s E ¼ X1 X2
X
i
s 12 ðX1 X2 Þi
(14.8)
i0
As many terms may be included in the series in Eq. (14.5) as are
required to fit the experimental data for a given solution. If the
series is truncated after the first term, then gE ¼ 0L12X1X2, and
the solution is regular. If the series is truncated after the second
term, then
(14.9)
g E ¼ X1 X2 0 L 12 + 1 L 12 ðX1 X2 ÞÞ
and the solution is called “subregular.”
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
The polynomial expansion is not a purely mathematical equation with no physical basis. It is clearly an extension of regular
solution theory. Hence, for a polynomial expansion to adequately
represent gE of a solution without requiring an excessive number
of terms and for it to permit reasonable interpolations and extrapolations to compositions where data are not available, the solution should approximately fulfill the basic physical assumptions
of regular solution theory. That is, the distribution of species on
the lattice sites should not deviate too far from a random distribution, and the pair-bond approximation should be reasonably
valid. This caveat is particularly important when one uses the
model to estimate thermodynamic properties of a multicomponent solution from the optimized properties of its binary subsystems as will be discussed in Section 14.8.
As an example, the available thermodynamic data for liquid
Au-Cu solutions are well represented (SGTE, 2017) by the following three-term equation:
g E =XAu XCu ¼ ð27900 1:000T Þ + 4730ðXAu XCu Þ
+ ð3500 + 3:500T ÞðXAu XCu Þ2 J mol1
(14.10)
Eq. (14.10) was used to generate the functions plotted in Fig. 4.2.
The equations for the partial excess Gibbs energies (also called
the excess chemical potentials) corresponding to Eqs. (14.4),
(14.5) are
h
i
X
i
L 12 ðX1 X2 Þi + 2iX 1 ðX1 X2 Þi1
(14.11)
μE1 ¼ X22
i0
μE2 ¼ X12
X
i
h
i
L 12 ðX1 X2 Þi 2iX 2 ðX1 X2 Þi1
(14.12)
i0
Similar equations give the partial enthalpies of mixing and excess
entropies in terms of the coefficients of Eqs. (14.7), (14.8).
14.3.1
Respecting the Gibbs-Helmholz Equation
It merits pointing out that the temperature derivatives of the
enthalpy of mixing and excess entropy are related by the GibbsHelmholtz equation, Eq. (2.58). Accordingly, the temperature
dependencies of the coefficients ih12 and is12 of Eqs. (14.7),
(14.8) are not independent of each other. Suppose, for example,
that the empirical temperature dependence of a coefficient iL12
is as follows:
i
L 12 ¼ a bT cT ln T
(14.13)
197
198
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
where the coefficients a, b, and c are independent of T. Clearly, if
c ¼ 0, then a ¼ ih12, and b ¼ is12. If c 6¼ 0, then from the GibbsHelmholtz equation,
i
h 12 ¼ a + cT
(14.14)
s 12 ¼ b + c ð ln T + 1Þ
(14.15)
and
i
14.3.2
Redlich-Kister Polynomials
The functions (X1 X2)i in Eq. (14.5) are called Redlich-Kister
polynomials.
Rather than using Redlich-Kister polynomials, one might
expand α12 in terms of the site fractions as follows:
X X ij
j
Q12 X1i X2
(14.16)
α12 ¼
i0 j0
Clearly, the set of coefficients Qij12 can be calculated from the set
L12 and vice versa.
That is, in principle, Eqs. (14.5), (14.16) are equivalent. However, a disadvantage of the expansion of Eq. (14.16) is that the base
functions Xi1Xj2 become very small for higher values of i or j when
X1 or X2 is small. As a result, the coefficients Qij12 tend to become
very large for larger values of i or j, thereby requiring the retention
of a large number of significant digits. Furthermore, and for the
same reason, the coefficients are highly correlated. That is, adding
an additional term to an expansion can completely change the
numerical values of the preceding coefficients even if the additional term only changes gE slightly. Redlich-Kister polynomials,
on the other hand, are much less strongly correlated, and as i
and j become larger, the values of the coefficients iL12 tend to
become smaller.
For a detailed treatment of this subject and the use of other
polynomial functions such as orthogonal Legendre polynomials,
see Pelton and Bale (1986).
i
14.3.3
A Shortcoming of Polynomial Expansions
Regular solution theory predicts a curve of gE versus composition that is parabolic with an extremum at X1 ¼ X2 ¼ 0.5.
For many solutions, the observed gE function is approximately
parabolic, but the extremum is shifted from the equimolar composition. In order to represent such a function, one might
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
consider adding a second “subregular” term to the expansion as in
Eq. (14.9). The problem with this is that as the ratio 1L12/0L12
becomes larger, the curve becomes distorted and can even change
sign. For example, suppose that 0L12 > 0, 1L12 > 0, and
1
L12/0L12 ¼ 1.5. The maximum value of gE is then shifted to
X1 ¼ 0.69, but gE becomes negative when X2 > 0.71.
One way to shift the composition of the extremum without
severely distorting the parabolic shape of the curve is to replace
the molar fractions in Eqs. (14.4), (14.5) by “coordination equivalent fractions” as discussed in Section 14.9. As will be discussed in
Section 14.9, for liquid solutions this can be rationalized as resulting from a variation of coordination number with composition.
Although this rationalization does not apply to solid solutions,
polynomial expansions in terms of equivalent fractions might
nevertheless be useful empirically in some cases for solid as well
as for liquid solutions.
14.3.4
A Caveat for Optimizations When Data Are
Available Only Over a Limited
Composition Range
If the coefficients iL12 of the integral property gE in Eq. (14.5)
are known, then the expressions for the partial properties μE1
and μE2 in Eqs. (14.11), (14.12) are complete because they are
obtained by differentiation.
Suppose, however, that the only available data in a solution are
for the partial property μE1 of component 1 over a limited composition range. For example, suppose that in a binary system, only the
liquidus in equilibrium with pure solid component 1 has been
measured over a composition range extending from X1 ¼ 1. If the
Gibbs energy of fusion is known and there is negligible solubility
in the solid phase, then Eq. (5.5) can be used to calculate μE1 in
the liquid as a function of composition along the liquidus. If we further assume that sE1 0, then μE1 is independent of temperature. We
can then fit μE1 to Eq. (14.11) to obtain the empirical parameters iL12.
We might now be tempted to calculate μE2 by substituting these
parameters into Eq. (14.12). However, this can give rise to very
serious errors. The Gibbs-Duhem equation may be written:
X1 dμE1 + X2 dμE2 ¼ 0
(14.17)
This is a differential equation. If μE1 is known as a function of composition, then μE2 can only be calculated within a constant of integration. That is, μE2 is in fact given by Eq. (14.12) but with
the addition of a constant C that can only be evaluated if a value
199
200
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
of μE2 is known from an independent measurement at some composition within the range over which μE1 has been measured.
Errors introduced by assuming that C ¼ 0 can often be very large.
In this regard, see also Section 14.6.
14.4 Solutions With Two or More
Sublattices But With Only One
Sublattice of Variable Composition
Systems with two or more sublattices are the subject of
Chapter 15. However, it may be noted that the equations of a multisublattice model reduce to those of a single-sublattice model when
only one sublattice is of variable composition. For example, solid
and liquid MgO-CaO solutions can be modeled by assuming an
anionic sublattice occupied only by O2– ions, while Mg2+ and Ca2+
ions mix on a cationic sublattice. In this case, the model is mathematically the same as that of a single-sublattice substitutional
model because the site fractions XMg and XCa of the ions on the cationic sublattice are numerically equal to the overall component
mole fractions XMgO and XCaO. Solid and liquid MgO-CaO solutions
have been shown (Wu et al., 1993) to be well represented by simple
polynomial expansions as in Eqs. (14.5), (14.16).
14.4.1
Asymmetric Common-Ion Molten Salt
Solutions: Temkin Model
Consider a molten salt solution such as MgCl2-KCl in which the
components share a common anion. Such a solution is termed
“asymmetrical” since Mg2+ and K+ ions have different charges.
In a solid asymmetrical salt solution, dissolution can only occur
with the formation of lattice defects such as cation vacancies or
interstitial ions, and the components cannot be completely miscible at all compositions since they have different crystal structures. However, liquid MgCl2 and KCl are miscible at all
compositions, and evidence shows (see, e.g., Blander, 1964) that
the Temkin model applies, whereby the Mg2+ and K+ ions mix randomly on a cationic sublattice with the number of lattice sites
varying with composition so as to be always equal to the number
of cations in solution. That is, there is no formation of lattice
defects. Hence, the model is mathematically the same as that of
a single-lattice substitutional model, since the cationic site fractions XMg and XK are numerically equal to the component mole
fractions XMgCl2 and XKCl.
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
14.5
Solutions With Limited Solubility:
Lattice Stabilities
The concept of a dilute Henrian solution was introduced in
Section 5.9 using the example of the Mg-rich hcp phase in the
Ag-Mg system. In this quite dilute solution, the activity coefficient
of Ag, γ oAg, relative to pure fcc Ag as standard state is approximately
independent of composition. By combining terms as in Eqs. (5.25),
(5.26), it was shown that the hcp solution is then ideal relative to a
.
hypothetical standard state of pure Ag with Gibbs energy μo(hcp-Mg)
Ag
as defined by Eq. (5.26)
As pointed out in Section 5.9, μo(hcp-Mg)
Ag
is not the same as, for
is solvent-dependent. That is, μo(hcp-Mg)
Ag
for Ag in dilute hcp Cd solutions. However,
example, μo(hcp-Cd)
Ag
for purposes of developing consistent thermodynamic databases
for multicomponent solution phases, it is advantageous to select
one single set of values for the chemical potentials of pure elements and compounds in metastable or unstable crystal struc) that are independent of the solvent. Such
tures (e.g., μo(hcp)
Ag
values, sometimes called lattice stabilities, may be obtained
(i) from measurements at high pressures if the structure is stable
at high pressure, (ii) as averages of the values for several solvents,
or (iii) from first-principle quantum mechanical or molecular
dynamics calculations. Tables (Dinsdale, 1991) of such values
for many elements in metastable or unstable states have been prepared and agreed upon by the international community working
in the field of phase diagram evaluation and thermodynamic
database optimization.
Hence, for the hcp solution in the Ag-Mg system, we would
write the following:
oðhcpÞ
oðhcpÞ
+ XMg μMg
+ RT XAg ln XAg + XMg ln XMg + g E
g ¼ XAg μAg
(14.18)
In this case, we say that hcp Mg and hcp Ag are the end-members
of the solution (as opposed to Eq. (5.24) where hcp Mg and fcc Ag
are the end-members).
gE in Eq. (14.18) can be expressed by a polynomial expansion,
and in this way the equation is extended to compositions beyond
the dilute Henrian region. However, even if the equation is only
required to apply to very dilute solutions, at least one nonzero
coefficient in the polynomial expansion of gE will usually be
is now solventrequired because the standard state μo(hcp)
Ag
independent. That is, it is no longer an adjustable model parameter. However, gE will still generally be smaller than if fcc Ag was
chosen as standard state.
201
202
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
14.6 Darken’s Quadratic Formalism
Some physical insight into regular solution theory and lattice
stabilities is provided by Darken’s quadratic formalism. In several
binary solutions with complete miscibility of the end-members 1
and 2, Darken (1967a, 1967b) observed that the Gibbs energy of
solutions relatively rich in component 1 is often well represented
by the following equation:
(14.19)
g ¼ X1 μo1 + X2 μo2 + C2 + ðω1 η1 T ÞX1 X2
where C2 is a constant. By substitution into Eq. (4.40), this gives
μE1 ¼ ðω1 η1 T ÞX22
(14.20)
μE2 ¼ ðω1 η1 T ÞX12 + C2
(14.21)
When C2 ¼ 0, Eqs. (14.20), (14.21) reduce to the regular solution
equations (Eq. 4.45).
From the discussion in Section 5.9, it can be seen that
Eq. (14.19) is equivalent to defining a new hypothetical standard
state for component 2 with Gibbs energy (μo2 + C2). Similarly, for
solutions rich in component 2, the Gibbs energy is given by
(14.22)
g ¼ X1 μo1 + C1 + X2 μo2 + ðω2 η2 T ÞX1 X2
which is equivalent to defining a new hypothetical standard state
for component 1 with Gibbs energy (μo1 + C1)..
As an example, the phase diagram of the AgCl-Ag2S system is
shown in Fig. 14.1. The experimental points in the figure and
experimental data for the enthalpy of liquid mixing are well reproduced with the following expressions for the Gibbs energy of the
liquid phase (Brookes et al., 1978):
When XAg2S < 0.45:
g ¼ XAgCl μoAgCl + XAg2 S μoAg2 S 1033 + 9100XAgCl XAg2 S J mol1
(14.23)
When XAg2S > 0.55:
g ¼ XAgCl μoAgCl + CAgCl + XAg2 S μoAg2 S + 9260XAgCl XAg2 S J mol1
(14.24)
There are insufficient data to permit the constant CAgCl to be
determined.
This can be interpreted as (μAg2So 1033) being the molar Gibbs
energy of hypothetical pure liquid Ag2S with the structure of AgCl.
That is, at AgCl-rich compositions, the solute Ag2S is incorporated
into a solution that retains the structure of liquid AgCl for Ag2S
contents up to approximately XAg2S ¼ 0.45. Similarly, at Ag2S-rich
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
900
800
Liquid
T (∞C)
700
600
500
Liquid + Ag2S
Liquid + AgCl
400
300
0
0.2
0.4
0.6
Mole fraction Ag2S
0.8
1
Fig. 14.1 AgCl-Ag2S phase diagram. Experimental points from Bell and Flengas
(1964). Calculated liquidus from Eqs. (14.23), (14.24).
compositions, the structure of liquid Ag2S is retained for AgCl contents up to approximately XAgCl ¼ 0.45. And both terminal phases
are approximately regular solutions. The narrow transition region
between these two terminal regular solutions is evident even from
a cursory examination of the phase diagram.
Eqs. (14.23), (14.24) do not apply in the transition region. That
is, Darken’s formalism does not provide a single equation for the
Gibbs energy that applies at all compositions. Hence, it is unfortunately not well suited for the development of general databases
of thermodynamic properties.
14.7
Introduction to Coupled
Thermodynamic/Phase Diagram
Optimization: Binary Systems
The concept and purpose of coupled thermodynamic/phase
diagram optimization were discussed in Chapter 13. In the present
section, we present an example of an optimization of a binary
system using single-sublattice Bragg-Williams models as discussed in Sections 14.3–14.5. Other examples, using other models,
will be given in later chapters. A great many optimizations
have been published since 1977 in the Calphad Journal
(Pergamon Press).
203
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
The NaNO3-KNO3 system has been evaluated and optimized
by Robelin et al. (2015) who provide full details of the procedure.
Only an outline will be presented here as an example of an
optimization.
This system has been the subject of several experimental investigations. The many measurements of the phase diagram, obtained
by a variety of experimental techniques, are shown in Fig. 14.2. (The
most recent studies indicate that the IIA ➔ IA transition is actually
second-order. However, Robelin et al. modeled it as first-order as
shown in the figure.) Enthalpies of liquid mixing (Kleppa, 1960)
are shown in Fig. 14.3. (These data were obtained at either 619 or
721 K, but the temperature dependence of Δhm was found to be
negligible.) Measurements of partial enthalpies of mixing in liquid
solutions and of heat contents (hT h298.15) for equimolar mixtures
between T ¼ 300 and 717 K are also available. These data were
also taken into account in the optimizations by Robelin et al.
As can be seen in Fig. 14.2, there are four equilibrium solid
solutions in the system (IA, IIA, IB, and IIB). All four solid phases
and the liquid phase were modeled assuming that NO
3 ions
occupy an anionic sublattice while Na+ and K+ ions are distributed
randomly on a cationic sublattice. Hence, as discussed in
Section 14.4, a single-sublattice model can be used.
The Gibbs energies of the stable end-members (μNaNO3o(LIQ),
o(IA)
, μNaNO3o(IIA), μKNO3o(LIQ), μKNO3o(IB), and μKNO3o(IIB)) were
μNaNO3
evaluated from enthalpy and Cp data for pure NaNO3 and KNO3
650
600
Liquid
550
T (K)
204
500
IA
IB
450
400
IIA
IIB
350
0.0
0.1
0.2
0.3
0.5
0.4
0.6
0.7
Mole fraction XKNO3
0.8
0.9
1.0
Fig. 14.2 Optimized phase diagram of the NaNO3-KNO3 system (Robelin et al., 2015.
For references for experimental points, see this reference).
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
0
Δhm (J·mol-1)
−100
−200
−300
−400
−500
−600
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Mole fraction XKNO3
0.8
0.9
1.0
Fig. 14.3 Enthalpy of liquid-liquid mixing in the NaNO3-KNO3 system (Kleppa, 1960)
(circles, 619 K; triangles, 721 K).
and were expressed as functions of T as described in Section 2.6.
The evaluated functions are given by Robelin et al. (2015).
The four solid solutions were modeled as ideal Henrian solutions as discussed in Section 5.9. For example, for the IB solution,
oðIBÞ
oðIBÞ
g ðIBÞ ¼ XKNO3 μKNO3 + XNaNO3 μNaNO3 + RT ðXKNO3 ln XKNO3
+ XNaNO3 ln XNaNO3 Þ
(14.25)
oðIBÞ
where μNaNO3 , the molar Gibbs energy of the unstable endmember NaNO3 with the IB crystal structure, was optimized as
an adjustable parameter:
oðIBÞ
oðIAÞ
μNaNO3 ¼ μNaNO3 + ð6171 3:347T Þ J mol1
(14.26)
The Gibbs energies of the unstable end-members of the other
three solid solutions were similarly optimized as
oðIIBÞ
oðIIAÞ
μNaNO3 ¼ μNaNO3 + 7950 J mol1
oðIAÞ
oðIBÞ
μKNO3 ¼ μKNO3 + ð690 + 9:623T Þ J mol1
oðIIAÞ
oðIIBÞ
μKNO3 ¼ μKNO3 + 8368 J mol1
(14.27)
(14.28)
(14.29)
For the liquid solution, Robelin et al. used the modified quasichemical model (to be discussed in Chapter 16) in order to take
short-range ordering into account. However, short-range ordering
205
206
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
is very small in this solution. Hence, we have reoptimized the liquid using a subregular Bragg-Williams model with
g E ¼ XNaNO3 XKNO3 ð1860 143ðXNaNO3 XKNO3 ÞÞ J mol1 (14.30)
The parameters of Eqs. (14.26)–(14.30) are those that best reproduce all the available data simultaneously. The phase diagram and
liquid enthalpy of mixing calculated from the optimized equations are compared with the experimental data in Figs. 14.2 and
14.3. The other calorimetric data mentioned above are also well
reproduced.
14.7.1
Optimization Algorithms
Several algorithms have been developed to perform thermodynamic optimizations using least squares, Bayesian, and other
techniques (e.g., see Bale and Pelton, 1983; Lukas et al., 1977;
€rner et al., 1980; Ko
€ nigsberger and Eriksson, 1995). In general,
Do
however, it is not possible to have a single universal objective
method. The preferred optimization procedure will vary from system to system depending upon the type and accuracy of the available data, the number of data points, the number of phases
present, etc. Furthermore, no purely objective statistical criterion
exists for determining when an optimization “best” represents the
data. Optimizations necessarily involve a large degree of subjective judgment and much trial and error.
14.7.2
Use of “Virtual Data”
In the absence of experimental thermodynamic data, “virtual
data” from first-principles quantum mechanical calculations
and molecular dynamics calculations can be used. As the accuracy of these calculations has improved in recent years, their
use in thermodynamic optimizations has increased apace. Such
calculations are most accurate, and have found their widest application, in estimating the enthalpies of compounds and unstable
end-members of solutions (see Section 14.5). However, estimates
of entropies and heat capacities of compounds and even of some
solution properties are becoming increasingly possible.
Several commercial software packages are available for performing first-principles and molecular dynamics calculations.
Density functional theory codes with pseudopotential include
VASP (Kresse and Joubert, 1999), Quantum ESPRESSO
(Giannozzi et al., 2009), ABINIT (Gonze et al., 2002), and CASTEP
(Segall et al., 2002). For a density functional theory code with full
potential, see WIEN2k (Blaha et al., 1990). For an ab initio
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
molecular dynamics code, see VASP (Kresse and Furthmuller,
1996), and for a classical molecular dynamics code, see LAMMPS
(Plimpton, 1995). For discussions of first-principles and molecular dynamics techniques, see these references and Turchi et al.
(2007).
Many optimizations employing virtual data have been published in recent years in the Calphad Journal (Pergamon Press).
14.8
Multicomponent Systems
Among the 70 metallic elements, 80!/3!67! ¼ 54,740 ternary systems and 916,895 quaternary systems are formed. In view of the
amount of work involved in measuring even one isothermal section of a relatively simple ternary phase diagram, it is of much
practical importance to have a means of estimating ternary and
higher-order phase diagrams. In most ternary systems, experimental data are very limited or entirely lacking.
For a solution phase in a ternary system with components 1-2-3,
one first performs evaluations/optimizations on the three binary
subsystems in order to obtain the binary model parameters. Next,
a model is used, along with reasonable assumptions, to estimate
the thermodynamic properties of the ternary solution. If some
experimental data are available for the ternary system, they can
be used to refine the calculations through the addition of optimized
ternary model parameters.
Suppose that gE of each binary subsystem of a ternary solution
has been modeled by a polynomial expansion as in Eqs. (14.4),
and either (14.5) or (14.16). Then, for the ternary solution, several
“geometric” models can be proposed. The concept of geometric
models was introduced by Larry Kaufman. For a general detailed
discussion, see Chartrand and Pelton (2000) and Pelton (2001).
Some geometric models are illustrated in Fig. 14.4. In each
model, gE in the ternary solution at a composition point p is estimated from the excess Gibbs energies in the three binary subsystems at points a, b, and c by the following equation:
g E ¼ X1 X2 α12ðaÞ + X2 X3 α23ðbÞ + X3 X1 α31ðcÞ + ðternary termsÞ (14.31)
where α12(a), α23(b), and α31(c) are the binary α-functions (see
Eqs. 14.4, 14.5) evaluated at points a, b, and c. The “ternary terms”
in Eq. (14.31) are polynomials that are identically zero in the three
binary subsystems (see Section 14.8.3). The empirical coefficients
of these ternary terms may be optimized in order to fit ternary
experimental data. However, the ternary coefficients should be
207
208
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
1
1
c
a
X2
= Constant
(X1 + X2)
p
b
2
3
Kohler model
(A)
X1 = Constant
p
a
c
b
2
3
Kohler / Toop model
(B)
1
1
(1 + X1 − X2)/2 = Constant
or (X1 − X2) = Constant
c
a
b
2
p
a
p
3
b
2
Muggianu model
3
Muggianu /Toop model
(C)
(D)
1
1
a
c
a
2
(E)
c
p
p
b
A weird model
c
3
2
(F)
b
3
Another weird model
Fig. 14.4 Some “geometric” models for estimating ternary thermodynamic properties from optimized binary data
(Pelton, 2001).
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
small. That is, Eq. (14.31) with no ternary terms should provide a
reasonable first estimate of gE in the ternary solution.
Eq. (14.31) is based on the polynomial expansion model
(Section 14.3) which is an extension of regular solution theory.
Hence, Eq. (14.31) without ternary terms can only be expected
to provide a good estimate of gE when the solution can reasonably
be approximated as a random substitutional solution on a single
sublattice with relatively weak interactions where the lattice
energy is approximately given as the sum of pair-bond energies
(see Section 4.9). If large ternary terms are required in order to
fit the ternary data, then this is an indication that the polynomial
model is not up to the task and interpolations and extrapolations
to compositions and temperatures where data are not available or
to quaternary and higher-order systems will probably be inaccurate. For more complex solutions involving, for example, shortrange order or long-range order, other models should be used
as will be discussed in Chapter 16.
That is, the “geometric” models are not purely mathematical
models as is sometimes stated. They are physical models based
upon regular solution theory. A purely “mathematical” model
can have very little predictive ability.
If, at a given temperature, all three binary α-functions in
Eq. (14.31) are constant, independent of composition, then all
the geometric models are clearly identical.
The Kohler (Kohler, 1960) and Muggianu (Muggianu et al.,
1975) models in Fig. 14.4A and C are “symmetrical” models,
whereas the Kohler/Toop (Toop, 1965) and Muggianu/Toop
models in Fig. 14.4B and D are “asymmetrical” models inasmuch
as one component (component 1 in Fig. 14.4B and D) is singled
out. In these two asymmetrical models, α12 and α31 are assumed
to be constant along lines where X1 is constant. That is, replacing
component 2 by component 3 is assumed to have no effect on the
energy, α12, of forming (1–2) pair-bonds and similarly for (3–1)
pairs. An asymmetrical model is thus more physically reasonable
than a symmetrical model if components 2 and 3 are chemically
similar while component 1 is chemically different, for example in
the systems SiO2-CaO-MgO, S-Fe-Cu, Na-Au-Ag, and AlCl3NaCl-KCl. (where the asymmetrical component has been written
first in each example).
When gE is large and α12 and α31 depend strongly upon composition, a symmetrical and an asymmetrical model will give very
different results. As an example, consider a solution with
α12 ¼ α31 ¼ 50(1 X1) kJ mol1 and α23 ¼ 0. That is, the binary
solutions 1–2 and 3–1 have identical properties, and the 2–3 solution is ideal. Clearly, one would expect gE in the ternary system to
209
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
be nearly constant at constant X1 as is predicted by the asymmetrical models. However, when the symmetrical Kohler model is
applied to this solution, an obviously incorrect region of
immiscibility is calculated as shown in Fig. 14.5 (Pelton, 2001).
Similar incorrect results are obtained with the symmetrical
Muggianu model.
Therefore, we disagree with the common tendency to use the
symmetrical Muggianu model for nearly all solutions. In many
systems, an asymmetrical model is more appropriate and can
result in very different ternary Gibbs energy functions.
As well as the four models illustrated in Fig. 14.4A–D, many
other geometric models might be proposed. Two of these are
shown in Fig. 14.4E and F. In general, rather than speaking of a single model for an entire ternary 1-2-3 solution, we should instead
define the approximation used for each of the three αmn functions.
For example, in Fig. 14.4E and F, a “Kohler-type” approximation is
used for α23, and a “Muggianu-type” approximation is used for α31.
For α12, a “Toop-type” approximation along lines of constant X1 is
used in Fig. 14.4E, while a “Toop-type” approximation along lines
of constant X2 is used in Fig. 14.4F. Since each function αmn can be
approximated in four different ways, there are 64 such possible
geometric ternary models. However, probably only the four
models in Fig. 14.4A–D are of practical importance.
Suppose that data for the three binary subsystems of a ternary
system have been optimized to give binary coefficients iLmn of
Eq. (14.5) or Qijmn of Eq. (14.16) and that we wish to estimate gE
in the ternary solution by means of Eq. (14.31) having chosen
which type of approximation we wish to use for each of the three
αij functions.
m
kJ
=−
50
X
2
1X 2
2 k
2
phases
X3
X1
50
=−
Jm
E
−
ol
12
1
2
g 13
ol −1
1
gE
210
gE
=0
23
3
Fig. 14.5 Spurious miscibility gap calculated with the Kohler model at 973 K in a
solution with a12 ¼ a31 ¼ 50(1 X1) kJ mol1 and a23 ¼ 0 (Pelton, 2001).
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
Say, for example, that we have chosen to use a Kohler-type
approximation for α12 as in Fig. 14.4A. Along the line ap in
Fig. 14.4A, the ratio X2/(X1 + X2) is constant. Furthermore, this
ratio is equal to X2 at point a where (X1 + X2) ¼ 1. Therefore, the
function α12(a) in Eq. (14.31) can be written:
X
X1 X2 i
i
α12ðaÞ ¼
L 12
(14.32)
X1 + X2
i0
where iL12 are the binary coefficients of Eq. (14.5). Alternatively, if
the 1–2 binary system was optimized with an expansion as in
Eq. (14.16), then α12(a) in Eq. (14.31) is written:
X X ij X1 i X2 j
α12ðaÞ ¼
Q12
(14.33)
X1 + X2
X1 + X2
i0 j 0
where Qij12 are the binary coefficients of Eq. (14.16). That is, α12(a),
as given in Eq. (14.32) or (14.33), is constant along the line ap in
Fig. 14.4A. Similarly, if we have chosen to use Kohler-type approximations for α23 and α31, then these would be written in terms of
the ratios X3/(X2 + X3) and X3/(X3 + X1), respectively.
Now, suppose instead that we have chosen to approximate α12
by a Toop-type approximation along lines of constant X1 as in
Fig. 14.4B, D, and E. In this case, α12(a) is set constant along these
lines of constant X1 by means of the substitution:
X
X
i
i
L 12 ðX1 ð1 X1 ÞÞi ¼
L 12 ð2X1 1Þi (14.34)
α12ðaÞ ¼
i0
i0
or
α12ðaÞ ¼
X X
i0
j 0
ij
Q12 X1i ð1 X1 Þj
(14.35)
Finally, suppose that we have chosen to approximate α12 by a
Muggianu-type approximation as in Fig. 14.4C. We note that
(X1 X2) is constant along the line ap in Fig. 14.4C. Hence, if α12
in the binary system has been expressed by a Redlich-Kister polynomial as in Eq. (14.5), then, as was pointed out by Hillert (1980),
Eq. (14.5) can be substituted directly into Eq. (14.31) with no
change. Because of this particularly simple substitution, the
Muggianu model is sometimes referred to as the “MuggianuRedlich-Kister model.” However, this is a misnomer. The use of
the Muggianu model does not require that the binary α-functions
be expressed as Redlich-Kister polynomials. For example, if α12 is
expressed as a general polynomial as in Eq. (14.16), then we note
that the functions (1 + X1 X2)/2 and (1 X1 + X2)/2 are both
211
212
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
constant along the line ap in Fig. 14.4C and that these are equal to
X1 and X2, respectively, in the 1–2 binary system. Hence, in order
that α12(a) be constant along the line ap, we make the substitution:
X X ij 1 + X1 X2 i 1 X1 + X2 j
α12ðaÞ ¼
Q12
(14.36)
2
2
i0 j 0
Conversely, the use of the Kohler or Kohler/Toop models does not
preclude expressing the binary α-functions as Redlich-Kister polynomials, as has just been shown in Eqs. (14.32), (14.34). In summary,
the choice among using the Kohler, Toop, or Muggianu approximations is not related to the use of Redlich-Kister polynomials.
14.8.1
Kohler Model vs Muggianu Model
The Kohler model assumes that the energy α12 of forming (1–2)
pair-bonds in the ternary solution is constant at constant X1/X2
ratio, whereas the Muggianu model sets this energy equal to its
value at the geometrically closest binary composition. From a
physical standpoint, the former assumption seems intuitively
more justifiable. Nevertheless, for most concentrated solutions,
the Kohler and Muggianu models will give quite similar results.
However, in dilute solutions, the Kohler model is to be preferred
for the reason illustrated in Fig. 14.6 (Chartrand and Pelton, 2000).
In a ternary solution dilute in component 1, the Kohler model
estimates values of α12(a) and α31(c) from values in the binary systems
at compositions that are also dilute in component 1, as seems reasonable. On the other hand, the Muggianu model uses values from
the binary systems at compositions that can be far from dilute.
Hence, the Kohler model presents an advantage and no obvious disadvantages vis-à-vis the Muggianu model. Therefore, we
1
1
c
a
a
2
(A)
p
b
Kohler
c
p
3
2
(B)
b
Muggianu
3
Fig. 14.6 The Kohler and Muggianu models applied at a composition dilute in
component 1 (Chartrand and Pelton, 2000).
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
prefer the Kohler model over the Muggianu model and the Kohler/
Toop model over the Muggianu/Toop model.
14.8.2
Sample Calculation: The KF-LiF-NaF System
All three binary subsystems of the NaF-KF-LiF system are simple eutectic systems with no intermediate compounds and limited solubility in the terminal solid solutions (<9 mol% LiF in
NaF, <5 mol% NaF in KF, and <1% in all other terminal solutions).
The three binary systems have been optimized (Sangster and
Pelton, 1987) taking into account all available data for the phase
diagrams and calorimetric measurements of enthalpies of mixing
in the liquid solutions. For the binary liquid phases, two- or threeterm polynomial expansions as in Eq. (14.16) were used.
The thermodynamic properties of the ternary liquid solution
were estimated from Eq. (14.31) with the symmetrical Kohler
approximation as in Fig. 14.4A with no ternary terms. The
ternary liquidus projection calculated in this way solely from the
binary model parameters is shown in Fig. 14.7. The calculated
0.2
0.8
0.1
0.9
KF
0.7
0.5
0.5
0.4
0.6
0.3
KF
500
0.7
0.4
0.3
0.6
o
467
600
700
LiF
800
0.9
900
0.1
NaF
0.8
0.2
NaF
0.9
0.8
0.7
0.6
0.5
0.4
Mole fraction
0.3
0.2
0.1
LiF
Fig. 14.7 Liquidus projection of the LiF-NaF-KF system calculated by the Kohler
approximation solely from the optimized binary model parameters (temperature in °C).
213
214
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
ternary eutectic liquid at XKF ¼ 0.42, XLiF ¼ 0.47, and XNaF ¼ 0.11 at
467°C is nearly within the experimental error limits of the reported
(Hoffman, 1958) value of XKF ¼ 0.41, XLiF ¼ 0.47, and XNaF ¼ 0.12 at
457°C.
Given the good agreement between the measured and calculated
eutectic liquid composition and temperature, it is safe to assume
that the calculated isotherms on Fig. 14.7 are of similar accuracy.
In this system, gE of the liquid phase is relatively small in
all binary subsystems (with a minimum of approximately
4.7 kJ mol1 in the NaF-LiF system.) Furthermore, the solution
in question is a liquid, the species are chemically similar, and
approximating the energy of the solution as the sum of pair-bond
energies is reasonable for coulombic interactions in an ionic liquid. That is, the criteria for a solution to be approximately regular
(see Section 14.2) are satisfied, and as a result, the calculations
based on Eq. (14.31) agree well with the measurements. In other
solutions where the criteria for regular solution behavior are not
so closely met, the use of a geometric model may be expected
to yield less accurate results. However, if a few experimental ternary data are available, they can be used to refine the model as
is discussed in the following section.
14.8.3
Ternary Polynomial Terms
If experimental ternary data are available, then they may be
included in the optimization to give empirical “ternary terms”
in Eq. (14.31). These terms are identically zero in all binary subsystems. Terms such as
ijk
j
Q123 X1i X2 X3k
(14.37)
where i 1, j 1, and k 1 and Qijk
123 is an empirical coefficient are
frequently used. However, such terms have little theoretical justification, and it is not clear how such ternary terms should be
extrapolated into systems of four or more components.
From a physical standpoint, such terms might be related to the
energies of triplet interactions. However, the regular solution
model is a pair-bond approximation model. Hence, it would seem
better to include ternary terms that are designed to represent the
effect of a third component, 3, upon the energy α12 of the pairbond formation energy.
Terms as in Expression 14.37 are ill-suited to this purpose
since, when i>1 or j > 1, the terms rapidly become very small as
X3 is increased. Consequently, if α12 is given by a Kohler-type
approximation in the 1-2-3 ternary system as in Fig. 14.4A, it
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
has been proposed (Chartrand and Pelton, 2000) to employ ternary terms of the following form:
!
i j
X1
X2
ijk
k
X3
(14.38)
X1 X2 Q12ð3Þ
X1 + X2
X1 + X2
(cf. Eq. 14.33) where i 0, j 0, k 1, and Qijk
12(3) are ternary coefficients. Alternatively, this could be written in Redlich-Kister form
(cf. Eq. 14.32):
"
#
X1 X2 i k
ik
L12ð3Þ
X3
(14.39)
X1 X2
X1 + X2
where i 0, k 1, and ikL12(3) are ternary coefficients.
If α12 is given in the 1-2-3 ternary system by a Toop-type
approximation along lines of constant X1 as in Fig. 14.4B and D,
then it has been proposed (Chartrand and Pelton, 2000) to represent the effect of component 3 upon α12 by terms of the form (cf.
Eq. 14.35)
k !
X3
ijk
i
j
X1 X2 Q12ð3Þ ðX1 Þ ð1 X1 Þ
(14.40)
X2 + X3
or in Redlich-Kister form (cf. Eq. 14.34) as
"
k #
X3
i
ik
L12ð3Þ ð2X1 1Þ
X1 X2
X2 + X3
(14.41)
In the case where α12 is given by a Muggianu-type approximation,
it has been proposed (Pelton, 2001) to include terms of the form
h
i
X1 X2 ik L12ð3Þ ðX1 X2 Þi X3k
(14.42)
or alternatively (cf. Eq. 14.36)
!
1 + X1 X2 i 1 X1 + X2 j k
ijk
X3
X1 X2 Q12ð3Þ
2
2
(14.43)
Note that in any ternary system 1-2-3, three types of ternary terms
may thus be included: terms giving the effect of component 3 upon
α12 as in Eqs. (14.38)–(14.43), terms giving the effect of component
1 upon α23, and terms giving the effect of component 2 upon α31.
Finally, it may be noted that it is a common practice when
using the Muggianu model for all three αmn functions to include
ternary terms of the form (Hillert, 2008)
i
L 123 X1 X2 X3 ðð1 X1 X2 X3 Þ=3 + Xi Þ
(14.44)
215
216
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
In a ternary system (where (X1 + X2 + X3) ¼ 1), these terms are simply a special case of the general form of Expression 14.37.
14.8.4
Extension to Systems of Four or More
Components
Suppose that all binary and ternary subsystems of a multicomponent solution of components 1-2-3-4-5- … have been optimized and we wish to estimate the properties of the
multicomponent solution.
In this regard, it may be noted that estimating the properties of
a C-component system from the optimized parameters of its
(C 1)-component subsystems becomes increasingly more exact
as C becomes larger. For example, if all the ternary subsystems of a
four-component system have been experimentally investigated
and if optimized ternary model parameters have been obtained,
then the estimation of the properties of the four-component solution is generally reasonable.
Binary Terms
Suppose that a Kohler-type approximation has been used for
α12 in all ternary subsystems 1-2-k (k ¼ 3, 4, 5, …) of a multicomponent solution as in Fig. 14.4A. It is then reasonable to assume
that α12 remains constant along lines of constant ratio
X1/(X1 + X2) in the multicomponent system just as in the ternary
subsystems. Hence, Eqs. (14.32) or (14.33) can be used directly
in the multicomponent system. Similarly, if a Muggianu-type
approximation has been used for α12 in all 1–2-k ternary subsystems as in Fig. 14.4C, then Eqs. (14.5) or (14.36) can be used
directly in the multicomponent solution.
Suppose instead that a Toop-type approximation with component 1 as the asymmetrical component has been used for α12 in all
subsystems 1–2-k as in Fig. 14.4B or D. In this case, it is reasonable
to assume that α12 remains constant along lines of constant X1 in
the multicomponent solution just as in the ternary subsystems.
Hence, Eqs. (14.34) or (14.35) can be used directly.
The situation is less clear when different geometric models
have been used for α12 in different 1–2-k ternary subsystems. In
the general case, the following rational method of extending the
Kohler, Toop, and Muggianu models to multicomponent solutions
has been proposed (Pelton, 2001). The equations reduce to the
model equations used for each ternary subsystem.
We define
X
Xk
(14.45)
ξij ¼ Xi +
k
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
where the summation is over the components k of all ternary subsystems i-j-k in which αij is given by a Toop-type approximation
along lines of constant Xj. We also define
X
σ ij ¼ σ ji ¼ 1 Xk
(14.46)
k
where the summation is over the components k of all ternary subsystems i-j-k in which αij is given by a Kohler-type approximation.
Then, αij in the multicomponent system is given by replacing Xi
and Xj in Eq. (14.5) by (ξij ξji)/σ ij to give (e.g., for the case of α12)
X
ξ ξ21 i
i
α12 ¼
L 12 12
(14.47)
σ 12
i0
or alternatively
Xi and Xj in Eq. (14.16) by
by replacing
ξ ξ
ξ ξ
1 + ijσ ij ji =2 and 1 + jiσij ij =2, respectively.
As an example, consider a five-component system in which the
models for α12 in the three 1–2-k subsystems are depicted schematically in Fig. 14.8.
From Eqs. (14.45), (14.46),
ξ12 ¼ X1
ξ21 ¼ X2 + X4
σ 12 ¼ σ 21 ¼ 1 X3
(14.48)
Hence, from Eq. (14.47),
α12 ¼
X
i0
i
X1 X2 X4 i
L 12
1 X3
(14.49)
It can be verified that Eq. (14.49) reduces to Eqs. (14.32), (14.34),
and (14.5) in the 1-2-3, 1-2-4, and 1-2-5 subsystems, respectively.
It can also be seen that α12 remains constant as X2 is replaced by X4
keeping X1, X3, and X5 constant in the five-component system just
as in the 1-2-4 subsystem.
1
2
Kohler
1
1
3
2
Toop
4
2
Muggianu
5
Fig. 14.8 Schematic of models used for a12 in the 1-2-3, 1-2-4, and 1-2-5 ternary subsystems of a hypothetical
five-component system.
217
218
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
These equations are incorporated into the FactSage thermodynamic software (see FactSage). One needs only to specify the geometric model (Kohler, Toop, or Muggianu) used for αij in each
ternary subsystem. This gives complete flexibility in the choice
of geometric models. In any multicomponent system, any one
of the 64 possible combinations (as in Fig. 14.4) can be used for
any ternary subsystem.
Ternary Terms
For purposes of extending the ternary terms of Eqs. (14.38)–
(14.43) into multicomponent systems, the following equations
have been proposed (Pelton, 2001):
(i) If α12 in the 1-2-3 system is given by a Kohler-type or a
Muggianu-type approximation, then the following ternary
terms may be included:
"
ik
X1 X2
L12ð3Þ
ξ12 ξ21
σ 12
#
i
X3 ð1 ξ12 ξ21 Þ
k1
(14.50)
or
2
0
1
ξ12 ξ21 i
1
+
6
B
C
σ 12
6 ijk
B
C
X1 X2 6Q
A
4 12ð3Þ @
2
3
1
ξ21 ξ12 j
7
C
σ 12
7
C X ð1 ξ
ξ21 Þk1 7
3
12
A
5
2
0
B1 +
B
@
(14.51)
(ii) If α12 in the 1-2-3 system is given by a Toop-type approximation along lines of constant X1, then the following ternary
terms may be included:
"
X1 X2
ik
L12ð3Þ
ξ12 ξ21
σ 12
i X3
ξ21
#
X2 k1
1 ξ21
(14.52)
or
2
1
0
ξ12 ξ21 i
1
+
6
C
σ 12
6 ijk B
C
B
X1 X2 6Q
A
4 12ð3Þ @
2
3
1
0
ξ21 ξ12 j
k1 7
1
+
C
B
X3
X
σ 12
7
C
B
7
1 2
A ξ
@
5
2
ξ21
21
(14.53)
Eqs. (14.50)–(14.53) reduce to Eqs. (14.38)–(14.43) in the 1-2-3 ternary system.
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
One might consider proposing a factor Xk3 rather than X3
(1 ξ12 ξ21)k1 in Eqs. (14.50), (14.51). Consider, however, a multicomponent system in which the effects of components 3, 4, and
5 upon α12 are identical. That is, ikL12(3) ¼ ikL12(4) ¼ ikL12(5) (or, alterijk
ijk
natively, Qijk
12(3) ¼ Q12(4) ¼ Q12(5)). In the multicomponent system,
the effects should to a first approximation be additive, and so
the sum of the three ternary terms should clearly contain the factor (X3 + X4 + X5)k not (Xk3 + Xk4 + Xk5). This will be the case only if the
ternary terms are defined as in Eqs. (14.50), (14.51). A similar argument justifies the form of Eqs. (14.52), (14.53) and the form of
Eq. (14.44) proposed by Hillert (2008).
These generalizations permit several geometric models to be
combined within one multicomponent database. For example,
very many ternary systems have been evaluated/optimized by
many researchers with the Muggianu model. If future optimizations of other ternary systems are performed with the Kohler or
Toop models, then all systems can be immediately combined
within one large multicomponent database. No reoptimization
is required. That is, the fact that certain subsystems have already
been optimized with one model does preclude other models being
used for other subsystems.
14.8.5
Corresponding Expressions for Partial
Properties
The equations derived in Section 14.8 are for the integral excess
Gibbs energy. Explicit equations for the corresponding partial
excess Gibbs energies of the components can be easily calculated
therefrom. These calculations may be facilitated by the following
particularly simple equation (Sundman and Agren, 1981):
C
X
Xj ∂g E =∂Xj
giE ¼ g E + ∂g E =∂Xi (14.54)
j¼1
where gE is expressed as a function of X1, X2, X3, …, XC in a
C-component system and the partial differentials are calculated
as if all C mole fractions were independent variables (even though,
in reality, only (C 1) are independent).
14.9
Liquid Solutions: Coordination
Equivalent Fractions
The excess Gibbs energy of liquid Mg-Sn solutions at 800°C
( Jung et al., 2007) is shown in Fig. 14.9. The minimum is not far
from the composition XSn ¼ 1/3, whereas a regular solution model
219
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
0
−2
−4
gE (kJ mol−1)
220
−6
−8
−10
−12
−14
0
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction XSn
0.7
0.8
0.9
1
Fig. 14.9 Molar excess Gibbs energy of liquid Mg-Sn solutions at 800°C from Jung
et al. (2007).
with one constant parameter gives an extremum at the equimolar
composition. As discussed in Section 14.3.3, it is difficult to represent a curve such as that in Fig. 14.9 by a polynomial expansion as
in Eqs. (14.5) or (14.16) without the use of many terms because
adding terms in order to shift the composition of the minimum
also distorts the shape of the curve.
In Section 4.9, regular solution theory was developed under the
assumption of the additivity of pair-bond energies. In this derivation, it was assumed that the coordination number Z of the solution is independent of composition. However, this is not
necessarily true in the case of liquid solutions. Let Z1 and Z2 be
the nearest-neighbor coordination numbers of species (atoms
or molecules) 1 and 2, respectively. We define coordination equivalent fractions Y1 and Y2 as
Y1 ¼ Z1 X1 =ðZ1 X1 + Z2 X2 Þ
(14.55)
Y2 ¼ Z2 X2 =ðZ1 X1 + Z2 X2 Þ
(14.56)
The total number of bonds in 1 mol of solution is equal to No
(Z1X1 + Z2X2)/2 where N° is Avogadro’s number. The contribution
to the molar enthalpy of mixing resulting from the change in energy
of the first-nearest-neighbor bonds upon mixing is then equal to
ΔhFNN ¼ ðN o ðZ1 X1 + Z2 X2 Þ=2Þ Y12 ε11 + Y22 ε22 + 2Y1 Y2 ε12
(14.57)
ðN o Z1 =2ÞX1 ε11 ðN o Z2 =2ÞX2 ε22
¼ ðZ1 X1 + Z2 X2 Þ½ε12 ðε11 + ε22 Þ=2Y1 Y2
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
where εij is defined as in Eq. (4.47). Assuming a similar equation
for sE(nonconfig) as in Eq. (4.49) and assuming that the ratios
Z2/Z1 for second-nearest-neighbor, third-nearest-neighbor, etc.
bonds are the same as for first-nearest-neighbors, the resultant
expression for gE is
g E ¼ ðZ1 X1 + Z2 X2 ÞY1 Y2 α012
(14.58)
When Z1 ¼ Z2 ¼ Z, Eq. (14.58) reduces to gE ¼ ZX1X2α012, which is
identical to Eq. (14.4) except that the constant factor Z is not
included in the function α012 as it was in the function α12 in
Eq. (14.4).
If α012 is independent of composition, then a plot of
E
g /(Z1X1 + Z2X2) versus Y2 will be parabolic with an extremum at
Y1 ¼ Y2 ¼ 0.5, that is, at the composition where X2 ¼ Z1/(Z1 + Z2).
In liquid Mg-Sn solutions, if we set (ZSn/ZMg) ¼ 2.0, the resulting plot of α0MgSnYMgYSn ¼ gE/(XMg + 2XSn) versus YSn shown in
Fig. 14.10 has a minimum very near YMg ¼ YSn ¼ 0.5. (Since the
composition of the minimum depends only on the ratio ZSn/
ZMg, we have arbitrarily set ZMg ¼ 1 and ZMg ¼ 2.)
The function α012 can be expressed as a power series as in
Eqs. (14.5) or (14.16) but with the mole fractions Xi replaced by Yi:
X
i
L12 ðY1 Y2 Þi
(14.59)
α012 ¼
i0
YMgYSn a′MgSn = gE/(XMg + 2XSn) (kJ mol−1)
0
−2
−4
−6
−8
−10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Coordination equivalent fraction YSn
Fig. 14.10 Fig. 14.9 replotted as gE/(XMg + 2XSn) ¼ a0MgSnYMgYSn versus coordination
equivalent fraction of Sn.
221
222
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
or
α012 ¼
XX
ij
j
Q12 Y1i Y2
(14.60)
i0 j0
Only a few empirical coefficients are now required to represent
the excess Gibbs energy function.
14.9.1
Extension to Multicomponent Solutions
In a ternary liquid solution, Eq. (14.31) becomes
g E =ðZ1 X1 + Z2 X2 + Z3 X3 Þ ¼ Y1 Y2 α012 + Y2 Y3 α023 + Y3 Y1 α031
+ ðternary termsÞ
where Zi are the coordination numbers and
X
Y i ¼ Z i X i = Zi X i
(14.61)
(14.62)
All equations of Sections 14.8, 14.8.1–14.8.5 now apply with all Xi
replaced by Yi and all αij replaced by αij0 .
14.10 Wagner’s Interaction Parameter
Formalism and the Unified Interaction
Parameter Formalism
In 1962, Wagner (1962) proposed the following well-known
first-order interaction parameter formalism for dilute solutions
of components 1-2-3- … -C where component 1 is the solvent:
ln γ i ¼ ln γ oi + εi2 X2 + εi3 X3 + … + εiC XC ði 2Þ
γ oi
(14.63)
where is the limiting Henrian activity coefficient of component
i at infinite dilution when X1 ¼ 1.
The formalism has been used extensively, particularly by
metallurgists for solid and liquid solutions of alloying elements
in Fe and other metals, due to its simplicity and straightforward
interpretation whereby the first-order interaction parameter εij
shows the effect of component j upon the activity coefficient
of component i. Tables of values of εij and γ oi for many elements
in molten and solid Fe and in other metal solvents can be
found in the literature (Sigworth and Elliott, 1974a, 1974b).
Second-order terms ρiiX2i , ρjiX2j , ρi,j
i XiXj are also sometimes
included.
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
From the Gibbs-Duhem equation (Eq. 2.82) applied at X1 ¼ 1, it
can be shown that
εij ¼ εji
(14.64)
and also that
ln γ 1 ¼ C X
C
1X
εij Xi Xj
2 i¼2 j¼2
(14.65)
Wagner’s formalism has sometimes been misapplied. Confusion
has arisen because Eqs. (14.63), (14.65) only obey the GibbsDuhem equation in the limit of infinite dilution and because
the formalism does not obey the following necessary thermodynamic relationship except at infinite dilution:
(14.66)
∂ ln γ i =∂nj ¼ ∂ ln γ j =∂ni
This problem has been resolved (Pelton and Bale, 1986; Bale and
Pelton, 1990; Pelton, 1997) by showing that Wagner’s formalism is
a limiting case of Darken’s quadratic formalism (see Section 14.6).
For a ternary system 1-2-3 with component 1 as solvent, Darken’s
formalism can be written as (Pelton, 1997)
g E =RT ¼ ðα12 X1 X2 + α23 X2 X3 + α31 X3 X1 Þ + ðC2 X2 + C3 X3 Þ
(14.67)
with partial properties given by
ln γ1 ¼ α12 X22 + α13 X32 + ðα12 + α13 α23 ÞX2 X3
(14.68)
ln γ2 ¼ ðα12 + C2 Þ 2α12 X2 ðα12 + α31 α23 ÞX3
+ α12 X22 + α31 X32 + ðα12 + α31 α23 ÞX2 X3
(14.69)
ln γ3 ¼ ðα31 + C3 Þ 2α31 X3 ðα12 + α31 α23 ÞX2
+ α12 X22 + α31 X32 + ðα12 + α31 α23 ÞX2 X3
(14.70)
where the constants C2 and C3 are related to changes in standard
states as explained in Section 14.6.
By comparing Eqs. (14.68)–(14.70) with Eqs. (14.63), (14.65), it
can be seen that Wagner’s formalism is identical to Darken’s formalism if the quadratic terms (terms in X22, X23, and X2X3) are
neglected. That is,
ln γoi ¼ ðα1i + Ci Þ
εij ¼ 2α1i
i ¼ 2, 3
i ¼ 2, 3
ε23 ¼ ðα12 + α31 α23 Þ
(14.71)
(14.72)
(14.73)
223
224
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
By substituting Eqs. (14.71)–(14.73) into Eqs. (14.68)–(14.70), the
following equations of the unified interaction parameter formalism are obtained:
ε
ε33 2
22 2
X2 +
X3 + ε23 X2 X3
(14.74)
ln γ 1 ¼ 2
2
ε
ε33 2
22 2
X2 +
X3 + ε23 X2 X3
ln γ 2 ¼ ln γ o2 + ε22 X2 + ε23 X3 2
2
(14.75)
¼ ln γ 02 + ε22 X2 + ε23 X3 + ð ln γ 1 Þ
ln γ 3 ¼ ln γ 03 + ε32 X2 + ε33 X3 + ð ln γ 1 Þ
(14.76)
where lnγ 1 in Eqs. (14.75), (14.76) is given by Eq. (14.74) and
ε23 ¼ ε32.
Eqs. (14.74)–(14.76) are identical to Wagner’s formalism in the
limit of infinite dilution (at X1 ¼ 1) where the quadratic terms
become vanishingly small relative to the linear terms. Existing
tabulated values of εij and γ 0i can be used directly in
Eqs. (14.74)–(14.76), and there are no additional parameters.
Maintaining the quadratic terms in Eqs. (14.74)–(14.76) ensures
that the equations are consistent with the Gibbs-Duhem equation
and with Eq. (14.66) at all compositions.
Eqs. (14.74)–(14.76) can clearly be extended to systems of more
than three components simply by adding additional terms as discussed by Bale and Pelton (1990) and Pelton (1997). The same
authors also extend the unified formalism to second-order and
higher-order terms.
Wagner’s formalism and the unified interaction parameter formalism are clearly based on regular solution theory, that is, upon
the assumption of random mixing of species. In solutions where
the distribution is far from random, the formalisms do not apply.
A case in point is solutions of a highly reactive metal M, such as Al
and Ca, along with oxygen in liquid Fe. In such solutions, the
M and O atoms are not distributed randomly, but have a very
strong tendency to form associated MO pairs. This topic will be
discussed in Section 18.4.
14.10.1
Approximation for e23
If ε22 and ε33 have been determined from measurements in the
1–2 and 1–3 binary systems and one wishes to estimate the properties of the ternary solution, then, as a first approximation, one
might assume that (2–3) interactions are negligible. Because ε23
is an “interaction parameter,” one might then propose setting
ε23 ¼ 0. However, it is clear from Eq. (14.67) that it is α23 that should
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
be set to zero. From Eqs. (14.72), (14.73), then, it follows that a better first approximation to ε23 is
ε23 ¼ ðε22 + ε33 Þ=2
14.10.2
(14.77)
Interaction Parameter Formalism With
Molar Ratios
Another means of rendering the interaction parameter formalism
consistent with the Gibbs-Duhem equation and with Eq. (14.66) is
to replace the mole fractions Xi in the expansions by the molar
ratios yi ¼ Xi/X1 (where 1 ¼ solvent), as proposed by Schuhmann
(1985):
ln γ i ¼ ln γ 0i +
C
X
εij yj ði 2Þ
(14.78)
j¼2
ln γ 1 ¼ 1=2
C
X
εij yi yj
(14.79)
i, j¼2
Eqs. (14.78), (14.74) are very similar to Eqs. (14.74)–(14.76) with
the important exception that the (lnγ 1) terms of Eqs. (14.75),
(14.76) do not appear in Eq. (14.78). It can easily be shown that
Eqs. (14.78), (14.79) satisfy Eq. (14.66) and the Gibbs-Duhem
equation. Furthermore, the simple relationship of Eq. (14.64)
can also be shown to apply.
Similar expansions are used in the well-known Pitzer equation
(Harvie et al., 1984) for aqueous solutions. Here, the composition
variables are the molalities mi ¼ (yi/y1)(1000/M1), where M1 is the
molecular weight of the solvent.
An advantage of the interaction parameter formalism with
molar ratios is that Eqs. (14.78), (14.79) can be written with weight
ratios (mi/m1) in place of the molar ratios yi, because weight ratios
vary directly as molar ratios: (mi/m1) ¼ yi(Mi/M1), where (Mi/M1)
is the ratio of molecular weights. It is only necessary to multiply
each interaction parameter by a constant. This conversion is
not so simple when mole fractions are used because weight fractions do not vary directly as mole fractions.
The disadvantage of the molar ratio representation is that it is
not consistent with the quadratic formalism and so, for simple
substitutional solutions at least, it is less likely to provide as
good a fit to experimental data and to extrapolate as well as
Eqs. (14.74)–(14.76).
225
226
Chapter 14 SINGLE-LATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
14.11 Thermal Vacancies
The introduction of vacancies on a lattice increases the
entropy. Consequently, even a pure substance at equilibrium will
contain vacancies.
Consider 1 mol of a metal A containing nVa moles of vacant lattice sites. If the concentration of vacancies is small, we may
assume that the interaction among vacancies is negligible and
that the system can thus be treated as an ideal solution of
1.0 mol of atoms A and nVa mol of vacancies on (1 + nVa) lattice
sites with a Gibbs energy:
∗
nVa
1
o
(14.80)
+ ln
g ¼ gA + nVa gVa + RT nVa ln
1 + nVa
1 + nVa
where g∗A is the Gibbs energy of hypothetical pure A containing no
vacancies and the parameter goVa is the molar nonconfigurational
Gibbs energy of forming vacancies at infinite dilution in pure A,
where goVa > 0.
The equilibrium number of vacancies is that which
minimizes g:
o
+ RT ln
dg=dnVa ¼ gVa
nVa
¼0
1 + nVa
(14.81)
The number of vacancies per mole of A at equilibrium is thus
given by
o
=RT
(14.82)
nVa =ð1 + nVa Þ ¼ exp gVa
Hence, the vacancy concentration varies exponentially with temperature. The Gibbs energy of pure A at equilibrium, goA, is then
given by substituting nVa from Eq. (14.82) into Eq. (14.80).
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Bale, C.W., Pelton, A.D., 1990. Metall. Trans. A. 21A, 1997–2002.
Bell, M.C., Flengas, S.N., 1964. J. Electrochem. Soc. 111, 569–575.
Blaha, P., Schwarz, K., Sorantin, P., Trickey, S.B., 1990. Comput. Phys. Commun.
59, 399–415.
Blander, M., 1964. Molten Salt Chemistry. InterScience, NY.
Brookes, H.C., Coppens, H.J., Pelton, A.D., 1978. Trans. Faraday Soc. 74, 2193–2199.
Chartrand, P., Pelton, A.D., 2000. J. Phase Equilib. 21, 141–147.
Darken, L.S., 1967a. Trans. Metall. Soc. AIME 239, 80–89.
Darken, L.S., 1967b. Trans. Metall. Soc. AIME 239, 90–96.
Dinsdale, A.T., 1991. Calphad J. 15, 317–425.
€rner, P., Henig, E.T., Krieg, H., Lukas, H.L., Petzow, G., 1980. Calphad J.
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Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C.,
Ceresoli, D., Chiarotti Guido, L., Cococcioni, M., Dabo, I., Dal Corso, A., de
Gironcoli, S., Fabris, S., Fratesi, G., Gebauer, R., Gerstmann, U.,
Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N.,
Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C.,
Scandolo, S., Sclauzero, G., Seitsonen Ari, P., Smogunov, A., Umari, P.,
Wentzcovitch Renata, M., 2009. J. Phys. Condens. Matter. 21(39).
Gonze, X., Beuken, J.M., Caracas, R., Detraux, F., Fuchs, M., Rignanese, G.M.,
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Harvie, C.E., Moeller, N., Weare, J.H., 1984. Geochim. Cosmochim. Acta
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Hillert, M., 1980. Calphad J. 4, 1–12.
Hillert, M., 2008. Phase Equilibria, Phase Diagrams and Phase Transformations –
Their Thermodynamic Basis, 2nd ed. Cambridge University Press, Cambridge.
Hoffman, H.W., 1958. ORNL-CF-58-2. Oak Ridge National Lab. Report, Oak Ridge,
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Jung, I.-H., Kang, D.-H., Park, W.-J., Kim, N.J., Ahn, S., 2007. Calphad J. 31, 192–200.
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Kohler, F., 1960. Monatsh. Chem. 91, 738–740.
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227
15
MULTIPLE-SUBLATTICE
RANDOM-MIXING (BRAGGWILLIAMS—BW) MODELS
CHAPTER OUTLINE
15.1 Case of a Two-Sublattice (A,B)(X,Y) Solution 231
15.1.1 Reciprocal Excess Terms 235
15.2 Activities of the End-Members 239
15.3 The Compound Energy Formalism (CEF) 240
15.3.1 Interstitial Solutions 240
15.3.2 Nonstoichiometric Compounds 241
15.3.3 Ceramic Solutions 246
15.4 Asymmetric Molten Ionic Solutions: Temkin Model
15.4.1 Solid vs Liquid Asymmetric Ionic Solutions 251
References 252
List of Website 252
249
In many compounds and solutions, one can distinguish two or
more sublattices. In a simple solid ionic salt solution, for example,
cations are found on one sublattice and anions on another. In
steel, the metal atoms reside on one sublattice, while an interstitial sublattice is occupied by C atoms and vacancies. In ceramic
oxides, one often finds more than two sublattices; in a spinel solution, for example, some cations occupy a tetrahedrally coordinated sublattice and others an octahedrally coordinated
sublattice, while oxygen ions reside on a third.
When one can distinguish two or more sublattices, the compound or solution is said to exhibit long-range ordering (LRO).
In principle, of course, a liquid cannot exhibit long-range
ordering. However, in ionic liquids, it is generally the case that cations are surrounded almost exclusively by anions as nearest neighbors and vice versa. Therefore, for practical purposes of modeling,
it is usually much simpler to treat such solutions as formally
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00015-4
# 2019 Elsevier Inc. All rights reserved.
229
230
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
consisting of cationic and anionic quasi-sublattices. In fact, the
sublattice models discussed in this chapter were first developed
in the 1960s (see, for instance, Blander, 1964) for applications to
molten salt solutions and were only later applied to solid solutions.
In this chapter, multiple sublattice models will be developed
under the assumption of random mixing of the species on each
sublattice (Bragg-Williams (BW) approximation). A polynomial
first approximation to account for first-nearest-neighbor shortrange ordering will also be developed, but a more complete treatment of short-range ordering will be left until later chapters. It will
be assumed throughout most of the chapter that the ratio of the
number of sites on the different sublattices is constant. This is
always true in solid solutions but is not necessarily the case in liquid solutions such as, for example, liquid NaCl-CaCl2 solutions.
The case of variable site ratios for liquid ionic solutions will be
discussed in Section 15.4.
As an example of a solution with multiple sublattices, consider
the following three-sublattice model: (A,B,C, …)3(X,Y,Z, …)2(P,Q,
R, …). In this standard notation, the species (atoms, ions, molecules, vacancies) A,B,C, … occupy the first sublattice, while
species X,Y,Z, … and P,Q,R, … occupy the second and third sublattices, respectively. Note that in general, the same species may
occupy more than one sublattice. For instance, A and P could
be the same species.
In this example, the number of sites on the sublattices is in the
ratio 3:2:1. For the sake of simplicity, we shall develop model equations for this specific example, but the extension to site ratios other
than 3:2:1 or to more than three sublattices should be obvious.
We define Xi0 as the site fraction of species i on the first sublattice:
Xi0 ¼ ni =ðnA + nB + nC + …Þ
(15.1)
where
ni is the number of moles of i on the first sublattice where
P
( Xi0 ¼ 1). Site fractions Xi00 and Xi000 of species on the second and
third sublattices are defined similarly. (Note that in the current literature, the symbol yi is often used for site fractions instead of Xi.)
The following equation is written for the molar Gibbs energy of
the solution:
h
i
g ¼ XA0 XX00 XP000 gAo3 X2 P + XA0 XY00 XP000 gAo3 Y 2 P + … + XC0 XZ00 XR000 qoC3 Z2 R + …
"
#
X
X
X
+ RT 3
Xi0 ln Xi0 + 2
Xj00 ln Xj00 +
Xk000 ln Xk000 +
h
i
j
k
E
E
E
E
XX00 XP000 gABC…:X:P
+ XY00 XP000 gABC…:Y:P
+ … + XA0 XP000 gA:XYZ…:P
+ … + XC0 XX00 gC:X:PQR…
+…
i
+ ½Reciprocal excess terms
(15.2)
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
Inside the first set of square brackets in Eq. (15.2), there is one term
for each end-member of the solution. An end-member is defined as
a “compound” in which each sublattice is occupied by only one
species. For example, in the end-member A3X2Q, the three sublattices are occupied solely by A, X, and Q species, respectively. If, for
instance, there are three species on each of the three sublattices,
that is, (A,B,C)3(X,Y,Z)2(P,Q,R), then there are exactly 27 endmembers. The goi are the Gibbs energies of the end-members.
Some end-members may be real stable compounds, but others
may be metastable, unstable, or even completely fictitious, in
which case the goi are adjustable model parameters. The factors
Xi0 Xj00 Xk000 are the probabilities that a set of randomly selected sites,
one on each
contain i, j, and k species, respectively.
PP
P sublattice,
(Note that i j k Xi0 Xj00 Xk000 ¼ 1.) The first set of square brackets
in Eq. (15.2) is called the mechanical mixing energy.
Inside the second set of square brackets in Eq. (15.2) are the
configurational entropy terms for random mixing of the species
on each sublattice.
The terms inside the third set of square brackets in Eq. (15.2)
are excess Gibbs energy terms resulting from interactions among
species on the same sublattice. For example, the factor gEABC…:X:P is
the excess Gibbs energy arising from interactions among the
species A,B,C, … on the first sublattice when the second and
third sublattices are occupied only by X and P species, respectively. This factor is weighted in Eq. (15.2) by XX00 XP000 which is
the probability of finding X and P species on the second and third
sublattices, respectively. If, for instance, there are three species on
each of three sublattices, that is, (A,B,C)3(X,Y,Z)2(P,Q,R), then
there are exactly 27 such excess Gibbs energy terms. The factor
E
is obtained from an evaluation/optimization of the
gABC…:X:P
(A,B,C, …)3(X)2(P) subsystem and is a function of XA0 , XB0 , XC0 ,
… as described in Chapter 14 where, in this subsystem, these site
fractions are numerically equal to the mole fractions of the endmembers A3X2P, B3X2P, C3X2P, …, respectively (see Section 14.4.)
15.1
Case of a Two-Sublattice (A,B)(X,Y)
Solution
In a simple two-sublattice solution (A,B)(X,Y) with constant
site ratio 1:1, Eq. (15.2) becomes
o
o
o
o
+ XA0 XY00 gAY
+ XB0 XX00 gBX
+ XB0 XY00 gBY
g ¼ XA0 XX00 gAX
+ RT XA0 ln XA0 + XB0 ln XB0 + XX00 ln XX00 + XY00 ln XY00
+ XX00 XA0 XB0 αAB:X + XY00 XA0 XB0 αAB:Y + XA0 XX00 XY00 αA:XY
+ XB0 XX00 XY00 αB:XY + ½Reciprocal excess terms
(15.3)
231
232
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
The factors XA0 XB0 αAB:X, etc. are the excess Gibbs energies in the (A,
B)(X), (A,B)(Y), (A)(X,Y), and (B)(X,Y) binary subsystems. For
example, in the AX-BX subsystem where XA ¼ XAX and XB ¼ XBX
(see Eq. 14.4),
E
XA0 XB0 αAB:X ¼ XAX XBX αAXBX ¼ gAXBX
(15.4)
where the α-function is given by a polynomial expansion as in
Eq. (14.5):
X
i
i
L AB:X X 0A XB0
(15.5)
αAB:X ¼
i
An example of such a ternary reciprocal solution (a solution with
two species on each of two sublattices) is the liquid (Li,K)(F,Br)
solution with Li+ and K+ ions on the cation sublattice and F
and Br ions on the anion sublattice. The reported phase diagram
of the LiF-NaF-LiBr-KBr system is shown in Fig. 15.1A. (See
Section 7.10 for a description of the coordinate system of such diagrams.) A liquid-liquid “island” miscibility gap is observed closely
centered along the LiF-KBr diagonal of the composition square
with two-phase liquid-liquid tie-lines approximately parallel to
this diagonal. The existence of the gap indicates positive deviations from ideal mixing behavior along the diagonal. As a second
example, the reported phase diagram of the NaF-KF-NaCl-KCl
system is shown in Fig. 15.2A. Although no liquid-liquid miscibility gap is observed, the liquidus isotherms along the NaF-KCl
diagonal are widely spaced. (That is, in the NaF-KCl pseudobinary
system, the liquidus is flat as in Fig. 5.8D.) Again, this is indicative
of positive deviations from ideality along the diagonal. Such positive deviations are a feature of all ternary reciprocal solutions and
may be explained as follows.
Consider only the first (“mechanical mixing energy”) term in
square brackets in Eq. (15.3). Along the diagonal line AX-BY,
XB0 ¼ XY00 ¼ (1 XA0 ) ¼ (1 XX00 ) ¼ XBY ¼ (1 XAX) where XAX and
XBY are the mole fractions along the diagonal line. Hence, along
this diagonal, the first term in Eq. (15.3) is equal to
o
o
XAX gAX
XAX XBY ΔG exchange
+ XBY gBY
(15.6)
where ΔGexchange is the Gibbs energy change for the following
exchange reaction among the pure end-member salts:
o
o
o
o
AY + BX ¼ AX + BY; ΔGexchange ¼ gBY
+ gAX
gBX
gAY
(15.7)
Therefore, when ΔGexchange < 0, AX and BY are called the stable
pair, the AX-BY diagonal is called the stable diagonal, the second
term in Eq. (15.6) is positive, and positive deviations will be
LiF
KF
e
80
0
70
0
0
60
0
50 00
5
0
70
0
60
0
84
0
87
0
90
0
93 3
95
0
80
465
E2
e
70
0
60
0
e
50
0
0
40
0
50
(A)
E
320
e
LiBr
KBr
LiF
KF
e
80
0
0
60
E
60
0
70
0
70
0
0
80
. .
. .
. .
70
0
.
0
.
. .
.
.
.
60
e
.
.
.
50
0
0
70
.
.
0
50
400
E
(B)
LiBr
KBr
Mole %
KF
e
LiF
0
80
0
70
800
E
0
0
70
60
0
60
e
70
0
70
0
0
60
e
0
50
50
0
(C)
LiBr
400
E
e
Mole %
KBr
Fig. 15.1 Liquidus projection of the (Li,K)(F,Br) reciprocal ternary system: (A) As reported by Margheritis et al. (1979).
(B) Calculated from Eqs. (15.12), (15.13) with De ¼DGexchange. (C) Calculated as in (B) but including the correction for the
effect of FNN SRO on the binary gE terms. Reprinted with permission from Dessureault, Y., Pelton, A. D., 1991. Contribution to the
quasichemical model of reciprocal molten salt solutions. J. Chim. Phys. 88, 1811–1830.
234
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
Fig. 15.2 Liquidus projection of the (Na,K)(F,Cl) reciprocal ternary system: (A) As reported by Berezina et al. (1963).
(B) Calculated from Eqs. (15.12), (15.13) with De ¼ DGexchange. Reprinted with permission from Dessureault, Y., Pelton, A. D.,
1991. Contribution to the quasichemical model of reciprocal molten salt solutions. J. Chim. Phys. 88, 1811–1830.
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
observed along this line. By a similar derivation, negative deviations will be observed along the AY-BX diagonal.
In
liquid
(Li,K)(F,Br)
solutions
(where
BY ¼ KBr),
ΔGexchange ¼ 75.2 kJ mol1 at 700°C (FactSage). Hence, LiF
and KBr are the stable pair. Mixing pure LiF and KBr is thermodynamically unfavorable because it involves the formation of Li+Br and K+-F first-nearest-neighbor pairs at the expense of the
energetically preferable Li+-F and K+-Br pairs. Since ΔGexchange
is sufficiently negative, an island miscibility gap centered on the
stable diagonal results as in Fig. 15.1.
In liquid (Na,K)(F,Cl) solutions, (where BY ¼ KCl),
ΔGexchange ¼ 23.5 kJ mol1 at 700°C (FactSage). Hence, NaF
and KCl are the stable pair, and positive deviations are found along
the stable diagonal. However, since ΔGexchange is less negative, no
miscibility gap is observed. It must be stressed that the occurrence
of positive deviations along the stable diagonal arises mainly
because of the ΔGexchange term and will be observed even if all
the excess Gibbs energy terms in Eq. (15.3) are equal to zero.
With the Gibbs energies of the pure end-member salts and the
gE terms for the binary subsystems taken from the literature
(FactSage; Dessureault and Pelton, 1991), the Gibbs energy of
the liquid (Li,K)(F,Br) solution was calculated from Eq. (15.3) with
the reciprocal excess terms set to zero. An island miscibility gap
centered along the LiF-KBr diagonal was indeed predicted as
expected, but the calculated consolute temperature was 2100°C,
which is much higher than the observed consolute temperature
of 953°C. That is, the positive deviations are overestimated. This
is a general result for all ternary reciprocal solutions that was first
explained by Blander and Braunstein (1960) as follows.
15.1.1
Reciprocal Excess Terms
As explained in the preceding section, positive deviations
along the stable AX-BY diagonal arise because the mixing of AX
and BY results in the formation of energetically unfavorable
(A-Y) and (B-X) first-nearest-neighbor (FNN) pairs. In a random
mixture of the species on their sublattices, the probabilities of
(A-Y) and (B-X) FNN pairs are XA0 XY00 and XB0 XX00 , respectively.
However, the Gibbs energy of the solution can be decreased if
the number of unfavorable pairs is decreased through the formation of first-nearest-neighbor short-range ordering (SRO).
Let the fractions of unfavorable (A-Y) and (B-X) pairs be
decreased to (XA0 XY00 y) and (XB0 XX00 y), while the fractions of
favorable (A-X) and (B-Y) pairs are increased accordingly to
(XA0 XX00 + y) and (XB0 XY00 + y) where y indicates the degree of SRO
235
236
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
and y > 0. Let the energies of the FNN (i-j) bonds be denoted by εij.
The Gibbs energy of the solution is then modeled as
0 00
0 00 g ¼ XA XX + y εAX + XA XY y εAY + XB0 XX00 y εBX + XB0 XY00 + y εBY
o
o
o
o
εAX + XA0 XY00 gAY
εAY + XB0 XX00 gBX
εBX + X 0B XY00 gBY
εBY
+ X 0A XX00 gAX
+ ZRT ½ XA0 XX00 + y ln XA0 XX00 + y + XA0 XY00 y ln XA0 XY00 y
(15.8)
+ XB0 XX00 y ln X 0B XX00 y + XB0 XY00 + y ln XB0 XY00 + y + ZRT ½ XA0 XX00 + y ln XA0 XX00 + XA0 XY00 y ln XA0 XY00 + XB0 XX00 y lnX 0B XX00
+ XB0 XY00 + y lnX 0B XY00 + RT XA0 ln XA0 + XB0 ln XB0 + XX00 ln XX00 + XY00 ln XY00 + g E
The first term in square brackets is the energy of the FNN pairbonds. The second term is the remaining part of the mechanical
mixing term due to second-nearest-neighbor, third-nearestneighbor, and other pair-bonds. (When y ¼ 0, the first two terms
in square brackets sum to the first bracketed term in Eq. (15.3).)
In 1 mol of solution (i.e., one mole of species on each sublattice), let there by Z FNN bonds. That is, Z is the FNN coordination
number. The third square-bracketed term in Eq. (15.8) is a configurational entropy term given by randomly distributing the (A-X),
(A-Y), (B-X), and (B-Y) FNN bonds over the Z “bond sites.” However, this term overestimates the actual entropy of the solution
since, for example, if one bond emanating from a central
A species is an (A-X) bond, then all other bonds emanating from
the same site must be either (A-X) or (A-Y) bonds. To approximately account for this constraint, the fourth square-bracketed
term in Eq. (15.8) is a correction term designed so that the two
entropic terms together reduce to the ideal configurational
entropy term of Eq. (15.3) when there is no SRO (when y ¼ 0).
(Although this correction is approximate in two and three dimensions, it can be shown to be exact for a one-dimensional lattice.)
Calculating the equilibrium amount of SRO by setting
dg/dy ¼ 0 in Eq. (15.8) and assuming that gE is not a function of
y, we obtain the following equation:
1 + y=XA0 XX00 1 + y=XB0 XY00
¼ exp ðΔε=ZRT Þ
(15.9)
1 y=XA0 XY00 1 y=XB0 XX00
where
Δε ¼ εAX + εBY εAY εBX
(15.10)
Taking the logarithm of both sides of Eq. (15.9) and approximating
the logarithmic terms as first-order Taylor expansions (i.e., setting
ln(1 + y/XAXX) y/XAXX) when y is small) gives the following
expansion for y:
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
y XA0 XB0 XX00 XY00 ðΔε=ZRT Þ
(15.11)
Substituting Eq. (15.11) into Eq. (15.8) and again approximating
the logarithmic terms as Taylor expansions then gives an
expression for the Gibbs energy of the solution identical to
Eq. (15.3) with
½Reciprocal excess term ¼ XA0 XB0 XX00 XY00 LAB:XY
(15.12)
where
LAB:XY ¼ Δε2 =2ZRT
(15.13)
Note that LAB:XY is always negative. Hence, this reciprocal excess
term that results from SRO always decreases the positive deviations from ideality along the stable diagonal.
The phase diagram for the (Na,K)(F,Cl) system shown in
Fig. 15.2B was calculated (Dessureault and Pelton, 1991) from
Eq. (15.3) for the liquid solution, with data for the end-member
Gibbs energies and binary gE terms of the liquid solution taken
from the literature (FactSage; Dessureault and Pelton, 1991), with
Z ¼ 6 and with Eqs. (15.12) and (15.13) for the reciprocal excess
term, further assuming, as did Blander and Braunstein (1960), that
Δε ΔGexchange. Agreement with the reported diagram in
Fig. 15.2A is excellent. Similar good agreement is observed in other
reciprocal ternary molten salt solutions when j ΔGexchange j is relatively small. Unfortunately, this is no longer true when j ΔGexchange j
is relatively large.
Fig. 15.1B shows the phase diagram of the (Li,K)(F,Br) system
calculated similarly, with Eqs. (15.12), (15.13) for the reciprocal
excess term and assuming that Δε ΔGexchange. The calculated
consolute temperature has indeed been reduced by the inclusion
of the reciprocal excess term, but in the central region of the composition square, it has been reduced far too much. In fact, it has
been decreased so much that the calculated miscibility gap has
been split in two. Closer to the edges of the composition square
where the factor XA0 XB0 XX00 XY00 in Eq. (15.12) is relatively small,
the agreement with the experimental phase diagram is better.
Similar results are seen (Dessureault and Pelton, 1991) in other
ternary reciprocal molten salt solutions when jΔGexchange j is large.
The failure of the model in the central composition region is due
partly to the use of the first-order Taylor expansion in evaluating y,
but as shown by Dessureault and Pelton, it is mainly due to the
effect of FNN SRO on the binary gE terms in Eq. (15.3) as follows.
Consider the term XA0 XB0 αAB:X in Eq. (15.3). This is the excess
Gibbs energy in the binary AX-BX system that is related in regular
solution theory (Section 4.9) to the energy change of the pair
237
238
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
exchange reaction among (A-A), (B-B), and (A-B) second-nearestneighbor pairs on the same sublattice:
ðA ðX Þ AÞ + ðB ðX Þ BÞ ¼ 2ðA ðX Þ BÞ
(15.14)
where the notation indicates that the other sublattice is occupied
solely by X species. Now, in Eq. (15.3), this term is multiplied by
XX00 because, in a randomly mixed solution, the probability of
an A-(X)-B grouping is XX00 XA0 XB0 . However, when j ΔGexchange j is
large, FNN SRO produces clusters of the energetically favorable
(A-X) and (B-Y) FNN pairs as has just been discussed, thereby
reducing the number of A-(X)-B groupings and increasing the
number of A-(X)-A and B-(X)-B groupings, the effect being greatest near the middle of the composition square. Hence, Eq. (15.3)
overestimates the importance of the binary excess Gibbs energy
terms, particularly in the central composition region. Empirically,
in molten salt systems, the binary excess terms are usually negative. Eq. (15.3) thus usually overestimates the stability of the ternary reciprocal solution, particularly in the central composition
region.
Dessureault and Pelton (1991) proposed that the term
ðX 0 X 00 + yÞ ðXB0 XX00 y Þ
XX00 XA0 XB0 αAB:X in Eq. (15.3) be replaced by XX00 A XX00
X 00
X
X
where y is the FNN SRO parameter from Eq. (15.11) and where
(XA0 XX00 + y)/XX00 and (XB0 XX00 y)/XX00 are the conditional probabilities that a cationic sublattice site is occupied by an A or a B cation,
respectively, given that a neighboring anionic sublattice site is
occupied by an X anion and similarly for the other binary excess
terms. With this replacement in Eq. (15.3) and solving for y exactly
by an iterative technique, Dessureault and Pelton calculated the
phase diagram of the (Li,K)(F,Br) system shown in Fig. 15.1C that
is in fair agreement with the reported diagram.
Unfortunately, this correction does not readily lend itself to
being approximated by explicit polynomial terms as was the case
for the reciprocal excess term in Eq. (15.12). Furthermore, it does
not take second-nearest-neighbor SRO into account, and extension to multicomponent solutions and to solutions with more
than two sublattices is complex. For systems with two sublattices,
it has been superseded by the modified quasichemical model in
the quadruplet approximation, which will be described in
Chapter 17. In the remainder of the present chapter, we shall
not consider this correction further. Clearly, however, as can be
seen from the foregoing example, the effect can be very important
when j ΔGexchange j and the binary α-functions in Eq. (15.3) are
large and when the site fractions of the species involved are all
of the same order (i.e., near the center of the composition square).
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
The value of the parameter LAB:XY in Eq. (15.12) can be optimized to reproduce available experimental data. However, in light
of the above arguments, even if no data are available, the parameter should not be set to zero but should be estimated by
Eq. (15.13) (and also, in general, by setting Δε ¼ ΔGexchange).
The preceding arguments can be extended to the more general
case of the first example in this chapter, an (A,B,C, …)3(X,Y,
Z, …)2(P,Q,R, …) solution, to show that the reciprocal excess
terms in Eq. (15.2) should take the approximate form
XP000 (XA0 XB0 XX00 XY00 )LAB:XY:P, etc., where, to a first approximation,
!
2
exchange
=2ZRT
(15.15)
LAB:XY:P exp ΔGAB:XY:P
where ΔGexchange
AB:XY:P is the Gibbs energy change of the following
exchange reaction among end-members:
A3 Y 2 P + B3 X 2 P ¼ A3 X 2 P + B3 Y 2 P
15.2
(15.16)
Activities of the End-Members
For an (A,B)(X,Y) solution, the extensive Gibbs energy is given by
G ¼ (nA + nB)g ¼ (nX + nY )g with g given by Eq. (15.3). The partial
molar Gibbs energy of the end-member AX is then calculated as
gAX ¼ ∂G/∂ nAX, noting that ∂ nAX ¼ ∂ nA ¼ ∂nX. This gives the
following:
o
E
gAX gAX
¼ RT ln aAX ¼ RT ln XA0 XX00 XB0 XY00 ΔGexchange + gAX
(15.17)
with ΔG
given by Eq. (15.7). When ΔG
< 0, AX is a
member of the stable pair, and the term ( XB0 XY00 ΔGexchange) contributes to positive deviations along the stable diagonal of the
reciprocal ternary system as has already been discussed. If
0
00
ΔGexchange ¼ 0 and gEAX ¼ 0, then aideal
AX ¼ XA XX , the product of the
site fractions.
In solutions rich in AX, it follows from Eq. (15.17) that
lim XAX !1 aAX ¼ XA0 XX00 , which may be considered to be the form of
Raoult’s law applicable to this two-sublattice solution.
Extension of these calculations to the more general example
of the (A,B,C, …)3(X,Y,Z, …)2(P,Q,R, …) solution given earlier gives
Raoult’s limiting law as
3 00 2 000 XB XP
lim aA3 X2 P ¼ XA0
(15.18)
exchange
exchange
XA3 X 2 P !1
and similarly for the other end-members.
239
240
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
15.3 The Compound Energy Formalism (CEF)
An end-member was defined earlier as a “compound” in which
each sublattice is fully occupied by only one species. In the example of the (Li,K)(F,Br) solution given in Section 15.1, all four endmembers are real stable compounds. However, in many solutions,
some end-members can be metastable, unstable, or even hypothetical compounds. In some cases, as will be shown in
Section 15.3.3, an end-member can even carry a net electric
charge. In such cases, the Gibbs energy of an end-member by
itself may have little or no direct physical significance. However,
combinations of the end-member Gibbs energies will have physical significance, and it is these combinations that are the actual
model parameters. Although models for any solution can be formulated explicitly in terms of the actual model parameters, this
results in a different model equation for each different type of
solution. On the other hand, as pointed out by Hillert and
coworkers (Hillert and Staffansson, 1970; Hillert et al., 1988;
Barry et al., 1992), formulating the Gibbs energy functions of solutions formally in terms of the end-member Gibbs energies as in
Eq. (15.2) provides a single formalism applicable to a wide variety
of types of solution.
In using this compound energy formalism (CEF), one must
always take care to distinguish between the formalism parameters
(the end-member Gibbs energies) and the actual physical model
parameters. In particular when estimating parameters, it is the
model parameters, never the formalism parameters, which
should be approximated.
15.3.1
Interstitial Solutions
Consider an fcc solution (Fe,Cr)(C,Va) where Fe and Cr are distributed randomly on the first (metal) sublattice and C atoms and
vacancies, denoted Va, are randomly distributed on the second
(octahedral interstitial) sublattice. In the framework of the CEF,
the molar Gibbs energy is written as
0 00 o
0
o
0
00 o
0
o
XVa gFeVa + XFe
XC00 gFeC
+ XCr
XVa
gCrVa + XCr
XC00 gCrC
g ¼ XFe
0
E
0
0
0
00
00
ln XFe
+ XCr
ln XCr
+ XC00 ln XC00 + XVa
ln XVa
+ RT XFe
+g
(15.19)
The end-members FeVa and CrVa are pure fcc Fe and Cr. The endmembers FeC and CrC are hypothetical compounds in which all
the interstitial sites are occupied by C atoms. The solution is
shown schematically on the composition square in Fig. 15.3. Since
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
CrC
X″C
FeC
Region of solubility
FeVa
X′Cr
CrVa
Fig. 15.3 Composition square of the fcc (Fe,Cr)(C,Va) solution.
C is only soluble to a limited extent in the fcc lattice as shown on
the figure, the end-member Gibbs energies, goFeC and goCrC, are
adjustable parameters that can be obtained by optimization using
data from this composition region.
It should be noted that the molar Gibbs energy in Eq. (15.19)
applies for 1 mol of sites on each sublattice and that XFe0 ¼ nFe/
(nFe + nCr) and XVa00 ¼ nVa/(nC + nVa) ¼ nVa/(nFe + nCr). Therefore, in
the binary (Fe)(C,Va) system (where nCr ¼ 0) at high concentration of Fe, the limiting activity of Fe from Eq. (15.17)
0
00
XVa
¼ nVa =nFe ¼ ðnFe nC Þ=nFe ¼ ð1 nC =nFe Þ,
is lim XFe !1 aFe ¼ XFe
which is the form of Raoult’s law applicable to the interstitial
solution.
15.3.2
Nonstoichiometric Compounds
Nonstoichiometric compounds are generally treated by sublattice models. Consider the compound AmδBn+δ. The sublattices
normally occupied by A and B atoms will be called, respectively,
the A-sublattice and the B-sublattice. Deviations from stoichiometry (where δ ¼ 0) can occur through the formation of point defects
such as substitutional defects (A atoms on B-sites and vice versa),
vacant sites, or atoms on interstitial sublattice sites. Generally, one
type of defect will predominate for solutions with an excess of A,
and another type will predominate for solutions with excess B.
These are called the majority defects.
241
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
Consider first a solution in which the majority defects are
A atoms on B-sites and B atoms on A-sites. An example is the
Laves C14 phase Mg2δY1+δ. In the framework of the CEF, the compound is represented as (A, B)m (B, A)n with
h
i
g ¼ XA0 XB00 gAom Bn + XA0 XA00 gAom An + XB0 XA00 gBom An + XB0 XB00 gBom Bn
+ mRT XA0 ln XA0 + XB0 ln XB0 + nRT XA00 ln XA00 + XB00 ln XB00 + g E
(15.20)
As an example, let us set m ¼ 2 and n ¼ 1 as for the A2δB1+δ
phase shown on the phase diagram in Fig. 15.4. We simplify the
notation by letting XB0 ¼ x and XA00 ¼ y, thereby representing the
phase as (A1xBx)2(AyB1y) with
h
i
g ¼ ð1 xÞð1 y ÞgAo2 B + ð1 xÞyg oA2 A + xð1 y ÞgBo2 B + xygBo2 A
+ 2RT ðx ln x + ð1 xÞ ln ð1 xÞÞ + RT ðy ln y + ð1 y Þ ln ð1 y ÞÞ + g E
(15.21)
The solution is represented schematically on the composition
square in Fig. 15.5.
In Eq. (15.21), g oA2B is the Gibbs energy of the hypothetical
defect-free compound (A)2(B). We define
Δg1 ¼ gAo2 A gAo2 B
(15.22)
Δg2 ¼ gBo2 B gAo2 B
(15.23)
and
A
A2-δ B1+δ
Liquid
T
242
A2B
Mole fraction B
B
Fig. 15.4 Phase diagram of a binary system with a nonstoichiometric A2dB1+d
compound.
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
B2A
X B′ = x
B2B
)
x(d
=0
2
y=
Equilibrium stoichiometric
compound
Equilibrium line
A2B
X A″= y
A 2A
Fig. 15.5 Composition square of a nonstoichiometric intermetallic compound
A2dB1+d with substitutional defects: (A,B)2(B,A).
These are the model parameters.
One sometimes sees gAo 2A set equal to (3goA + (a bT)) where goA is
the molar Gibbs energy of pure A (and a and b are adjustable
parameters.) However, there is no real justification for this. The
end-member A2A does not have the crystal structure of pure A,
but is a hypothetical two-sublattice compound (A)2(A) with the
crystal structure of A2B. It is preferable to set the model parameter
Δg1 ¼ (a1 b1T) so that from Eq. (15.22),
gAo2 A ¼ gAo2 B + ða1 b1 T Þ
(15.24)
and similarly, with Δg2 ¼ (a2 b2T),
gBo2 B ¼ gAo2 B + ða2 b2 T Þ
(15.25)
Similarly, gBo 2A should be set by identifying (gBo 2A gAo2B) ¼
(Δgo1 + Δgo2 ) as the Gibbs energy of forming of both majority defects
simultaneously, whereby
gBo2 A ¼ gAo2 A + gBo2 B gAo2 B
(15.26)
Substituting into Eq. (15.21) and rearranging terms then yields the
following equation written in terms of the model parameters:
g ¼ gAo2 B + yΔg1 + xΔg2 + 2RT ðx ln x + ð1 xÞ ln ð1 xÞÞ
(15.27)
+ RT ðy ln y + ð1 y Þ ln ð1 y ÞÞ + g E
243
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
g
244
goA B(defect free)
2
A
A2B
Mole fraction B
B
Fig. 15.6 Gibbs energy function of nonstoichiometric A2dB1+d (at constant T)
corresponding to the phase diagram of Fig. 15.4.
with Δg1 ≫ 0 and Δg2 ≫ 0, a Gibbs energy function as in Fig. 15.6
results, rising steeply on either side of its minimum and resulting
in the phase diagram of Fig. 15.5 (see Section 5.8). If, furthermore,
Δg2 > Δg1, then the function rises more steeply on the B-rich
side of the minimum (as shown schematically in Fig. 15.6), and
hence, the range of stoichiometry of the phase is wider on the
A-rich side than on the B-rich side (as shown schematically in
Fig. 15.4).
It is instructive to calculate the equilibrium defect concentrations by setting (∂ g/∂ x)δ ¼ 0 at a constant overall A/B ratio
(i.e., keeping δ constant in A2δB1+δ.) It can be seen from the formula (A1xBx)2(AyB1y) that δ ¼ (2x y). Hence, (∂ y)δ ¼ 2(∂ x)δ.
Assuming that d gE/dx 0, this gives the following equation for
the equilibrium defect concentrations:
xy=ð1 xÞð1 y Þ ¼ exp ðð2Δg1 + Δg2 Þ=2RT Þ
(15.28)
which is the equilibrium constant for the site exchange reaction:
AAsite + BBsite ¼ ABsite + BAsite
(15.29)
Solutions of Eq. (15.28) lie along the “equilibrium line” in Fig. 15.5.
Solving Eq. (15.28) when δ ¼ 0 (y ¼ 2x) gives the equilibrium defect
concentrations at the stoichiometric A2B composition. Substitution into Eq. (15.27) then gives the Gibbs energy of the actual stoichiometric compound at equilibrium which is less than the Gibbs
energy gAo2B of the hypothetical defect-free end-member as shown
in Fig. 15.6 (cf. Section 14.11).
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
In the second example, the majority defects in A2–δ B1+δ are
assumed to be vacancies on the B-sublattice and substitutional
B atoms on interstitial sublattice sites, that is (A)2(B1–xVax)
(ByVa1–y) where the third sublattice is the interstitial sublattice.
(The number of interstitial sublattice sites is assumed to be equal
to the number of B-lattice sites.) When y ¼ 0, the interstitial sublattice contains only vacancies. The Gibbs energy in the framework of the CEF is written:
h
i
g ¼ ð1 xÞð1 y ÞgAo2 BVa + ð1 xÞyg oA2 BB + xð1 y ÞgAo2 VaVa + xyg oA2 VaB
+ RT ðx ln x + ð1 xÞ ln ð1 xÞ + y ln y + ð1 y Þ ln ð1 y ÞÞ + g E
(15.30)
The model parameters in this case are the Gibbs energy of the
defect-free stoichiometric A2B, gAo2BVa, and the parameters:
Δg1 ¼ gAo2 BB gAo2 BVa and Δg2 ¼ gAo2 VaVa gAo2 BVa
(15.31)
As in the preceding example, gAo2VaB of the hypothetical endmember containing both defects should be set to
gAo2 VaB ¼ gAo2 BB + gAo2 VaVa gAo2 BVa
(15.32)
Eq. (15.30) can be rearranged to give an equation very similar to
Eq. (15.27), and setting (∂ g/∂ x)δ ¼ 0 gives an equilibrium constant
very similar to Eq. (15.28).
Other defect models such as (A1xVax)2(B1yVay) or (A)2
(B1xAx)(ByVa1y) also all result in very similar Gibbs energy functions (like Fig. 15.6) and very similar phase diagrams (like
Fig. 15.4). Therefore, it is generally not possible to deduce the correct defect model solely from the experimental phase diagram and
thermodynamic properties.
It should be mentioned that for a solution such as (A,Va)(B,Va)
with vacancies on both sublattices, one end-member Gibbs
energy is necessarily goVaVa. One often sees this formalism parameter set equal to zero since (Va)(Va), which is sometimes humorously called “vacanconium,” contains “nothing.” In light of the
preceding discussion, however, it can be appreciated that it
should rather be given by
o
o
o
o
¼ gAVa
+ gVaB
gAB
gVaVa
(15.33)
Clearly, the foregoing examples can be generalized to other stoichiometries AmBn or to different ratios of available interstitial
sites, etc., and the extension to multicomponent solutions is relatively straightforward.
245
246
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
15.3.3
Ceramic Solutions
As shown by Hillert et al. (1988), Sundman (1991), and Barry
et al. (1992), the CEF is particularly powerful when applied to
ceramic solutions in which there are two or more distinct cationic
sublattices.
As an example, consider the spinel solution (A,B,E)(A,B,E)2O4
in which A2+ and B2+ are divalent cations and E3+ is a trivalent cation. The spinel structure can be derived from the fcc close packing
of the oxygen anions. The first, or tetrahedral, sublattice consists
of one-eighth of the tetrahedral interstices, and the second, or
octahedral, sublattice consists of half of the octahedral interstices.
There are thus twice as many octahedral as tetrahedral sublattice
sites. A simple spinel AE2O4 is called “normal” if all the tetrahedral
sites are occupied by A cations and all the octahedral sites by
E cations: (A)(E)2O4. If all the tetrahedral and half the octahedral
sites are occupied by E cations, the spinel is termed “inverse”:
(E)(AE)2O4. In a real spinel, the A and E cations are distributed
on both sublattices. Since all anionic sites are occupied by oxygen
ions, we need only to concern ourselves with the distribution of
cations between the two cationic sublattices. (Note that A and
E could be the same element; for example, A ¼ Fe2+, and E ¼ Fe3+.)
Within the framework of the CEF, the Gibbs energy of (A,B,E)
(A,B,E)2O4 is given as
!
XX
X
X
g¼
Xi0 Xj00 gijo + RT
Xi0 ln Xi0 + 2
Xj00 ln Xj00
i
+
j
XXX
i
j
i
Xi0 Xj0 Xk00 Lij:k
j
+ Xk0 Xi00 Xj00 Lk:ij
(15.34)
k
(where reciprocal excess terms have been neglected).
The following treatment follows that of Degterov et al. (2001)
which is similar to that of Sundman (1991) and of Barry et al. (1992).
Consider first a simple spinel (A,E)(A,E)2O4 with four endmembers:
+
(A)(E)2O4, (A)(A)2O2
4 , (E)(E)2O4 , and (E)(A)2O4 , abbreviated as
AE, AA, EE, and EA. Three of these end-members bear a net electric charge and are thus clearly fictional “compounds.”
The reciprocal composition square is shown in Fig. 15.7. Only
points along the neutral line are physically realizable. The actual
equilibrium spinel composition lies at the point of minimum
Gibbs energy on this line.
The Gibbs energy of the purely inverse spinel (E)(A,E)2O4 at
one end of the neutral line is equal to 0.5(goEE + goEA) +
2RT(XA00 ln XA00 + XE00 ln XE00 ).
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
+
EA–
X′E
Ne
ut
line ral
EE
(inverse)
E(AE)
Equilibrium AE2O4
AE
(normal)
X″A
AA2–
Fig. 15.7 Composition square of a simple spinel (A,E)(A,E)2O4.
Degterov et al. (2001) proposed the following physically meaningful model parameters:
o
IAE ¼ gEE
o
gAE
(15.35)
o
o
+ gEA
2gAE
(15.36)
o
o
o
o
+ gEE
gEA
gAE
ΔAE ¼ gAA
(15.37)
Since all physical solutions lie on the neutral line, the Gibbs
energy of one of the charged end-members can be chosen arbitrarily. Let us set
o
o
¼ gAE
gEA
(15.38)
The parameter IAE is the main factor in determining the degree of
inversion of the real spinel. (It may be noted that older terminology for spinels refers to the “site exchange energy” which is the
energy change of the “site exchange reaction” AT + EO ¼ ET + AO
where the subscripts T and O indicate tetrahedral and octahedral
sites. The parameter IAE is equal to twice this site exchange
energy.)
The parameter ΔAE is the exchange Gibbs energy of the following reciprocal reaction among end-members (cf. Eq. 15.7):
AE + EA ¼ AA + EE
(15.39)
Given values of the model parameters, the end-member Gibbs
energies (the formalism parameters) can be calculated from
247
248
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
Eqs. (15.35)–(15.38). Substitution into Eq. (15.34) and minimization of the Gibbs energy then permits the composition of the equilibrium real spinel on the neutral line to be calculated.
For an (A,B,E)(A,B,E)2O4 spinel solution, Degterov et al. proposed the following model parameters in addition to those in
Eqs. (15.35)–(15.37):
o
gBE
(15.40)
o
o
o
+ gEB
2gBE
IBE ¼ gEE
(15.41)
o
o
o
o
+ gEE
gEB
gBE
ΔBE ¼ gBB
(15.42)
o
o
o
o
+ gEB
gEA
gAB
ΔEAB ¼ gAA
(15.43)
o
ΔEBA ¼ gBB
(15.44)
o
o
o
+ gEA
gEB
gBA
The nine end-member Gibbs energies can be calculated from the
eight model parameters in Eqs. (15.35)–(15.37) and (15.40)–
(15.44) along with Eq. (15.38). Note that goEB is fully determined
by these nine equations. That is, in a general multicomponent spinel solution, only one charged end-member Gibbs energy can be
chosen arbitrarily. Degterov et al. adopted the suggestion of
Sundman (1991) to set goAE ¼ goEA when A ¼ Fe2+ and E ¼ Fe3+.
In Eq. (15.34), there are 9 end-member Gibbs energy parameters and 18 excess Lij:k and Lk:ij parameters (and even more if the
latter are composition-dependent.) In multicomponent solutions,
the number of possible parameters increases rapidly as the number of components increases. The amount of experimental data is
almost always insufficient to permit all possible parameters to be
optimized. Hence, the most important parameters should be optimized first, followed by less important parameters as the availability of data permits. Finally, the least important parameters can be
estimated. Degterov et al. recommend the following list of types of
parameters in order of decreasing priority: (i) goAE of normal endmembers, (ii) inversion energy parameters IAE, (iii) reciprocal
exchange parameters ΔAE, (iv) reciprocal exchange parameters
ΔEAB, and finally (v) L parameters. And of course, when parameters
are being estimated, one should always estimate model parameters, never formalism parameters; for example, whereas one might
of necessity set a ΔEAB parameter to zero, one should never set an
end-member Gibbs energy to zero.
Barry et al. (1992) demonstrated that different sets of parameters can give identical descriptions of the properties of a simple
AE2O4 spinel. However, as pointed out by Degterov et al. (2001),
when the properties of a simple spinel are extrapolated into a multicomponent spinel solution, these different sets of parameters
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
may no longer necessarily give equivalent Gibbs energy functions,
thus resulting in different models. For example, for the simple spinel (A, E) (A, E)2O4, if we set LAE:A ¼ LAE:E and LA:AE ¼ LE:AE ¼ 0, substitute Eqs. (15.35)–(15.38) into Eq. (15.34), and collect terms, then
along the neutral line in Fig. 15.7 where XA00 ¼ XE0 /2,
1
o
g ¼ gAE
+ IAE XE0 =2 + ΔAE XE0 1 XE0 + LAE:A XE0 1 XE0
2
(15.45)
The final two terms in Eq. (15.45) are equivalent in the simple spinel, but will not give equivalent Gibbs energy functions when
extrapolated into a multicomponent solution.
Degterov et al. also propose a consistent set of model parameters for the case of oxygen-deficient spinels with vacancies on the
octahedral sublattice, (A,B,E)(A,B,E,Va)2O4, and they discuss the
case where the spinel solution can encompass the composition
(E)(E5/6Va1/6)2O4 that corresponds to 4/3 mol of γ-E2O3 that has a
spinel-related structure. (The best known example being γ-Al2O3.)
The considerations discussed in the present section have been
applied by Degterov et al. (2001) to the evaluation and optimization of the Fe-Zn-O system.
The CEF has found broad applications in many ceramic solutions other than spinels. The considerations and caveats presented in the present section for spinel solutions also apply in
these solutions. Ceramic solutions to which the CEF has been
applied include
olivines (Ca,Mg,Fe,Mn,Co,Zn,Ni,Cr)(Ca,Mg,Fe,Mn,Co,Zn,Ni,
Cr)(Si)(O)4,
pyroxenes
(Ca,Mg,Fe,Ni,Zn,Mn,Na)(Mg,Fe2+,Fe3+,Al,Ni,Zn,
3+
Mn)(Al,Fe ,Si)(Si)(O)6,
melilites
(Ca,Ba,Pb,Na)2(Zn,Mg,Ni,Fe2+,Fe3+,Al,B)(Fe3+,Al,B,
Si)2(O)7,
mullite (Al,Fe)2(Al,Si,B,Fe)(O,Va)5.
15.4
Asymmetric Molten Ionic Solutions:
Temkin Model
The liquid (Li,K)(F,Br) and (Na,K)(F,Cl) solutions in Figs. 15.1
and 15.2 are called symmetrical salt solutions since all ions on
the same sublattice have the same charge. An example of an asymmetrical salt solution is the liquid (Na,Ca)(F,Cl) solution shown in
Fig. 7.19 and discussed in Section 7.10 in which the Na+ and Ca2+
ions have different charges. The definition of a mole of such a
solution can be made in different ways. Hence, in order to avoid
249
250
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
ambiguity, it is preferable to define composition variables per
charge equivalent of solution rather than per mole, where one
charge equivalent of a solution is defined as containing 1 mol of
positive charge on the cation sublattice and 1 mol of negative
charge on the anion sublattice. To this end, we define charge
equivalent site fractions as
X
(15.46)
Yi0 ¼ qi = qi Xi
where qi is the charge on ion i and the summation is over all ions
on the same sublattice. In a symmetrical salt solution, Yi0 ¼ Xi.
Note that charge equivalent site fractions Yi0 must not be confused
with the coordination equivalent fractions Yi defined in Eqs.
(14.55), (14.56).
Hence, in the (Na,Ca)(F,Cl) solution, YCa0 ¼ 2nCa/(2nCa + nNa) ¼
(1 YNa0 ), and YCl0 ¼ nCl/(nF + nCl) ¼ (1 YF0 ) where ni is the number of moles of ion i and where, from charge balance considerations, (2nCa + nNa) ¼ (nF + nCl).
As discussed in Section 14.4.1, the liquid solution is modeled
with the Temkin model, whereby the cations and anions mix randomly on their respective quasi-sublattices regardless of their
charges, and the ratio of the number of cation sites to the number
of anion sites varies with composition.
Consider a general asymmetrical ternary reciprocal salt solution (A,B)(X,Y) with ionic charges qi, which are not all the same.
The probability of an (A-X) nearest-neighbor pair-bond is then
YA0YX0 . The Gibbs energy per charge equivalent of solution is then
given, as originally proposed by Blander (1964), as
o
o
o
o
+ YA0 YY0 gA=Y
+ YB0 YX0 gB=X
+ YB0 YY0 gB=Y
g ¼ YA0 YX0 gA=X
h
i
+ RT ðqA XA + qB XB Þ1 ðXA ln XA + XB ln XB Þ + ðqX XX + qX XY Þ1 ðXX ln XX + XY ln XY Þ
+ ½YX0 YA0 YB0 αAB:X + YY0 YA0 YB0 αAB:Y + YA0 YX0 YY0 αA:XY
+ YB0 YX0 YY0 αB:XY + YA0 YB0 YX0 YY0 LAB:XY
(15.47)
goi/j
where
is the Gibbs energy of one equivalent of the chargeneutral end-member i/j. For example, goCa/Cl is the Gibbs energy
of 0.5 mol of CaCl2.
When all ions are monovalent, Eq. (15.47) reduces to Eq. (15.3).
Note that the ideal entropy of mixing terms are written in terms
of the site fractions Xi, not in terms of Yi0 , and that
(qAXA + qBXB) ¼ (qXXX + qYXY ) is the number of charge equivalents
per mole of lattice sites on the sublattices. The binary excess terms
in Eq. (15.47) have been written in terms of the charge equivalent
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
fractions Yi0 as in the original formulation by Blander (1964), but
this is not necessary in the general case. Furthermore, the αfunctions may also be expanded as polynomials in Yi0 , but again,
this is not necessary in general; expansions in terms of the site
fractions Xi or the coordination equivalent fractions Yi could also
be used.
The coefficient LAB:XY in the reciprocal excess term in
Eq. (15.47) may be optimized from experimental data or may be
estimated from Eqs. (15.12) and (15.13) with Δε ΔGexchange,
which is the Gibbs energy of the exchange reaction involving
one charge equivalent of each end-member, that is,
A=Y + B=X ¼ A=X + B=Y
The extension to multicomponent solutions is straightforward
and has been given in detail by Pelton (1988).
15.4.1
Solid vs Liquid Asymmetric Ionic Solutions
The phase diagram of the BaF2-LaF3 system is shown in
Fig. 15.8. The maximum in the solidus/liquidus at BaF2-rich compositions can be explained as resulting from entropic stabilization
of the solid phase relative to the liquid phase due to the formation
of vacant cation sites in the solid. That is, the dissolution of two
moles of LaF3 in solid BaF2 occurs by the replacement of three
moles of BaF2 through the substitution of two La3+ ions for two
Ba2+ ions and the formation of a vacant cation site. In relatively
dilute solutions, the two La3+ ions and the vacancy will be approximately randomly distributed, resulting in a larger entropy and
hence lower Gibbs energy, than in the liquid solution in which
no lattice vacancies are formed (Temkin model). At higher LaF3
concentrations, the La3+ ions and vacancies will tend to form
Liquid
T (oC)
1450
1400
Solid
1350
1300
0
BaF2
20
40
60
Mole percent LaF3
80
100
LaF3
Fig. 15.8 BaF2-LaF3 phase diagram (Nafziger and Riazance, 1972) illustrating
entropic stabilization of the solid phase relative to the liquid.
251
252
Chapter 15 MULTIPLE-SUBLATTICE RANDOM-MIXING (BRAGG-WILLIAMS—BW) MODELS
associates due to coulombic attraction, thereby lowering the
entropy.
A similar phenomenon is observed in other asymmetrical molten salt systems.
References
Barry, T.I., Dinsdale, A.T., Gisby, J.A., Hallstedt, B., Hillert, M., Jansson, B.,
Jonsson, S., Sundman, B., Taylor, J.R., 1992. J. Phase Equilib. 13, 459–475.
Berezina, S.I., Bergman, A.G., Bakumskaya, E.L., 1963. Zh. Neorg. Khim.
8, 2140–2143.
Blander, M., 1964. Molten Salt Chemistry. Interscience, New York.
Blander, M., Braunstein, J., 1960. Ann. N. Y. Acad. Sci. 79, 838–852.
Degterov, S.A., Jak, E., Hayes, P.C., Pelton, A.D., 2001. Metall. Mater. Trans.
32B, 643–657.
Dessureault, Y., Pelton, A.D., 1991. J. Chim. Phys. 88, 1811–1830.
Hillert, M., Jansson, B., Sundman, B., 1988. Z. Metallkd. 79, 81–87.
Hillert, M., Staffansson, L.-I., 1970. Acta Chem. Scand. 24, 3618–3626.
Margheritis, C., Flor, G., Sinistri, C., 1979. Z. Naturforsch. 34A, 836–839.
Nafziger, R.H., Riazance, N., 1972. J. Am. Ceram. Soc. 55, 130–134.
Pelton, A.D., 1988. Calphad J. 12, 127–142.
Sundman, B., 1991. J. Phase Equilib. 12, 127–140.
List of Website
FactSage, Montreal, www.factsage.com.
16
SINGLE-LATTICE MODELS
WITH SHORT-RANGE
ORDERING (SRO)
CHAPTER OUTLINE
16.1 Associate Models 257
16.2 The Modified Quasichemical Model (MQM) 260
16.2.1 Combining the Quasichemical and Bragg-Williams Models 264
16.2.2 Expansion of ΔgAB as a Polynomial 268
16.2.3 Composition-Dependent Coordination Numbers 269
16.2.4 Example of an Optimization Using the MQM—The Mg-Sn
System 270
16.3 Second-Nearest-Neighbor Short-Range Ordering in Ionic
Liquids 272
16.3.1 Complex Anion Model 273
16.3.2 Modified Quasichemical Model 274
16.4 Short-Range Ordering and Positive Deviations From Ideal
Mixing 274
16.5 Approximating Short-Range Ordering with a Polynomial
Expansion 277
16.6 Modified Quasichemical Model—Multicomponent Systems 278
16.6.1 Interpolation Formulae 280
16.7 The MQM Equations in Closed Explicit Form 286
16.8 Combining Bragg-Williams and MQM Models in One
Multicomponent Database 287
16.9 Comparison of the Bragg-Williams, Associated and Modified
Quasichemical Models in Predicting Ternary Properties From
Binary Properties 288
16.10 The Two-Sublattice “Ionic Liquid” Model 292
References 293
List of Websites 294
The molar enthalpy of mixing of liquid Al-Ca alloys is shown in
Fig. 16.1. The function is negative and “V-shaped” with nearly linear terminal segments, whereas solutions that are approximately
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00016-6
# 2019 Elsevier Inc. All rights reserved.
253
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Δh (kJ mol−1)
0
Notin et al. (1982)
680°C
765°C
Sommer et al. (1983)
852°C
857°C
907°C
917°C
−10
−20
−30
0
0.2
0.4
0.6
0.8
1
Ca
Al
Mole fraction Ca
Fig. 16.1 Enthalpy of mixing in liquid Al-Ca solutions (FactSage).
2
Kawakami, 1930: claorimetry, 800°C
Eremenko and Lukashenko, 1963: emf, 650°C
Steiner et al., 1964: vapor, 770°C
Eldridge et al., 1967: vapor, 770°C
Sharma, 1970: vapor, 800°C
Nayak and Oelsen, 1971: calorimetry, 800°C
Sommer et al., 1980: calorimetry, 800°C
0
−2
−4
Dh (kJ mol-1)
254
−6
−8
−10
−12
800°C
−14
650°C
−16
−18
−20
0.0
Mg
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction Sn
0.7
0.8
0.9
1.0
Sn
Fig. 16.2 Optimized ( Jung et al., 2007) enthalpy of liquid Mg-Sn solutions along with
experimental data. (For references for the data, see Jung et al., 2007.)
regular have approximately parabolic enthalpy functions. As a second example, the molar enthalpy and entropy of mixing of liquid
Mg-Sn alloys are shown in Figs. 16.2 and 16.3. Again, the enthalpy
function is negative and V-shaped, while the entropy curve is
“m-shaped.” A third example is shown in Figs. 16.4 and 16.5 for
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
6.00
Δs (J mol−1 K−1)
5.00
4.00
3.00
2.00
1.00
0
Mg
0.20
0.40
0.60
Mole fraction Sn
0.80
Sn
Fig. 16.3 Optimized ( Jung et al., 2007) entropy of mixing of liquid Mg-Sn solutions
at 800°C.
0
Litovskii et al. (1986), 1600oC
Δh (kJ mol−1)
−10
−20
−30
−40
0
Al
0.2
0.4
0.6
Mole fraction Sc
0.8
1
Sc
Fig. 16.4 Calculated optimized enthalpy of mixing in liquid Al-Sc solutions at 1600°C.
Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
liquid Al-Sc alloys in which the V-shape of the enthalpy function is
somewhat less pronounced. The partial enthalpy functions corresponding to Fig. 16.4 are shown in Fig. 16.6; they have the typical
form of titration curves.
These shapes of the thermodynamic functions in Figs. 16.1–
16.6 are characteristic of solutions exhibiting short-range ordering
255
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
6
Δs (J mol−1 K−1)
Ideal (random mixing)
4
2
0
0.2
0
Al
0.4
0.6
0.8
Mole fraction Sc
1
Sc
Fig. 16.5 Calculated optimized entropy of mixing in liquid Al-Sc solutions at 1600°C.
Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
0
ΔhAl and ΔhSc (kJ mol−1)
256
−50
−100
Litovskii et al. (1986), 1600°C
−150
0
Al
0.2
0.4
0.6
Mole fraction Sc
0.8
1
Sc
Fig. 16.6 Partial optimized enthalpies of mixing in liquid Al-Sc solutions at 1600°C.
Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
(SRO), with the minima in Δh and Δs occurring near the compositions of maximum SRO (Al2Ca, Mg2Sn, and AlSc in these
examples).
It is difficult to reproduce such curves with a random-mixing
Bragg-Williams model and a polynomial expansion for gE as in
Eqs. (14.4), (14.5). Furthermore, even if a reasonable fit is obtained
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
by employing a large number of adjustable parameters in the
expansion, subsequent estimations of the properties of ternary
and higher-order solutions from the binary model parameters will
generally be poor. If the degree of SRO is appreciable, it should be
accounted for explicitly in the solution model.
16.1
Associate Models
To account for SRO, associate (also called “molecular”) models
are often used. In the simplest case, a binary liquid is considered
to contain AB “associates” (or molecules) and unassociated A and
B atoms. The species are at internal equilibrium:
o
o
¼ ΔhoASS T ΔsASS
(16.1)
A + B ¼ AB; ΔgASS
where ΔgoASS is the enthalpy change of this association reaction per
mole of AB associates:
o
o
(16.2)
¼ gAB
gAo gBo
ΔgASS
where ΔgoAB is the Gibbs energy of 1.0 mol of a hypothetical solution consisting only of AB associates.
The mass balance equations are
nA ¼ n0A + n0AB
(16.3)
nB ¼ n0B + n0AB
(16.4)
where nA and nB are the number of moles of the components (i.e.,
the end-members) and nAB0 , nA0 , and nB0 are the number of moles
of AB associates and of unassociated A and B atoms. We define XA0
as the mole fraction of unassociated A atoms according to
XA0 ¼ n0A = n0A + n0B + n0AB
(16.5)
and similarly for XB0 and XAB0 .
The model assumes that the three species are randomly distributed on the sites of a single lattice, or quasi-lattice in the case
of liquids. Hence, the configurational entropy of mixing is given by
0
(16.6)
ΔSconfig ¼ R n0A ln XA0 + n0B ln XB0 + n0AB ln XAB
The Gibbs energy of the solution is given by
o
G ¼ nA gAo + nB gBo T ΔSconfig + n0AB ΔgASS
+ GE
(16.7)
Minimizing the Gibbs energy at constant overall composition (i.e.,
constant nA and nB) under the constraints of Eqs. (16.3), (16.4)
then permits the calculation of the equilibrium amounts nA0 ,
nB0 , and nAB0 . The molar enthalpy and entropy of mixing per mole
of components are then given as
Δh ¼ n0AB ΔhoASS =ðnA + nB Þ
(16.8)
257
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
and
o
Δs ¼ ΔSconfig + n0AB ΔsASS
= ð nA + n B Þ
(16.9)
and the following equilibrium constant for the reaction in
Eq. (16.1) may be written:
0
o
= XA0 XB0 ¼ exp ΔgASS
=RT
(16.10)
XAB
As ΔgoASS becomes progressively more negative, the equilibrium in
the reaction in Eq. (16.1) is displaced progressively to the right, and
the amount of SRO increases. Calculated curves of Δh at 1000°C are
shown for several negative values of ΔhoASS in Fig. 16.7 for the case
when ΔsoASS ¼ 0 and when ΔgoASS ¼ ΔhoASS is constant. For relatively
small negative values of ΔhoASS, the curves of Δh are approximately
parabolic and close to those calculated with the Bragg-Williams
regular solution model with no associates. For large negative
ΔhoASS values, the Δh curves take on the V-shape characteristic of
SRO. When ΔhoASS ¼ 80 kJ mol1, for example, a solution at the
equimolar composition XA ¼ XB ¼ 0.5 consists almost entirely of
AB associates with very few unassociated A and B atoms. At such
a high degree of SRO, every atom of B added to an A-rich solution
produces on average nearly exactly one AB associate. Hence, XAB0
increases linearly with XB, and through Eq. (16.7), the graph of Δh
versus XB is nearly a straight line.
The corresponding curves of Δsconfig are shown in Fig. 16.8. For
very negative values of ΔgoASS ¼ ΔhoASS, the solution at XA ¼ XB ¼ 0.5
10
0
Δh ASS = −5 kJ mol−1
o
−20
−10
Δh (kJ mol−1)
258
−40
−20
−30
−40
−50
−80 kJ mol−1
0
A
0.2
0.4
0.6
Mole fraction B
0.8
1
B
Fig. 16.7 Enthalpy of mixing of a solution A-B at 1000°C calculated from the
associate model with the constant values of DhoASS shown and with DsoASS ¼ 0.
Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
8
Δh ASS = 0 kJ mol−1
o
−5
−20
Δsconfig (J mol−1 K−1)
6
Ideal
(random mixing)
4
−40
2
−80 kJ mol−1
0
0
A
0.2
0.4
0.6
Mole fraction B
0.8
1
B
Fig. 16.8 Configurational entropy of mixing of a solution A-B at 1000°C calculated
from the associate model with the constant values of DhoASS shown and with
DsoASS ¼ 0. Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res.
98, 907–917.
is highly ordered, and Δsconfig is close to zero. As ΔhoASS becomes
less negative, the entropy becomes more positive as expected.
However, it can be seen that the associate model does not reduce
to the ideal random-mixing model when ΔgoASS ¼ 0, because when
ΔgoASS ¼ 0, XAB0 is not equal to zero. The model only reduces to an
ideal solution when ΔgoASS ¼ ∞. This has sometimes been termed
€ ck et al., 1989).
an “entropy paradox” (Lu
Substituting Eqs. (16.2)–(16.4) into Eq. (16.7) and rearranging
terms give
o
T ΔSconfig + G E
(16.11)
G ¼ n0A gAo + n0B gBo + n0AB gAB
with ΔSconfig from Eq. (16.6). Hence, the solution can be treated
formally as a “ternary” random Bragg-Williams solution (A, B,
AB) with the end-members being unassociated A, unassociated
B, and AB and with goAB given by Eq. (16.2). Excess Gibbs energy
terms can be included to account for pairwise interactions among
unassociated A, unassociated B, and AB species:
0
0
αA, AB + XB0 XAB
αB,AB + XA0 XB0 αA,B + ðternary termsÞ
g E ¼ XA0 XAB
where gE is the excess Gibbs energy per mole of end-members (not
per overall mole of A + B). The α-functions can be expanded as
polynomials in XA0 , XB0 , and XAB0 , and the model can be extended
to multicomponent solutions (A,B,C, …,AB, AC, BC, …) with the
same equations and software as described in Chapter 14.
259
260
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
In many binary liquid solutions, the Δ h and Δ sconfig functions
often exhibit minima near compositions other than the equimolar
composition. Examples were given for the Al-Ca and Mg-Sn solutions in Figs. 16.1–16.4. Such solutions can be modeled by postulating associates with other AmBn stoichiometries (Al2Ca and
Mg2Sn in these examples).
Despite its mathematical simplicity, the associate model has
serious weaknesses. Firstly, although solution properties can usually be well represented by an associate model when ΔgoASS < 0, the
model is not applicable to solutions with positive deviations from
ideality as has just been discussed. Secondly, predictions of the
properties of ternary and higher-order solutions from binary
model parameters are generally poor as will be demonstrated in
Section 16.9. These weaknesses result from the fact that, in most
real solutions, the associate model is not physically realistic. That
is, SRO in most solutions does not actually result from the formation of AmBn “molecules.”
16.2 The Modified Quasichemical Model
(MQM)
A more physically realistic treatment of SRO in most solutions
is provided by the modified quasichemical model (MQM) in the
pair approximation. The quasichemical model as first proposed
by Guggenheim (1935) and by Fowler and Guggenheim (1939)
has been extended and modified in a series of articles by Pelton
and Blander (1984), Blander and Pelton (1987); Pelton et al.
(2000), Pelton and Chartrand (2001), Chartrand and Pelton
(2001), and Pelton et al. (2001).
In this model, in a liquid binary solution, atoms or molecules
A and B are distributed over the sites of a lattice (or quasi-lattice in
the case of a liquid). The following pair-exchange reaction is
considered:
ðA AÞ + ðB BÞ ¼ 2ðA BÞ; ΔgAB ¼ ðΔhAB T ΔsAB Þ
(16.12)
where (i-j) represents a first-nearest-neighbor (FNN) pair. The
nonconfigurational Gibbs energy change for the formation of
two moles of (A-B) pairs is ΔgAB. Let nA and nB be the number
of moles of A and B and nij be the number of moles of (i-j) pairs.
(nAB and nBA represent the same quantity and can be used interchangeably.) Let ZA and ZB be the first-nearest-neighbor (FNN)
coordination numbers of A and B. It follows that
ZA nA ¼ 2nAA + nAB
(16.13)
ZB nB ¼ 2nBB + nAB
(16.14)
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
(Clearly, in the case of solid solutions, ZA ¼ ZB necessarily.)
We also introduce pair fractions Xij:
Xij ¼ nij =ðnAA + nBB + nAB Þ
(16.15)
overall mole (or site) fractions:
X A ¼ n A = ð nA + nB Þ ¼ 1 X B
(16.16)
and “coordination-equivalent” fractions defined as in Eqs. (14.55),
(14.56):
YA ¼ ZA nA =ðZA nA + ZB nB Þ ¼ ZA XA =ðZA XA + ZB XB Þ ¼ 1 YB (16.17)
If ZA ¼ ZB, then YA ¼ XA and YB ¼ XB.
Substitution of Eqs. (16.13), (16.14) into Eqs. (16.15), (16.7)
gives
YA ¼ XAA + XAB =2
(16.18)
YB ¼ XBB + XAB =2
(16.19)
The Gibbs energy of the solution is given by
G ¼ nA gAo + nB gBo T ΔSconfig + ðnAB =2ÞΔgAB
goA
(16.20)
and goB are
config
the molar Gibbs energies of the pure compowhere
is the configurational entropy of mixing given
nents and ΔS
by randomly distributing the (A-A), (B-B), and (A-B) pairs:
ΔSconfig ¼ RðnAA ln XAA + nBB ln XBB + nAB ln XAB Þ + RðnAA ln YA2
+ nBB ln YB2 + nAB ln 2YA YB Þ RðnA ln XA + nB ln XB Þ
(16.21)
Because no exact mathematical expression is known for the
entropy of this distribution in three dimensions, Eq. (16.21) is an
approximate equation. The first bracketed term in Eq. (16.21) is
the entropy of an unrestricted random distribution of the pairs.
However, this term overestimates the entropy since, for example,
if an (A-A) pair emanates from a central A atom, then all other pairs
emanating from this atom can only be either (A-A) or (A-B) pairs.
An approximate correction is applied by noting that when ΔgAB
is equal to zero, there should be no SRO, and the A and B atoms
should be randomly distributed over the lattice sites. Since the
numbers of pairs emanating from A and B atoms are ZAnA and ZBnB,
respectively, the probability in a random Bragg-Williams solution
that a given pair has both ends emanating from an A atom is given
by XAA ¼ [ZAnA/(ZAnA + ZBnB)]2 ¼ YA2. Similarly, XBB ¼ Y2B, and
XAB ¼ 2YAYB. Hence, the second and third bracketed terms in
Eq. (16.21) are added to ensure that the entropy expression reduces
261
262
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
to that of a randomly mixed Bragg-Williams solution in the limit
when there is no SRO.
Although Eq. (16.21) is only approximate in three dimensions,
it can be shown to be exact (Ising model) for a one-dimensional
lattice (Pelton and Blander, 1984; Pelton and Kang, 2007).
Therefore, one must choose between using a one-dimensional
model (i.e., by setting Z ¼ 2) with an exact mathematical expression for the entropy and using a three-dimensional model with
an approximate entropy expression. This is an important consideration that will be examined in later sections in this chapter.
The equilibrium distribution is calculated by setting
ð∂G=∂nAB ÞnA ,nB ¼ 0
(16.22)
at constant nA and nB, subject to the constraints of Eqs. (16.13),
(16.14). This gives (Pelton and Blander, 1984)
XAB =ðXAA XBB Þ ¼ 4 exp ðΔgAB =RT Þ
(16.23)
For a given value of ΔgAB, the solution of Eq. (16.23) together with
Eqs. (16.13), (16.14) gives nAA, nBB, and nAB that can then be
substituted into Eqs. (16.20), (16.23). Eq. (16.23) is the equilibrium
constant for the “quasichemical reaction” in Eq. (16.12). When
ΔgAB ¼ 0, Eq. (16.23) is satisfied by the Bragg-Williams limit:
XAA ¼YA2, XBB ¼ Y2B, and XAB ¼ 2YAYB. As ΔgAB becomes progressively
more negative, the reaction in Eq. (16.12) is shifted progressively to
the right, and the calculated enthalpy and configurational entropy
of mixing assume, respectively, the negative “V”- and “m”-shape
characteristic of SRO.
Calculated enthalpies and configurational entropies of mixing
for the case when ZA ¼ ZB ¼ 2 and ΔgAB is constant, independent
of temperature and composition, are shown in Figs. 16.9 and
16.10. (Note that when ΔgAB is independent of temperature, the
entropy of mixing is necessarily entirely configurational since
ΔsAB ¼ 0 in Eq. (16.12).)
Furthermore, when ΔgAB is small, the MQM can be shown to
approach the regular solution model as follows. From
Eqs. (16.13), (16.14), the total number of pairs in the solution,
(nAA + nBB + nAB), is equal to (ZAnA + ZBnB)/2. When ΔgAB is small
and there is little SRO, XAB 2YAYB as discussed above. Hence,
nAB is approximately equal to (ZAnA + ZBnB)/2)2YAYB. Also, in this
case, the configurational entropy is approximately equal to the
random Bragg-Williams value. Substitution into Eq. (16.20) and
division by (nA + nB) then gives, per mole of solution, for small
ΔgAB,
ðX Z + X B Z B Þ
g XA gAo + XB gBo + RT ðXA lnXA + XB lnXB Þ + A A
YA YB ΔgAB
2
(16.24)
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
263
10
0
Δh (kJ mol−1)
ΔgAB = −5 kJ mol−1
−10
−20
−20
−40
−30
−40
−50
0
A
−80 kJ mol−1
0.2
0.4
0.6
Mole fraction B
0.8
1
B
Fig. 16.9 Enthalpy of mixing of a solution A-B at 1000°C calculated from the modified quasichemical model with the
constant values of DgAB shown when ZA ¼ ZB ¼ 2. Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater.
Res. 98, 907–917.
8
Ideal (ΔgAB = 0 kJ mol−1) random mixing
Δsconfig (J mol−1 K−1)
6
−5
−20
4
−40
2
−80 kJ mol−1
0
0
A
0.2
0.4
0.6
Mole fraction B
0.8
1
B
Fig. 16.10 Configurational entropy of mixing of a solution A-B at 1000°C calculated from the quasichemical model with
the constant values of DgAB shown when ZA ¼ ZB ¼ 2. Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J.
Mater. Res. 98, 907–917.
264
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
For a rigorous proof, it must be shown that the ratio of the first two
term in Eq. (16.21) to the last term in Eq. (16.20) approaches zero
as ΔgAB approaches zero. By setting XAB ¼ (2YAYB + y), expanding
the logarithmic terms in Eq. (16.21) as a Taylor series, and letting
y ! 0, this ratio can be shown to be of the order of (YAYBΔgAB/RT).
Eq. (16.24) is the regular solution polynomial expansion in
Eq. (14.58) for a solution with coordination numbers ZA and ZB,
and ΔgAB/2 is then the α0 -function that may be expanded as a
polynomial as in Eqs. (14.59), (14.60).
The composition, XB, of maximum SRO is determined by
the ratio (ZB/ZA). For example, in liquid Al-Sc solutions,
Figs. 16.4–16.6, the maximum SRO occurs near XAl ¼ XSc ¼ 1/2
where Al atoms are predominantly surrounded by Sc atoms and
vice versa. Hence, we set ZAl ¼ ZSc. In Al-Ca solutions, Fig. 16.1,
on the other hand, maximum SRO occurs near the composition
corresponding to Al2Ca where XCa ¼ 1/3. Hence, we set ZCa ¼ 2ZAl.
As a result, YCa ¼ 1/2 when XCa ¼ 1/3. That is, in general, we choose
a ratio ZB/ZA such that the composition of maximum SRO occurs
at YA ¼YB ¼ 1/2. Of course, this only applies to liquid solutions.
In solid solutions, necessarily, ZA ¼ ZB.
When ZA ¼ ZB ¼ 2 and ΔgAB ≪ 0, the solution exhibits virtually
complete SRO at XA ¼ XB ¼ 0.5. That is, at this composition, there
are only (A-B) pairs in solution, and from Eq. (16.21), Δsconfig ¼ 0.
When ZA 6¼ ZB and ΔgAB ≪ 0, the solution exhibits virtually complete SRO when XA/XB ¼ ZB/ZA. It can be shown (Pelton and
Blander, 1984) from Eq. (16.21) that Δsconfig is equal to zero at this
composition when
ZB ¼ ð ln ð1 + XA =XB ÞÞ + ðXA =XB Þ ln ð1 + XB =XA ÞÞ= ln 2
(16.25)
For example, when the composition of maximum SRO is at
XA/XB ¼ 2 (i.e., at the composition A2B), Eq. (16.25) gives
ZB ¼ 2.754 and ZA ¼ 1.377.
16.2.1
Combining the Quasichemical and BraggWilliams Models
Experience from examining a great many systems shows that
the quasichemical model with Z ¼ 2 frequently gives calculated
Δh functions in which the negative V-shape is too sharp. (The
same observation applies for the associate model of
Section 16.1.) In Fig. 16.11, the curve labeled as Z ¼ 2 with (αAB0 /
ΔgAB ¼ 0) was calculated with one constant quasichemical coefficient ΔgAB chosen so that the curve reproduces the experimental
points at low values of XSc. At higher values of XSc, however, this
calculated curve clearly gives a function with too sharp a minimum. This can be attributed to the fact that Eq. (16.20) of the quasichemical model assumes that the enthalpy of mixing is due only
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Litovskii et al. (1986), 1600°C
Δh (kJ mol−1)
0
−20
−40
Z = 12
(a′AB/ΔgAB = 0)
(a′AB/ΔgAB = 1.25)
(Z = 2)
Z=6
(a′AB/ΔgAB = 0)
(a′AB/ΔgAB = 0.5)
(Z = 2)
Z=2
(a′AB/ΔgAB = 0)
−60
0
Al
0.2
0.4
0.6
Mole fraction Sc
0.8
1
Sc
Fig. 16.11 Enthalpy of mixing in liquid Al-Sc solutions at 1600°C. Curves calculated
from the quasichemical model from Eq. (16.26) for various ratios (aAB0 /DgAB) with
Z ¼ 2 and for various values of Z with aAB0 ¼ 0. Reprinted with permission from
Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
to changes in FNN interactions upon mixing. In reality, all lattice
interactions contribute to the enthalpy of mixing. If we assume
that SRO is significant only between first-nearest-neighbors, then
the remaining lattice contributions to Δh can be approximated by
a Bragg-Williams term of the form of Eq. (14.58). That is, the quasichemical expression of Eq. (16.20) is replaced by
G ¼ nA gAo + nB gBo
T ΔSconfig + ðnAB =2ÞΔgAB + ðZA nA + ZB nB ÞYA YB α0AB (16.26)
with ΔSconfig from Eq. (16.21), where the term in ΔgAB accounts for
first-nearest-neighbor interactions and SRO and the term in αAB0
accounts for the remaining lattice interactions. The other equations of the quasichemical model remain unchanged. Note that
the quasichemical equilibrium constant, Eq. (16.23), still depends
only upon ΔgAB. Hence, when ΔgAB ¼ 0, the configurational
entropy reduces to the Bragg-Williams expression.
Figs. 16.12 and 16.13 show curves of Δh and Δsconfig calculated
with Z ¼ 2 for various ratios αAB0 /ΔgAB where the (constant) values
of αAB0 and ΔgAB in each case were chosen to give the same value of
Δh at its minimum. The trends in these figures are clear. In
Fig. 16.11 are shown curves calculated with Z ¼ 2 for the Al-Sc system for different ratios (αAB0 /ΔgAB). With (αAB0 /ΔgAB) ¼ 1.25
(αAB0 ¼ 42,630 J mol1 and ΔgAB ¼ 34,104 J mol1), a very good
fit to all the experimental points is obtained.
265
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
0
Δh (kJ mol−1)
−20
Z=2
(a′AB/ΔgAB = 0) = 0.0
0.25
0.5
1.0
2.0
40
∞ (Bragg-Williams limit)
−40
−60
0
A
0.2
0.4
0.6
0.8
Mole fraction B
1
B
Fig. 16.12 Enthalpy of mixing of a solution A-B at 1000°C calculated from Eq. (16.26)
with constant parameters aAB0 and DgAB in the ratios shown. Reprinted with
permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
Z=2
a′AB/ΔgAB = ∞ ideal (Bragg-Williams limit)
6
4.0
Δsconfig (J mol−1 K−1)
266
4
2.0
1.0
‘
0.5
0.25
2
0.0
0
0
A
0.2
0.4
0.6
Mole fraction B
0.8
1
B
Fig. 16.13 Configurational entropy of mixing of a solution A-B at 1000°C calculated
from Eq. (16.26) with constant parameters aAB0 and DgAB in the ratios shown.
Reprinted with permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
An alternative strategy for making the calculated Δh function
less sharply V-shaped is to choose values of Z > 2 with αAB0 ¼ 0.
As Z is made larger, the number of bonds per mole of solution
increases, the entropy thereby becomes more important relative
to the enthalpy (through the factor Z in Eq. (16.24)), and the
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
0
Δh (kJ mol−1)
Z=2
6
12
60
∞ (Bragg-Williams limit)
−20
−40
−60
0
A
0.2
0.4
0.6
0.8
Mole fraction B
1
B
Fig. 16.14 Enthalpy of mixing of a solution A-B at 1000°C calculated from Eq. (16.26)
with various constant parameters DgAB with different values of Z. Reprinted with
permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
8
Z = ∞ ideal (Bragg-Williams limit)
Δsconfig (J mol−1 K−1)
6
Z = 60
4
2
Z = 12
Z=2
0
−2
Z=6
0
A
0.2
0.4
0.6
0.8
Mole fraction B
1
B
Fig. 16.15 Configurational entropy mixing for a solution A-B at 1000°C calculated
from the quasichemical model with different values of Z with the same constant
parameters DgAB as in the corresponding curves in Fig. 16.14. Reprinted with
permission from Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
degree of SRO for a given values of ΔgAB decreases. Figs. 16.14 and
16.15 show curves of Δh and Δsconfig calculated with different
values of Z, with a constant value of ΔgAB ¼ ΔhAB chosen in each
case to give the same value of Δh at its minimum. The curves for
267
268
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Δh in Fig. 16.14 are very similar to those in Fig. 16.12. However,
the entropy functions in Fig. 16.15 differ from those in Fig. 16.13.
As Z is increased above 2.0, the entropy at first decreases and can even
become negative, before eventually increasing for large values of Z.
Clearly, a negative configurational entropy of mixing is physically
unrealistic and arises from the aforementioned fact that Eq. (16.24)
is exact only when Z ¼ 2 (i.e., for a one-dimensional lattice).
In general, because the entropy expression of Eq. (16.24) is an
approximation in three dimensions, the coordination numbers ZA
and ZB are nonphysical and should be considered to be parameters of the model.
In Fig. 16.11 are shown Δh curves calculated for the Al-Sc system with Z ¼ 6 and Z ¼ 12 with constant values of ΔgAB ¼ ΔhAB
with αAB0 ¼ 0. These are almost identical to the curves calculated
with Z ¼ 2 and (αAB0 /ΔgAB) ¼ 0.5 and 1.25, respectively.
16.2.2
Expansion of DgAB as a Polynomial
In order to permit the MQM to be used for fitting data in real
systems, additional empirical parameters may be introduced by
expanding ΔgAB as a polynomial in the fractions Yi as
X ij j
o
+
qAB YAi YB
(16.27)
ΔgAB ¼ ΔgAB
ði + jÞ1
P
o
+ i1 i l AB ðYA YB Þi .
or in Redlich-Kister form as ΔgAB ¼ ΔgAB
Where ΔgoAB, ΔgijAB, and ilAB are the adjustable parameters of the
model and are, in general, functions of temperature.
As has been discussed above, when ΔgAB is small and there is
consequently little SRO, Eqs. (16.20), (16.21) reduce to Eq. (16.24),
and so, in the limit, the coefficients ΔgoAB/2, qijAB/2, and ilAB/2
become numerically equal to the coefficients QijAB and iLAB of a
polynomial expansion of gE as in Eqs. (14.59), (14.60).
Alternatively, ΔgAB can be expanded as a polynomial in terms
of the pair fractions XAA and XBB rather than in terms of the equivalent fractions. That is,
X
X 0j j
o
i0 i
ΔgAB ¼ ΔgAB
+
gAB
XAA +
gAB XBB
(16.28)
i1
j1
0j
where, ΔgoAB, Δgi0
AB, and ΔgAB are the parameters of the model and
may be functions of temperature. In Eq. (16.28), ΔgAB is “configuration-dependent” rather than simply composition-dependent
as in Eq. (16.27). This presents a significant practical advantage
for solutions with a large degree of SRO. When YB < 1/2, XBB is very
j
small in such solutions. Hence, the terms (g0j
ABXBB) in Eq. (16.28)
have little effect on the Gibbs energy, and so, in this composition
region, only the parameters gi0
AB are important. Similarly, for
YB > 1/2, only the parameters g0j
AB are important. Hence, for
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
curve-fitting purposes, the solution can effectively be split into
two nearly independent subsystems. This generally results in significantly improved
with fewer coefficients.
P fits,
Cross terms
gijABXiAAXjBB with i 1 and j 1 can also be
included in Eq. (16.28). Usually, however, this provides no advantage in practice.
When ΔgAB is small, XAA YA2, XBB Y2B, XAB 2YAYB, and
Eq. (16.28) reduces to Eq. (14.60) in the limit. For example, if we
01
set g10
AB ¼ gAB and set all coefficients for i 2 and j 2 equal to
zero, then for small ΔgAB,
2
o
o
01
01
01
+ 2gAB
+ gAB
YB YA2 ¼ ΔgAB
gAB
YB
(16.29)
ΔgAB ΔgAB
This is equivalent to Eq. (14.60) for a two-coefficient (subregular)
polynomial expansion.
16.2.3
Composition-Dependent Coordination
Numbers
The quasichemical model can be rendered more flexible
through the introduction of composition-dependent coordination numbers. There are a number of drawbacks to the use of constant coordination numbers. If a solution exhibits a high degree of
SRO at, for example, XB ¼ 1/3, then ZB must equal 2ZA near this
composition. However, this should not require that ZB in pure liquid B be necessarily equal to 2ZA in pure A. Furthermore, from a
more practical standpoint, for the development of databases for
multicomponent solutions, consider a ternary liquid solution
A-B-C in which we have chosen the ratios (ZB/ZA) and (ZC/ZB)
to correspond to the observed compositions of maximum SRO
in the A-B and B-C binary subsystems. If we are restricted to constant coordination numbers, then the ratio (ZC/ZA) is now fixed.
However, this may not necessarily correspond to the observed
composition of maximum SRO in the C-A subsystem.
Finally, as discussed above, Eq. (16.21) is an approximate
expression for the configurational entropy that is only exact in
one dimension. Consider the case when ZA ¼ ZB ¼ Z and ΔgAB ≪ 0.
The solution is then completely ordered at XA ¼ XB ¼ 1/2 (where
XAB ¼ 1), and the configurational entropy should be zero at this
composition. However, as discussed above, this is only true when
Z ¼ 2. Therefore, for highly ordered solutions, better results are
expected if coordination numbers approximately equal to 2.0 are
used, even though this value is nonphysical in three dimensions.
On the other hand, for solutions with only a small degree of ordering and particularly for solutions with ΔgAB > 0 that exhibit clustering (as will be discussed in Section 16.4), larger values of ZA and ZB
of the order of 6 have been found to be necessary to yield good fits.
269
270
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
However, if one is restricted to constant coordination numbers,
then a problem again arises for multicomponent solutions. If
binary solutions A-B are highly ordered while binary solutions
B-C exhibit a tendency to immiscibility, then we should choose
Z 2 in the A-B system but Z 6 in the B-C system. However, with
constant coordination numbers, this is not possible if we want one
consistent set of parameters for the A-B-C ternary solution.
Accordingly, we permit ZA and ZB to vary with composition as
follows:
1
1
2nAA
1
nAB
+ A
(16.30)
¼ A
2nAA + nAB
ZA ZAA
ZAB 2nAA + nAB
1
1
2nBB
1
nAB
+ B
(16.31)
¼ B
2nBB + nAB
ZB ZBB
ZBA 2nBB + nAB
where ZAAA and ZAAB are the values of ZA when all nearest neighbors
of an A are As and when all nearest neighbors of an A are Bs,
respectively, and where ZBBB and ZBBA are defined similarly. Substitution into Eqs. (16.13), (16.14) gives
A
A
nA ¼ 2nAA =ZAA
+ nAB =ZAB
(16.32)
B
B
+ nAB =ZBA
nB ¼ 2nBB =ZBB
(16.33)
Eqs. (16.17)–(16.19) remain unchanged. The composition dependence of Eqs. (16.30), (16.31) was chosen because it results in the
simple linear relationships of Eqs. (16.32), (16.33). This simplifies
subsequent calculations, particularly calculations of chemical
potentials and calculations in multicomponent solutions.
The composition of maximum short-range ordering is determined by the ratio (ZBBA/ZAAB). Values of ZBBA and ZAAB are unique
to the A-B binary system, while the value of ZAAA is common to
all systems containing A as a component.
Clearly, of course, in the case of solid solutions,
ZAAA ¼ ZBBB ¼ ZAAB ¼ ZBBA necessarily.
16.2.4
Example of an Optimization Using the
MQM—The Mg-Sn System
As an example of an optimization using the MQM, the optimization of the Mg-Sn binary system by Jung et al. (2007) is
reproduced here.
The phase diagram with data points from several investigators
is shown in Fig. 16.16. The data have been thoroughly reviewed by
Nayeb-Hashemi and Clark (1988). Enthalpies of liquid-liquid mixing and activities of Mg in the liquid phase as reported by several
investigators are shown in Figs. 16.2 and 16.17, respectively. These
data have been reviewed by Jung et al. (2007).
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
271
1000
Grube, 1905
Kurnakow and Stepanow, 1905
Navak and Oelsen, 1968
Raynor, 1940
783
900
700
600
Liquid
563
500
a-Mg
Mg2Sn
Temperature (°C)
800
Steiner et al., 1964
Heycock and Neville, 1890
Ellmer et al., 1973
Vosskuhler, 1941
Grube and Vosskuhler, 1934
Willey, 1948
Hume-Rothery, 1926
400
300
L + Sn
200
200
100
0.0
0.1
0.2
0.3
Mg
0.4
0.5
0.6
0.7
0.8
0.9
Mole fraction Sn
1.0
Sn
Fig. 16.16 Optimized ( Jung et al., 2007) Mg-Sn phase diagram with experimental data points. (For references for the
data, see Jung et al., 2007.)
1.0
0.9
Eldridge et al., 1966: Isopiestic
Moser et al., 1990: EMF
Eckert et al., 1982: EMF
Sharma, 1970: EMF
Egan, 1974, EMF
0.8
0.7
Activity
Sn
Mg
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
Mg
0.1
0.2
0.3
0.4
0.5
0.6
Mole fraction Sn
0.7
0.8
0.9
1.0
Sn
Fig. 16.17 Optimized ( Jung et al., 2007) activities of Mg and Sn in liquid Mg-Sn solutions and experimental data points
at 800°C. (For references for the data, see Jung et al., 2007.)
272
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Data for the elements were taken from the compilation of
Dinsdale (1991). The entropy of Mg2Sn at 298.15 K was taken as
108.815 J mol1 K from the recommended value of Jelinek et al.
(1967), which was derived from their low-temperature heatcapacity data obtained by adiabatic calorimetry. Experimental
values of Δho298.15 of Mg2Sn as reported by six different investigators
vary from 81 to 66 kJmol1. Jung et al. selected an optimized
value of 80.000 kJ mol1 as giving the best simultaneous reproduction of all the data for the system. The temperature of fusion was
taken as 783°C with an enthalpy of fusion of 58.100kJ mol1 in agreement with the most recent experimental data. The heat capacity of
Mg2Sn was approximated by the Neumann-Kopp rule:
cP ðMg2 SnÞ ¼ 2cP ðMgÞ + cP ðSnÞ
(16.34)
This gives good agreement with the measurements of Sommer
et al. (1983) over the range 700 < T < 900°C.
The α-Mg solid solution was modeled with a one-sublattice
Bragg-Williams model:
o
o
2
+ XSn gSn
+ 20920 80333XMg
XSn Jmol1
g ðα MgÞ ¼ XMg gMg
(16.35)
A sharp negative peak in the liquid enthalpy of mixing near
XSn ¼ 1/3 is evident in Fig. 16.2. Accordingly, the MQM was used
Mg
with ZSn
SnMg/ZMgSn ¼ 2.0. No additional Bragg-Williams term was
used; that is, αMgSn0 in Eq. (16.26) was set to zero. The following
values of the coordination numbers were selected:
Sn
Sn
Mg
ZMg
MgMg ¼ ZSnSn ¼ 6, ZSnMg ¼ 6, and ZMgSn ¼ 3. The MQM pairexchange Gibbs energy, ΔgMgSn, was optimized as in Eq. (16.28) as
ΔgMgSn ¼ ð18409:6 3:7656 T Þ
2
+ 4602:4 XMgMg 2719:6 XMgMg
2510:4 XSnSn Jmol1
(16.36)
The calculated optimized phase diagram, enthalpy of liquid-liquid
mixing, and activities in the liquid are compared with the experimental data in Figs. 16.16, 16.2, and 16.17, respectively. The calculated optimized entropy of liquid-liquid mixing is shown in Fig. 16.3.
16.3 Second-Nearest-Neighbor
Short-Range Ordering in Ionic Liquids
Many ionic liquids exhibit second-nearest-neighbor (SNN)
SRO. For example, the enthalpy of liquid-liquid mixing of KClCoCl2 molten salt solutions shown in Fig. 16.18 exhibits the negative “V-shape” characteristic of SRO with a minimum near
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
0.0
−2.0
Δh (kJ mol-1)
−4.0
−6.0
−8.0
−10.0
−12.0
−14.0
−16.0
−18.0
−20.0
0.0
KCl
0.1
0.2
0.3
0.4 0.5 0.6 0.7
Mole fraction CoCl2
0.8
0.9
1.0
CoCl2
Fig. 16.18 Enthalpy of liquid-liquid mixing in KCl-CoCl2 solutions at 800°C
(FactSage).
XCoCl2 ¼ 1/3. Many other molten ACl-MCl2 solutions (where
A ¼ alkali and M ¼ Mg, Fe, Ni, etc.) exhibit similar V-shaped
enthalpy functions with minima near XMCl2 ¼ 1/3. Other examples
are liquid ACl-AlCl3 solutions (A ¼ alkali) with minima near
XAlCl3 ¼ 0.5, molten ACl-ZrCl4 (A ¼ alkali) solutions with minima
near XZrCl4 ¼ 1/3, MO-SiO2 solutions (M ¼ alkaline earth, Fe, Mn,
Co, etc.) with strong V-shaped minima near XSiO2 ¼ 1/3, and
A2O-SiO2 (A ¼ alkali) solutions with minima near XSiO2 ¼ 1/5. In
all these solutions, the entropies of liquid-liquid mixing exhibit
the “m-shape” characteristic of SRO.
16.3.1
Complex Anion Model
This behavior has sometimes been attributed to the formation
of “complex anions” such as CoCl4 2 , AlCl4 , ZrCl6 2 , and SiO4 4 .
For example, the KCl-CoCl2 solution in Fig. 16.18 could be
modeled as a Bragg-Williams reciprocal solution (K+,Co2+)
(Cl, CoCl4 2 ) with K+ and Co2+ ions mixing randomly on a cationic sublattice and Cl and CoCl4 2 ions mixing randomly on
an anionic sublattice. In this model, in the case of very strong
¼ 1/3 will consist mainly of
complexing, the solution at XCoCl2 K+ and CoCl4 2 ions, that is, K2 2 + CoCl4 2 . This model thus
accounts qualitatively for the V-shaped enthalpy and m-shaped
entropy of mixing. For a quantitative treatment, one must postulate partial dissociation of the complex ions:
CoCl4 2 ¼ Co2 + + 4Cl
(16.37)
273
274
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
The model thus necessitates the postulate of “two kinds of cobalt”
in solution: complexed cobalt as CoCl4 2 and uncomplexed Co2+
ions. Pure CoCl2 must be modeled as (Co2+)(CoCl4 2 ). That is, we
must suppose that some cobalt ions form complex ions with their
surrounding chlorine ions while other cobalt ions, which also
have chlorine ions as nearest neighbors, are not complexed.
Hence, although the concept of a complex anion may be physically realistic in a neutral solvent such as water, it cannot be satisfactorily defined in a molten salt solution.
16.3.2
Modified Quasichemical Model
As an alternative to the complex anion model, molten KCl-CoCl2
solutions can be modeled with the MQM as (K,Co)(Cl) with SRO
between SNN cation-cation pairs on the cationic sublattice. That
is, the anionic sublattice is occupied solely by Cl ions, and SNN
(K-(Cl)-Co) bonds are favored over (K-(Co)-K) and (Co-(Cl)-Co)
bonds. The pair-exchange reaction in Eq. (16.12) becomes.
ðK ðClÞ KÞ + ðCo ðClÞ CoÞ ¼ 2ðK ðClÞ CoÞ; ΔgKCo (16.38)
K
Co
K
with ZCo
CoK/ZKCo ¼ 2, where ZCoK and ZKCo are the SNN coordination
numbers. Good optimizations in many AX-MX2 solutions
(A ¼ alkali, X ¼ halogen, and M ¼ alkaline earth, Fe, Co, Ni, etc.) have
K
M
A
been obtained (FactSage) with ZM
MM ¼ ZKK ¼ ZMA ¼ 6 and ZAM ¼ 3.
Since ΔgKCo of the reaction in Eq. (16.38) is very negative, most
Co ions in solution when XCoCl2 1/3 (and even at somewhat
higher CoCl2 contents) are surrounded mainly by K+ ions in their
second coordination shells as illustrated schematically in
Fig. 16.19. With a first-nearest-neighbor cation-anion coordination number of 4, it can be seen from the figure that relatively isolated CoCl2
4 groupings occur, even though the model does not
treat these groupings explicitly as complex anions.
Similarly, the MQM involving SNN cation-cation SRO has been
applied to many other ionic liquids such as molten silicate solutions like CaO-SiO2 that are treated in Section 18.1 and to solutions of AlF3 with alkali and alkaline earth fluorides that are
discussed in Section 18.3.
16.4 Short-Range Ordering and Positive
Deviations From Ideal Mixing
As discussed in Sections 5.4 and 5.6, in a binary solution with
components A-B, if gE is sufficiently positive, then, at lower temperatures, the Gibbs energy function exhibits two minima resulting in a miscibility gap. With a Bragg-Williams model as in
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
K+
K+
Cl−
Co2+
Cl−
Co2+
Cl−
K+
K+
Cl−
K+
Co2+
K+
Fig. 16.19 Short-range ordering in molten KCl-CoCl2 solutions.
Section 14.3, a miscibility gap results if gE is sufficiently positive.
The solution then separates into A-rich and B-rich phases in order
to decrease the number of energetically unfavorable nearestneighbor (A-B) pairs.
In Fig. 16.20 are shown experimental data points along the
boundary of the liquid-liquid miscibility gap in the Ga-Pb system.
Also shown is the miscibility gap calculated (Kang and Pelton,
2013) with the Bragg-Williams model, Eq. (14.5) with
αGaPb ¼ 17950 + 1506ðXGa XPb Þ J mol1
(16.39)
where the two parameters were chosen so as to reproduce the
measured monotectic temperature and compositions. Clearly,
the consolute temperature of the calculated gap is much too high.
This is the case in most systems with miscibility gaps. The calculated gaps can only be made lower and flatter by adding many
empirical terms to the expansion for αAB, including
temperature-dependent (i.e., excess entropy) terms. However, in
most systems with liquid miscibility gaps, only limited data are
available. Very often, the boundaries of the gaps have been measured only at temperatures near the monotectic, not near the consolute temperature, and calorimetric data for the mixing
enthalpies are lacking. Hence, if empirical parameters are optimized, based only upon the measured compositions of the
boundaries of a miscibility gap at lower temperatures, the
275
276
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Ga-Pb System
700
Liquid
600
Temperature (°C)
500
Liquid1+ Liquid2
BW model (Eqn16.39)
MQM (Eqn16.40)
BW model corrected for sro (Eqn16.41)
400
300
Liquid +Pb(s)
200
100
Puschin et al. (1932)
Mathon et al. (1996), Onset of Cp change
Greenwood (1958–59)
Mathonet al. (1996), Inflection of Cp change
Predel (1959)
Plevachuk et al. (2003), Acoustic
Viscometric
Electrical conductivity
0
0.0
0.2
0.4
0.6
Mole fraction Pb
0.8
1.0
Fig. 16.20 Equilibrium diagram of the Ga-Pb system with liquid miscibility gap optimized with different models. (For
references for experimental data points, see Kang and Pelton, 2013.)
resultant calculated critical temperature will be much too high.
Many such examples can be found in the literature. Moreover,
even if a complete set of experimental data for a binary system
is available and these data have been adequately represented by
an empirical expansion for αAB with many terms, subsequent
attempts to use these binary parameters to estimate thermodynamic properties of ternary and higher-order systems will usually
give unsatisfactory results.
With the modified quasichemical model, the flattening of the
gap can be attributed to SRO. If the Gibbs energy change ΔgAB
of the pair-exchange reaction in Eq. (16.12) is positive, then (A-A)
and (B-B) pairs are favored over (A-B) pairs. In a random-mixing
BW model, the probabilities of (A-A), (B-B), and (A-B) pairs are
necessarily always YA2, Y2B, and 2YAYB, respectively. Hence, the
system can only reduce the number of energetically unfavorable
(A-B) pairs by separating into an A-rich and a B-rich phase. In
the MQM, however, clustering of A and B can occur within a
single-phase solution, thereby permitting an increase in the number of favorable (A-A) and (B-B) pairs without necessitating separation into two phases. Such clustering will be most pronounced
and have the greatest effect in lowering the Gibbs energy, in the
central composition region where XA XB. In the dilute terminal
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
composition regions, the configurational entropy terms predominate, and so, the solution tends toward random mixing. As a
result, the SRO has the largest effect on lowering the gap in the
central region, thereby producing the observed flattened shape.
The miscibility gap in the Ga-Pb system, calculated (Kang and
Pelton, 2013) with the MQM with ZGa ¼ ZPb ¼ 6 and with
ΔgGaPb ¼ ð1=3Þð18263 + 1506ðXGa XPb ÞÞ J mol1
(16.40)
is shown in Fig. 16.20. The two parameters were chosen so as to
reproduce the observed monotectic temperature and compositions. The observed gap at higher temperatures is well predicted,
as are experimental calorimetric enthalpies of liquid-liquid mixing
(see Kang and Pelton, 2013). Note that the coefficients of Eq. (16.40)
are temperature-independent. This shows that the entropy is well
represented by the configurational entropy expression of the
MQM; nonconfigurational terms are not required. It may also be
noted that the parameters of Eqs. (16.39), (16.40) are very similar
apart from the factor of 1/3 ¼ Z/2 that results from the fact that
ΔgAB in Eq. (16.12) is the Gibbs energy for the formation of two
moles of (A-B) pairs, whereas αAB in Eq. (14.5) is the Gibbs energy
for the formation of Z ¼ 6 mol of (A-B) pairs (see Eq. 4.47).
Examples for other binary systems with miscibility gaps are
given by Kang and Pelton (2013).
It is interesting that the MQM with Z ¼ 2 never results in a calculated miscibility gap, even for extremely large positive values of ΔgAB.
As ΔgAB becomes very large, the Gibbs energy of mixing approaches
zero at all compositions, but never exhibits two minima. See Pelton
and Kang (2007) for a demonstration. That is, in a one-dimensional
lattice, clustering within a single phase is sufficient to minimize the
Gibbs energy without separation into two phases. If one imagines a
one-dimensional lattice with long clusters (strings) of A and
B atoms, then the boundary between an A-cluster and a B-cluster
consists of a single (A-B) pair. In a two- or three-dimensional lattice
on the other hand, boundaries between A and B clusters contain
many (A-B) pairs, even when the clusters are large. Hence, separation into two phases is required in order to reduce sufficiently the
number of energetically unfavorable (A-B) pairs.
16.5
Approximating Short-Range Ordering
with a Polynomial Expansion
Kang and Pelton (2013) have derived a polynomial expansion
for gE within the Bragg-Williams model that approximates SRO
in binary systems as long as deviations from ideal random mixing
are not too large. Following the method of Førland (1964), the pair
277
278
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
fractions
are
written
as
XAA ¼ (X2A + y),
XBB ¼ (X2B + y)
and XAB ¼ (2XAXB 2y) (for the case where ZA ¼ ZB). These functions are then substituted into the MQM Eqs. (16.20), (16.21),
and the logarithmic terms are expanded as second-order Taylor
expansions. The resultant approximate quasichemical expression
for the molar Gibbs energy is
g ¼ XA gAo + XB gBo + RT ðXA ln XA + XB ln XB Þ + αAB XA XB
α2AB XA2 XB2 =ZRT
(16.41)
which is in Bragg-Williams form. The final term in Eq. (16.41),
which is always negative, is an approximate correction for SRO.
The function αAB may be expanded as a polynomial as in
Eq. (14.5). The factor Z should generally be set equal to 6.
The miscibility gap in the Ga-Pb system, calculated with
Eq. (16.41) with the expression for αAB of Eq. (16.39), is shown
in Fig. 16.20.
16.6 Modified Quasichemical
Model—Multicomponent Systems
In a multicomponent solution, there is a pair-exchange reaction like Eq. (16.12) for each nearest-neighbor pair:
ðm mÞ + ðn nÞ ¼ 2ðm nÞ; Δgmn ðm, n ¼ 1, 2, 3, …, N Þ
(16.42)
Let nm and Zm be the number of moles and coordination number
of component m, and let nmn be the number of moles of (m-n)
pairs (where nmn and nnm represent the same quantity and can
be used interchangeably). Then, by generalizing Eqs. (16.13),
(16.14),
Zm nm ¼ 2nmm +
X
nmn
(16.43)
n>m
Pair fractions Xmn, overall mole (or site) fractions Xm, and
coordination-equivalent fractions Ym are defined by generalizing
Eqs. (16.15)–(16.17) as
X
Xmn ¼ nmn = nij
(16.44)
X
(16.45)
X m ¼ n m = ni
X
X
Ym ¼ Zm nm = ðZi ni Þ ¼ Zm Xm = ðZi Xi Þ
(16.46)
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
(where Xmn and Xnm represent the same quantity and can be used
interchangeably). Substitution of Eq. (16.43) into Eqs. (16.44),
(16.46) gives, by generalizing Eqs. (16.18), (16.19),
X
Xmn =2
(16.47)
Ym ¼ Xmm +
n>m
The Gibbs energy of the solution is given by
X
X
o
G¼
nm gm
T ΔSconfig +
nmn ðΔgmn =2Þ
(16.48)
n>m
where, by generalizing Eq. (16.21), ΔSconfig is given as
X
ΔSconfig ¼ R nm ln Xm
R
X
nmm ln
Xmm =Ym2
+
X
!
nmn ln ðXmn =2Ym Yn Þ
m>n
(16.49)
In each m-n binary system, the energy parameter Δgmn of
Eq. (16.48) may be expanded as an empirical polynomial in the
component fractions Ym as in Eq. (16.27):
X
o
j
Δgmn ¼ Δgmn
+
qimn
Ymi Ynj
(16.50)
ð i + jÞ1
or
o
+
Δgmn ¼ Δgmn
X
i
lmn ðYn Ym Þi
(16.51)
i1
or by an expansion in terms of the pair fractions Xmm and Xnn as in
Eq. (16.28):
X
o
ij
i
j
+
gmn
Xmm
Xnn
(16.52)
Δgmn ¼ Δgmn
ði + jÞ 1
By generalizing Eqs. (16.30), (16.31), composition-dependent
coordination numbers are introduced:
!
1
1
2n11 X n1j
X
¼
+
(16.53)
1
Z1
Z11
Z1
n1j
2n11 +
j6¼1 1j
j6¼1
1
1
X
¼
Z2
n2j
2n22 +
j6¼2
2n22 X n2j
+
2
Z22
Z2
j6¼2 2j
!
(16.54)
etc.
(Note that Z112 and Z121 represent the same quantity and can be
used interchangeably.)
279
280
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Zm
mm is a constant for each pure component m and is the same
for all solutions containing m. The composition of maximum
short-range ordering in each binary subsystem is determined by
n
the ratio Zm
mn/Znm. Substitution into Eq. (16.43) gives
X
m
m
nm ¼ 2nmm =Zmm
+
nmn =Zmn
(16.55)
n>m
We now define standard Gibbs energies gomm and gomn as
o
o
m
¼ 2gm
=Zmm
gmm
o
gmn
o
¼ Δgmn
=2
+
o
m
gm
=Zmn
(16.56)
+
n
gno =Znm
(16.57)
where Δgomn is the binary parameter from Eqs. (16.50), (16.51), or
(16.52). Substitution into Eq. (16.48) then gives
o
o
o
o
+ n12 g12
+ n22 g22
+ n13 g13
+⋯
G ¼ n11 g11
X
o
ðnmn =2Þ Δgmn Δgmn
T ΔSconfig +
(16.58)
n>m
with ΔS
given by Eq. (16.49)
Pand Δgmn by Eq. (16.50), (16.51),
or (16.52). Dividing though by nij gives
o
o
o
o
+ X12 g12
+ X22 g22
+ X13 g13
+⋯
g ðper mole of pairsÞ ¼ X11 g11
config
+ RT ðX11 ln X11 + X12 ln X12 + X22 ln X22 + ⋯Þ
X
2Xm
ln Xm X11 ln Y12 X22 ln Y22
+ RT
Zm
X12 ln ð2Y1 Y2 Þ ⋯ + g E
(16.59)
where
gE ¼
X12 X13 o
o
+
+ :::
Δg12 Δg12
Δg13 Δg13
2
2
(16.60)
Eqs. (16.50)–(16.52) only apply in the binary subsystems. It is now
required to write expressions for Δgmn for compositions within the
N-component system for use in Eq. (16.58). To this end, we
employ a strategy similar to that described in Section 14.8.
16.6.1
Interpolation Formulae
When Dgmn in a Binary System is Given by Eq. (16.50) or (16.51)
Suppose Δg12 in the 1-2 binary subsystem has been expressed
as a polynomial in the fractions Y1 and Y2 by Eq. (16.50). We now
want an expression for Δg12 in a multicomponent solution. We
consider first a ternary system 1-2-3.
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
281
Symmetric Model
With the symmetrical model illustrated in Fig. 16.21A (cf.
Fig. 14.4A), Δg12 in the ternary solution is given by
2
o
Δg12 ¼ 4Δg12
+
+
X
X
ði + jÞ1
ijk
q12ð3Þ
k1
ij
q12
Y1
Y1 + Y2
Y1
Y1 + Y2
i i Y2
Y1 + Y 2
Y2
Y1 + Y2
j
j
3
5
Y3k
(16.61)
i0
j0
where the first term on the right-hand side of Eq. (16.61) is constant
along the line 3-a in Fig. 16.21A and equal to Δg12 in the 1-2 binary
system at point a (where (Y1 + Y2) ¼ 1). That is, it is assumed that the
binary 1-2 pair-exchange energy is constant at constant Y1/Y2 ratio.
The second summation in Eq. (16.61) consists of “ternary terms”
that are all zero in the 1-2 binary system and that give the effect
of the presence of component 3 upon the energy Δg12 of the 1-2 pair
interactions. The empirical ternary coefficients qijk
12(3) are found by
optimization of experimental ternary data.
Similar equations give Δg23 and Δg31 with the binary terms
equal to their values at points b and c, respectively, in
ijk
Fig. 16.21A and with ternary coefficients qijk
23(1) and q31(2) that give
the effect of the presence of component 1 upon the pair energy
Δg23 and of component 2 upon Δg31, respectively. As discussed
in Section 14.8.3, this functional form for the ternary terms is preferable to expressions such as Yi1Yj2Yk3.
Alternatively, if Δg12 is expressed in the 1-2 binary system by a
Redlich-Kister expansion as in Eq. (16.51), then the first
1
1
c
a
p
2
(A)
b
Y2
(Y1 + Y2)
a
= constant
3
2
Y1 = constant
p
b
c
3
(B)
Fig. 16.21 (A) Symmetrical (left) and (B) asymmetrical (right) schemes for interpolation from binary to ternary solutions.
282
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
summation in Eq. (16.61) is replaced by
Xi
i1
Y1 Y2 i
l12
, which
Y1 + Y2
is also constant along the line 3-a and equal to Δg12 at point a.
Asymmetric Model
With the asymmetrical model illustrated in Fig. 16.21B (cf.
Fig. 14.4B), Δg12 in the ternary solution is given by
2
o
Δg12 ¼ 4Δg12
+
+
X
3
X
ij
ði + jÞ1
q12 Y1i ð1 Y1 Þj 5
ijk
q12ð3Þ Y1i ð1 Y1 Þj
k1
Y3
Y2 + Y3
k
(16.62)
i0
j0
where the binary terms are constant along the line ac and equal to
their values at point a in Fig. 16.21B. A similar expression is written
for Δg31, while Δg23 is given by an expression similar to Eq. (16.61).
If Δg12 is expressed in the binary system by a Redlich-Kister
expansion,
then the binary terms in Eq. (16.62) are replaced by
Xi
l 12 ð2Y1 1Þi :
i1
Multicomponent Solutions
In order to extend this symmetrical/asymmetrical dichotomy
to N-component solutions while still maintaining complete flexibility to treat any ternary subsystem as symmetrical or asymmetrical, we define, as in Eq. (14.45),
X
Yk
(16.63)
ξij ¼ Yi +
k
where the summation is over all components k in asymmetrical
i-j-k ternary subsystems in which j is the asymmetrical component. This is best presented by an example. For the five-component
system in Fig. 16.22, the ternary subsystems 1-2-3 and 1-2-5 are
asymmetrical with 1 and 5, respectively, as asymmetrical components, while the system 1-2-4 is symmetrical and so on, as illustrated schematically in the figure. In this example, then,
ξ21 ¼ Y2 + Y3
ξ12 ¼ Y1
ξ13 ¼ Y1 + Y4
ξ31 ¼ Y3 + Y2
ξ14 ¼ Y1
ξ41 ¼ Y4
ξ15 ¼ Y1 + Y2
ξ51 ¼ Y5
ξ23 ¼ Y2 + Y5
ξ32 ¼ Y3 + Y4
(16.64)
ξ24 ¼ Y2
ξ42 ¼ Y4 + Y3
ξ25 ¼ Y2 + Y1 + Y4 ξ52 ¼ Y5
ξ34 ¼ Y3
ξ43 ¼ Y4 + Y1
ξ35 ¼ Y3
ξ53 ¼ Y5 + Y2
ξ45 ¼ Y4 + Y2
ξ54 ¼ Y5
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
1
2
1
3
2
1
4
1
4
2
2
5
3
1
5
3
2
4
1
4
3
2
3
5
4
283
5
3
5 4
5
Fig. 16.22 A five-component example solution illustrating which ternary subsystems are to be treated with a
symmetrical model and which are to be treated with an asymmetrical model. The dots indicate the asymmetrical
components.
Then, in the multicomponent solution,
X ξ12 i ξ21 j i j
X ξ12 i ξ21 j
o
Δg12 ¼ Δg12 +
q12 +
ξ12 + ξ21
ξ12 + ξ21
ξ12 + ξ21
ξ12 + ξ21
ði + jÞ1
i0
j0
"
k1
X
l
ijk
q12ðlÞ Yl ð1 ξ12 ξ21 Þk1
#
X ijk Ym Y2 k1 X ijk
Yn
Y1 k1
1
1
+
q12ðmÞ
+
q12ðnÞ
ξ21
ξ21
ξ12
ξ12
m
n
(16.65)
where the summations of ternary terms are over (i) all components l in 1-2-l ternary subsystems that are either symmetrical
or in which l is the asymmetrical component, (ii) all components
m in 1-2-m subsystems in which 1 is the asymmetrical component, and (iii) over all components n in 1-2-n subsystems in which
2 is the asymmetrical component.
Eq. (16.65) reduces to Eq. (16.61) in symmetrical ternary
subsystems (where (Y1 + Y2 + Y3) ¼ 1) and to Eq. (16.62) in asymmetrical ternary subsystems, as can be verified by substitution of
Eq. (16.64). The form of the ternary terms in Eq. (16.65) has been
chosen such that if two components, say 3 and 4, have the same
ijk
effect upon Δg12 (i.e., if qijk
12(3) ¼ q12(4)), then the effect of the presence of both 3 and 4 will be additive as was discussed in
Section 14.8.4. If a Redlich-Kister expansion as in Eq. (16.51) is used
284
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
in the 1-2 binary system, then the binary
X i terms (the first summation) in Eq. (16.65) are replaced by
l 12 ½ðξ12 ξ21 Þ=ðξ12 + ξ21 Þi.
i1
As a special example of Eq. (16.63), suppose all components are
grouped as either “acids” (group I) or “bases” (group II), where a
ternary system is symmetrical if all three components are in the
same group, and asymmetrical otherwise. It then follows from
the definition of ξij that ξ12 ¼ Y1 and ξ12/(ξ
P 12 + ξ21 ) ¼
Y1/(Y1 + Y2 ) if 1 and 2 are in the same group, while ξ12 ¼ group I Yi ¼
ξI and also ξ12/(ξ12 + ξ21) ¼ ξI if 1 and 2 are in groups I and II,
respectively.
When Dgmn in a Binary System is Given by Eq. (16.52)
Suppose that Δg12 in the binary system has been expressed as
in Eq. (16.52) by a polynomial in terms of the pair fractions. Let us
consider first a ternary system 1-2-3.
Symmetric Model
In a symmetrical ternary system as in Fig. 16.21A, the following
equation for Δg12 is proposed:
i j
X ij X11
X22
o
Δg12 ¼ Δg12 +
g12
X11 + X12 + X22
X11 + X12 + X22
ði + jÞ1
+
X
i0
ijk
g12ð3Þ
X11
X11 + X12 + X22
i X22
X11 + X12 + X22
j
Y3k
j0
k1
(16.66)
As with Eq. (16.52), in practice, it is usually sufficient to include only
terms with i ¼ 0 or with j ¼ 0. The form of this expression has been
chosen for the following reason. As Δg12, Δg23, and Δg31 become
small, the solution approaches ideality and Y11 ! Y21 , Y22 ! Y22
and Y12 ! 2Y1Y2. In this case, X11/(X11 + X12 + X22) ! [Y1/(Y1 + Y2)]2,
and Eq. (16.66) approaches Eq. (16.61) that, in the limit, becomes
the Kohler equation (Section 14.8) for symmetrical ternary systems.
From Eq. (16.47), it can be seen that the factor Y3 in the ternary
terms in Eq. (16.66) is equal to (X33 + X31/2 + X23/2). In principle,
the effect of these three terms upon Δg12 could easily be represented by three independent ternary coefficients. However, this
additional complexity is probably not required.
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Asymmetric Model
In an asymmetrical ternary system as in Fig. 16.21B, the following equation is proposed:
X ij
o
i
Δg12 ¼ Δg12
+
g12 X11
ðX22 + X23 + X33 Þj
ði + jÞ1
+
X
i0
j0
k1
ijk
i
g12ð3Þ X11
ðX22
j
+ X23 + X33 Þ
Y3
Y2 + Y3
k
(16.67)
In the limit of ideality, this equation reduces to Eq. (16.62) that, in
the limit, becomes the Kohler/Toop equation (Section 14.8) for
asymmetrical ternary systems.
Multicomponent Solutions
To extend this symmetrical/asymmetrical dichotomy to multicomponent solutions, we define
X X
X
X
Xij =
Xij
(16.68)
χ 12 ¼
i¼1, k j¼1, k
i¼1, 2, k , l j¼1, 2, k , l
where k represents all values of k in 1-2-k asymmetrical ternary
subsystems in which 2 is the asymmetrical component and l represents all values of l in asymmetrical 1-2-l subsystems in which 1
is the asymmetrical component. χ ij is analogous to the ratio
ξij/(ξij + ξji) defined by Eq. (16.63) and used in Eq. (16.65). Taking
the example of Fig. 16.22, from Eq. (16.64),
ξ23
ðY 2 + Y 5 Þ
¼
ξ23 + ξ32 ðY2 + Y5 Þ + ðY3 + Y4 Þ
(16.69)
whereas
χ 23 ¼
ðX22 + X55 Þ + X25
ðX22 + X55 Þ + ðX33 + X44 Þ + X25 + X23 + X24 + X53 + X54 + X34
(16.70)
That is, starting from Eq. (16.69), we can write the expression for
χ 23 in Eq. (16.70) by replacing the sums (Yi + Yj + Yk + …) in the
numerator and denominator of Eq. (16.69) by the sum of
(Xii + Xjj + Xkk + ⋯) plus all cross terms (Xij + Xik + Xjk + ⋯).
In the case where all components are grouped as either “acids”
(group I) or “bases” (group II), it follows from the definitions and
+ X12 + X22) if components 1 and
from Eq. (16.47) that χ 12 ¼ X11/(X11 P
2 are in the same group and χ 12 ¼ group I Yi ¼ ξI if components 1
and 2 are in groups I and II, respectively.
285
286
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
In the multicomponent system,
o
Δg12 ¼ Δg12
+
X
ði + jÞ 1
j
ij
χ i12 χ 21 g12 +
X
"
j
χ i12 χ 21
i0
X
l
ijk
g12ðlÞ Yl ð1 ξ12 ξ21 Þk1
j0
k1
+
X
m
ijk
g12ðmÞ
Ym
ξ21
Y2
1
ξ21
k1
+
X
n
ijk
g12ðnÞ
Yn
ξ12
Y1
1
ξ12
k1 #
(16.71)
where the summations over l, m, and n are as was described following Eq. (16.65). Eq. (16.71) reduces to the correct symmetrical
or asymmetrical ternary equation, either Eq. (16.66) or (16.67), in
any ternary subsystem. The analogy to Eq. (16.65) is evident.
If the quasichemical and Bragg-Williams models are combined
as discussed in Section 16.2.1, then the following additional
Bragg-Williams excess term is added to Eq. (16.48):
X
X
E
¼
Zi n i
Ym Yn α0mn
(16.72)
GBW
m6¼n
Interpolation formulas to estimate the binary αmn0 parameters
within the multicomponent solution are then as discussed in
Section 14.8.
16.7 The MQM Equations in Closed
Explicit Form
Substitution of Eq. (16.65) or (16.71) into Eq. (16.58) gives an
equation for G in terms of the pair fractions Xmn. For any overall
composition given by (n1, n2, …, nN), the equilibrium pair fractions are then calculated by minimizing G subject to the mass balance constraints of Eq. (16.55). This Gibbs energy minimization is
greatly facilitated by the form of Eq. (16.58). When expressed per
mole of pairs as in Eq. (16.59), the Gibbs energy equation is identical, apart from the second configurational entropy term to the
equations of Chapter 14 for the Gibbs energy of a randomly mixed
solution of the end-members mn whose Gibbs energies are given
by Eqs. (16.56), (16.57). The “excess” terms in Eq. (16.60) are polynomials in the fractions Xmn since Δgmn from Eq. (16.65) or (16.71)
is expressed only in terms of Xmn (Ym being given in terms of Xmn
by Eq. 16.47). Therefore, the same existing algorithms and computer subroutines that are used for Bragg-Williams polynomial
models can be used directly for the quasichemical model by simply including the extra entropy terms. That is, a significant
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
simplification is achieved by formally considering the mn pairs as
the end-members of the solution.
A further computational simplification can be achieved by
formally treating these end-members as “associates” or
n
m n1=Z n .
For example, if Zm
“molecules” m1=Zmn
mm ¼ 6, Znn ¼ 6,
nm
m
n
Zmn ¼ 3, and Znm ¼ 6, then the formal components would be
m2/6, n2/6, and m1/3n1/6. Setting
nm1=Z m
mn
n1=Z n
nm
¼ nmn
(16.73)
and substituting into Eq. (16.55) gives
X
m
nm ¼ 2nm2=Z m =Zmn
+
n
n6¼m m1=Z m
mn
mn
n1=Z n
nm
(16.74)
which then becomes a “true” chemical (as opposed to a quasichemical) mass balance in that the number of moles of m is the
same on both sides of the equation.
The fact that Eq. (16.59) is written solely in terms of the
fractions Xmn of the end-members mn permits the chemical
potentials to be easily calculated in closed explicit form. The
chemical potential of m is given by Pelton et al. (2000):
m
=2 ð∂G=∂nmm Þnij
(16.75)
μm ¼ ð∂G=∂nm Þni ¼ Zmm
Hence,
o
μm ¼ gm
m Zmm
Xmm
E
RT ln 2 + gmm
+ RT ln XA +
2
Ym
(16.76)
where the partial excess term gEmm can be calculated in the usual
way from the polynomial expansion for gE in Eq. (16.60):
X
E
gmm
¼ g E + ∂g E =∂Xmm Xij ∂g E =∂Xij
(16.77)
ðij6¼mmÞ
where Eq. (16.47) is used to express Ym in terms of Xmn.
16.8
Combining Bragg-Williams and MQM
Models in One Multicomponent
Database
If the Bragg-Williams excess term of Eq. (16.72) is added to
Eq. (16.48) and if further all quasichemical parameters Δgmn are
set to zero, the Bragg-Williams terms then will have no effect upon
the calculated pair distributions, and so, the Gibbs energy minimization will result in a random mixture. That is, the second entropy
term in Eq. (16.59) will automatically become zero, and the result
287
288
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
will be the same as if a Bragg-Williams expression had been used
for G. Hence, the same model algorithm that is used for the quasichemical model can be used for the random-mixing BraggWilliams model.
The major importance of this fact is that it is now possible to
mix models in one multicomponent solution database. This is
of much practical value. If some binary subsystems of a multicomponent solution have been optimized with the MQM while
others have been optimized with a Bragg-Williams model, all
the model parameters can be combined in one database for the
multicomponent solution without necessitating the reoptimization of any systems. For further discussion on this point, see
Pelton and Chartrand (2001).
16.9 Comparison of the Bragg-Williams,
Associated and Modified
Quasichemical Models in Predicting
Ternary Properties From Binary
Properties
The choice of a model must be based not only on how well it
reproduces data in binary systems but also on how well it predicts
and represents properties of ternary solutions.
Consider a ternary solution with components A, B, and C in
which the binary solution A-B exhibits a strong tendency to
SRO at the AxBy composition, while the B-C and C-A binary solutions are less strongly ordered. In such a case, positive deviations
from ideal mixing behavior will be observed, centered along the
AxBy-C join. An example is the Al-Sc-Mg system in which Al-Sc liquid solutions exhibit SRO centered around the AlSc composition
(see Figs. 16.4–16.6). The liquidus surface (Pelton and Kang, 2007)
of this system is shown in Fig. 16.23. Positive deviations along and
near the AlSc-Mg join are manifested by the flat liquidus surfaces
(witnessed by the widely spaced isotherms) of the AlSc, AlSc2, and
€ bner et al.,
Al2Sc compounds. Experimental liquidus points (Gro
1999) along the Al2Sc-Mg join are shown in Fig. 16.24.
Indeed, if the SRO in the A-B binary solution is sufficiently
stronger than in the B-C and C-A binary solutions, a miscibility
gap will be observed, centered along the AxBy-C join. The positive
deviations (tendency to immiscibility) along the join are simply
explained as follows. Since (A-B) nearest-neighbor pairs are energetically favored, the solution exhibits a tendency to separate into
an AxBy-rich and a C-rich solution in order to maximize the number of such pairs.
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
289
Sc
0.2
0.4
AlSc2
0.6
0.4
0.6
0.8
(Sc)
AlSc
0.8
0.2
Al2Sc
Al3Sc
0.2
Al
0.4
0.6
0.8
Mg
Mole fraction
Fig. 16.23 Optimized polythermal liquidus projection of the Al-Sc-Mg system. Reprinted with permission from Pelton, A.D.,
Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
Temperature (°C)
1500
Liquid
Bragg-Williams
model
Quasichemical
model (Z = 6)
1000
Al2Sc + Liquid
Associate
model
500
Al2Sc + Mg
Gröbner et al. (1999)
00
Al2Sc
0.2
0.4
0.6
Atomic fraction Mg
0.8
1
Mg
€bner et al., 1999) compared
Fig. 16.24 Al2Sc-Mg join in the Al-Mg-Sc phase diagram. Experimental liquidus points (Gro
with calculations from optimized binary parameters with various models. Reprinted with permission from Pelton, A.D., Kang,
Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Consider the simplest case of a solution A-B-C in which the
B-C and C-A binary solutions are ideal while the A-B binary solution has strong negative derivations centered at the equimolar
composition AB with a minimum value of gEAB ¼ 40 kJ mol1 in
the binary solution at XA ¼ XB ¼ 0.5.
If a Bragg-Williams regular solution model is used for this
example, then in Eq. (14.4) αAB ¼ 40/(0.5)(0.5) ¼ 160 kJ mol1
while αBC ¼ αCA ¼ 0. In the ternary solution, we set
g E ¼ XA XB αAB + XB XC αBC + XC XA αCA
(16.78)
combined with an ideal entropy of mixing. The resultant calculated miscibility gap, centered on the AB-C join, with tie-lines
aligned with the join, is shown as the “Bragg-Williams model”
in Fig. 16.25.
It is instructive to calculate the expression for the excess Gibbs
energy along the join when XC moles of liquid C are mixed with
(1 XC) moles of the equimolar solution A0.5B0.5. In the present
example, this becomes
gAE0:5 B0:5 C ¼ XA XB αAB ð1 XC Þ XA XB =ðXA + XB Þ2 αAB
¼ XC ð1 XC ÞðαAB =4Þ
(16.79)
0.6
0.4
0.8
0.2
A
Bragg-Williams model
Z = 60
Z = 12
Z=6
Z=4
0.4
0.6
AB
B
0.2
0.8
290
0.2
0.4
0.6
0.8
Mole fraction
C
Fig. 16.25 Miscibility gaps calculated for an A-B-C system at 1100°C from the
Bragg-Williams model and from the MQM when the B-C and C-A binary solutions
are ideal and the A-B binary solution has a minimum excess Gibbs energy of
40 kJ mol1 at the equimolar composition. Calculations from the MQM for various
values of Z. Tie-lines are aligned with the AB-C join. Reprinted with permission from
Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
where the substitution XA ¼ XB ¼ (1 XC)/2, which holds along the
AB-C join, has been made. That is, along the join, the solution
behaves like a regular solution of A0.5B0.5 and C, with a regular
solution parameter equal to ( αAB/4).
Experience with many systems has shown that, in general, the
positive deviations (and tendency to immiscibility) along the join
as predicted by the Bragg-Williams model are too large. For example,
in Fig. 16.24 is shown the Al2Sc-Mg phase diagram section calculated
solely from optimized binary parameters when the Bragg-Williams
model is used. The model predicts a miscibility gap where none is
observed. The reason for this overestimation of the positive deviations is that SRO is ignored. In reality, through ordering, the solution
can increase the number of (A-B) pairs above that of a randomly
mixed solution, thereby decreasing its Gibbs energy and hence
decreasing the driving force to separate into two phases.
One might therefore conclude that accounting for SRO through
the use of the associate model would improve the predictions in the
ternary system. Exactly the opposite is true; if the SRO in the A-B
binary solution is modeled with the associate model, no positive
deviations whatsoever are predicted along the AxBy-C ternary join.
In fact, in the limit of very strong SRO in the A-B binary solution
and ideal mixing in the B-C and C-A liquids, the associate model
simply predicts that, along the AxBy-C join, the solution is an ideal
mixture of AxBy associates and C atoms randomly distributed on
the quasi-lattice. For example, in Fig. 16.24 is shown the Al2Sc-Mg
phase diagram section calculated from optimized binary parameters when the associate model is used.
This failure of the associate model to predict the observed positive deviations and tendency to immiscibility in ternary systems
with SRO is the strongest argument against its use.
When the modified quasichemical model is applied to the
example of Fig. 16.25, the parameter ΔgAB ≪ 0, and ΔgBC ¼ ΔgCA ¼ 0.
In Fig. 16.25 are shown miscibility gaps calculated with various
values of Z where the value of the (constant) parameter ΔgAB was
adjusted in each case to give the same minimum value of
ΔgEAB ¼ 40 kJ mol1 in the A-B system. As Z is decreased, the critical temperature of the calculated gap decreases, and the
gap changes shape, becoming narrower and elongated along the
AB-C join in accordance with the shape of observed gaps. At
the same time, the calculated gap becomes “flatter” similar to
the case of binary miscibility gaps discussed in Section 16.4.
Fig. 16.24 shows the Al2Sc-Mg phase diagram section calculated solely from optimized binary parameters when the quasichemical model with Z ¼ 6 is used.
Similar examples for ternary oxide slags have been shown and
discussed by Pelton (2004).
291
292
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
16.10 The Two-Sublattice “Ionic
Liquid” Model
A different model for treating SRO in liquid solutions was introduced by Hillert et al. (1985) and has become known as the “ionic
liquid model,” although it is applicable to nonionic liquids as well.
Although it is a two-sublattice model, it can be used to treat SRO in
binary metallic solutions and so is introduced in this chapter.
The model is best introduced by an example. Molten Mg-Bi
solutions show a strong tendency to SRO around the composition
Mg3Bi2, with negative V-shaped enthalpy of mixing and m-shaped
entropy of mixing functions. In the ionic liquid model, the solu2+
2
0
tion is modeled as (Mg2+)2+x2y(Bi3
x Va1xy Biy )2. That is, Mg
3
ions occupy a cationic sublattice, while Bi ions, charged vacancies, and neutral Bi0 atoms occupy an anionic sublattice. The
number of lattice sites varies with composition. In this example,
the number of anionic sites is constant, but in the general case,
the number of sites on both sublattices can vary with composition. The species on each sublattice are distributed randomly.
The overall mole fraction of Bi is
XBi ¼ ð2x + 2y Þ=ðð2x + 2y Þ + ð2 + x 2y ÞÞ ¼ 2ðx + y Þ=ð3x + 2Þ (16.80)
Hence, when XBi ¼ 1, y ¼ 1, and x ¼ 0. When XBi ¼ 0, x ¼ 0, and
2
(pure Mg), (2)
y ¼ 0. The end-members are thus (1) Mg2+
2 Va
2+
3
Mg0Bi2 (pure Bi), and (3) Mg3 Bi2 (completely ordered Mg3Bi2
with x ¼ 2 and y ¼ 0).
Let the anionic site fractions of the end-members be
X1 ¼ XVa2 ¼ ð1 x y Þ
(16.81)
X2 ¼ XBi0 ¼ y
(16.82)
X3 ¼ XBi3 ¼ x
(16.83)
Then, the Gibbs energy of the solution per mole of atoms is given by
g ¼ X1 g10 + X2 g20 + X3 g30 + 2RT ðX1 ln X1 + X2 ln X2 + X3 ln X3 Þ
+ ðα12 X1 X2 + α23 X2 X3 + α31 X3 X1 Þ
(16.84)
goi
where the are the Gibbs energies of the end-members per mole
of atoms and the final term in parentheses is an excess Gibbs
energy.
At any given overall composition, the variables x and y are
related by Eq. (16.80). The equilibrium values of x and y are found
by minimizing the Gibbs energy. When the model parameter
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
go3 ≪ 0, the solution at XBi ¼ 2/5 consists almost entirely of
3–
Mg2+
3 Bi2 (i.e., X1 0 and X2 0), and the enthalpy and entropy
of mixing have the V-shape and m-shape characteristic of SRO.
Similarly to the case of the associate model in Section 16.1, the
ionic liquid model does not reduce to an ideal solution expression
when go3 ¼ 0 but only when go3 ¼ ∞ (when x ¼ 0 at all compositions).
If one sets go3 ¼ ∞ (or simply removes Bi3 as a species), the model
0
reduces to (Mg2+)22y(Va2
1y Biy )2. The configurational entropy of
mixing then formally results from the random distribution of
Va2 and Bi0 species on the anionic sublattice. While numerically
identical to the ideal entropy expression for the random mixing of
Mg and Bi atoms on a single sublattice, this is physically unrealistic, as is the requirement that there are “two kinds of Bi,” namely,
Bi0 atoms and Bi3 ions.
The model is developed in detail by Hillert et al. (1985) and by
Sundman (1991) who give examples of applications to multicomponent solutions. An example of the application to a ternary molten oxide solution will be given in Section 18.1.3.
References
Blander, M., Pelton, A.D., 1987. Geochim. Cosmochim. Acta 51, 85–95.
Chartrand, P., Pelton, A.D., 2001. Metall. Mater. Trans. 32A, 1397–1407.
Dinsdale, A.T., 1991. Calphad J. 15, 317.
Førland, T., 1964. Fused Salts. In: Sundheim, B.R. (Ed.), McGraw Hill, NY, pp. 86–89.
Fowler, R.H., Guggenheim, E.A., 1939. Statistical Thermodynamics. Cambridge
University Press, Cambridge, p. 350.
€ bner, J., Schmid-Fetzer, R., Pisch, A., Cacciamani, G., Riani, P., Parodi, N.,
Gro
Borzone, G., Saccone, A., Ferro, R., 1999. Z. Metallkd. 90, 872–880.
Guggenheim, E.A., 1935. Proc. R. Soc. Lond. A148, 304–312.
Hillert, M., Jansson, B., Sundman, B., Agren, J., 1985. Metall. Trans. 16A, 261–266.
Jelinek, F.J., Shickell, W.D., Gerstein, B.C., 1967. J. Phys. Chem. Solids 28, 267–270.
Jung, I.-H., Kang, D.-H., Park, W.-J., Kim, N.J., Ahn, S., 2007. Calphad J. 31, 192–200.
Kang, Y.-B., Pelton, A.D., 2013. J. Chem. Thermodyn. 60, 19–24.
Litovskii, V.V., Valishev, M.G., Esin, Y.O., Gel’d, P.V., Petrushevskii, M.S., 1986. Russ. J.
Phys. Chem. 60, 1385–1386.
€ ck, R., Gerling, U., Predel, B., 1989. Z. Metallkd. 80, 270–275.
Lu
Nayeb-Hashemi, A.A., Clark, J.B., 1988. Phase Diagrams of Binary Magnesium
Alloys. ASM International, Materials Park, OH, pp. 293–304.
Notin, M., Gachon, J.C., Hertz, J., 1982. J. Chem. Thermodyn. 14, 425–434.
Pelton, A. D., 2004. Proceedings of the 7th International Conference on Molten
Slags, Fluxes and Salts, Johannesburg: SAIMM, pp. 607–614.
Pelton, A. D., Blander, M., 1984. Proceedings of the 2nd International Symposium
on Metallurgical Slags and Fluxes, Eds. H. A. Fine and D. R. Gaskell, Warrendale,
PA: TMS-AIME, pp. 281–294.
Pelton, A.D., Chartrand, P., 2001. Metall. Mater. Trans. 32A, 1355–1360.
Pelton, A.D., Chartrand, P., Eriksson, G., 2001. Metall. Mater. Trans. 32A, 1409–1416.
293
294
Chapter 16 SINGLE-LATTICE MODELS WITH SHORT-RANGE ORDERING (SRO)
Pelton, A.D., Decterov, S.A., Eriksson, G., Robelin, C., Dessureault, Y., 2000. Metall.
Mater. Trans. 31B, 651–659.
Pelton, A.D., Kang, Y.-B., 2007. Int. J. Mater. Res. 98, 907–917.
Sommer, F., Lee, J.-J., Predel, B., 1983. Z. Metallkd. 74, 100–104.
Sundman, B., 1991. Calphad J. 15, 109–119.
List of Websites
FactSage, Montreal. www.factsage.com.
17
MODELING SHORT-RANGE
ORDERING WITH TWO
SUBLATTICES
CHAPTER OUTLINE
17.1 Introduction 295
17.2 Definitions, Coordination Numbers 296
17.3 Formal Treatment of Quadruplets as “Complexes” or
Molecules” 300
17.3.1. Choice of Zmij/kl 302
17.4 Gibbs Energy Equation 303
17.5 The Configurational Entropy 305
17.5.1. Default Values of ζi/j 307
17.6 Second-Nearest-Neighbor Interaction Terms 308
17.6.1. The Quasichemical Model in a “Complex” Formalism
17.6.2. Combining Different Models in One Database 313
17.7 Summary of Model 314
17.8 Sample Calculations 317
References 318
17.1
312
Introduction
Chapter 16 discussed the modified quasichemical model
(MQM) in the pair approximation for short-range ordering (SRO)
between species on a single sublattice. As was shown, the singlesublattice model also applies to solutions with two sublattices
when one sublattice is occupied by only one species as in (A,B)
(X) solutions. In this case, the SRO is between second-nearestneighbor (SNN) pairs on the same sublattice.
In Chapter 15, Section 15.1.1, a two-sublattice model was
developed in which SRO between first-nearest-neighbor (FNN)
pairs was treated approximately.
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00017-8
# 2019 Elsevier Inc. All rights reserved.
295
296
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
In the present chapter, the MQM is developed in a quadruplet
approximation for solutions with two sublattices to account simultaneously for FNN (intersublattice) and SNN (intrasublattice) SRO
and the coupling between them (Pelton et al., 2001). When one
sublattice is occupied by only one species or is empty, the model
equations reduce exactly to the single-sublattice MQM equations
of Chapter 16. As in previous chapters, the coordination numbers
of sites on the sublattices are permitted to vary with composition,
thereby making the model suited to the treatment of liquid solutions such as molten salts. However, the model also applies to
solid solutions if the numbers of sublattice sites and coordination
numbers are held constant.
17.2 Definitions, Coordination Numbers
The solution consists of two sublattices, I and II. Let A,B,C, …
and X,Y,Z, … be the species that reside on sublattices I and II,
respectively: (A,B,C, …)(X,Y,Z, …). In a salt solution, for example,
A,B,C, … are the cations, and X,Y,Z, … are the anions, and in this
chapter, we shall refer to them as “cations” and “anions.” However,
the model is also applicable to other solutions. For example, in a
solid spinel solution, sublattices I and II would be associated with
the tetrahedral and octahedral cationic sublattices. Although
there is a third anionic sublattice, as long as this is occupied
by only one species, O2, the present model can be applied. In
other applications, lattice vacancies could also be considered as
“species,” or the same chemical species could occupy both sublattices. For instance, in an ordered Cu-Au alloy, Cu and Au reside
mainly on the I and II sublattices, respectively. However, due to
substitutional disordering, some Cu is found on the II sublattice
and some Au on the I sublattice. That is, in this example, A and
X would both be Cu, and B and Y would both be Au (see
Section 18.2).
When sublattice II is occupied only by X species, then the
single-sublattice MQM, as developed in Chapter 16, considers
the formation of SNN (A-[X]-B) pairs from SNN (A[X]A) and
(B-[X]-B) pairs according to
ðA ½X AÞ + ðB ½X BÞ ¼ 2ðA ½X BÞ; ΔgAB=X
(17.1)
The entropy expression in Chapter 16 was obtained by distributing
the pairs over “pair positions.” ΔgAB/X is the empirical parameter
of the model, which may be composition-dependent. If ΔgAB/X ¼ 0,
then ideal random mixing results. If ΔgAB/X < 0, then the reaction in
Eq. (17.1) is displaced to the right, and SRO results with (A[X]B)
pairs predominant. Similarly, when sublattice I is occupied only
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
by A species, the model considers the formation of (X-[A]-Y) SNN
pairs according to
ðX ½A X Þ + ðY ½A Y Þ ¼ 2ðX ½A Y Þ; ΔgA=XY
(17.2)
As discussed in Section 15.1, FNN (intersublattice) SRO involves
the following exchange reaction among FNN pairs:
exchage
ðA X Þ + ðB Y Þ ¼ ðA Y Þ + ðB X Þ; ΔgAB=XY
(17.3)
< 0,
The FNN pairs are distributed over pair positions. If Δgexchange
AB/XY
then there is FNN SRO with (AY) and (BX) pairs predominating. As a result, the probability of an (A[X]B) SNN pair is less
than in a random mixture, and so, the contribution of the
(A[X]B) SNN energy ΔgAB/X to the total Gibbs energy of the
solution is reduced. As shown in Section 15.1.1, this effect
becomes very important when |Δgexchange | is greater than about
50 kJ mol1.
With the quasichemical model in the pair approximation
(either FNN or SNN pairs), it is not possible to treat both FNN
and SNN SRO simultaneously. For example, for systems such as
(K,Co)(Cl,F), a large degree of SNN SRO is observed in the binary
KCl-CoCl2 and KF-CoF2 subsystems (see Section 16.3), and this
cannot be neglected in a quantitative model. In the present
chapter, therefore, we consider the distribution, not of pairs, but
of “quadruplets”: A2X2, ABX2, A2XY, ABXY, etc. As illustrated in
Fig. 17.1, each quadruplet consists of two SNN cations and two
SNN anions, the cations and anions being mutual first-nearest
neighbors. Both FNN and SNN SRO can thereby be treated
simultaneously.
In a solid, all possible configurations of the quadruplets on the
sublattices could be considered, as is done in the cluster variation
method (CVM) (see Section 18.2), and this is necessary for the full
quantitative modeling of order-disorder phenomena. However,
this additional complexity is neither necessary nor possible in
the case of liquid solutions and can also be reasonably neglected
in the modeling of solid solutions with a limited amount of SRO.
A
X
A
X
A
Y
X
A
X
B
X
B
A2X2
ABX2
Fig. 17.1 Some quadruplets (Pelton et al., 2001).
ABXY
297
298
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
Let ni (i ¼ A,B, … X,Y …) be the number of moles of species i;
nA/X the number of moles of FNN (A-X) pairs; and nA2/X2, nAB/X2,
nAB/XY, etc. the number of moles of A2X2, ABX2, ABXY, and other
quadruplets. (Note that nAB/X2 and nBA/X2 represent the same
quantity and can be used interchangeably.) Let ZA be the SNN
coordination number of A (i.e., the number of SNN pairs emanating from an A). Since each quadruplet contains just one SNN pair,
ZA is also the number of quadruplets emanating from an A. ZX is
defined similarly. Then,
ZA nA ¼ 2nA2 =X2 + 2nA2 =Y 2 + 2nA2 =XY + 2nAB=X2 + nAB=Y 2 + nAB=XY + …
(17.4)
ZX nX ¼ 2nA2 =X2 + 2nB2 =X2 + 2nAB=X2 + nA2 =XY + nB2 =XY + nAB=XY + …
(17.5)
Overall mole (or site) fractions Xi, FNN pair fractions Xi/j, and
quadruplet fractions Xij/kl are defined as
X X ¼ n X = ð nX + n Y + … Þ
(17.6)
X A ¼ n A = ð nA + nB + … Þ
XA=X ¼ nA=X = nA=X + nB=X + nA=Y + …
(17.7)
X
(17.8)
XAB=XY ¼ nAB=XY = nij=kl
P
where nij/kl is the total number of moles of quadruplets. From
Eqs. (17.4), (17.5),
X
2 nij=kl ¼ ðZA nA + ZB nB + …Þ ¼ ðZX nX + ZY nY + …Þ
(17.9)
Note that in the (A,B,C, …)(X) subsystem, nij/X2 and Xij/X2 are identical to the mole numbers and mole fractions nij and Xij of SNN
pairs as defined in Chapter 16.
Coordination equivalent site fractions, Yi are defined as
YA ¼ ZA nA =ðZA nA + ZB nB + …Þ
YX ¼ ZX nX =ðZX nX + ZY nY + …Þ
(17.10)
These are called “coordination equivalent” fractions to distin0
guish them from “charge equivalent” site fractions Yi in which
the ni are weighted by the ionic charges qi (or valences) rather
than by the coordination numbers:
YA0 ¼ qA nA =ðqA nA + qB nB + …Þ
(17.11)
YX0 ¼ qX nX =ðqX nX + qY nY + …Þ
(17.12)
Substitution of Eq. (17.10) into Eqs. (17.4), (17.5) gives
YA ¼ XA2 =X2 + XA2 =Y 2 + XA2 =XY + 0:5XAB=X2 + 0:5XAB=XY + …
(17.13)
YX ¼ XA2 =X2 + XB2 =X2 + XAB=X2 + 0:5XA2 =XY + 0:5XAB=XY + …
(17.14)
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
In a solid solution, it is clearly required that ZA ¼ ZB ¼ ZC ¼ … and
that ZX ¼ ZY ¼ ZZ ¼ …. However, in a liquid, this is not necessary,
and furthermore, the coordination numbers can vary with composition. As shown in Section 16.2.3, the use of variable coordination numbers provides the necessary flexibility to fix the
compositions of maximum SRO in each (A,B)(X) and (A)(XY)
subsystem.
Let ZAAB/XY be the SNN coordination number of an A species
when (hypothetically) all A species exist in ABXY quadruplets.
We then let
2nA2 =X2 2nA2 =Y 2 2nA2 =XY nAB=X2 nAB=XY
+ A
+ A
+ A
+ A
ZAA2 =X2
ZA2 =Y 2
ZA2 =XY ZAB=X
ZAB=XY
2
1
¼
ZA 2nA2 =X2 + 2nA2 =Y 2 + 2nA2 =XY + nAB=X2 + nAB=XY + …
2nA2 =X2 2nB2 =X2 2nAB=X2 nA2 =XY nAB=XY
+ X
+ X
+
+ X
ZAX2 =X2
ZB2 =X2
ZAB=X2 ZAX2 =XY ZAB=XY
1
¼
ZX 2nA2 =X2 + 2nB2 =X2 + 2nAB=X2 + nA2 =XY + nAB=XY + …
(17.15)
(17.16)
This composition dependence is chosen because substitution of
Eqs. (17.15), (17.16) into Eqs. (17.4), (17.5) yields the following
simple relations:
nA ¼
2nA2 =X2 2nA2 =Y 2 2nA2 =XY nAB=X2 nAB=XY
+ A
+ A
+ A
+ A
+…
ZAA2 =X2
ZA2 =Y 2
ZA2 =XY ZAB=X
ZAB=XY
2
(17.17)
nX ¼
2nA2 =X2 2nA2 =Y 2 2nA2 =XY nAB=X2 nAB=XY
+ X
+ X
+
+ X
+…
ZAX2 =X2
ZB2 =X2
ZAB=X2 ZAX2 =XY ZAB=XY
(17.18)
Note that in the (A,B,C, …)(X) subsystem, where sublattice II is
occupied only by X species, the Ziij/X2 are identical to the SNN
coordination numbers Ziji as defined in Chapter 16.
We now define parameters ζi/j as the number of quadruplets
emanating from an i/j FNN pair. Hence,
ζ A=X nA=X ¼ 4nA2 =X2 + 2nAB=X2 + 2nA2 =XY + nAB=XY + …
(17.19)
and similarly for the other ζ i/j. In general, ζ A/X, ζ B/X, ζ A/Y, … can
have different values and can even vary with composition, but this
makes the model unnecessarily complex. Hence, for the moment
and for the development of most of the terms of the model, we
shall assume that all the ζ i/j are equal to the same constant
value ζ. However, in Section 17.5, we shall permit the ζ i/j to have
different values in order to provide additional flexibility in the
model terms most affected by the ζ i/j values.
It follows from Eq. (17.19) that when ζ ¼ constant, the total
numbers of pairs and quadruplets are related by
299
300
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
ζ
X
X
ni=j ¼ 4 nij=kl
(17.20)
and since each quadruplet contains two FNN pairs and one SNN
pair emanating from a given ion, it follows that
Zi =zi ¼ ζ=2
(17.21)
where zi is the FNN coordination number of ion i, and that the
coordination equivalent fractions of Eqs. (17.13), (17.14) are also
given by
YA ¼ zA nA =ðzA nA + zB nB + …Þ
(17.22)
and similarly for YX.
17.3 Formal Treatment of Quadruplets as
“Complexes” or “Molecules”
This section is a generalization of the strategy discussed in
Section 16.7 for the single-sublattice MQM.
The quadruplet ABXY may be treated formally as the
A
Y
B1/ZBAB/XYX1/ZXAB/XYY1/ZAB/XY
. Similarly,
“complex” or “molecule” A1/ZAB/XY
A
X2/ZXA2/X2, etc. For
an A2X2 quadruplet is formally treated as A2/ZA2/X2
example, in molten KCl-CoCl2 solutions, if we were to choose
Cl
ZKKCo/Cl2 ¼ 3, ZCo
KCo/Cl2 ¼ 6, and ZKCo/Cl2 ¼ 3, then the quadruplet
KCoCl2 is formally treated as a K1/3Co1/6Cl2/3 complex.
It must be stressed that this is only a formalism. The entropy
expression in the quasichemical model (see Section 17.4) is not
obtained by distributing the quadruplets as if they were molecules
(as is, in fact, done in “associate models”). It is not essential that this
formalism be used. However, it aids in the conceptualization and in
many derivations. For example, if nA2/X2, nA2/XY, etc. in Eq. (17.17)
are replaced by nA2=ZA X2=ZX , nA2=ZA =X1=ZX Y 1=ZY , etc., then
A2=X
2
A2=X
2
A2=XY
A2=XY
A2=XY
the number of moles of A is the same on both sides of
the equation which thereby becomes a “normal” mass balance
equation.
Furthermore, it can be seen that all quadruplets such as
K1/3Co1/6Cl2/3 in a molten salt solution must be electrically
neutral. This stoichiometry represents the composition of a hypothetical solution formed exclusively of these quadruplets and
containing one mole of quadruplets. That is, one mole of KCoCl2
quadruplets contains 1/3 mol K+, 1/6 mol Co2+, and 2/3 mol Cl.
In general, for a molten salt solution, if qA, qB, …, qX are
the absolute cationic and anionic charges (or valences), then
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
A
the charge neutrality condition for ABXY (i.e., A1/ZAB/XY
B1/ZBAB/XY
Y
X1/ZXAB/XYY1/ZAB/XY
) quadruplets is as follows:
qA
A
ZAB=XY
+
qB
B
ZAB=XY
¼
qX
X
ZAB=XY
+
qY
Y
ZAB=XY
(17.23)
This equation also holds when A ¼ B and/or when X ¼ Y. The
previous example of K+1/3 Co2+
1/6 Cl2/3 clearly satisfies Eq. (17.23).
In the most general form of the model, it is not essential for
Eq. (17.23) to be satisfied. However, in practice, for liquid solutions that can be described as consisting of two quasi sublattices,
this equation will almost always be satisfied.
Fig. 17.2 shows a traditional composition square of a reciprocal
ternary (A,B)(X,Y) system. One charge equivalent of each pure
component is shown at each corner. For example, in the (Na,
Ca)(F,SO4) system, where the absolute ionic charges are qA ¼ 1,
qB ¼ 2, qX ¼ 1, and qY ¼ 2, the components would be NaF, Ca1/2F,
Na(SO4)1/2, and Ca1/2(SO4)1/2. The axes are the charge equivalent
0
0
fractions YA and YX of Eqs. (17.11), (17.12) (not to be confused with
YA and YX in Eq. (17.10)). The formal compositions of quadruplets
A
B1/ZBAB/X2X2/ZXAB/X2 are found on the sides of the square
such as A1/ZAB/X2
as shown in Fig. 17.2.
A1/q Y1/q
A
“A2XY”
Y
“A2Y2”
A1/q X1/q
A
X
“A2X2” = A2
a
X2
A
X
“ABX2”= A1
y
“ABY2”
Y¢A = qAnA/(qAnA+ qBnB )
ZA2/X2
“B2Y2”
“ABXY”
= A1 B1 X1 Y1
ZA2/X2
X2
B1
A
x
ZAB
B
/X2
ZAB
X
/X2
ZAB
/X2
A
B
Y
X
ZAB/XY
ZAB/XY
ZAB/XY
ZAB/XY
b
B1/q Y1/q “B2XY”
B
Y
Y¢X = qXnX/(qXnX+qYnY)
“B2X2”
B1/q X1/q
B
X
Fig. 17.2 Composition square of an (A,B)(X,Y) reciprocal ternary system showing
formal “compositions” of the various quadruplets (Pelton et al., 2001).
301
302
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
17.3.1
Choice of Zm
ij/kl
For a solid solution, it is required that all cationic coordination
i
be equal and that all anionic coordination numbers
numbers Zij/kl
k
A
and
Zij/kl be equal. For liquid solutions on the other hand, ZAB/X
2
B
ZAB/X2 are chosen to correspond to the composition of maximum
SNN SRO in the (A,B)(X) binary subsystem as discussed in
Section 16.2.3. For example, for KCl-CoCl2 solutions, SRO is at
a maximum near the composition K2CoCl4 as was shown in
K
Fig. 16.18. Hence, we set ZCo
KCo/Cl2/ZKCo/Cl2 ¼ 2.0.
As discussed in Chapter 16, since the quasichemical expression for the entropy (see Section 17.4) is, of necessity, only
approximate, it is not necessary that the absolute values of
m
used in the model correspond exactly to the actual coordiZij/kl
nation numbers. In fact, it is found that better representations
are frequently obtained with intentionally nonphysical values
m
since one can thereby partially compensate for the error
of Zij/kl
caused by the entropy approximation. However, it is essential
m
correspond to the compositions of
that the ratios of the Zij/kl
maximum SRO as discussed above.
In principle, the coordination numbers Zm
AB/XY for the
“reciprocal” ABXY quadruplets (A 6¼ B and X 6¼ Y) could be chosen
to reflect a tendency to SRO at some particular composition in the
reciprocal (A,B)(X,Y) solutions. However, in most cases, there will
be no such tendency, and one can set the value of Zm
AB/XY as an
“average” of the values in the A2XY, B2XY, ABX2, and ABY2 quadruplets. It is suggested that this be done by defining the “composition”
0
of the ABXYquadruplets as lying at the average of the values of YX of
0
the quadruplets A2XYand B2XYand at the average of the values of YA
of the quadruplets ABX2 and ABY2 as illustrated on Fig. 17.2. This
construction corresponds to the following “default” values:
1
A
ZAB=XY
1
X
ZAB=XY
¼
¼
X
ZAB=X
2
A
qX ZAB=X
2
ZAA2 =XY
qA ZAX2 =XY
+
+
Y
ZAB=Y
2
A
qY ZAB=Y
2
ZBB2 =XY
!
F
!
qB ZBX2 =XY
(17.24)
F
and similarly for ZBAB/XY and ZYAB/XY, where
F ¼ 1=8
qX
X
ZAB=X
2
+
qY
Y
ZAB=Y
2
+
qA
ZAA2 =XY
+
qB
ZBB2 =XY
!
(17.25)
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
17.4
Gibbs Energy Equation
We now define
gAo2 =X2
!
!
2qA
2qX
o
¼
¼
g
go
ZAA2 =X2 A1=qA X1=qx
ZAX2 =X2 A1=qA X1=qx
(17.26)
where goA1/qAX1/qx is the standard Gibbs energy of the pure component per charge equivalent. For Al2O3, for example,
!
!
6
4
o
o
o
gAl2 =O2 ¼
(17.27)
gAl1=3 =O1=2 ¼
gAl
O
Al
1=3 =O1=2
Z
ZAl
Al2 =O2
2 =O2
That is, goAl2/O2 is the standard Gibbs energy of Al2O3 per mole
of Al2O2 quadruplets.
If goA/X is defined as the standard Gibbs energy per mole of
FNN pairs, then it follows that
o
(17.28)
gAo2 =X2 ¼ 4=ζ A=X gA=X
The Gibbs energy of the solution is given by the model as
G ¼ nA2 =x2 gAo2 =X2 + nB2 =x2 gBo2 =X2 + nA2 =Y 2 gAo2 =Y 2 + …
(17.29)
+ nAB=X2 gAB=X2 + nAB=Y 2 gAB=Y 2 + nA2 =XY gA2 =XY + …
+ nAB=XY gAB=XY + … T ΔSconfig
where gAB/X2, gAB/XY, etc. are the Gibbs energies of one mole
of the quadruplets.
Consider the following reaction for the formation of
quadruplets:
ðA2 X 2 Þquad + ðB2 X 2 Þquad ¼ 2ðABX 2 Þquad ; ΔgAB=X2
(17.30)
for which the Gibbs energy change is ΔgAB/X2. If we do not use the
shorthand notation, then Eq. (17.30) is written as
!
!
ZAA2 =X2
ZBB2 =X2
A2=ZA X 2=ZX +
B2=ZB X 2=ZX
A
B
A2 =X 2
A2 =X 2
B2 =X 2
B2 =X 2
ZAB=X
ZAB=X
2
2
(17.31)
¼ 2A1=ZA B1=ZB X 2=ZX
AB=X 2
AB=X 2
AB=X 2
When ΔgAB/X2 ¼ 0, the binary (A,B)(X) solution is ideal. In this case,
!
!
ZAA2 =X2
ZBB2 =X2
o
2gAB=X2 ¼
(17.32)
gA2 =X2 +
gBo2 =X2
A
B
ZAB=X
Z
AB=X 2
2
Now, ΔgAB/X2 is an empirical parameter of the model. It may be
expressed as
303
304
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
o
o
ΔgAB=X2 ¼ ΔgAB=X
+
Δg
Δg
AB=X 2
AB=X 2
2
(17.33)
where ΔgoAB/X2 is a constant, independent of composition, and
(ΔgAB/X2 ΔgoAB/X2) is expanded as an empirical polynomial in
the quadruplet fractions Xij/kl.
As will be discussed in Section 17.5, ΔgAB/X2 in the (A,B,C, …)
(X) subsystem is identical to ΔgAB/X of the reaction in Eq. (17.1)
in this subsystem, and ΔgoAB/X2 and the coefficients of the polynomial expansion of (ΔgAB/X2 ΔgoAB/X2) are all numerically equal to
the coefficients obtained from optimization of data in this subsystem as described in Section 16.2.2.
We now define the “standard molar Gibbs energy of the ABX2
quadruplets” as
o
2gAB=X
2
¼
ZAA2 =X2
A
ZAB=X
2
!
gAo2 =X2
+
ZBB2 =X2
!
o
gBo2 =X2 + ΔgAB=X
2
B
ZAB=X
2
(17.34)
Similarly, for the quadruplet formation reaction,
ðA2 X 2 Þquad + ðA2 Y 2 Þquad ¼ 2ðA2 XY Þquad ; ΔgA2 =XY
(17.35)
we define
2gAo2 =XY
¼
ZAX2 =X2
ZAX2 =XY
!
gAo2 =X2
+
ZAY2 =Y 2
ZAY2 =XY
!
gAo2 =Y 2 + ΔgAo2 =XY
(17.36)
In order to now define ΔgoAB/XY for the reciprocal ABXY quadruplet,
consider that in an ideal solution, gAB/XY (when normalized per
charge equivalent) would vary linearly with composition on
Fig. 17.2 between points x and y and between points a and b. This
linear variation is added to the sum of ΔgoAB/X2, ΔgoAB/Y2, ΔgoA2/XY,
and ΔgoB2/XY (which were included in Eqs. (17.34), (17.36)) corrected
to the same molar basis, and finally, we add the compositionindependent term ΔgoAB/XY of the Gibbs energy change ΔgAB/XY of
the quadruplet formation reaction:
1=2ðABX 2 + ABY 2 + A2 XY + B2 XY Þ ¼ 2ðABXY Þ; ΔgAB=XY
o
o
¼ ΔgAB=XY
+ ΔgAB=XY ΔgAB=XY
(17.37)
ΔgAB/XY is an empirical parameter of the model, obtained from
optimization of data for the reciprocal (A,B)(X,Y) system as
discussed in Section 17.5. In the ideal case, ΔgAB/XY ¼ 0. The resultant definition of goAB/XY is
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
o
gAB=XY
¼
qX
qY
!1
+ Y
X
ZAB=XY
ZAB=XY
"
qX ZAA2 =X2
qX ZBB2 =X2
o
:g
+
:gBo2 =X2
=X
A
A
X
B
X
2
2
2ZAB=XY
ZAB=XY
2ZAB=XY
ZAB=XY
#
B
qY ZAA2 =Y 2
q
Z
Y
B
=Y
2
2
:gAo2 =Y 2 +
:gBo2 =Y 2
+
A
Y
B
Y
2ZAB=XY
ZAB=XY
2ZAB=XY
ZAB=XY
" X
Y
ZAB=X
ZAB=Y
ZAA2 =XY
2
o
o
+
Δg
+
ΔgAo2 =XY
+ 1=4 X 2 ΔgAB=X
AB=Y
Y
A
2
2
ZAB=XY
ZAB=XY
ZAB=XY
#
ZBB2 =XY
o
o
+ B
Δg
(17.38)
+ ΔgAB=XY
ZAB=XY B2 =XY
Eq. (17.38) also applies if all goij/kl and Δgoij/kl factors are replaced
by gij/kl and Δgij/kl.
Note that Eq. (17.23) applies and that the definition in
Eq. (17.7) holds, even if one does not choose to define ZiAB/XY as
in Eq. (17.24).
Substitution of Eqs. (17.34), (17.36), and (17.38) into Eq. (17.29)
gives
2
3
o
o
G ¼4nA2 =X 2 gAo =X + … + nAB=X 2 gAB=X
+ nA2 =XY gAo =XY + … + nAB=XY gAB=XY
+ …5
2
2
2
2
2
+ 1=24nAB=X 2 +
2
X
ZAB=X
2
0
2
@
nAB=XY
X
ZAB=XY
nAB=XZ
+ X
ZAB=XZ
13
+ …A5 Δg
AB=X 2
o
ΔgAB=X
2
0
13
ZAA =XY nAB=XY nAC=XY
o
@
A5 Δg
+ 1=24nA2 =XY + 2
+
+
…
Δg
A
=XY
2
=XY
A
A
A
2
2
ZAB=XY
ZAC=XY
+ 1=2…
"
+ 1=2
o
nAB=XY ΔgAB=XY ΔgAB=XY
o
+ nAB=YZ ΔgAB=YZ ΔgAB=YZ
T ΔSconfig
17.5
#
+…
(17.39)
The Configurational Entropy
In Eq. (17.39), ΔSconfig is given by distributing all the quadruplets
randomly over “quadruplet positions.” Unfortunately, an exact mathematical expression for this is unknown, and
P so, approximations
must be made. Letting ΔSconfig equal R (nij/kl lnXij/kl) would
305
306
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
clearly overcount the number of possible configurations. The
following approximate expression is proposed:
ΔSconfig =R ¼ðnA ln XA + nB ln XB + … + nX ln XX + nY ln XY + …Þ
XA=X
XB=X
XA=Y
+ nB=X ln
+ nA=Y ln
+…
+ nA=X ln
FA FX
FB FX
FA FY
XA2 =X 2
XAB=X 2
+ nA2 =X 2 ln
+ … + nAB=X 2 ln
3
3=2
3=2
1=2 1=2
X
=YA YX
2X
X
=Y
Y
Y
A=X
A=X B=X
A
B
XA2 =XY
.
+ nA2 =XY ln
3=2 3=2
1=2 1=2
2X
X
YA YX YY
A=X A=Y
+ …nAB=XY ln
XAB=XY
.
+…
3=4 3=4 3=4 3=4
1=2 1=2 1=2 1=2
4X
X
X
X
YA YB YX YY
X
!
A=X B=X A=Y B=Y
(17.40)
where
FA ¼ XA=X + XA=Y + XA=Z + …
FX ¼ XA=X + XB=X + XC=X + …
(17.41)
Within the final set of parentheses in Eq. (17.40), the quadruplet
fractions Xij/kl are divided by their values when there is no SNN
SRO. In a one-dimensional lattice, there are four distinguishable
types of quadruplets: AXBY, AYBX, BXAY, and BYAX. The probability of an AXBY quadruplet when there is no SNN SRO is equal
to XA/X(XB/X/YX)(XB/Y/YB) where XA/X is the probability of an A/X
FNN pair, (XB/X/YX) is the conditional probability of a B/X pair
given that one member of the pair is an X, and (XB/Y/YB) is the
conditional probability of a B/Y pair given that one member
of the pair is a B. Similarly, the probability of an AYBX quadruplet
when there is no SNN SRO is XA/Y(XB/Y/YY )(XB/X/YB) and similarly for the BXAY and BYAX quadruplets. In the onedimensional case, this give rise to four configurational entropy
terms:
nAXBY ln XAXBY = XA=X XBX XBY =YX YB
+ nAYBX ln XAYBX =ðXAY XBY XBX =YY YB Þ + …
(17.42)
In three dimensions, the four quadruplets are indistinguishable.
Therefore, we set
nAXBY ¼ nAYBX ¼ nBXAY ¼ nBYAX ¼ nAB=XY =4
(17.43)
XAXBY ¼ XAYBX ¼ XBXAY ¼ XBYAX ¼ XAB=XY =4
(17.44)
and
Substitution into Eq. (17.42) then gives an entropy term,
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
3/4 3/4 3/4
1/2 1/2 1/2 1/2
nAB/XY ln[XAB/XY/(4X3/4
A/XXB/XXA/YXB/Y/YA YB YX YY )], which is
the final term within the final set of parentheses in Eq. (17.40).
The other terms in Eq. (17.40) within the final set of parentheses
are derived similarly for the cases where X ¼ Y and/or
A ¼ B. For the sake of simplicity in these terms, it is assumed that
all the ζ i/j have the same constant value ζ. The actual numerical
value of ζ is then not required since all the ζ factors are cancel out.
Consequently, the terms within the final set of parentheses in
Eq. (17.40) sum to zero when there is no SNN SRO.
The terms within the second set of parentheses in Eq. (17.40)
are related to the configurational entropy when FNN pairs are
distributed randomlyPover FNN pair positions. Letting this
entropy equal to R nA/X ln XA/X would clearly overcount the
number of possible configurations. Each pair fraction Xi/j within
the second set of parentheses is thus divided by its value in an
ideal random solution. In the case where all ζ i/j have the same
value ζ, the P
terms within the second set of parentheses would
be given by
nij ln Xij/YiYj, and all factors ζ would cancel out.
However, in order to provide more flexibility, the values of ni/j
are calculated from Eq. (17.19) permitting different values of
ζ i/j, and the Xi/j factors are given by
X
Xi=j ¼ ni=j
ni=j
(17.45)
and the factors Fi are given by Eq. (17.41). If all ζ i/j have the same
value, then Fi ¼ Yi.
Hence, the terms within the second set of parentheses in
Eq. (17.40) sum to zero when there is no FNN SRO. Finally, the terms
within the first set of parentheses in Eq. (17.39) give the ideal configurational entropy when there is neither FNN nor SNN SRO.
Eq. (17.40), while still necessarily an approximation, is an
improvement on the expression given for the configurational
entropy in the original article by Pelton et al. (2001).
17.5.1
Default Values of zi/j
If the exchange Gibbs energy for the reaction in Eq. (17.3) is
very positive, then at the midpoint of the stable diagonal of the
compositions square (Fig. 17.2), there will be only A2X2 and
B2Y2 quadruplets and only A/X and B/Y FNN pairs. Hence, ΔSconfig
should be equal to zero at this point. By making the appropriate
substitutions in Eq. (17.40), it can be shown that this condition
is satisfied when
ZAA2 =X2 ¼ ZBB2 =Y 2
(17.46)
307
308
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
ZAX2 =X2 ¼ ZBY2 =Y 2
ζ A=X ¼ 2ZAA2 =X2 ZAX2 =X2 = ZAA2 =X2 + ZAX2 =X2
and
(17.47)
(17.48)
ζ B=Y ¼ 2ZBB2 =Y 2 ZBY2 =Y 2 = ZBB2 =Y 2 + ZBY2 =Y 2
(17.49)
Even when Eqs. (17.45), (17.47) do not apply, it is proposed that
Eqs. (17.48), (17.49) still be used as initial default values of ζ A/X
and ζ B/Y and similarly for ζ A/Y and ζ B/X.
17.6 Second-Nearest-Neighbor
Interaction Terms
ΔgAB/X2 is an empirical parameter of the model, obtained by
optimization of data in the (A,B,C, …)(X) subsystem, which is
related to the Gibbs energy of the formation of SNN pairs according to the reactions in Eqs. (17.30) or (17.1). As in Eq. (16.71), in
this subsystem, ΔgAB/X2 is expanded as
o
ΔgAB=X2 ¼ΔgAB=X
+
2
X
j
ði + jÞ1
ij
χ iAB=X2 χ BA=X2 gAB=X2 +
X
j
χ iAB=X2 χ BA=X2
i0
j0
"
k1
X
k1
ijk
gABðlÞ=X2 Yl 1 ξAB=X2 ξBA=X2
l
+
X
m
+
X
n
ijk
gABðmÞ=X2
ijk
gABðnÞ=X2
!
Ym
ξBA=X2
Yn
ξAB=X2
1
!
1
!k1
YB
ξBA=X2
YA
(17.50)
!k1 #
ξAB=X2
where the summations over l, m, and n are as described previously. Eq. (17.50) may be compared with Eq. (17.33). The empirical coefficients ΔgoAB/X2 and gijAB/X2 are found by optimization of
data in the (A,B)(X) binary system, the other terms in Eq. (17.50)
all being equal to zero in this binary system. For example, if
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
all gijAB/X2 are zero and if ΔgoAB/X2 is small, then the model
approaches regular solution behavior, with ΔgoAB/X2 as the regular solution parameter. The coefficients gAB(C)/X2ijk are ternary
parameters, which should not be large, that give the influence
of the presence of a third cation, C, upon the energy of formation of SNN (A-[X]-B) pairs. These are found by optimization of
available data in the (A,B,C)(X) ternary system. In the absence of
such data, these coefficients can be set to zero. The variables,
χ AB/X2, χ BA/X2, ξAB/X2, and ξBA/X2 were defined in Eqs. (16.63),
(16.68). They are functions of the quadruplet fractions Xij/X2,
which are equal to the SNN bond fractions Xij in the (A,B,C,
…)(X) subsystem.
In the multicomponent (A,B,C, …)(X,Y,Z, …) system, the
definition of χ 12 in Eq. (16.68) is replaced by
χ 12=X2 ¼
X X
i¼1, k j¼1, k
X
Xij=X2 + 0:5
X
i¼1, 2, k , l j¼1, 2, k, l
X
!
Xij=Xm
m
Xij=X2 + 0:5
X
!
Xij=Xm
(17.51)
m
(where in this notation, A, B, C, … are replaced by 1, 2, 3, ….) (Note
that
P in the original article by Pelton et al. (2001), the factors
m Xij=Xm were not included in this equation.)
The coefficients ΔgoAB/X2, gijAB/X2, and gijk
AB(C)/X2 are identical to the
coefficients ΔgoAB, gijAB, and gijk
AB(C) of Chapter 16 in which the quasichemical model was developed and applied for the case where
sublattice II is occupied solely by X species.
Eq. (17.50) applies to the (A,B,C, …)(X) subsystem. The factors
χ AB/X2 and χ BA/X2 remain unchanged in the multicomponent
system.
In the (A,B,C, …)(X) subsystem, from Eq. (17.13), the equivalent fraction Yi is equal to (Xi2/X2 + 0.5 (XiA/X2 + XiB/X2 + …)), which
is equal to Xi/X in this subsystem. In the multicomponent system,
we assume that the ternary terms in Eq. (17.50) are constant
along lines of constant Xi/X/YX (where YX ¼ 1 in the (A,B,C, …)
(X) system). Therefore, the factors Yi in Eq. (17.50) are replaced
by Xi/X/YX. The factors ξAB/X2 and ξBA/X2 were defined in
Eq. (17.18) as sums of YA, YB, YC…. Hence, in Eq. (17.50), these
are replaced by the corresponding sums of XA/X/YX, XB/X/YX,
XC/X/YX, etc.
309
310
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
Finally, then, ΔgAB/X2 is given in the multicomponent system by
X
j
ij
o
+
χ iAB=X2 χ BA=X2 gAB=X2
ΔgAB=X2 ¼ΔgAB=X
2
ði + jÞ1
+
X
"
j
χ iAB=X2 χ BA=X2
X
l
i0
ijk
gABðlÞ=X2
k1
Xl=X 1 ξAB=X2 ξBA=X2
YX
j0
k1
+
X
m
+
X
n
+
ijk
gABðmÞ=X2
ijk
gABðnÞ=X2
X
Y6¼X
!
!k1
Xm=X
XB=X
1
YX ξBA=X2
YX ξBA=X2
!
!k1
Xn=X
XA=X
1
YX ξAB=X2
YX ξAB=X2
#
ijk
gAB=X2 ðY Þ YY ð1 YX Þk1
(17.52)
where χ AB/X2 and χ BA/X2 are ratios as defined in Eq. (17.51) and where
ξAB/X2 and ξBA/X2, which were defined in Eq. (16.63) as sums of YA, YB,
YC…, are now the corresponding sums of XA/X/YX, XB/X/YX, etc.
The final term in Eq. (17.52) is zero in the (A,B,C, …)(X) system.
The empirical coefficients gijk
AB/X2(Y) are reciprocal ternary coefficients giving the effect of the presence of a Y anion upon the
energy of the formation of ABX2 quadruplets. These coefficients,
which should not be large, are found by optimization of available
data for the (A,B)(X,Y) reciprocal ternary subsystem. In the
absence of such data, these coefficients may be set to zero.
A similar expansion of ΔgA2/XY for the formation of A2XY quadruplets may be written:
X
j
ij
χ iA2 =XY χ A2 =YX gA2 =XY
ΔgA2 =XY ¼ΔgAo2 =XY +
ði + jÞ1
+
X
χ iA2 =XY
i0
j0
k1
+
X
m
+
ijk
gA2 =XY ðmÞ
X
n
+
X
B6¼A
ijk
"
j
χ A2 =XY
gA2 =XY ðnÞ
X
l
ijk
gA2 =XY ðlÞ
k1
XA=l 1 ξA2 =XY ξA2 =YX
YA
!
!k1
XA=m
XA=Y
1
YA ξA2 =YX
YY ξA2 =YX
!
!k1
XA=n
XA=X
1
YA ξA2 =XY
YA ξA2 =XY
#
ijk
gA2 ðBÞ=XY YB ð1 YA Þk1
(17.53)
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
In Eq. (17.50), ΔgAB/X2 in the (A,B,C, …)(X) system is expanded
as a polynomial in the quadruplet fractions Xij/X2 that are equal to
the SNN pair fractions in this subsystem. Alternatively, as discussed in Chapter 16, Eq. (16.65), ΔgAB/X2 may be expanded in
terms of the equivalent fractions YA, YB, YC…. The general
expression is
!i
!j
ξAB=X2
ξBA=X2
ij
+
qAB=X2
ξ
+
ξ
ξ
+
ξ
AB=X
BA=X
AB=X
BA=X
2
2
2
2
ði + jÞ1
!i
!j
X
ξAB=X2
ξBA=X2
+
ξAB=X2 + ξBA=X2
ξAB=X2 + ξBA=X2
o
ΔgAB=X2 ¼ΔgAB=X
2
X
i0
j0
"
k1
X
l
+
k1
ijk
qABðlÞ=X2 Yl 1 ξAB=X2 ξBA=X2
X
m
+
X
n
ijk
qABðmÞ=X2
ijk
qABðnÞ=X2
!
Ym
ξBA=X2
Yn
ξAB=X2
1
!
1
!k1
YB
ξBA=X2
YA
ξAB=X2
!k1 #
(17.54)
If a Redlich-Kister polynomial was used in the (A,B)(X) binary system,
in Eq. (17.54) are replaced by
Pi then the binary terms
l12[(ξ12 ξ21)/(ξ12 + ξ21)]i.
Eq. (17.54) is clearly very similar to Eq. (17.50). If the (A,B,C, …)
(X) system was optimized using this expansion, then Eq. 17.54 can
be substituted into Eq. (17.39) after first replacing all Yi by Xi/X/YX,
replacing all ξAB/X2 and ξBA/X2 by the corresponding sums of Xi/X/
YX, and after adding reciprocal ternary terms if required, just as
was done in Eq. (17.52).
In order to provide more flexibility in optimizing data in reciprocal systems, the parameter ΔgAB/XY for the reaction in Eq. (17.37)
can also be expanded as follows and substituted into Eq. (17.39):
X
o
i
i
i
i
ΔgAB=XY ¼ΔgAB=XY
+
ðgAB=XY
ðAX Þ XA2 =X 2 + gAB=XY ðBX Þ XB2 =X 2
i1
(17.55)
i
i
i
+ gAB=XY ðAY Þ XAi 2 =Y 2 + gAB=XY
ðBY Þ XB2 =Y 2 Þ
where ΔgoAB/XY, giAB/XY(AX), etc. are additional empirical parameters
obtained by optimization of available data for the reciprocal
ternary (A,B)(X,Y) system. In the absence of such data, these terms
should be set to zero.
311
312
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
17.6.1
The Quasichemical Model in a “Complex”
Formalism
As discussed in Section 17.2, if the quadruplets ABXY are
formally
treated
as
“complexes”
or
“molecules”
A1=Z A B1=Z B X 1=Z X Y 1=Z Y , then the mass balances of
AB=XY
AB=XY
AB=XY
AB=XY
Eqs. (17.17), (17.18) become “normal” mass balances rather than
quasichemical mass balances. Furthermore, all the parameters
Δgij/kl in Eq. (17.39) were shown in Section 17.5 to be expressible
solely in terms of the quadruplet fractions Xij/kl. (The pair fractions Xi/j and equivalent fractions of Eqs. (17.52), (17.53) can be
expanded in terms of the Xij/kl through the use of Eqs. (17.13),
(17.14), (17.19), and (17.20).) Hence, apart from the ( TΔSconfig)
term, Eq. (17.39) is of the same form as an expression for the Gibbs
energy of a mixture of molecules A1/ZAAB/XYB1/ZBAB/XYX1/ZXAB/XYY1/ZYAB/XY on
a single lattice, with the nonconfigurational “excess” terms
expressed as polynomials in the mole fractions XAB/XY of these
molecules. Thus, the same existing algorithms and computer subroutines that are commonly used for simple polynomial solution
models can be used directly for the quasichemical model merely
by including the additional configurational entropy terms.
Furthermore, the fact that Eq. (17.39) can be written solely
in terms of the fractions Xij/kl of the quadruplet “end-members”
permits the chemical potentials to be easily calculated in closed
explicit form. By the same argument as that of Section 16.7,
the chemical potential of the actual component A1/qAX1/qX is given
in terms of the chemical potential μA2/X2 of the quadruplet A2X2 by
μA1=q X1=q ¼ μA2 =X2 ZAA2 =X2 =2qA ¼ μA2 =X2 ZAX2 =X2 =2qX
(17.56)
A
X
where
μA2 =X2 ¼ ð∂G=∂nA2 X2 Þnij=kl
(17.57)
Substitution of Eq. (17.39) into Eq. (17.57) then gives
0
1
X
X
2
lnX
2
lnX
4
A
=X
A=X
2
2
A
X
A + gE
μA2 =X 2 ¼ gAo =X + RT @ A
+ X
+ ln 3
+ ln
A2 X 2
2
2
FA FX
ZA =X
ZA =X
X A=X =YA YX ζ
2
2
2
2
(17.58)
where gEA2/X2 is calculated from the nonconfigurational “gE”
polynomial terms of Eq. (17.39) in the usual way:
X
gAE2 =X2 ¼ g E + ∂g E =∂X A2 =X2 Xij=kl ∂g E =∂Xij=kl
(17.59)
ij=kl6¼A2 =X 2
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
17.6.2
Combining Different Models in One Database
Many binary and ternary common-ion systems (A,B,C, …)(X)
have already been evaluated and optimized by using a randommixing (Bragg-Williams) expression for ΔSconfig and by expanding
the excess Gibbs energy as a simple polynomial in the mole fractions as discussed in Chapter 14. For example, in a binary (A,B)
(X) system, this is equivalent to writing
o
o
+ nB=X gB=X
+ RT ðnA ln X A + nB ln X B Þ + nAB=X2 ΔgAB=X2
G ¼ nA=X gA=X
(17.60)
P
with nAB/X2 now equal to ( nij/kl)XA/XXB/X and with ΔgAB/X2
expanded as in Eq. (14.60) as
X ij
j
qAB=X2 YAi YB
(17.61)
ΔgAB=X2 ¼ qoo
AB=X 2 +
ði + jÞ1
or as in Eq. (14.59) as
ΔgAB=X2 ¼ o l AB=X2 +
X
i
lAB=X2 ðYA YB Þi
(17.62)
i1
For example, if all qijAB/X2 or ilAB/X2 coefficients are zero, then this is
a simple regular solution. Ideally, of course, all these systems
should be reoptimized with the quasichemical model. However,
this would entail a great deal of work. For systems in which
ΔgAB/X2 is relatively small, the neglect of SRO involving SNN
(A-[X]-B) pairs will give rise only to small errors. Hence, it would
be very useful to be able to combine the large existing databases of
evaluated simple polynomial coefficients for such subsystems
with the quasichemical coefficients obtained by optimization
of other subsystems where SRO is more important, in order to
produce one large database for the multicomponent solution. It
was shown in Section 16.8 how this combination of coefficients
can be achieved for (A,B,C, …)(X) systems. In the present case, if
a binary (A,B)(X) system has been optimized using a simple
Bragg-Williams entropy and a polynomial expansion as in
Eq. (17.61) or (17.62), then the coefficients of these equations
can be substituted directly into Eq. (17.54) with ΔgoAB/X2 replaced
o
by qoo
AB/X2 or lAB/X2. This is also true for simple polynomial ternary
coefficients qijk
AB(C)/X2 as described in Chapter 14. It is required that
A
A
¼
Z
and
that
ZBAB/X2 ¼ ZBB2/X2.
The
term
ZAB/X
A2/X2
2
X
ZAB=X
n
o
+…
ΔgAB=X2 ΔgAB=X
0:5 nAB=X2 + 2 2 ZAB=XY
in Eq. (17.39)
X
2
AB=XY
P
XA=X XB=X
ΔgAB=X2, and the term ΔgoAB/X2 must
is replaced by
nij=kl
YX
313
314
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
be removed from Eqs. (17.34), (17.38). It is also recommended that
if any binary subsystem of the (A,B)(X,Y) system has been
optimized with a simple polynomial expansion, then all coefficients in the expansion for ΔgAB/XY in Eq. (17.55) be set to zero.
17.7 Summary of Model
A quasichemical model for treating short-range ordering (SRO)
in the quadruplet approximation has been proposed for solutions
with two sublattices. Both SRO of first-nearest-neighbor pairs and
SRO of second-nearest-neighbor pairs are taken into account. If
one sublattice is occupied by only one species or is empty, then
the present model reduces exactly to the quasichemical model
for SRO on one sublattice in the pair approximation as developed
in Chapter 16. Also, by means of a minor alteration to the entropy
expression, the Gibbs energy expression can be made identical to
that of a randomly mixed (Bragg-Williams) solution with a simple
polynomial expansion for the excess Gibbs energy as in
Chapter 14. This is of much practical importance because the
large existing databases of evaluated simple polynomial coefficients of certain subsystems can thereby be combined in one
database with the quasichemical coefficients of other subsystems,
in order to produce one large database for a multicomponent
solution.
The model is well suited to liquid solutions where the ratio of
the numbers of sites on the two sublattices can vary with composition. Further flexibility is provided by permitting coordination
numbers to vary with composition. Nevertheless, the model also
applies to solid solutions if the numbers of lattice sites and coordination numbers are held constant. The model can thus be combined with the compound energy formalism (Section 15.3) to treat
a wide range of types of solution (slags, mattes, ceramics, salts,
and alloys), point defects, order-disorder phenomena, nonstoichiometric phases, etc. If SRO is not included (by assuming
Bragg-Williams random-mixing entropy as just mentioned), the
model reduces exactly to the compound energy formalism for
two (or one) sublattices.
That is, several different models are limiting cases of the present model. These models can thus all be treated with the same
algorithms, the coefficients can all be stored in the same multicomponent databases, and different models for different subsystems can be combined in many cases.
NaCl
LiCl
(606o)
NaCl.LiCl
o
620 576
546
o
Na
β
0
α
570o
60
676o
LiCl
50
53 o
6
49 o
4
70
0
.
aC
Cl
. Li
l
iC
lL
Cl
2N
NaCl
55
4o
o
46
2NaCl.LiCl
(800o)
0
498o
0
o
670
70
65
0
0
LiF
70
0
80
582o
0
90
NaF
80
0
0
652o
NaF
(990o)
LiF
(848o)
Mol. %
Fig. 17.3 (Li,Na)(F,Cl) system. Experimental liquidus projection of Bergman et al. (1964) (temperature in °C).
900
Gabcova et al., Chem. Zvesti, 30(6), 796–804, 1976
850
Haendler et al., J.Electrochem. Soc., 106, 264–268, 1959
848°C
Vereshchagina, Zh. Neorg. Khim., 7, 873, 1962
Bergman et al., Zh. Neorg. Khim., 8(9), 2144–2147, 1963
Temperature (°C)
Bergman et al., Russ. J. Inorg. Chem., 9(5), 663, 1964
800
801°C
Margheritis et al., Z. Naturforsch. A, 28(8), 1329–1334, 1973
Liquid
750
Liquid + LiF(s)
700
Liquid + NaCl-LiCl(ss)
681°C
650
Liquid + NaCl-LiCl(ss) + LiF(s)
NaF-[LiF](ss) + NaCl-LiCl(ss) + LiF(s)
600
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mole fraction NaCl
Fig. 17.4 LiF-NaCl join of the (Li,Na)(F,Cl) system optimized with the MQM (Chartrand and Pelton, 2001). (For references
for data points, see Chartrand and Pelton, 2001.)
NaCl
0.2
(801°C)
0.4
0.6
0.8
0.8
494
500
55
0
NaCl-LiCl(ss)
0.6 685
60
0
00
65
0
7
681
70
0
0.4
0.4
65
0
LiF(s)
0
70 0
75
0
75
0.2
610
85
0
80
0.2
0.6
NaF-[LiF](ss) Equivalent fraction Cl / (F + Cl)
(610°C)
550
600
650
0
75
LiCl
min. 554
0.8
0
0
90
80
0
0
95
0.2
NaF
0.4
646 0.6
0.8
Equivalent fraction Li / (Na + Li)
(996°C)
LiF
(848°C)
Fig. 17.5 (Li,Na)(F,Cl) system. Liquidus projection optimized with the MQM (Chartrand and Pelton, 2001).
(NaF)2
CaF2
814o
(996o)
(1418o)
13
00
0
95
12
00
β
0
α
11
00
0
735o
80
0
85
700
10
00
652o
680o
664o
CaFCl
Equivalent fraction
90
650
602o
90
600
(NaCl)2
(801o)
504o
Equivalent fraction
70
0
486o
60
0
650
700
784o
750
0
CaCl2
(772°)
Fig. 17.6 (Na,Ca)(F,Cl) system. Experimental liquidus projection of Bukhalova (1959) (temperature in °C).
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
0.2
1151
1100
0
75
750
0.2
600 600
550
550
65
0
600
70
0
0
80
735
70
0
CaFCl(s)
650
700 650
50
60
0
70
0
0.4
0.2
0
90
0
85
0
70
7
0
485
652
0.8
0.6 504
CaCl2(s)
Equivalent fraction 2Ca / (Na + 2Ca)
0.4
0.2
(801°C)
00
10
0
95
668
785 NaCl-[CaCl2](ss)
(NaCl)2
1 05
CaF2(s)
0.6 CaFCl 0.4
0.6
80
0
0.8
0.8
Equivalent fraction F / (Cl + F)
CaF2(s2)
NaF(s)
685
12
50
00
0
85
(1418°C)
13
0
90
CaF2
0.8 1151
0.6
1200
0
95
0.4 814
65
0
70
75 0
0
(NaF)2
(996°C)
CaCl2
(772°C)
Fig. 17.7 (Na,Ca)(F,Cl) system. Liquidus projection optimized with the MQM
(Chartrand and Pelton, 2001).
By formally treating the quadruplets as the “components” of
the solution, a significant computational simplification is
realized.
17.8
Sample Calculations
An experimental liquidus projection of the (Li,Na)(F,Cl) system
is shown in Fig. 17.3. The phase diagram along the LiF-NaCl join
has also been measured by several authors as shown in Fig. 17.4.
The corresponding diagrams calculated with the two-sublattice
MQM (Chartrand and Pelton, 2001) are shown in Figs. 17.4 and
17.5. A very small ternary reciprocal parameter was introduced
in the optimization:
ΔgLiNa=FCl ¼ 393:3 + 3765:6XLi2 F2 J mol1
(17.63)
(The binary compounds 2NaClLiCl and NaClLiCl shown on the
experimental diagram of Fig. 17.3 have not been observed in more
recent studies of the NaCl-LiCl binary system.)
317
318
Chapter 17 MODELING SHORT-RANGE ORDERING WITH TWO SUBLATTICES
Experimental and calculated liquidus projections of the (Na,
Ca)(F,Cl) system are shown in Figs. 17.6 and 17.7, respectively.
A small ternary reciprocal parameter was introduced:
ΔgNaCa=FCl ¼ 2719:6 + 5020XCa2 F2 J mol1
(17.64)
The two-sublattice MQM has been used successfully in the optimization of the liquid phases in a variety of systems including
metals, oxides, salts, and sulfides (see FactSage).
References
Bergman, A.G., Kozachenko, E.L., Berezina, S.I., 1964. Russ. J. Inorg. Chem. 9, 663.
Bukhalova, G.A., 1959. Zh. Neorg. Chem. 4, 121.
Chartrand, P., Pelton, A.D., 2001. Metall. Mater. Trans. 32A, 1417–1430.
Pelton, A.D., Chartrand, P., Eriksson, G., 2001. Metall. Mater. Trans. 32A, 1409–1416.
18
SOME APPLICATIONS
CHAPTER OUTLINE
18.1 Molten Oxide Solutions 319
18.1.1 The Modified Quasichemical Model (MQM) 324
18.1.2 Associate Model 325
18.1.3 Reciprocal Ionic Liquid Model 326
18.1.4 Cell Model 327
18.1.5 The Charge Compensation Effect: Amphoteric Oxides 327
18.1.6 Other Network Forming (Acidic) Oxides 328
18.1.7 Nonoxide Anions in Oxide Melts 330
18.2 Order-Disorder Transitions 333
18.3 Liquid Solutions With More Than One Composition of Short-Range
Ordering 341
18.4 Deoxidation Equilibria in Steel 342
18.5 Magnetic Contributions to the Thermodynamic Properties of
Solutions 346
18.6 Limiting Liquidus Slope in Dilute Solutions 346
References 349
List of Websites 350
18.1
Molten Oxide Solutions
Molten oxides are generally classified as basic, acidic, or
amphoteric. So-called basic oxides (Na2O, CaO, FeO, Cu2O, …)
are ionic liquids like molten salts, consisting of metal cations
and O2– anions. Acidic oxides (SiO2, B2O3, P2O5, …) contain networks of metal atoms joined by oxygen “bridges.” In liquid SiO2,
each Si is bonded through an oxygen “bridge” to four other
silicons as illustrated in Fig. 18.1, which is a two-dimensional
schematic; the four SidO bonds are actually oriented tetrahedrally around each silicon in a three-dimensional network. The
bonds are mainly covalent.
The structure of binary liquid solutions of a basic oxide like
CaO with SiO2 is illustrated in Fig. 18.2. Basic melts (those rich
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00018-X
# 2019 Elsevier Inc. All rights reserved.
319
320
Chapter 18 SOME APPLICATIONS
O
O Si O
O
O
O
O
O
O Si O Si O Si O Si O Si O
O
O
O
O
O
O
O Si O Si O Si O Si O Si O
O
O
O
O
O
O
O
O Si O Si O Si O Si O Si O Si O Si O
O
O
O
O
O
O Si O
O
Fig. 18.1 Schematic of structure of liquid SiO2. (The oxygens are actually
tetrahedrally oriented around each Si.)
Composition
CaO
Ca2SiO4
SiO2
Mainly
Ca2+, O2-, SiO44- Mainly
Ca2+,
SiO44- (orthosilicate ions)
O
Si
Si
O
O
Si
Si
O
Linear + Branched
chain polymers
O-
Ca2+
O-
OSi
O-
O-
O2Ca2+
Ca2+
O2-
Ring polymers
OSi
O-
Ca2+
O-
Si O Si O Si O Si O
3-Dimensional
network
O
Si
O
Si
Fig. 18.2 Structure of liquid CaO-SiO2 solutions.
in CaO) consist mainly of Ca2+, O2, and SiO4
4 (orthosilicate) ions.
As the SiO2 content increases, linear dimers, O3SidOdSiO3;
linear trimers, O3SidOdSiO2dOdSiO3; and eventually longer
linear and branched chain polymers are formed. At even higher
SiO2 contents, ring polymers and eventually a three-dimensional
network structure are formed. If we consider the evolution of the
structure starting with pure SiO2 and progressively adding CaO,
then at first each addition of a Ca2+ O2– pair breaks a SidOdSi
bridge as illustrated in Fig.18.3. The reaction can be written as
Chapter 18 SOME APPLICATIONS
O0 + O2− = 2 O−
Q4 structural units (black)
Q3 structural units (red/bold)
O
O Si O
O
O
O
O
−
OO−
O
O
Si O Si O
O
O
O Si O Si O Si O Si
+
Ca 2+
O Na− − O
O
O
OO
O Si
Si O Si O Si O Si O
O Na + O
O
O
O
O
O
O Si O Si O Si O Si O Si O Si O Si O
O
O
O
O
O
O Si O
O
Fig. 18.3 SiO2 network modification by basic oxides.
ðSi O SiÞ + Ca2 + O2 ¼ 2 Si O + Ca2 +
(18.1)
or simply as
O0 + O2 ¼ 2 O
0
2–
(18.2)
where O and O represent a bridging oxygen and an oxygen
anion, respectively, and O– represents an oxygen bonded to a single Si in a “broken bridge.” The broken bridge is then a center of
negative charge, and the Ca2+ cation will tend to remain in its
vicinity. As more CaO is added, more bridges are broken until
eventually the three-dimensional network collapses.
The terms “basic” and “acidic” are borrowed from aqueous
chemistry. From Eq. (18.1), CaO can be considered to be an
“electron donor” or base, and SiO2 can be considered to be an
“electron acceptor” or acid. Acidic and basic oxides are also called
network formers and network modifiers, respectively.
The structural changes with composition can be characterized
by the amounts of so-called Qi species, which are silicons bonded
to i bridging and (4-i) nonbridging oxygens. Thus, a Si in an SiO4
4
ion is a Q0 species, and all silicons in pure SiO2 are Q4 species. The
fractions of Qi species in PbO-SiO2 melts as measured by NMR
(Fayon et al., 1998) are shown in Fig. 18.4.
The Gibbs energy of the reaction in Eq. (18.2) is very negative.
Consequently, the enthalpy of mixing is negative as shown in
Fig. 18.5 with a minimum near XSiO2 ¼ 1/3 which is the composition Ca2SiO4. The entropy of mixing, Fig. 18.6, also exhibits a minimum near XSiO2 ¼ 1/3 as this is the composition of maximum
ordering. The phase diagram of the CaO-SiO2 system is shown
321
Chapter 18 SOME APPLICATIONS
1.0
PbO-SiO2
1998, Fayon et al.
by NMR
Q0-species
Q1-species
Q2-species
Q3-species
Q4-species
0.9
Fraction of Qi-species
0.8
0.7
Q0
0.6
Q4
0.5
0.4
0.3
0.2
Q1
0.0
0.0
Q3
Q2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mole fraction SiO2
Fig. 18.4 Fractions of Qi species in PbO-SiO2 melts measured (Fayon et al., 1998) by
NMR and calculated (Kim et al., 2012) from the modified quasichemical model.
0
Enthalpy of mixing (kJ mol−1)
322
−20
MgO-SiO2
−40
−60
CaO-SiO2
−80
O2
Si
rO
S
−100
−120
0.0
BaO-SiO2
0.2
0.4
0.6
Mole fraction SiO2
0.8
1.0
Fig. 18.5 Enthalpy of mixing in liquid alkaline-earth silicates (FactSage).
in Fig. 18.7. The large negative deviations from ideal liquid mixing
result in a very low minimum eutectic temperature of 1463°C. In
SiO2-rich solutions, there is a liquid miscibility gap that can be
attributed to the difference in bonding characteristics between
Chapter 18 SOME APPLICATIONS
10
Entropy of mixing (J mol−1 K−1)
8
CaO-SiO2
6
4
2
SiO
O-
MgO-SiO2
Sr
2
0
−2
−4
−6
−8
−10
BaO-SiO2
0
0.20
0.40
0.60
0.80
1.00
Mole fraction SiO2
Fig. 18.6 Entropy of mixing in liquid alkaline-earth silicates (FactSage).
2700
2400
2572
o
Liquid + CaO
2100
2154
o
2017°
1800
1500
1437°
2 Liquids
0.423
1464°
0.718
1540°
1463°
0.613
Ca2SiO4
900
847°
1465°
1437°
SiO2(trid) + CaSiO3
1125°
0.988
1689
ASlag-liq + SiO2(crist)
1300°
1200
1895°
1723°
Liquid
o
Ca2SiO4
1799
Ca3SiO5
Temperature (°C)
0.296
CaO + Ca2SiO4
CaSiO3 + SiO2(trid)
1125°
867°
CaSiO3
Ca3Si2O7
Ca2SiO4
SiO2(s2) + CaSiO3(quartz)
600
575°
CaSiO3(s) + SiO2(quartz)
300
0
0.2
0.4
0.6
Mole fraction SiO2
Fig. 18.7 Calculated CaO-SiO2 phase diagram (FactSage).
0.8
1
323
324
Chapter 18 SOME APPLICATIONS
solutions very rich in SiO2 that are mainly covalently bonded and
solutions less rich in SiO2 that have an increasingly ionic character. This results in the positive “hump” in the enthalpy of mixing
curve at high SiO2 contents seen in Fig. 18.5 that, in turn, gives rise
to the miscibility gap.
18.1.1
The Modified Quasichemical Model (MQM)
Molten MO-SiO2 solutions (M ¼ Mg, Ca, Fe, Mn, …) can be
modeled by the single-sublattice MQM (Chapter 16) as (M,Si)
(O) with short-range ordering (SRO) between second-nearestneighbor (SNN) pairs on the metal sublattice with the following
pair-exchange reaction:
ðM O MÞ + ðSi O SiÞ ¼ 2ðM Si OÞ; ΔgMSi
(18.3)
Note the similarity to Eq. (18.2). When the ratio of SNN coordination numbers ZSi/ZM ¼ 2, the composition of maximum SRO is at
XSiO2 ¼ 1/3. The model is analogous to that discussed for liquid
KCl-CoCl2 solutions in Section 16.3.2. Since ΔgMSi is very negative,
most Si atoms in solutions where XSiO2 1/3, and even at somewhat higher SiO2 contents, are surrounded in their second coordination shells mainly by M2+ ions as illustrated in Fig. 18.2. It
can be seen that relatively isolated orthosilicate SiO4
4 groupings
occur even though the model does not treat these groupings
explicitly as complex anions.
Many binary oxide systems have been optimized with the
MQM being used for the liquid phase (FactSage). In the CaOSiO2 system, all available data have been optimized (Eriksson
et al., 1994) with the following expression for the quasichemical
parameter of Eq. (18.3) for the liquid phase:
5
ΔgMSi ¼ ð158218 + 19:456T Þ 37932YSiO2 901485YSiO
2
7
+ ð439893 133:888ÞYSiO
J mol1
2
(18.4)
with ZSi ¼ 2.7548 and ZCa ¼ 1.3774 (see Eq. 16.25). The terms in Y5
and Y7 are required to reproduce the miscibility gap at high XSiO2.
The enthalpy and entropy in Figs. 18.5 and 18.6 and the phase diagram of Fig. 18.7 were calculated from this optimization.
Optimized enthalpies and entropies of liquid mixing for other
alkaline earth-SiO2 systems (FactSage) are also shown in Figs. 18.5
and 18.6. For systems A2O-SiO2 such as Na2O-SiO2, the MQM has
been applied with a second coordination number ratio ZSi/ZA ¼ 4.
This model assumes that all A+ ions are distributed independently.
However, this is not true at very high SiO2 contents where two A+
ions will tend to congregate around a broken oxygen bridge
Chapter 18 SOME APPLICATIONS
because of coulombic attraction as illustrated in Fig. 18.3.
The observed limiting slope of the SiO2 liquidus at XSiO2 ¼ 1.0 thus
corresponds to a “one-new-particle limiting slope” (See
Section 18.6), whereas the MQM necessarily gives a two-particle
limiting slope. However, this weakness of the model only occurs
at the highest SiO2 contents.
From the optimized parameters of the MQM model, one can
calculate the fraction XSiSi of SNN (SidOdSi) pairs (i.e., oxygen
bridges) at any composition, and from this, the number of Qi species can be calculated. For details of the procedure, see Kim et al.
(2012). Plots of Qi vs composition calculated from the optimized
(FactSage) MQM parameters for PbO-SiO2 liquid solutions are
compared with the experimental data in Fig. 18.4. The only data
used in the optimization were phase equilibrium and thermodynamic data. The NMR measurements of Qi were not used to
obtain the optimized model parameters.
The liquidus surface of the ternary CaO-MgO-SiO2 system, calculated from optimized MQM parameters, is shown in Fig. 18.8.
Very good agreement with the measured phase diagram was
obtained with three small ternary model parameters. Short-range
ordering is stronger in CaO-SiO2 melts than in MgO-SiO2 melts as
witnessed by the relative values of Δh in Fig. 18.5. Hence, as discussed in Section 16.9, there are positive deviations from ideality
and a resultant tendency to immiscibility along the Ca2SiO4MgO join due to the relative stability of (CadSi plus MgdMg)
SNN pairs compared with (MgdSi plus CadCa) pairs. This is witnessed by the widely spaced isotherms along this join in Fig. 18.8
and is predicted by the MQM as discussed in Section 16.9. It is also
instructive to consider that since Ca2SiO4 and Mg2SiO4 liquids are
both very highly ordered, the orthosilicate compositions consist
2+
and Mg2+ cations. Hence, the
primarily of SiO4
4 anions and Ca
trapezoidal region outlined in Fig. 18.8 can be considered to be
approximately a reciprocal ionic system (Ca,Mg)(O,SiO4) with
Ca2SiO4-MgO as the stable diagonal along which a tendency to
immiscibility occurs as discussed in Section 15.1.
18.1.2
Associate Model
Liquid MO-SiO2 binary solutions can be modeled by a singlelattice associate model as (CaO, Ca2SiO4, SiO2). While good optimizations can be obtained in binary systems, the associate model
gives poor predictions of the properties of ternary systems for the
reasons enunciated in Section 16.9. In Fig. 18.8, for example, the
associate model predicts that Ca2SiO4 associates mix randomly
with MgO along the Ca2SiO4–MgO join. That is, the positive
325
326
Chapter 18 SOME APPLICATIONS
SiO2
(1723)
2 liquids
1689
Crist
1600
1687
CaSiO3
P-Woll
(1540)
Ca3Si2O7
1411
14631461
1391 1392
1464
α-Ca2SiO4
(2154)
α
2100
2017
1923
200 1900
0
0
1962 2002100
Woll
1336 Trid
1331
1395
1369
1370
1409
Diop
1438
1382
(1392)
1370 1363
1410
1388
1361
1366
Pig opx
1366
1370
00
4
Aker
00
1
15
(1454)
00
16
1438
1429
1453
1576
1507
ppx
1548
MgSiO3
00
17
Fors
1516
1558
18
00
1465
1437
Mg2SiO4
(1888)
1863
Mont
Merw
2200
Ca3SiO5
2300
(1799)
2400
24
00
MgO
CaO
2500
2600
25
00
2700
2374
CaO
MgO
Mole fraction
(2572)
(2825)
Fig. 18.8 Calculated CaO-MgO-SiO2 phase diagram (FactSage) (temperatures in °C).
deviations and resultant tendency to immiscibility along the join
are not predicted, resulting in a calculated liquidus temperature
up to several hundred degrees too low, which can only be remedied by including large ternary model parameters.
18.1.3
Reciprocal Ionic Liquid Model
A variation of the ionic liquid model discussed in Section 16.10
has been applied (see Thermo-Calc) to molten silicate
solutions. For example, the CaO-MgO-SiO2 system is modeled as
Chapter 18 SOME APPLICATIONS
0
(Ca2+,Mg2+)(O2,SiO4
4 , SiO2) with Ca and Mg cations distributed
randomly on one sublattice and O2– ions, SiO4
4 ions, and neutral
SiO02 species randomly distributed on a second sublattice. The following equilibrium is established:
SiO02 + 2O2 ¼ SiO4
4 ; ΔG < 0
(18.5)
Note the similarity to Eq. (18.2). For basic solutions within the
trapezoidal region outlined in Fig. 18.8, the model reduces
approximately to that of a reciprocal ionic system (Ca,Mg)(O,
SiO4) with the ions mixing randomly on their sublattices and with
Ca2SiO4-MgO as the stable diagonal. Hence, the positive deviations and tendency to immiscibility along the join are predicted.
However, they are overestimated because SRO is neglected. The
effect of SRO can be accounted for approximately by the inclusion
of ternary parameters such as XCa XMg XO XSiO4 i LCa,Mg:O, SiO4 (see
Section 15.1.1). The model has been used with success to prepare
large optimized databases for multicomponent oxide systems
(see Thermo-Calc).
18.1.4
Cell Model
A “cell model” for molten silicates has been described by Gaye
and Welfringer (1984) and has been applied with success to
develop databases for multicomponent systems (IRSID-Arcelor
database. See SGTE). The solution is considered to consist of
“cells” [M1-O-M2] that mix essentially ideally, with equilibria
among the cells such as
½Mg O Mg + ½Si O Si ¼ 2½Mg O Si; ΔG < 0
(18.6)
The close similarity to Eqs. (18.2), (18.3) is evident. The model is
conceptually quite similar to the MQM. It correctly predicts the
positive deviations along the Ca2SiO4-MgO join in Fig. 18.8 and
accounts for SRO.
18.1.5
The Charge Compensation Effect:
Amphoteric Oxides
Binary liquid Al2O3-SiO2 solutions exhibit small positive deviations from ideal mixing behavior. However, when a basic oxide
such as Na2O or CaO is added to the binary solution, the phase
is greatly stabilized. Part of the liquidus surface of the Na2OAl2O3-SiO2 system is shown in Fig. 18.9. The precipitous drop of
the liquidus temperature as Na2O is added to binary Al2O3-SiO2
solutions is a result of this stabilization. This phenomenon is
327
Chapter 18 SOME APPLICATIONS
SiO2
(1723°)
Cristobalite
1465°
160
0
1595°
1400
Tridymite
1200
1078°
867°
1000
Quartz
803°
Na6Si8O19
Na2Si2O5
1120°
Albite
3
NaAlSiO4
O
Al 2
Weight %
0
160
1090°
773°
0
150
Al)O 2
0
140 (N a
1200
Na2SiO3
720°
723°
NaAlSi3O8
llite
Mu
875°
800
809°
1076°
Na
2O
328
Fig. 18.9 Na2O-SiO2-Al2O3 phase diagram (liquidus projection) (FactSage)
(temperatures in °C).
known as the charge compensation effect and is explained as follows.
Normally, Al, which is trivalent, cannot replace tetravalent Si in the
SiO2 network. However, when Na+ ions are present, they can compensate the charge difference as illustrated in Fig. 18.10, thereby
allowing Al to replace Si in the network. Essentially, NaAl pairs act
as an independent species. In this sense, Al2O3 can be considered
to be “amphoteric,” sometimes behaving like an acid and entering
the network and sometimes behaving like a base. Along the
SiO2-NaAlSi3O8 join, Fig. 18.11, the SiO2-liquidus can be calculated
very closely by modeling the liquid as an ideal solution of SiO2 and
(NaAl)O2. The ternary Na2O-Al2O3-SiO2 system has been optimized
by modeling the liquid phase as (Na,Al,Si,NaAl)(O) with the singlesublattice MQM (Chartrand and Pelton, 1999).
18.1.6
Other Network Forming (Acidic) Oxides
Liquid solutions of B2O3 with basic oxides can be modeled in
the basic composition regions as containing mainly O2– and BO3
3
ions, that is, as (Ca)(O,BO3) or (Na)(O,BO3). At more acidic compositions, richer in B2O3 than Ca3(BO3)2 or (Na)3(BO3), the liquid
Chapter 18 SOME APPLICATIONS
+
Al 3+ Na
Si4+
Bridging oxygen
Al 3+ can replace Si 4+
Na+ or K + compensates the charge
2 Al 3+ replace 2 Si 4+
Ca 2+ or Mg 2+ compensates the charge
Fig. 18.10 Silicate systems with amphoteric Al2O3. The charge compensation effect.
1800
1723°
1700
Schairer and Bowen
Temperature (°C)
1600
1500
crist.
trid.
1465°
Liquid
1400
1300
1200
1120°
1178°
1100
Albite + Tridymite
1000
0
0.1
0.2
0.3
0.4
0.5
0.6
Weight fraction silica
0.7
0.8
0.9
1
Fig. 18.11 Albite (NaAlSi3O8)-SiO2 join of the Na2O-Al2O3-SiO2 phase diagram (Schairer and Bowen, 1956).
structures become very complex; boron can be trigonally bonded
to three oxygens and can also at least partially be bonded to four
oxygens as at the composition NaBO2. Certain compositions such
as the tetraborate composition Na2B8O13 are particularly stable,
and charge compensation effects play a role. In the presence of
329
330
Chapter 18 SOME APPLICATIONS
SiO2 and Al2O3, the structure becomes even more complex.
Progress in modeling liquid borate solutions through a combination of the MQM and associate models has been reported
(Decterov, 2018).
Molten phosphate solutions are equally complex. Recent progress in modeling has been reported by Jung (2018).
18.1.7
Nonoxide Anions in Oxide Melts
Most solutes in molten oxide solutions dissolve ionically. For
2
example, CO2 and SO3 dissolve by forming CO2
3 and SO4 ions.
Anions that have been studied in molten silicates include S2,
2
2
3
3
4
SO2
4 , CO3 , P , N , CN , C , C2 , F , Cl , and I . Molten
solutions such as MnO-SiO2-MnS have been modeled over the
entire composition range with the two-sublattice MQM as (Mn,
Si)(O,S) (Kang and Pelton, 2009). Molten oxyfluoride solutions
have similarly been modeled ( Jung, 2018) up to high fluoride
concentrations.
For relatively dilute solutions of anions in molten oxides, a simplified model, the Reddy-Blander Capacity Model (Reddy and
Blander, 1987), has proved successful. We take as example sulfide
solutes in molten binary AnO-SiO2 solutions. The exchange of
oxygen and sulfur between an oxide melt and other phases at
low oxygen potentials can be written in terms of the following
equilibrium:
An OðliqÞ + 0:5 S2 ðgasÞ ¼ An SðliqÞ + 0:5 O2 ðgasÞ; ΔG o ¼ RT ln K
(18.7)
where AnO and AnS are melt components, A being a cation such as
Ca, Mg, Na or Al. An equilibrium constant may be written for the
following reaction:
K ¼ ðaAn S =aAn O ÞðpO2 =pS2 Þ0:5 ¼ exp ðΔG o =RT Þ
(18.8)
where aAnS and aAnO are the component activities.
The standard Gibbs energy change of the reaction can be
obtained from data for pure liquid AnO and AnS. The activity of
AnO is known as a function of composition and temperature in
many binary AnO-SiO2 systems as a result of extensive experimental data and critical evaluations and optimizations. If the amount
of sulfide in the solution is relatively small, the oxide activities
remain close to their values in the sulfide-free melt. Hence,
Eq. (18.8) can be used to calculate the sulfide activity. The sulfide
Chapter 18 SOME APPLICATIONS
solubility can then be calculated from the capacity model that
relates the sulfide activity to the sulfide concentration as follows.
For AnO-AnS-SiO2 solutions dilute in AnS and in the basic composition region where XSiO2 1/3, an extremely simple approximate model is proposed, namely, that the only species in
solution are A(2/n)+ cations and O2– and SiO4
4 anions, along with
a small amount of S2 anions. The three kinds of anions are
assumed to form an ideal solution on an anionic sublattice:
2 2
(A(2/n)+)(SiO4
4 ,O ,S ). In one mole of AnO-SiO2 solution, there
2
ions, and
are thus XSiO2 moles of SiO4
4 , XAnS moles of S
2–
(XAnO 2XSiO2) moles of O ions for a total of (XAnO XSiO2 + XAnS)
moles of anions. The activity of AnS relative to pure liquid AnS
as standard state is thus
(18.9)
aAn S ¼ XAn S γ oAn S =ðXAn O XSiO2 + XAn S Þ
where γ oAnS is a Henrian activity coefficient that, in general, can
vary with composition. However, an ideal solution is assumed,
and so, γ oAnS ¼ 1 is assumed at all compositions.
For acidic solutions where XSiO2 1/3, a very simple approximate model is again assumed (Pelton, 1999). The concentration
of O2– ions in acid solutions is considered to be negligible. There
are thus two types of oxygen: first, “bridging” oxygens bonded to
two silicon atoms (Si-O-Si) and, second, oxygens singly bonded to
one silicon atom. Every silicon is bonded to four oxygens to form a
silicate tetrahedron. There are thus (2XSiO2 XAnO) moles of bridging oxygens per mole of solution. For instance, at XSiO2 ¼ 1/3, there
are no bridging oxygens, all silicate tetrahedra being SiO4
4 anions,
while at XSiO2 ¼ 1, all oxygens are bridging oxygens. Sulfur is
assumed to enter the solution as S2 ions that share sites with
the silicate tetrahedra. The molar configurational entropy of
mixing is then calculated as follows. In a first step, XAnO moles
of S2 ions and XSiO2 moles of silicate tetrahedra are distributed
randomly on a quasi-lattice. In a second step, the oxygen bridges
are distributed randomly between neighboring silicate tetrahedra.
In one mole of solution, the number of quasi-lattice sites is
(XSiO2 + XAnS). The probability that two neighboring sites are
both occupied by silicate tetrahedra is (XSiO2/(XSiO2 + XAnS))2. The
number of potential bridge positions is thus given by
n1 ¼ 2X2SiO2/(XSiO2 + XAnS), where the coordination number has been
taken as 4. The second step consists of distributing the
n2 ¼ (2XSiO2 XAnO) moles of oxygen bridges randomly over these
n1 positions. The molar configurational entropy of mixing is
therefore
331
332
Chapter 18 SOME APPLICATIONS
XA n S
XSiO2
+ XSiO2 ln
Δs ¼ R XAn S ln
ðXSiO2 + XAn S Þ
ðXSiO2 + XAn S Þ
n2
ð n 1 n2 Þ
(18.10)
+ ðn1 n2 Þ ln
R n2 ln
n1
n1
Eq. (18.10) can be differentiated to give the partial entropy of AnS,
from which aAnS can be calculated in the limit as XAnS ! 0 as follows. Let the solution consist of nSiO2 moles of SiO2, nAnO moles
of AnO, and nAnS moles of AnS. The total entropy is then given
by ΔS ¼ (nSiO2 + nAnO + nAnS) Δs, where Δs is the molar entropy from
Eq. (18.10). By substituting Xi ¼ ni/(nSiO2 + nAnO + nAnS), where
i ¼ SiO2, AnO, or AnS, an expression for ΔS can be written in terms
of the mole numbers. The partial molar entropy of AnS is then
given by the thermodynamic relationship ΔsAnS ¼ (∂ ΔStotal/∂ nAnS),
where the partial derivative is calculated at constant nSiO2 and
nAnO. Assuming ideal behavior, we set RT ln aAnS ¼ TΔsAnS. In
the limit as XAnS ! 0, this gives
aA n S ¼
XA n S
2XSiO2
ðXSiO2 + XAn S Þ 1 XSiO2
2
γ oAn S
(18.11)
for XSiO2 1/3, with the factor γ oAnS to account for deviations
from ideal behavior. Again, ideality is assumed, and γ oAnS is set
equal to 1.0.
It may be noted that Eqs. (18.9) and (18.11) are identical when
XSiO2 ¼ 1/3.
Eqs. (18.9), (18.11) can thus be used to calculate the sulfide solubility XAnS when the sulfide activity has been calculated from
Eq. (18.8). It has been shown (Pelton et al., 1993) that predicted
sulfide solubilities agree within experimental undertainties with
measurements in binary AnO-SiO2 melts when A ¼ Mn, Fe, Ca,
and Mg. The calculations for MnO-SiO2 melts are reproduced
and compared with measurements in Fig. 18.12. Data for ΔGo
of Eq. (18.7) and aMnO were taken from FactSage. It should be
stressed that these are a priori predictions of the model since it
was assumed that γ oMnS ¼ 1.0 at all compositions. That is, the
model has no adjustable parameters.
In Fig. 18.12, the sulfide solubilities have been normalized as
sulfide capacities CS as defined by Fincham and Richardson (1954):
0:5
CS ¼ ðwt % SÞ pO2 =pS2
(18.12)
where CS is constant at a given oxide composition as long as the
sulfide content is small.
The sulfide capacity model has been applied with success to
the binary melts AnO-Al2O3, AnO-TiO2, and AnO-ZrO2 by simply
Chapter 18 SOME APPLICATIONS
Fig. 18.12 Sulfide capacities in MnO-SiO2 melts predicted from Reddy-Blander
Capacity Model compared with measured points. (For references to experimental
points, see Pelton et al., 1993.)
replacing XSiO2 in Eqs. (18.9) and (18.11) by XAlO1.5, XTiO2, or XZrO2
and has been extended to multicomponent oxide solutions dilute
in sulfide (Pelton et al., 1993).
Finally, the capacity model has been modified to permit similar
a priori predictions of the solubilities of sulfates, phosphates, carbonates, and halides in oxide melts (Pelton, 1999).
18.2
Order-Disorder Transitions
Order-disorder transactions were introduced in Chapter 10. An
example is the B2 ! A2 transition in the bcc phase of the Al-Fe system in Fig. 10.2. At low temperatures, the bcc phase exhibits longrange ordering (LRO). Two sublattices can be distinguished, one
consisting of the corner sites of the unit cells and the other of
the body-centered sites. The Al atoms preferentially occupy one
sublattice, while the Fe atoms preferentially occupy the other. This
is the ordered B2 (CsCl) structure. As the temperature increases,
the phase becomes progressively disordered, with more and more
Al atoms moving to the Fe-sublattice and vice versa until, at the
transition line, the disordering becomes complete with the Al
and Fe atoms distributed equally over both sublattices. At this
point the two sublattices become indistinguishable, the LRO disappears, and the structure becomes the A2 (bcc) structure.
The transition is second order, with no two-phase (ordered +
disordered) region in the phase diagram.
333
Chapter 18 SOME APPLICATIONS
Consider a binary solution of components A and B with two
sublattices, with an equal number of sites on each and with
A and B atoms on both sublattices: (A,B)(A,B).
Using the compound energy formalism (CEF), Section 15.3, we
write in the simplest case
o
o
o
o
+ XB0 XB00 gBB
+ XA0 XB00 gAB
+ XB0 X 0 0A gBA
g ¼ XA0 XA00 gAA
(18.13)
+RT XA0 ln XA0 + XB0 ln XB0 + XA00 ln XA00 + XB00 ln XB00
The similarity to the two-sublattice (A,B)(X,Y) solution discussed
in Section 15.1 is evident. Site fractions can be represented on a
“reciprocal” composition square as in Fig. 18.13. Lines of constant
overall component mole fractions XB ¼ (1 – XA) are shown on the
figure.
Since the two sublattices are identical, we set goAB ¼ goBA and
define
o
o
o
o
ΔG exchange ¼ gAB
+ gBA
gAA
gBB
<0
(18.14)
(B)(A)
0.2
0.1
0.3
0.4
0.5
0.7
0.6
0.8
(B)(B)
0.9
XB
0.9
0.8
0.8
9
0.
0.9
=
b
XB
=
0.5
0.7
0.5
0.6
0.6
0.7
8
0.
h
=
0.4
7
0.
XB
0.4
=
6
0.
XB
XB
X'B
=
0.3
0.3
5
0.
XB
g
D
is
or
de
re
d
0.1
=
2
0.
3
0.
0.2
a
XB
=
1
0.
0.1
0.2
ered
=
(A)(A)
4
0.
0.2
=
Ord
XB
XB
0.1
334
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(A)(B)
X''B
Fig. 18.13 Composition square for a solution (A,B)(A,B). Axes are site fractions.
Lines of constant overall component fraction XB are shown. Phase boundary
shown by heavy line is the calculated order/disorder transition composition at
800 K from Eq. 18.15 when DGexchange ¼ 33.3 kJ mol1.
Chapter 18 SOME APPLICATIONS
335
Hence, (A-B) and (B-A) nearest-neighbor pairs are energetically
favored, and so, at lower temperatures where the entropic terms
are less important, the solution is ordered, with one sublattice
occupied principally by A atoms and the other by B atoms. As
the temperature increases, the entropic terms become progressively more important with more and more B atoms on A-sites
and vice versa.
Consider the stoichiometric AB composition where
XA ¼ XB ¼ 0.5 and where through the mass balance, XB0 ¼ XA00 .
At this composition, let us define an LRO parameter α ¼ XB0 ¼
(1 XA0 ) ¼ XA00 ¼ (1 X B00 ). Substitution into Eq. (18.13) then gives,
along the stoichiometric line,
o
o
+ 2RT ðα ln α + ð1 αÞ ln ð1 αÞÞ
+ ð1 αÞgAB
g ¼ αgBA
(18.15)
αð1 αÞ ΔG exchange
(The formal similarity of Eq. (18.15) to the Gibbs energy equation
of a simple regular solution is noteworthy.) When α ¼ 0, the solution is completely ordered as (A)(B). (Equivalently, when α ¼ 1, the
solution is completely ordered as (B)(A).) When α ¼ 0.5, the solution is completely disordered as (A0.5B0.5)(A0.5B0.5). Plots of g
from Eq. (18.15) vs α along the equimolar line XA ¼ XB ¼ 0.5
when ΔGexchange ¼ 33.3 kJ mol1 are shown in Fig. 18.14.
2.0
1.5
60
1.0
0K
g - (αg°BA + (1 − α)g°AB) (kJ mol-1)
0.5
0.0
-0.5
d
c
800 K
-1.0
-1.5
a
b
-2.0
-2.5
-3.0
e
1000 K
-3.5
-4.0
-4.5
-5.0
-5.5
-6.0
(A)(B)
f
0.1
0.2
0.3
1200 K
0.5
0.6
0.4
(A0.5B0.5)(B0.5A0.5)
0.7
a
0.8
0.9
(B)(A)
Fig. 18.14 Relative g from Eq. (18.15) vs a at the equimolar composition XA ¼ XB ¼ 0.5 in a solution (A,B)(B,A) when
DGexchange ¼ 33.3 kJ mol1.
Chapter 18 SOME APPLICATIONS
The equilibrium value of α is that that minimizes g: that is, at
800 K, either point a or point b (which are equivalent); at 600 K,
either of the equivalent points c or d; and at 1000 and 1200 K,
points e and f at α ¼ 0.5. (The formal similarity of Fig. 18.14 to
Fig. 5.9 for a simple regular solution is noteworthy.) A plot of
the equilibrium value of α vs Tat the equimolar composition when
α 0.5 is shown in Fig. 18.15. At 300 K, the solution is virtually
completely ordered. As T increases, it becomes progressively more
disordered until it becomes completely disordered above
T ¼ 1000 K which is the second-order order-disorder transition
temperature when XA ¼ XB ¼ 0.5.
Unlike short-range order (SRO) that theoretically only disappears completely at T ¼ ∞, LRO disappears completely above a
finite temperature.
Taking now Eq. (18.13) with ΔGexchange ¼ 33.3 kJ mol1 and
calculating the locus of points of minimum Gibbs energy at 800 K,
the heavy line on Fig. 18.13 is generated. The formal similarity of
Figs. 18.13–18.15 may be noted. If the order-disorder transition
temperatures are plotted as a function of XB, the phase diagram
of Fig. 18.16 is generated with a second-order transition boundary.
Points a, e, g, and h on Fig. 18.16 correspond to those on
Figs. 18.13–18.15. Fig. 18.16 may be compared with the secondorder B2 ! A2 transition line in the Fe-Al system in Fig. 10.2.
0.5
e
f
0.4
a
336
0.3
a
0.2
0.1
c
0.0
300
400
500
600
700
800
900
1000
1100
1200
T(K)
Fig. 18.15 Plot of equilibrium value of a at the equimolar composition XA ¼ XB ¼ 0.5
in a solution (A,B)(B,A) vs T setting dg/da ¼ 0 in Eq. 18.15 when
DGexchange ¼ 33.3 kJ mol1.
Chapter 18 SOME APPLICATIONS
1050
1000
e
Disordered
950
Ordered
T(K)
900
850
g
800
a
h
750
700
650
600
A
0.2
0.4
0.6
0.8
Overall component mole fraction XB
B
Fig. 18.16 Second-order order-disorder transition boundary in a solution (A,B)(B,A)
from Eq. 18.13 with DGexchange ¼ 33.3 kJ mol1.
1200
Liquid
1000
T(°C)
800
Disordered fcc
600
400
L10
200
L12
L12
0
Au
0.2
0.4
0.6
Mole fraction cu
0.8
Cu
Fig. 18.17 Au-Cu phase diagram (FactSage).
Order-disorder transitions can also be first order. The phase
diagram of the Au-Cu system is shown in Fig. 18.17. There are
three ordered fcc solutions, each exhibiting a first-order transition
to the disordered fcc phase.
337
338
Chapter 18 SOME APPLICATIONS
In the L10 phase, the corner sites and the sites on one pair of
opposing faces of the fcc unit cell constitute one sublattice, while
the sites on the other two pairs of opposing faces constitute a second sublattice with the same number of lattice sites. In the simplest
case, we model the phase as (A,B)(B,A) with the Gibbs energy
given by Eq. (18.13) with one additional term: XA0 XA00 XB0 XB00 LAB:BA,
with LAB:BA < 0. This term is sharply negative near the center
of the composition square in Fig. 18.13. At the stoichiometric composition XA ¼ XB ¼ 0.5, g is then given by Eq. (18.15) with the additional term α2(1 α)2 LAB:BA. With ΔGexchange ¼ 70 kJ mol1 and
LAB:BA ¼ 100.0 kJ mol1, plots of g vs α are shown in Fig. 18.18. At
950 K, an ordered phase at point a (or, equivalently, at point b) has
the lowest Gibbs energy. At 1050 K, a disordered phase at point f has
the lowest Gibbs energy. At 1000 K, an ordered phase at point c (or
equivalently at point d) transforms isothermally to a disordered
phase with α ¼ 0.5 at point e by a first-order transition. The calculated phase diagram of T vs overall mole fraction XB is shown
in Fig. 18.19.
The term XA0 XA00 XB0 XB00 LAB:BA may be compared with the reciprocal excess term in Eq. (15.12) that was shown in Section 15.1.1
to be an approximate correction to account for SRO. Through
1000
800
950 K
G − (αg°BA + (1− α)g°AB) (J mol−1)
600
400
200
1000 K
0
−200
−400
b
a
c
e
d
1050 K
−600
−800
f
−1000
0
(A)(B)
0.1
0.2
0.3
0.4
0.5
(A0.5B0.5)(B0.5A0.5)
0.6
a
0.7
0.8
0.9
1
(B)(A)
Fig. 18.18 Relative g from Eq. (18.15) with the additional term XA0 XA00 XB0 XB00 LAB:BA vs a at the equimolar composition
XA ¼ XB ¼ 0.5 in a solution (A,B)(B,A) when DGexchange ¼ 70 kJ mol1 and LAB:BA ¼ 100 kJ mol1.
Chapter 18 SOME APPLICATIONS
1050
Disordered
1000
T(K)
950
900
850
Ordered
800
A
0.2
0.4
0.6
0.8
Overall component mole fraction XB
B
Fig. 18.19 First-order order-disorder transition in a solution (A,B)(B,A) from
Eq. 18.13 with DGexchange ¼ 70 kJ mol1 and with the additional term
XA0 XA00 XB0 XB00 LAB:BA with LAB:BA ¼ 100 kJ mol1.
the formation of SRO, the number of energetically favorable
nearest-neighbor (A-B) bonds can be increased within the disordered solution, thereby stabilizing the disordered phase relative to
the ordered phase in the central composition region. However, the
magnitude of the factor LAB:BA required to produce a first-order
transition (100 KJ mol1 when ΔGexchange ¼ 70 kJ mol1 in
the present example) is approximately twice than that given by
Eq. (15.13) (assuming Δε ¼ ΔGexchange) if simple pairwise FNN
SRO alone were responsible for the stabilization of the
disordered phase.
In real systems such as Fe-Al or Au-Cu, the bcc B2 and fcc L10
phases can be quantitatively modeled with a two-sublattice CEF
model through the addition of several empirical interaction terms
for interactions within a sublattice (i LAB:B, i LAB:A , etc.) and between
sublattices (LAB:BA). However, it is better to use the two-sublattice
MQM in the quadruplet approximation (Chapter 17) in which
FNN and SNN SRO and the coupling between them are explicitly
taken into account. When this model is used, go of the reciprocal
quadruplet end-member ABAB must be chosen to be more negative than that given by the default expression of Eq. (17.38) if a
first-order transition is to result.
In the ordered L12 phases in Fig. 18.17, the corner sites of the
fcc unit cells are occupied predominantly by one element, while
the face-centered sites are occupied predominantly by the other.
339
340
Chapter 18 SOME APPLICATIONS
The phases could then each be modeled as two-sublattice solutions as in the case of the L10 phase but with the number of sites
on the sublattices in the ratio 3:1.
The foregoing discussion has been intended only as an introduction to the modeling of order-disorder transitions. Clearly,
modeling each of the three ordered fcc phase in the Au-Cu system
with a two-sublattice model is incorrect. In a proper treatment,
the three ordered phases and the disordered solution must all
be treated as a single fcc phase with four sublattices, namely,
the corner sites of the unit cells and the sites on each pair of
opposing faces. (Similarly, correct modeling of both the B2 and
DO3 ordered bcc phases requires a four-sublattice model.) Successful four-sublattice models have been developed with the
CEF with many empirical interaction terms, including polynomial
terms to account for SRO. See, for example, Kusoffsky et al. (2002)
and Abe and Sundman (2003). However, a better modeling
requires that SRO between and within sublattices and their coupling be explicitly taken into account.
In this regard, the cluster variation method (CVM) of Kikuchi
(1951), Kikuchi and Sato (1974) has proved successful. See also
Inden (2001). This model is conceptually similar to the MQM
discussed in Chapter 17. “Clusters” (tetrahedra, octahedra, …)
involving atoms on the sites of four sublattices are distributed,
taking explicit account of their relative positions in the fcc structure, (i.e., taking account of whether they share corners, edges, or
faces.) The entropy expression is derived by a procedure similar to
that used in the two-sublattice MQM, Section 17.5. That is,
roughly speaking, an entropy term is written for the distribution
of the largest structural units (tetrahedra, octahedra, …) with
the fraction of each unit divided by its value in a random mixture
of smaller structural units (pairs, triplets, …); a second entropic
term is then written for the distribution of these smaller units,
with the fraction of each unit divided by its value in a random
mixture of even smaller units; etc. The resultant entropy expression is necessarily approximate as in the case of the MQM.
Due to its mathematical complexity, particularly for multicomponent systems, the CVM has only been applied so far to a few
systems. The CEF is used for most multicomponent ordered
phases in most current large database computing systems.
A simplified CVM model called the cluster-site approximation
(CSA) has been proposed by Oates and coworkers (Oates and
Wenzl, 1996; Oates et al., 1999; Cao et al., 2007). In this model,
the clusters are not permitted to share edges or bonds. The resultant approximate entropy expression is significantly simpler and
more mathematically tractable than that of the CVM.
Chapter 18 SOME APPLICATIONS
18.3
Liquid Solutions With More Than One
Composition of Short-Range Ordering
As discussed in Chapter 14, in a binary solution, a composition
of maximum SRO is characterized by a relatively sharp minimum
in a plot of gE vs composition. The presence of such minima can
be rendered more evident by calculating the excess stability function proposed by Darken (1967):
(18.16)
ES ¼ ∂2 g E =∂X 2 T ,P
Each relatively sharp minimum in gE translates into a maximum
in the ES function.
The excess stability function of liquid K-Te alloys is shown in
Fig. 18.20 (Petric et al., 1988). Peaks are observed near the compositions KTe and KTe7. A small hump near XK ¼ 0.37 is also seen.
That is, there is clearly more than one composition of SRO.
550
500
450
400
Excess stability (kJ)
350
300
250
200
150
100
50
0
−50
−100
0
10
20
30
40
50
60
Atom % K
Fig. 18.20 Excess stability function of liquid K-Te solutions at 773 K
(Petric et al., 1988).
341
342
Chapter 18 SOME APPLICATIONS
Liquid Fe-O solutions exhibit a strong tendency to SRO near
the compositions corresponding to FeO and Fe2O3. The solutions
have been modeled (Hidayat et al., 2015) over the entire composition range by the single-sublattice MQM as (FeII, FeIII, O), with
two oxidation states of iron. The species are not considered to
be charged so that the condition of electroneutrality is not
imposed. Hence, for example, pure metallic iron consists almost
entirely of FeII. The number of (FeII-O) pairs passes through a
maximum close to the FeO composition, while (FeIII-O) pairs
are most abundant near the Fe2O3 composition. That is, there is
a drastic shift from the FeII to the FeIII oxidation state near the
FeO composition. The model has been extended (Shishin et al.,
2015) to molten FedCudOdS solutions.
Molten cryolite solutions, NaF-AlF3, have been modeled in a
similar fashion (Chartrand and Pelton, 2002) with a singlesublattice MQM and assuming both four-coordinate and fivecoordinate Al. (In a complex anion model, this would correspond
2
to the formation of AlF
4 and AlF5 complex ions in contrast to the
common assumption of primarily AlF3
6 ions.) In order to reproduce the data in solutions very rich in AlF3, an additional dimeric
Al2 species was included. This is an example of combining the
MQM with an associate model. Another such example was given
previously in Section 18.1.5 in regard to the charge compensation
effect in Na2O-Al2O3-SiO2 melts that were modeled as (Na,Al,Si,
NaAl)(O) with NaAl associates.
Other examples of liquid solutions modeled with more than
one composition of SRO are complex aluminoborosilicates
(Decterov, 2018) and molten phosphate solutions ( Jung, 2018).
18.4 Deoxidation Equilibria in Steel
Dissolved oxygen is removed from molten steel by the addition
of deoxidants such as Al and Mg that precipitate the oxygen as
solid oxides. In Al deoxidation, for example, for the equilibrium
among dissolved Al and O and solid Al2O3,
2Al + 3O ¼ Al2 O3
(18.17)
the equilibrium constant is
K ¼ ðXAl γ Al Þ2 ðXO γ O Þ3
(18.18)
where XAl and XO are the concentrations of dissolved Al and O. If
the solution of Al and O in Fe is an ideal Henrian solution, then
γ Al ¼ γ oAl and γ O ¼ γ oO where γ oAl and γ oO are the limiting Henrian activity coefficients at infinite dilution. XO then continually decreases
as XAl increases.
Chapter 18 SOME APPLICATIONS
Deviations from ideal Henrian behavior are usually treated by
the interaction parameter formalism (Section 14.10). The problem
with this approach is that the deoxidizing metals have a strong
affinity for oxygen, and even in dilute solutions, there is a strong
tendency toward SRO with metal and oxygen atoms as preferential nearest neighbors. That is, much of the observed deviations
from ideal behavior are due to deviations of the configurational
entropy from ideality, whereas the interaction parameter formalism attributes the deviations entirely to enthalpy and nonconfigurational entropy.
The SRO can be taken into account by means of the single-lattice
MQM (Section 16.2). However, a simple alternative approach as proposed by Jung et al. (2004) postulates the formation of dissolved
metal oxide associates and treats the solution with an associate
model as in Section 16.1. In Al deoxidation, for example, the formation of dissolved Al*O and Al2*O associate “molecules” is proposed:
o
Al + O ¼ Al∗ O; ΔgAl
∗O
(18.19)
o
2Al + O ¼ Al2 ∗ O; ΔgAl
2∗O
(18.20)
The solution is then considered to be a Henrian solution of dissolved
Al, O, Al*O, and Al2*O species with an ideal configurational entropy.
The parameters of the model are the Gibbs energies of the association reactions in Eqs. (18.19), (18.20) and interaction parameters
between the dissolved species such as εM,M, εM,O, εM,M∗O, and εO,M∗O.
Curves for the Al deoxidation of Fe are shown in Fig. 18.21.
Total dissolved oxygen (as Al + Al*O + Al2*O) and total dissolved
Al (as Al + Al*O + 2Al2*O) in equilibrium with Al2O3 are plotted.
The calculations were performed with the associate model with
constant (i.e., temperature-independent) values of ΔgoAl∗O and
ΔgoAl2∗O of Eqs. (18.19), (18.20) relative to the ideal Henrian
standard states as the only adjustable model parameters. Small
interaction parameters εAl,Al and εO,O were taken from the literature. No other interaction parameters were required. Agreement
with experimental data is very good. The assumption of the higher
associate Al2*O only has a visible effect on the calculated curves
when log (wt% total Al) > 0.3.
Previously, Sigworth and Elliott (1974) modeled the solution
without considering the formation of associates. A very negative
temperature-dependent parameter εO,Al was required. The deoxidation equilibrium at 1600 °C calculated by their model is also
shown in Fig. 18.21. Clearly, their model cannot be extrapolated
beyond the composition range of the data. On the other hand,
with the associate model, extrapolations both in temperature
and composition are reasonable.
343
Chapter 18 SOME APPLICATIONS
0.0
Saturated with Al2O3
−0.5
−2.0
16
95
16
Gokcen and Chipman [11]
1866°C
McLean and Bell[12]
1760°C
1823°C
1696°C
1723°C
66
−1.5
18
17 17
23
23 60
18
−1.0
log[wt% total O]
344
00
°C
−2.5
−3.0
−3.5
1600°C
Fruehan [13]
Seo et al [18]
Janke and Fischer [15]
Dimitrov et al [17]
−4.0
−4.0 −3.5 −3.0 −2.5
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
log[wt% total Al]
Fig. 18.21 Total dissolved oxygen and total dissolved Al contents of liquid Fe in
equilibrium with solid Al2O3. Lines calculated from associate model ( Jung et al.,
2004). Dashed line from Sigworth and Elliott (1974). For references for experimental
data, see Jung et al. (2004).
An interesting feature of Fig. 18.21 is the “deoxidation
minimum.” Beyond a certain point, further additions of Al actually increase the dissolved oxygen content. This can be explained
as follows. At higher total Al contents, most of the dissolved oxygen is in the form of Al*O associates because ΔgoAl∗O in Eq. (19.18)
is very negative. The main deoxidation equilibrium then becomes
3Al∗ O ¼ Al2 O3 + Al
(18.21)
Consequently, an increase in the concentration of dissolved Al
results in an increase in the concentration of Al*O and hence in
the total amount of dissolved oxygen.
Another example of the application of the associate model is
shown in the equilibrium curve for Mg deoxidation in
Fig. 18.22. Previous attempts to model this equilibrium and the
similar equilibrium for Ca deoxidation, with the interaction
parameter formalism under the assumption of ideal mixing of dissolved Mg and O, have failed, even with many large first- and
second-order interaction parameters. With the associate model,
however, the curves in Fig. 18.22 were calculated with only one
temperature-independent adjustable parameter, namely, ΔgoMg∗O,
for the formation of dissolved Mg*O associates:
o
Mg + O ¼ Mg∗ O; ΔgMg
∗O
No interaction parameters were required.
(18.22)
Chapter 18 SOME APPLICATIONS
100
90
Han et al. [30]
70
wt ppm total O
Seo and Kim [31]
1550°C
1600°C
1650°C
80
60
50
Saturated with MgO
40
30
20
1650°C
10
0
0
50
100
200
250
150
Wt ppm total Mg
1600°C
1550°C
350
300
400
Fig. 18.22 Total dissolved oxygen and total dissolved Mg contents of liquid Fe in
equilibrium with solid MgO. Lines calculated from associate model ( Jung et al.,
2004). For references for experimental data, see Jung et al. (2004).
The shape of the curves may be explained as follows. Because
ΔgoMg∗O < < 0, the reaction in Eq. (18.22) is displaced very strongly
to the right. Hence, a solution will contain either dissolved Mg*O
and O species (but virtually no Mg species) or dissolved Mg*O and
Mg species (but virtually no O species). Suppose we start with Fe
containing a high concentration of dissolved oxygen at 1650°C
and begin adding Mg. At first, the Mg reacts with the oxygen to
form Mg*O associates, leaving virtually no free Mg in solution.
When the concentration of Mg*O reaches approximately
25 ppm, it attains equilibrium with solid MgO:
Mg∗ O ¼ MgO
(18.23)
As more Mg is added, it reacts with dissolved oxygen to precipitate
MgO. The concentration of dissolved oxygen thus decreases, while
the concentration of Mg*O remains constant, and consequently,
the concentration of total dissolved Mg remains nearly constant.
This results in the nearly vertical section of the curve in
Fig. 18.22. When the total dissolved oxygen content has been
reduced to approximately 13 ppm, the concentrations of free dissolved Mg and O are both extremely low, and Mg*O associates
are virtually the only species in solution. Further addition of Mg
thus serves only to increase the free Mg concentration, with virtually no further precipitation of MgO; hence the nearly horizontal
section of the curve in Fig. 18.22.
345
346
Chapter 18 SOME APPLICATIONS
18.5 Magnetic Contributions to the
Thermodynamic Properties of Solutions
An expression for the magnetic contributions to the Gibbs
energy of ferromagnetic and antiferromagnetic compounds was
given in Section 10.1. For binary solutions with components
A and B, the same equation can be applied with Tcurie and β varying between their values in pure A and B with purely empirical
excess terms expressed as Redlich-Kister polynomials:
X
i
m AB ðX A X B Þi (18.24)
Tcurie ¼ XA TcurieðAÞ + XB TcurieðBÞ + XA XB
i0
where the mAB are empirical parameters, and similarly for β.
For ternary and higher-order solutions, Tcurie and β can be estimated from the binary parameters by the same interpolation
equations as discussed in Chapter 14.
i
18.6 Limiting Liquidus Slope in Dilute
Solutions
The limiting slope of the liquidus line at XB ¼ 0 in a binary system of components A and B can provide useful information on the
structure of the liquid solution since, as will be shown, it is independent of the details of the particular solution model and of the
excess properties.
The liquidus of a binary solution A-B in the A-rich concentration region is shown in Fig. 18.23. We assume no solubility of B in
solid A. It will be shown that the limiting liquidus slope at XB ¼ 0 is
given by
2
(18.25)
lim ðdXB =dT Þ ¼ ΔhofðAÞ = TfoðAÞ νtotal
X B ¼0
where
and Δhof(A) are the melting point and enthalpy of fusion
of A at the melting point and νtotal is the total number of “new”
independently distributed species (atoms, molecules, ions, point
defects, etc.) not already in the solvent A that are introduced per
formula unit of solute B.
Let the solution consist of nA moles of A and nB moles of B.
Suppose that the nB moles of solute introduce n1 moles of new
species 1, n2 moles of new species 2, etc. that are not already in
the solvent, and let
Tof(A)
ni ¼ νi nB
(18.26)
Chapter 18 SOME APPLICATIONS
T°f(A)
Liquid
Temperature
Limiting liquidus slope at XB = 0
Liquid + Solid A
A
Mole fraction XB
Fig. 18.23 Limiting liquidus slope in a binary system A-B.
As examples, for liquid solvents:
Example 1:
Example 2:
Solvent ¼ KNO3
Solute ¼ CaCl2
n1 ¼ nCa2 + ¼ nCaCl2
ν1 ¼ 1
n2 ¼ nCl ¼ 2nCaCl2
ν2 ¼ 2
νtotal ¼ 3
Solvent ¼ KCl
Solute ¼ CaCl2
Assume no vacancies in the liquid solution
(Temkin model)
n1 ¼ nCa2 + ¼ nCaCl2
ν1 ¼ 1
νtotal ¼ 1
(Cl is already in the solvent, so it is not “new”)
Example 3:
Solvent ¼ KCl
Solute ¼ CaCl2
Assume that cation vacancies are introduced and
that the Ca2+ ions and vacancies are not associated
n1 ¼ nCa2 + ¼ nCaCl2
ν1 ¼ 1
n2 ¼ nVa ¼ nCaCl2
ν2 ¼ 1
νtotal ¼ 2
347
348
Chapter 18 SOME APPLICATIONS
Example 4:
Example 5:
Solvent ¼ Fe
Solute ¼ O2 (dissolves
monatomically)
n1 ¼ nO ¼ 2nO2
ν1 ¼ 2
νtotal ¼ 2
Solvent ¼ H2O
Solute ¼ O2 (dissolves molecularly)
n 1 ¼ nO 2 ¼ nO 2
ν1 ¼ 1
νtotal ¼ 1
If Δhof(A), Tof(A), and the limiting slope are known and if it is known
that the solubility of B in solid A is negligible, then Eq. (18.25) permits νtotal to be determined. It is not necessary to know the values of
any excess properties nor to assume any details of the solution
model. (For example, it does not matter if the species are distributed substitutionally or interstitially.) For instance, measurements
of the limiting slope of the KCl-liquidus in the KCl-CaCl2 system
permit one to determine rapidly whether Example 2 or Example
3 is correct. (Example 2 is correct. CaCl2 additions do not introduce
vacancies into liquid KCl as they do when added to solid KCl.)
Alternatively, if Δhof(A), Tof(A), and νtotal are known, then
Eq. (18.25) permits one to verify whether or not the solubility of
B in solid A is negligible. If it is not, then the measured limiting
liquidus will lie above that calculated by Eq. (18.25).
Finally, if Tof(A), νtotal, and the limiting slope are known and if it is
known that the solubility in the solid is negligible, then Eq. (18.25)
permits Δhof(A) to be calculated. This is a particularly useful technique
when Tof(A) is very high and calorimetric measurements are difficult.
The following derivation of Eq. (18.25) is that of Blander (1964).
In very dilute solutions, the solute species are randomly distributed and so far apart that they do not interact with one
another. Hence, Henry’s law applies with the enthalpy and nonconfigurational entropy given by
H ¼ nA hoA + nB h∗B
(18.27)
Sconfig ¼ nA sAo + nB sB∗
(18.28)
where hoA and soA are the molar enthalpy and entropy of pure A and
h∗B and s∗B are constants. The partial molar and relative partial
molar enthalpy of A are given by.
hA ¼ ∂H=∂nA ¼ hoA
ΔhA ¼ hA hoA ¼ 0
(18.29)
Chapter 18 SOME APPLICATIONS
and similarly, Δsnonconfig
¼ 0.
A
Let
NA ¼ nA N 0
(18.30)
N B ¼ nB N 0
(18.31)
N i ¼ ni N 0
(18.32)
0
where N is Avogadro’s number. Let the number of ways of placing
a (new) species i in solution be βiNA where βi ¼ constant. Then, the
multiplicity of the very dilute solution is given by
h
i
(18.33)
t ¼ ðβ1 NA ÞN1 ðβ2 NA ÞN2 ðβ3 NA ÞN3 … =ðN1 !N2 !N3 !…Þ
The configurational entropy of mixing is given by
X
X
X
X
ΔSconfig =kB ¼ ln t ¼
Ni ln NA +
Ni ln βi Ni ln Ni +
Ni
X
X
X
X
¼
νi NB ln NA +
νi NB ln βi νi NB ln ðνi NB Þ +
ðνi NB Þ
(18.34)
where Stirling’s approximation has been applied. The relative partial configurational entropy of A is
X
config
¼ ∂ΔSconfig =∂nA nB ¼ kN 0 ðnB =nA Þ νi
(18.35)
ΔsA
and
config
lim ΔsA
X B ¼0
¼ RX B νtotal
(18.36)
Since, in the limit, ΔhA and Δsnonconfig
are both zero, it follows that
A
(18.37)
lim μA μoA ¼ lim RTX B νtotal ¼ RT ofðAÞ νtotal dX B
X B ¼0
X B ¼0
Note that all the constants βi have dropped out.
From Eq. (5.5) along the liquidus when there is no solubility in
the solid phase,
lim μA μoA ¼ lim ΔhofðAÞ =R T TfoðAÞ =TfoðAÞ
X B ¼0
X B ¼0
¼ ΔhofðAÞ =RT ofðAÞ dT
(18.38)
Substitution of Eq. (18.37) into Eq. (18.38) then gives Eq. (18.25).
References
Abe, T., Sundman, B., 2003. Calphad J. 27, 403–408.
Blander, M., 1964. Molten Salt Chemistry. Interscience, New York.
Cao, W., Chang, Y.A., Zhu, J., Chen, S., Oates, W.A., 2007. Intermetallics
15, 1438–1446.
349
350
Chapter 18 SOME APPLICATIONS
Chartrand, P., Pelton, A.D., 1999. Calphad J. 23, 219–230.
Chartrand, P., Pelton, A.D., 2002. Light Metals 2002, 245–252.
Darken, L.S., 1967. Trans. Metall. Soc. AIME 239, 80–89.
Decterov, S.A., 2018. Ecole Polytechnique de Montreal, Private Communication.
Eriksson, G., Wu, P., Blander, M., Pelton, A.D., 1994. Can. Metall. Q. 33, 13–21.
Fayon, F., Bessada, C., Massiot, D., Farnan, I., Coutures, J.P., 1998. J. NonCryst.
Solids 232–234, 403–408.
Fincham, C.J.B., Richardson, F.D., 1954. Proc. Roy. Soc. (London) 223A, 40–62.
Gaye, H., Welfringer, J., 1984. Proceedings of the Second International Symposium
on Metallurgical Slags and Fluxes. TMS-AIME, Warrendale, PA, pp. 357–376.
Hidayat, T., Shishin, D., Jak, E., Decterov, S., 2015. Calphad J. 48, 131.
Inden, G., 2001. Kostorz, G. (Ed.), Atomic Ordering in Phase Transformations in
Materials. Chapter 8. Wiley VCH, Weinheim.
Jung, I.-H., Decterov, S.A., Pelton, A.D., 2004. Metall. Mater. Trans. B Process Metall.
Mater. Process. Sci. 35, 493–507.
Jung, I.H., 2018. McGill University, Montreal, Private Communication.
Kang, Y.-B., Pelton, A., 2009. Metall. Mater. Trans. 40B, 979–994.
Kikuchi, R., 1951. Phys. Rev. 81, 988–1003.
Kikuchi, R., Sato, H., 1974. Acta Metall. 22, 1099–1112.
Kim, W.Y., Pelton, A.D., Decterov, S., 2012. Metall Mater. Trans. 43B, 325–336.
Kusoffsky, A., Dupin, N., Sundman, B., 2002. Calphad J. 25, 549–565.
Oates, W.A., Wenzl, H., 1996. Scr. Mater. 35, 623–627.
Oates, W.A., Zhang, F., Chen, S.L., Chang, Y.A., 1999. Phys. Rev. B 59, 11221–11225.
Pelton, A.D., 1999. Glastech. Ber. 72, 214–226.
Pelton, A.D., Eriksson, G., Romero-Serrano, A., 1993. Metall. Trans. 24B, 817–825.
Petric, A., Pelton, A.D., Saboungi, M.L., 1988. J. Chem. Phys. 89, 5070–5077.
Reddy, R.G., Blander, M., 1987. Metall. Trans. 18B, 591–596.
Schairer, J.F., Bowen, N.L., 1956. Am. J. Sci. 254, 129–195.
Shishin, D., Jak, E., Decterov, S.A., 2015. Calphad J. 50, 144–160.
Sigworth, G.K., Elliott, J.F., 1974. Can. Metall. Q. 13, 455–461.
List of Websites
FactSage, Montreal. www.factsage.com.
SGTE, 2017. Solution database. http://www.sgte.net/en/thermochemicaldatabases.
Thermo-Calc, Stockholm. www.thermocalc.com.
19
EXERCISES WITH SOLUTIONS
CHAPTER OUTLINE
19.1. Exercises 351
19.2. Solutions to Exercises
19.1
363
Exercises
Exercise 1 (Chapter 2)
A closed bottle of volume 10.0 L is filled with humid air. The
bottle is slowly cooled. Water is observed to begin to condense
inside the bottle at 25.0°C. The bottle is subsequently heated
above 25°C, and its contents are flushed over P2O5 that absorbs
all the water. The weight of the P2O5 increases by 0.230 g. Calculate
the vapor pressure of water at 25.0°C.
Exercise 2 (Chapter 2)
The internal energy change of a chemical reaction can be
measured in a bomb calorimeter in which one measures the heat
evolved during the reaction in a sealed container at constant
volume. From Eq. (2.12), this heat is equal to ΔU of the reaction.
For the following reaction, ΔUo298 was measured in a bomb
calorimeter and was found to be equal to 1140 kJ per mole
of Fe3C.
Fe3 C ðsolÞ + 3 O2 ðgÞ ¼ Fe3 O4 ðsolÞ + CO2 ðgÞ
o
Calculate ΔH298
of the reaction which is the heat evolved if the
reaction were to occur at constant pressure (Eq. 2.15).
Exercise 3 (Chapter 2)
Limestone, CaCO3, at 25°C is charged into an electric kiln
where calcination (decomposition into CaO and CO2) occurs.
The products leave the kiln at 800°C. Ignoring heat losses,
Phase Diagrams and Thermodynamic Modeling of Solutions. https://doi.org/10.1016/B978-0-12-801494-3.00019-1
# 2019 Elsevier Inc. All rights reserved.
351
352
Chapter 19 EXERCISES WITH SOLUTIONS
calculate how much electric energy must be supplied to produce
1.00 mol of CaO.
Data:
o
¼ 177:8 kJ
CaO + CO2 ¼ CaCO3 ΔH298
cP ðCaOÞ ¼ 48:83 + 4:52 103 T 6:53 105 T 2 Jmol1 K1
cP ðCO2 Þ ¼ 44:14 + 9:04 103 T 8:54 105 T 2 Jmol1 K1
Exercise 4 (Chapter 2)
A bath of pure liquid Al is supercooled to 900 K. Solidification
begins at 900 K and proceeds adiabatically. That is, the process is
sufficiently rapid that heat losses are negligible. Hence, during
solidification, the temperature of the bath increases until it
reaches the equilibrium melting point Tof ¼ 932 K, at which point
solidification ceases.
Calculate the fraction of the Al that solidifies.
Data:
Δhof ¼ 10:750 kJ mol1 at Tfo ¼ 932 K
cP ðliquidÞ ¼ 29:29 J mol1 K1 :
Exercise 5 (Chapter 2)
3.00 mol of an ideal gas are at 27°C, and P ¼ 20.0 bar. The gas
expands to a final pressure of 2.0 bar. Calculate ΔS of the gas
and ΔStotal for each of the following paths:
(a) Reversible isothermal expansion.
(b) Isothermal expansion against an external pressure of 2.0 bar.
Exercise 6 (Chapter 2)
(a) 1.00 mol Zn solidifies isothermally at its equilibrium melting
point (420°C) in equilibrium with its surroundings.
(b) 1.00 mol of supercooled liquid Zn at 400°C solidifies
isothermally.
For each case, calculate ΔS, ΔSsurr, and ΔStotal.
Data:
Δhof ¼ 7280Jmol1 at Tfo ¼ 420°C
cP ðsolÞ ¼ 29:25Jmol1 K1
cP ðliqÞ ¼ 31:38Jmol1 K1
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 7 (Chapter 2)
A mixture of SrCO3 (sol) and SrO (sol) is placed in a sealed
evacuated container and heated to 850°C. The pressure in the
container is 3.25 103 bar. The experiment is repeated with
C (sol) as well as SrCO3 (sol) and SrO (sol) in the container.
The pressure is now 0.225 bar. Calculate ΔGo of the following
reaction at 850°C:
C ðsolÞ + CO2 ðgÞ ¼ 2 CO ðgÞ
Exercise 8 (Chapter 2)
What is the molar percentage of NO (g) in air at equilibrium at
2200 K at a total pressure of 1.00 bar? (Air contains 79% N2 and
21% O2 at 25°C).
Data:
N2 ðgÞ + O2 ðgÞ ¼ 2 NO ðgÞ K2200 ¼ 1:10 103
Exercise 9 (Chapter 2)
It is proposed to reduce Fe2O3 at 1273 K in an atmosphere
where pH2/pH2O ¼ 1.00.
What will be the reaction product, Fe, FeO, Fe3O4, or Fe2O3?
Data:
Fe + 0:5 O2 ¼ FeO
3 FeO + 0:5 O2 ¼ Fe3 O4
2 Fe3 O4 + 0:5 O2 ¼ 3 Fe2 O3
H2 + 0:5 O2 ¼ H2 O
o
ΔG1273
¼ 189:78kJ
o
ΔG1273
¼ 138:13kJ
o
ΔG1273
¼ 68:49kJ
o
ΔG1273
¼ 178:13kJ
Exercise 10 (Chapter 2)
Ignoring kinetic constraints, calculate the minimum pressure
required to transform graphite to diamond at 25°C?
Data:
CðgraphiteÞ ¼ CðdiamondÞ
o
¼ 2866Jmol1
ΔG298
Densities : ρ ðdiamondÞ ¼ 3:52gmL1
ρ ðgraphiteÞ ¼ 2:25gmL1
353
354
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 11 (Chapter 2)
The ratio pH2S/pH2 at equilibrium with pure solid Ni and pure
solid Ni3S2 has been measured at several temperatures:
T (°C):
pH2S/pH2 103:
700
1.90
800
2.69
900
3.35
1000
3.99
1100
4.41
Calculate ΔGo of the following reaction at each temperature.
3 Ni ðsolÞ + 2 H2 S ðgÞ ¼ Ni3 S2 ðsolÞ + 2 H2 ðgÞ
Calculate ΔHo and ΔSo of the reaction over each hundred degree
temperature interval.
Is ΔCoP of the reaction positive or negative?
Exercise 12 (Chapter 2)
In the P-T phase diagram of Fe, is the slope dP/dT of the
univariant line between the phase fields of α (bcc) and γ (fcc)
positive or negative?
Exercise 13 (Chapter 3)
A copper matte and slag are in equilibrium with a gas phase at a
constant temperature and total pressure. Assume that the matte is
a liquid solution Cu-Fe-S, that the slag is a liquid ionic solution
Cu2O-FeO-SiO2 and that the gas phase consists of O2, N2, S2,
SO2, and other oxysulfide species. At equilibrium, there will be
a small amount of Si and O dissolved in the matte and some
S in the slag. Suppose that all the thermodynamic properties of
all phases are precisely known as functions of temperature and
composition. Show that it is possible, at least in principle, to
calculate the compositions of all phases if pSO2, pO2, and the Cu
content of the slag have been measured.
Exercise 14 (Chapter 4)
In a gas phase at equilibrium with pure C at 900°C, pCO2 ¼
0.0295 bar, and pCO ¼ 0.9705 bar. A mixture of CO and CO2 is
circulated slowly at 900°C over a sample of plain carbon steel at
a total pressure of 1.00 bar. It is observed that the steel is decarburized if the CO2 content is >6.6% and is carburized if it is <6.6%.
Calculate the activity of C in the steel.
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 15 (Chapter 4)
Aluminum is produced by electrolysis at 1293 K in a bath of
liquid cryolite which is a solution of NaF and AlF3. If the activities
of NaF and AlF3 in a given bath are 0.414 and 4.562 104, respectively (with respect to the pure liquids as standard states),
calculate the Na content of the liquid Al that is produced in
equilibrium with the cryolite.
Data:
Na ðliqÞ + 0:5 F2 ¼ NaF ðliqÞ
Al ðliqÞ + 1:5 F2 ¼ AlF3 ðliqÞ
o
ΔG1293
¼ 440:37 kJ
o
ΔG1293
¼ 1117:13 kJ
γ oNa ðHenrian activity coefficientÞ ¼ 165 in liquid Al at 1293 K
Exercise 16 (Chapter 4)
In a certain Cu-Ni alloy at 700 K, the activities of Cu and Ni are,
respectively, 0.88 and 0.172. The alloy is placed in an atmosphere
of S2 at a pressure of 1.0 bar at 700 K. Which sulfide is formed, Cu2S
or Ni3S2? The sulfides are solid and immiscible.
Data:
4 Cu ðsolÞ + S2 ðgÞ ¼ 2 Cu2 S ðsolÞ
o
ΔG700
¼ 219:24 kJ
3 Ni ðsolÞ + S2 ðgÞ ¼ Ni3 S2 ðsolÞ
o
ΔG700
¼ 216:73 kJ
Exercise 17 (Chapter 4)
The activity coefficient of Zn in liquid Cd-Zn alloys has been
measured at 435°C and fitted by the following equation:
2
3
0:30XCd
ln γ zn ¼ 0:87XCd
Find an expression for lnγ Cd as a function of composition.
Exercise 18 (Chapters 2 and 4)
You read in a paper that a binary solution A-B has been
evaluated/optimized and that a regular solution model has been
applied as follows:
Δhm ¼ XA XB ð5000 + 20:0 T Þ J mol1
sE ¼ XA XB ð3:5Þ J mol1 K1
What is wrong with this?
355
356
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 19 (Chapter 4)
40.00 mol of pure liquid A and 60.00 mol or pure liquid B are
mixed in a calorimeter at constant T to form a homogeneous solution. A heat effect (exothermic) of 100.00 kJ is observed. Next,
1.00 mol of pure B is added to the solution at the same constant
T, and a heat effect (exothermic) of 670 J is observed.
(i) What is the integral molar enthalpy of mixing?
(ii) Calculate, approximately, the partial relative enthalpy of B.
(iii) Calculate, approximately, the partial relative enthalpy of A.
Exercise 20 (Chapter 4)
Is the boiling point of a solution of NaCl in H2O higher or lower
than that of pure H2O?
Exercise 21 (Chapter 4)
A homogeneous liquid solution at 1000°C contains
500.0 mol Mg and 500.0 mol Al. The volume of the solution is
13.7728 L. When 10.0 mol of pure Mg is added to the solution,
the volume increases to 13.9341 L. A further addition of 20.0 mol
of pure Al is made. What is the final volume?
Exercise 22 (Chapters 4 and 5)
LiCl-KCl is a simple eutectic system with no solubility in the
terminal solid phases and no intermediate compounds. The liquid
phase is a regular solution with
Δhm ¼ 17:86 XLiCl XKCl kJ mol1
sE ¼ 1:46 XLiCl XKCl J mol1 K1
Calculate the temperature and liquid composition at the
eutectic point.
Data:
0
ðLiClÞ ¼ 883 K
Δhofusion ðLiClÞ ¼ 19:87 kJ mol1 at Tfusion
0
ðKClÞ ¼ 1045 K
Δhofusion ðKClÞ ¼ 26:57 kJ mol1 at Tfusion
Assume that Δhofusion is constant, independent of T, for both components. That is, assume that cP(sol) cP(liq).
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 23 (Chapters 4 and 5)
Hf and Zr are miscible at all compositions in both the solid and
liquid state, and the solutions are nearly ideal. Consequently, the
phase diagram is of the simple lens shape like in Fig. 5.1. Calculate
the compositions of the liquidus and solidus at 2273 K.
Data:
o
ð Hf Þ ¼ 2488 K
Δhofusion ð Hf Þ ¼ 25:10 kJ mol1 at Tfusion
o
ðZrÞ ¼ 2125 K
Δhofusion ðZrÞ ¼ 23:00 kJ mol1 at Tfusion
Assume that Δhofusion is constant, independent of T, for both components. That is, assume that cP(sol) cP(liq).
Exercise 24 (Chapters 4 and 5)
The eutectic temperature in the Pb-Sn system is 453 K. At this
temperature, a liquid of composition XSn ¼ 0.73 is in equilibrium
with a Sn-rich terminal solid phase containing a small amount of
dissolved Pb. The liquid solution is regular with Δhm ¼ 5520
XPbXSn and sE ¼ 0. Assume that the Sn-rich solid phase obeys
Raoult’s law. Calculate the composition of the solid phase in
equilibrium with the liquid at the eutectic temperature.
Data:
0
ðSnÞ ¼ 505 K
Δh0fusion ðSnÞ ¼ 6:99 kJ mol1 at Tfusion
Assume that Δh0fusion is constant, independent of T. That is,
assume that CP (sol) CP (liq).
Exercise 25 (Chapter 5)
Nitrogen dissolves in liquid Fe monatomically. At pN2 ¼
1.00 bar, the solubility is 0.18 atom % at 1600°C. What is the nitrogen content at 1600°C when pN2 ¼ 5.00 bar? Assume that Henry’s
law applies.
Exercise 26 (Chapter 5)
When a metal A is only slightly soluble in another metal B, one
finds that a plot of ln XA (where XA is the solubility of B in A) versus
1/T (K1) is nearly linear. Explain this observation and derive an
equation for the slope. Assume that the solid in equilibrium with
the saturated liquid is pure A.
357
Chapter 19 EXERCISES WITH SOLUTIONS
Temperature
358
Liquid
Liquid
+β
Ttr
p(XB = X′B)
Liquid + α
Mole fraction B
Fig. 19.1 Phase diagram of a system A-B.
Exercise 27 (Chapter 5)
A partial phase diagram of a system A-B is shown in Fig. 19.1.
Calculate an expression relating the change in slope of the liquidus at point p to the transition temperature Ttr, the composition
XB0 at point p, and the enthalpy of the α ! β transition of B at Ttr.
Exercise 28 (Chapter 5)
A phase diagram of the Co-Ni-O2 system at 1600 K is shown in
Fig. 19.2.
Assume that the oxide and metal solutions are ideal. Calculate
the molar ratio Ni/(Co + Ni) on the phase boundaries when log
pO2 ¼ 107.
Exercise 29 (Chapter 5)
A phase diagram of the Ag-Cu-Cl2 system at constant
T ¼ 1000 K is shown in Fig. 19.3. The liquid phase is an ideal
solution. Calculate the activity coefficient of Cu in the saturated
Ag-rich solid alloy and of Ag in the saturated Cu-rich solid alloy.
Exercise 30 (Chapter 14)
In a solution of components A-B-C, the integral excess Gibbs
energies in the three binary subsystems are
E
gAB
¼ XA XB ða1 + b1 ðXB XA ÞÞ
Chapter 19 EXERCISES WITH SOLUTIONS
pO2 = 3.47 × 10−7
−6.4
CoO-NiO solid solution
−6.6
log10 p(O2) (bar)
−6.8
−7
−7.2
−7.4
−7.6
−7.8
Co-Ni solid solution
−8
pO2 = 1.10 × 10−8
−8.2
0
0.2
0.4
0.6
Molar ratio Ni/(Co+Ni)
0.8
1
Fig. 19.2 Co-Ni-O2 phase diagram at T ¼ 1600 K.
−8
AgCl-CuCl (liquid)
−8.5
−9.5
−9.62
log pCl2 = −9.78
−10
0.808
0.104
−10.5
−11
0.967
Cu (solid)
Ag (solid)
log10 p(CI2) (bar)
−9
−11.5
−12
0
0.2
0.4
0.6
Molar ratio Cu/(Ag+Cu)
Fig. 19.3 Phase diagram of the Ag-Cu-Cl2 system at T ¼ 1000 K.
0.8
1
359
360
Chapter 19 EXERCISES WITH SOLUTIONS
E
gBC
¼ XB XC ða2 + b2 ðXC XB ÞÞ
E
gCA
¼ XC XA ða3 + b3 XC Þ
Using the Toop-Muggianu approximation with A as the
asymmetrical component, give an equation for gE in the ternary
solution.
Exercise 31 (Chapter 14)
You are using a simple geometric model to estimate the properties of a ternary solution from the optimized properties of the
binary subsystems. You find that your estimation predicts a ternary solution that is somewhat too stable. That is, the activities
of the components predicted by your model are too low. Accordingly, you add a ternary term aX1X2X3 to your expression for gE
with a positive adjustable parameter a in order to increase the
activities. You are subsequently disappointed to find that in certain composition regions, this term actually decreases the activity
of a component. Explain this by calculating an equation giving the
contribution of this ternary term to the partial excess Gibbs energy
of component 1, gE1 .
Exercise 32 (Chapter 14)
In a dilute solution of component 1 in an ideal binary solution
of components 2 and 3, the partial excess Gibbs energy of component 1, gE1 , varies linearly with the ratio X3/(X2 + X3) between its
value when X2 ¼ 1.0 and when X3 ¼ 1.0.
Suppose now that the 2-3 binary solution is not ideal but is
regular, with
E
¼ α23 X2 X3
g23
Show that the deviation of gE1 from the linear variation is opposite
in sign to the parameter α23. Explain this in words.
Exercise 33 (Chapter 14)
Cu-Mn alloys are miscible at all compositions in both the solid
and liquid state. The solidus/liquidus exhibits a minimum as in
Fig. 5.5 at 1140 K at a composition XMn ¼ 0.37. Assume that the
liquid solution is ideal and that the solid solution is subregular
with sE ¼ 0 and Δhm ¼ XCuXMn(0L + 1L(XCu XMn)). Calculate
the coefficients oL and 1L.
Chapter 19 EXERCISES WITH SOLUTIONS
Data:
o
ðCuÞ ¼ 1357 K
Δhofusion ðCuÞ ¼ 13:054 kJ mol1 at Tfusion
o
ðMnÞ ¼ 1517 K
Δhofusion ðMnÞ ¼ 14:644 kJ mol1 at Tfusion
Assume that Δhofusion is constant, independent of T, for both
components. That is, assume that cP(sol) cP(liq).
Exercise 34 (Chapter 14)
Using the Temkin model, calculate the liquidus temperature of
an ideal ionic liquid solution containing 90 mol% Na2CO3 and
10 mol% Li2CO3. The liquid is in equilibrium with pure solid
Na2CO3.
Data:
o
¼ 1131 K
Δhofusion ðNa2 CO3 Þ ¼ 29:665 kJ mol1 at Tfusion
cP ðsolÞ ¼ 192:094 J mol1 K1
cP ðliqÞ ¼ 189:535 J mol1 K1
Exercise 35 (Chapter 15)
Consider a reciprocal ternary solution (A,B)(X,Y) with a Gibbs
energy given by Eq. (15.3) with all binary and ternary excess terms
equal to zero. That is, only the first two terms in square brackets
are nonzero. As shown in Section 15.1, if ΔGexchange as given by
Eq. (15.7) is sufficiently negative, then a miscibility gap will result
centered along the AX-BY stable diagonal of the composition
square. Derive an equation giving the consolute temperature of
the miscibility gap as a function of ΔGexchange.
Exercise 36 (Chapter 15)
Ferropseudobrookite, FeTi2O5, is a ceramic with two cation
sublattices. The first sublattice is occupied by Fe2+ ions and the
second sublattice by Ti4+ ions as (Fe2+)(Ti4+)2O5.
The ceramic Ti3O5 has the same structure and can be represented as (Ti3+)(Ti3+, Ti4+)2O5. Ti3+ and Ti4+ are randomly distributed
on the second sublattice.
Solid solutions of FeTi2O5 and Ti3O5 are represented as (Fe2+,
3+
Ti )(Ti4+, Ti3+)2O5 with the ions mixing randomly on each
sublattice.
361
362
Chapter 19 EXERCISES WITH SOLUTIONS
Write an expression for the Gibbs energy of FeTi2O5-Ti3O5 solutions as a function of composition using the compound energy
formalism. Ignore excess terms. Draw the reciprocal composition
square and show the path of charge neutrality. Relate the go values
of the end-members to go of pure real FeTi2O5 and go of pure real
Ti3O5. Write an expression for the Gibbs energy of mixing of
FeTi2O5 and Ti3O5.
Exercise 37 (Chapter 16)
Using an ideal Temkin “complex anion” model as in
Section 16.3.1, calculate the activity of Na2O in a molten oxide
solution of composition XCaO ¼ 0.75, XSiO2 ¼ 0.15, XNa2O ¼ 0.05,
and XFeO ¼ 0.05.
Exercise 38 (Chapter 16)
Liquid salt solutions are usually modeled with a two-sublattice
model. Strictly speaking, this is an approximation because a
liquid, lacking long-range ordering, cannot be said to consist of
more than one sublattice.
(i) Write an equation for the extensive Gibbs energy of a molten
salt solution of NaCl and KCl, in the Bragg-Williams approximation using a two-sublattice model, as a function of the
numbers of moles and mole fractions of NaCl and KCl
(nNaCl, nKCl, XNaCl, and XKCl) and of the molar Gibbs energies
of the pure salts, goNaCl and goKCl. Ignore excess Gibbs
energy terms.
(ii) Write an equation, using the quasichemical model, for the
extensive Gibbs energy of a solution of Na, K, and Cl atoms
on one sublattice as a function of the numbers of moles of
the atoms (nNa, nK, nCl); the numbers of moles and mole
fractions of the nearest-neighbor pairs (nNaNa, nNaCl,
nKK, nKCl, nClCl, nNaK; XNaNa, XNaCl, XKK, XKCl,
XClCl, XNaK); the molar Gibbs energies goNa, goK, and goCl;
the FNN coordination number Z; and the Gibbs energy
changes ΔgNa-Cl and ΔgK-Cl of the pair-exchange reactions:
ðNa NaÞ + ðCl ClÞ ¼ 2 ðNa ClÞ; ΔgNaCl
ðK KÞ + ðCl ClÞ ¼ 2 ðK ClÞ;
ΔgKCl
(For the pair-exchange reaction (Na Na) + (K K) ¼ 2
(Na K), assume that ΔgNa-K ¼ 0.)
(iii) In the limit where ΔgNa-Cl and ΔgK-Cl are extremely negative
and along the stoichiometric join between NaCl and KCl
Chapter 19 EXERCISES WITH SOLUTIONS
(i.e., where (nNa + nK) ¼ nCl) show, by making the
appropriate substitutions, that the quasichemical equation
of part (ii) reduces to the two-sublattice Bragg-Williams
approximation of part (i) when Z ¼ 2.
19.2
Solutions to Exercises
Exercise 1
The vapor pressure is defined as the partial pressure in
equilibrium with the liquid.
Hence, 25.0°C is the temperature where the partial pressure
becomes equal to the vapor pressure.
pH2 O V ¼ nH2 O RT
where nH2O is the number of moles of H2O in the container.
pH2 O ð10:0Þ ¼ ð0:230=18:00Þð0:008314Þð298:15Þ ¼ 0:0317 bar
Exercise 2
H ¼ U + PV
ΔH ¼ ΔU + ΔðPV Þ
ΔðPV Þ of the solid phases is negligible:
ΔðPV Þ of the gas phase ¼ ΔðnRT Þ ¼ RT ð1 3Þ
ΔH ¼ ΔU 2RT ¼ 1140 2ð0:008314Þ298:15 ¼ 1145 kJ
The gas phase has been assumed to be ideal. Since the pressure in
the bomb is generally very high, the gas is not actually ideal.
However, as the total correction is only 5 kJ out of 1140 kJ, the
assumption is justified.
Exercise 3
The reaction occurs over a range of temperature as the
limestone is heated. However, enthalpy is a state property, and
so, the total ΔH is independent of the process path. Hence, for
purposes of the calculation, we choose a hypothetical path
whereby the calcination occurs at 298.15 K and the products are
subsequently heated to 800°C.
1073
ð
ΔHtotal ¼ + 177:8 +
ðcP ðCaOÞ + cP ðCO2 ÞÞdT ¼ 253:4kJ
298
363
364
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 4
Solidification occurs over a range of temperature from 900 to
932 K. However, enthalpy is a state property, and so, the total
ΔH is independent of the process path. Hence, for purposes of
the calculation, we choose a hypothetical path whereby the liquid
is first heated from 900 to 932 K and then a fraction x of the Al
solidifies at 932 K. The total process is adiabatic. Hence, the total
enthalpy is zero.
932
ð
ΔHtotal ¼ 0 ¼
cP ðliquidÞdT xΔhof
900
x ¼ 0:087
Exercise 5
Calculate initial and final volumes.
Vi ¼ nRT=Pi ¼ 3ð0:08314Þð300Þ=20 ¼ 3:74L
Vf ¼ 37:4L
Ð
Ð
(a) Wrev ¼ PdV ¼ nRT (1/V)dV ¼ nRT ln(Vf/ Vi) ¼
17,230 J
In an ideal gas, U is a function only of the kinetic energy of
the molecules. Hence,
ΔU ¼ 0 for the isothermal expansion of an ideal gas
ΔU ¼ Q + W ¼ 0
Qrev ¼ 17, 230 J
ΔS ¼ Qrev =T ¼ 57:45JK1 :
The surroundings see only a reversible transfer of heat at
constant temperature.
ΔSsurr ¼ Qsurr =T ¼ 17, 230=300 ¼ 57:45 J K1
Hence, ΔStotal ¼ 0 as is the case for all reversible processes.
(b) Since the initial and final states are the same as in (a) and
since S is a state property, ΔS ¼ 57:45JK 1
ð
Wnonrev ¼ Pexternal dV ¼ 2:0ð37:4 3:74Þ ¼ 67:32 L bar
¼ 6732 J
ΔU ¼ Q + W ¼ 0
Q ¼ 6732 J Qsurr ¼ 6732 J
Chapter 19 EXERCISES WITH SOLUTIONS
Although the process is irreversible, the surroundings
still see only a reversible transfer of heat. Hence,
ΔSsurr ¼ Qsurr/T ¼ 22.44 J K1
ΔS total ¼ 57:45 22:44 ¼ 35:01JK1
ΔStotal > 0 as is the case for all spontaneous processes:
Exercise 6
(a) At the equilibrium melting point, the process is reversible.
ΔS ¼ Qrev =T ¼ 7280=693:15 ¼ 10:50Jmol1 K1
ΔSsurr ¼ Qsurr =Tsurr ¼ + 7280=693:15 ¼ +10:50Jmol1 K1
ΔStotal ¼ 0
(b) At 400°C, the process is not reversible. ΔS must be calculated
by a hypothetical reversible path. We first heat the liquid
reversibly from 400 to 420°C, then solidify it reversibly at
420°C, and then cool the solid reversibly to 400°C.
693:15
ð
ΔS ¼ 10:50 +
½ðcP ðliqÞ cP ðsolÞÞ=T dT ¼ 10:44Jmol1 K1
673:15
Similarly, at 400 °C,
693:15
ð
ΔH ¼ 7280 +
ðcP ðliqÞ cP ðsolÞÞdT ¼ 7237 J mol1 :
673:15
The surroundings see only a reversible transfer of heat at
constant T. Hence,
ΔSsurr ¼ Qsurr =T ¼ + 7237=673:15 ¼ 10:75Jmol1 K1
ΔStotal ¼ 10:44 + 10:75 ¼ +0:31Jmol1 K1 > 0 as for all
spontaneous processes.
Note that the terms in (cP(liq)-cP(sol)) are of secondary
importance. The main difference between the total entropy at
420 and 400°C is due to the denominators T in the terms Qsurr/T.
Exercise 7
When only SrCO3 and SrO are present, the following calcination equilibrium is established:
SrCO3 ¼ SrO + CO2 ; K ¼ pCO2 ¼ 0:00325 bar
365
366
Chapter 19 EXERCISES WITH SOLUTIONS
If C is also present, pCO2 is still equal to 0.00325 bar because it is
fixed by the above equilibrium. A second equilibrium is also
established:
C + CO2 ¼ 2 CO; K ¼ p2CO =pCO2
where pCO ¼ Ptotal pCO2 ¼ 0:225 0:00325
ΔG o ¼ RT ln K ¼ 25; 390 J
Exercise 8
N2 + O2 ¼ 2NO
Start with 79 mol N2 and 21 mol O2 at 25°C and heat to 2200 K
where the reaction attains equilibrium with the formation of
2x mol NO, leaving (21 x) mol O2 and (79 x) mol N2. For an
ideal gas, from Dalton’s law, pi ¼ XiPtotal ¼ (ni/ntotal)Ptotal
pNO ¼ ð2x=100Þð1:00Þ
pO2 ¼ ð21 xÞ=100
pN2 ¼ ð79 xÞ=100
K ¼ p2NO =pN2 pO2 ¼ 2x=ð79 xÞð21 xÞ ¼ 1:10 103
x ¼ 0:66
XNO ¼ 2x=100 ¼ 0:0132
%NO ¼ 1:32%
Exercise 9
The H2/H2O ratio fixes pO2.
¼ RT ln K
H2 ¼ 0:5 O2 ¼H2 O; ΔG o ¼ 178:13
0:5
¼ RT ln pH2 O =pH2 pO
2
pO2 ¼ 2:42 1015 bar
0:5
Fe + 0:5 O2 ¼ FeO; ΔG ¼ ΔG o + RT ln pO
¼ 11, 620 J < 0
2
Hence, FeO is more stable than Fe.
0:5
3 FeO + 0:5 O2 ¼ Fe3 O4 ; ΔG ¼ ΔG o + RT ln pO
2
¼ 40; 030 J > 040030 J > 0
Hence, Fe3O4 is less stable than FeO.
Therefore, FeO will be the only reaction product. It can
be seen from this calculation that only one solid phase can be
stable at equilibrium. This can also be demonstrated from the
Phase Rule.
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 10
The molar volumes of graphite and diamond are.
vgr ¼ 12:00=0:00225 ¼ 0:005333L=mol
vdi ¼ 0:003409L=mol:
At 25°C and P ¼ 1.0 bar, ΔGo > 0. That is, graphite is stable. As P
increases, diamond becomes progressively more stable because
it has the smaller molar volume. The minimum pressure P at
which diamond becomes more stable is the temperature at which
ΔG becomes zero. Since G is a state property independent of the
process path, we select a hypothetical path in which the pressure
on the graphite is first reduced from P to 1.0 bar, the reaction then
occurs at 1.0 bar, and the pressure on the diamond is then
increased to P. From Eq. (2.71) for a process at constant, T:
dG ¼ VdP. Hence,
ð1
o
ΔGP ¼ ΔGP¼1
+
vgr vdi dP ¼ 2866 vgr vdi ðP 1Þ ¼ 0
P
P ¼ 14:7 kbar
Exercise 11
ΔG o ¼ RT ln K ¼ RT ln ðpH2 =pH2 S Þ2
From the Gibbs-Helmholtz equation, Eq. (2.59), d(ΔGo)/
dT ¼ ΔSo.
Approximate d(ΔGo)/dT by Δ(ΔGo)/ΔT for each interval of
ΔT ¼ 100.
Then, ΔHo ¼ ΔGo + TΔSo.
T o ðC Þ
ΔG o ðkJ Þ
700
101.37
800
105.59
900
111.15
1000
116.92
1100
o
ΔSaverage
kJK1
o
ΔHaverage
ðkJÞ
0.042
60.5
0.056
60.1
0.058
43.1
0.069
29.1
123.82
ΔCP ¼ dðΔH o Þ=dT > 0
367
368
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 12
From Eq. (2.71),
dgα ¼ vα dP sα dT
dgγ ¼ vγ dP sγ dT :
Along the univariant line, α and γ are in equilibrium. Therefore,
dgα ¼ dgγ.
Hence, dP/dT ¼ (sγ sα)/(vγ vα).
This is known as the Clausius-Clapeyron equation.
For Fe, the γ-phase is stable at higher T, and the α-phase is
stable at lower T.
Therefore, (sγ sα) > 0.
The close-packed fcc phase is denser than the bcc phase.
Therefore, (vγ vα) < 0.
Consequently, dP=dT < 0:
Exercise 13
Take the components as the elements (Cu, Fe, Si, O, S, and N).
Hence, C ¼ 6.
There are three phases (matte, slag, and gas) at equilibrium.
Hence, P ¼ 3.
From the Phase Rule, Eq. (3.1), F ¼ C P + 2 ¼ 5.
The total pressure and temperature are fixed and pSO2, pO2, and
one composition variable of the slag are known. This fixes five
variables leaving zero degrees of freedom.
Exercise 14
C + O2 ¼ 2CO; K ¼ p2CO =ðpCO2 aC Þ
With pure C, aC ¼ 1.0; K ¼ (0.9705)2/0.0295 ¼ 31.928.
With the steel, K ¼ 31.928 ¼ (1 0.066)2/(0.066 aC)
aC ¼0.414.
Exercise 15
The exchange reaction between the metal phase and the
cryolite bath is.
3 Na + AlF3 ¼ 3 NaF + Al; ΔG o ¼ 1117:13 3ð440:37Þ ¼ RT ln K
3
K1293 ¼ 1:742 108 ¼ aAl a3NaF = γ oNa XNa aAlF3
Setting aAl 1:00, XNa ¼ 5:8 107 ¼ 58 ppm:
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 16
4 Cu + Ni3 S2 ¼ 2 Cu2 S + 3 Ni
ΔG ¼ ΔG o + RT ln a3Ni a2Cu2 S = a4Cu aNi3 S ¼ ð219:24 + 216:73Þ
+RT ln 0:1723 1:0= 0:884 1:0 ¼ 52:9 kJ
where the activities of the sulfides are set to 1.0 because they are
immiscible.
Because ΔG < 0, Cu2S is stable. Note that only one sulfide can
be formed at equilibrium because Cu2S and Ni3S2 are immiscible.
Compare with Exercise 15 where Na and Al form a solution.
Exercise 17
From the Gibbs-Duhem equation,
XCd d ln γ Cd + XZn d ln γ Zn ¼ 0
2 dX
XCd d ln γ Cd ¼ ð1 XCd Þ 1:74XCd 0:90XCd
Cd
ð
2
3
ln γ Cd ¼ ðXCd 1Þð1:74 0:90XCd Þ ¼ 1:74XCd + 1:32XCd
0:30XCd
+C
where C is a constant of integration. If the equation for lnγ Zn is
valid up to XCd ¼ 1, then we can use the fact that ln γ Cd ¼ 0.0 when
XCd ¼ 1.0 to evaluate C ¼ 0.72. Otherwise, some independently
measured value of ln γ Cd at some composition is required in order
to evaluate C.
Exercise 18
The Gibbs-Helmholtz equation, Eq. (2.61), is not satisfied.
Exercise 19
(i) Δhm ¼ 100/100 ¼ 1.00 kJ mol1
(ii) ΔhB ¼ d(ΔH)/dnB Δ(ΔH)ΔnB ¼ 6.70/1.0 ¼ 670 J mol1
(iii) Δh ¼ XA ΔhA + XB ΔhB
1000 ¼ 0.4 ΔhA + 0.6(670)
Δh A ¼ 1495 J mol 1
369
370
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 20
H2 O ðliqÞ ¼ H2 O ðgasÞ; K ¼ pH2 O =aH2 O
Adding NaCl lowers aH2O and thus lowers the equilibrium vapor
pressure at any temperature. Hence, the boiling point (defined
as the temperature where the vapor pressure equals 1.0 bar) is
increased.
Exercise 21
vMg ¼ ∂V =∂nMg ð13:9341 13:7728Þ=10:0 ¼ 0:0161300Lmol1
v ¼ 13:7728=1000 ¼ XAl vAl + XMg vMg ¼ 0:5 vAl + 0:5ð0:0161300Þ
vAl ¼ 0:0114156 ¼ ∂V =∂nAl ΔV =20:0
ΔV ¼ 0:2283
Final V ¼ 13.9341 + 0.2283 ¼ 14.162 L.
Exercise 22
Along the LiCl-liquidus from Eq. (5.6),
RT ln aLiCl ¼ ΔhofðLiClÞ + T ΔsfoðLiClÞ
RT ln XLiCl + RT ln γ LiCl ¼ ΔhofðLiClÞ 1 T =TfoðLiClÞ
From Eq. (4.45), μELiCl ¼ RT ln γ LiCl ¼ X2KCl (17,860 + 1.46 T) J mol1.
RT ln XLiCl + X2KCl(17,860 + 1.46 T) ¼ 19,870(1 T/883).
Similarly, along the KCl-liquidus,
2
RT ln XKCl + XLiCl
ð17, 860 + 1:46 T Þ ¼ 26, 570ð1 T =1045Þ
The two liquidus curves meet at the eutectic point. Therefore,
these two equations in the two unknowns T and XLiCl ¼ (1 XKCl)
can be solved simultaneously to give TE ¼629 K and XE(LiCl) ¼0.58.
Exercise 23
For equilibrium along the solidus/liquidus from Eq. (5.6),
RT ln aHf ðliqÞ RT ln aHf ðsolÞ ¼ Δhofð Hf Þ + T Δsfoð Hf Þ
Δhofð Hf Þ 1 T =Tfoð Hf Þ
Evaluating at T ¼ 2273 K and setting aHf(liq) ¼ XHf(liq) and
aHf(sol) ¼ XHf(sol) give
Rð2273Þ lnðXHf ðliqÞ=XHf ðsolÞÞ ¼ 25:10ð1 2273=2488Þ
Chapter 19 EXERCISES WITH SOLUTIONS
Similarly
for
Zr,
R(2273) ln(XZr(liq)/XZr(sol)) ¼ 23.00
(1 2273/2125)
Solving the two equations (with XZr ¼ 1 XHf ) gives
X Hf ðliqÞ ¼ 0:4008
X Hf ðsolÞ ¼ 0:4496
Exercise 24
For equilibrium along the solidus/liquidus from Eq. (5.6),
RT ln aSn ðliqÞ RT ln aSn ðsolÞ ¼ Δhof + T Δsfo Δhof 1 T =Tfo
The solid is Raoultian, so aSn(sol) ¼ XSn(sol).
The liquid is a regular solution with gE ¼ Δhm TsE ¼ 5520
XPbXSn.
From Eq. (4.45), μESn ¼ RT ln γ Sn ¼ 5520X2Pb(liq).
RT ln XSn ðliqÞ + 5520ð1 X Sn ðliqÞÞ2 RT ln XSn ðsolÞ
¼ 6:99ð1 T =505Þ
Substituting XSn(liq) ¼ 0.73 and T ¼ 453 K gives XSn(sol) ¼ 0.983.
Exercise 25
2
2
N2 ðgasÞ ¼ 2 N; K ¼ γ oN XN =pN2 ¼ γ oN ð0:0018Þ =1:00
2
¼ γ oN XN =5:00
XN ¼ ð0:0018Þð5=1Þ0:5 ¼ 0:0040
X N ¼ 0:40%
Exercise 26
The solubility curve is the liquidus of A. That is, it is the
composition of the liquid in equilibrium with pure solid A. From
Eq. (5.6), in the saturated liquid,
RT ln aA ¼ ΔhofðaÞ T ΔsfoðaÞ ¼ RT ln XA + ΔhA TsEA
where ΔhA and sEA are the relative partial molar properties.
ln XA ¼ ΔhofðaÞ + ΔhA =RT + ΔsfoðaÞ + sAE =R
In dilute solution, Henry’s law applies. That is, ΔhA and sEA are approximately independent of composition. If ΔhA and sEA are
approximately independent of T and Δhof(a) and Δsof(a) are also
approximately independent of T (that is, if cP(liq) cP(sol)), then a
plot of ln XA versus 1/T will be linear.
371
372
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 27
From Eq. (5.6), on the liquidus,
At T > Ttr, ln aB ¼ ln XB + ln γ B ¼ Δgof (β)/RT.
ln aB ¼ ln XB + ln γ B ¼ Δgof (α)/RT ¼ (Δgof (β) +
At
T < Ttr,
o
Δgtr(α ! β))/RT.
Hence, at T ¼ Ttr,
ðd ln XB =d ð1=T ÞÞT>Ttr ðd ln XB =d ð1=T ÞÞT<Ttr
¼ d Δgtro =RT =d ð1=T Þ ¼ Δhotr =R
(from the Gibbs-Helmholtz equation, Eq. 2.61). Hence,
ðdXB =d T ÞT>Ttr ðdXB =d T ÞT<Ttr ¼ XB0 Δhotr =RT 2tr
Exercise 28
Co + 0:5 O2 ¼ CoO;
0:5
0:5
K ¼ 1:10 108
¼ XCoO =XCo 1:0 107
Ni + 0:5 O2 ¼ NiO;
0:5
0:5
K ¼ 3:47 107
¼ XNiO =XNi 1:0 107
XNi ¼ ð1 XCo Þ ¼ 0:813
XNiO ¼ ð1 XCoO Þ ¼ 0:439
Exercise 29
0:5
Ag + 0:5 Cl2 ¼ AgCl; K ¼ aAgCl =aAg pCl
2
For pure Ag and AgCl, log K ¼ 8.42/2 ¼ 4.21.
At the invariant, aAgCl ¼ (1 0.808) ¼ 0.192.
Therefore, 4.21 ¼ log(0.192) log aAg + 9.78/2
aAg ¼ 0.919
Cu + 0:5 Cl2 ¼ CuCl
log K ¼ ð9:62=2Þ ¼ log ð0:808Þ log aCu + 9:78=2
aCu ¼ 0:971
In the saturated Ag-rich solid, aCu ¼ γ Cu XCu γ Cu ¼ 0:971=0:104 ¼
9:34.
In
the
saturated
Cu-rich
solid,
aAg ¼ γ Ag XAg
γ Ag ¼ 0:919=ð1 0:967Þ ¼ 27:85.
Exercise 30
g E ¼ XA XB ða1 + b1 ð1 2XA ÞÞ + XC XA ða3 + b3 ð1 XA ÞÞ
+ XB XC ða2 + b2 ðXC XB ÞÞ
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 31
g E ¼ aX 1 X2 X3 G E ¼ ðn1 + n2 + n3 Þg E ¼ aðn1 n2 n3 Þ=ðn1 + n2 + n3 Þ2
g1E ¼ ∂G E =∂n1 n2,n3 ¼ an2 n3 ðn1 + n2 + n3 Þ2 2n1 ðn1 + n2 + n3 Þ1
¼ aX 2 X3 ð1 2X1 Þ
g1E < 0 when X1 > 0:5
Exercise 32
The binary term makes the following contribution to the excess
Gibbs energy of the ternary solution:
g E ¼ α23 X2 X3 G E ¼ α23 ðn1 + n2 + n3 ÞX2 X3 ¼ α23 n2 n3 =ðn1 + n2 + n3 Þ
g1E ¼ ∂G E =∂n1 n2, n3 ¼ α23 n2 n3 =ðn1 + n2 + n3 Þ2 ¼ α23 X2 X3
Exercise 33
For equilibrium along the solidus/liquidus from Eq. (5.6),
assuming Δhof is independent of T:
RT lnðaCu ðliqÞ=aCu ðsolÞÞ ¼ ΔhofðCuÞ 1 T =TfoðCuÞ
¼ RT lnðXCu ðliqÞ=XCu ðsolÞγ Cu ðsolÞÞ
RT lnðaMn ðliqÞ=aMn ðsolÞÞ ¼ ΔhofðMnÞ 1 T =TfoðMnÞ
¼ RT lnðXMn ðliqÞ=XMn ðsolÞγ Mn ðsolÞÞ
In the solid solution, gE ¼ Δhm TsE ¼ XCuXMn(0L + 1L(XCu XMn)) 0.0.
From Eqs. (14.11) and (14.12),
0
2
L + 1 LððXCu XMn Þ + 2XCu Þ
RT ln γ Cu ðsolÞ ¼ XMn
2 0
RT ln γ Mn ðsolÞ ¼ XCu
L + 1 LððXCu XMn Þ + 2XMn Þ
At the solidus/liquidus minimum,
XMn ðsolÞ ¼ ð1 XCu ðsolÞÞ ¼ XMn ðliqÞ ¼ ð1 XCu ðliqÞÞ ¼ 0:37
Substitution into the above equations then gives 0L ¼ 10.63 J mol–1
and 1L ¼ 3.04 J mol–1.
373
374
Chapter 19 EXERCISES WITH SOLUTIONS
Exercise 34
Na+ and Li+ ions form an ideal solution on the cationic sublattice. Each mole of Na2CO3 contributes 2 mol of Na+ ions. Hence, it
is convenient to take the components as Na(CO3)0.5 and Li(CO3)0.5
with a(Na(CO3)0.5) ¼ X(Na(CO3)0.5) ¼ 0.90.
Along the liquidus, from Eq. (5.6),
RT ln a NaðCO3 Þ0:5 ¼ RT ln 0:9 ¼ Δgfo ðNaðCO3 Þ0:5 ¼ 0:5Δgfo ðNa2 CO3 Þ
2
3
ðT
¼ 0:5 Δhof T Δsfo ¼ 0:5 4Δhofð1131Þ +
ðcP ðliqÞ cP ðsolÞÞdT 5
1131
2
0:5T 4Δhofð1131Þ =1131 +
ðT
3
ððcP ðliqÞ cP ðsolÞÞ=TÞdT 5
T ¼ 1064K
1131
Exercise 35
00
00
Along the AX-BY diagonal, let XA0 ¼ XX ¼ x and XB0 ¼ XY ¼ (1 x).
Then, along the diagonal,
o
o
o
o
+ ð1 xÞ2 gBY
+ xð1 xÞ gAY
+ gBX
g ¼ x2 gAX
+ 2RT ðx ln x + ð1 xÞ ln ð1 xÞÞ
o
o
o
o
+ 2gBY
2 gAY
+ gBX
∂2 g=∂x2 ¼ 2gAX
+ 2RT ð1=x + 1=ð1 xÞÞ
¼ 2ΔGexchange + 2RT ð1=x + 1=ð1 xÞÞ
By symmetry, TC is at x ¼ 0.5 when ∂2g/∂ x2 ¼ 0
T C ¼ ΔG exchange =4R
Exercise 36
See Fig. 19.4.
Let X0 Ti3+ (on first sublattice) ¼ x X00 Ti3+ (on second sublattice) ¼ y.
Abbreviate the four end-members as A, B, C, and D as shown
on the figure.
g ¼ ð1 xÞð1 y ÞgAo + ð1 xÞyg oB + xð1 y ÞgCo + xyg oD
+ RT ðx ln x + ð1 xÞ ln ð1 xÞ + 2y ln y + 2ð1 y Þ ln ð1 y ÞÞ
Along the neutral line, x ¼ 2y. Substitute into the equation for g.
Chapter 19 EXERCISES WITH SOLUTIONS
D = [(Ti3+)(Ti3+)2O5]−1
B = [(Fe2+)(Ti3+)2O5]−2
(Ti3+)(Ti3+,Ti4+)2O5
y
ine
ll
tra
eu
N
A = (Fe2+)(Ti4+)2O5
x
C = [(Ti3+)(Ti4+)2O5]+1
Fig. 19.4 Composition square for FeTi2O5-Ti3O5 solutions.
i
g o =2 + g o =2 + 2RT ð0:5 ln 0:5 + 0:5 ln 0:5Þ
D
C
+ ð1 xÞðx=2Þ g o + g o g o g o
B
A
D
C
g ¼ ð1 xÞg o + x
A
h
+ RT ðx ln x + ð1 xÞ ln ð1 xÞ + x ln x + ð2 xÞ ln ð2 xÞÞ
+ RT ðx ln x + ð1 xÞ ln ð1 xÞ + x ln x + ð2 xÞ ln ð2 xÞÞ
¼ ð1 xÞg o
+ xg o
FeTi2 O5
Ti3 O5
+ xð1 xÞΔG exchange =2
One of goB, goC, or goD can be selected arbitrarily.
The model parameters are goFeTi2O5, goTi3O5, and ΔGexchange.
Exercise 37
n(Ca2+) ¼ 0.75
n(Na+) ¼ 2(0.05) ¼ 0.10
n(Fe2+) ¼ 0.05
ntotal (cations) ¼ 0.90
n SiO4 4 ¼ 0:15
n(O2) ¼ (0.75 + 0.05 + 0.05) 2(0.15) ¼ 0.55
ntotal (anions) ¼ 0.70
2
aðNa2 OÞ ¼ XNa+
XO2 ¼ ð0:10=0:90Þ2 ð0:55=0:70Þ ¼ 0:636
Exercise 38
Let nij ¼ moles of component and ni-j ¼ moles of (i-j) pair bonds.
Let Xij ¼ mol fraction of component and Xi-j ¼ mol fraction of
(i-j) pair bonds.
375
376
Chapter 19 EXERCISES WITH SOLUTIONS
o
o
(i) G ¼ (n
NaClgoNaCl + noKClgKCl)o+RT(nNaCl ln XNaCl + nKCl ln XKCl)
(ii) G ¼ nNa gNa + nK gK + nCl gCl + ðΔgNaCl =2ÞnNaCl
+ ðΔgKCl =2ÞnKCl + ðΔgNaK =2ÞnNaK
+ RT ðnNa ln XNa+ nK ln XK +nCl ln XCl Þ
2
=XNa
+ nKK ln XKK =XK2
+ RT nNaNa
ln XNaNa
2
+ nNaCl ln ðXNaCl =2XNa XCl Þ
+ nClCl ln XClCl =XCl
+ nKCl ln ðXKCl =2XK XCl Þ + nNaK ln ðXNaK =2XNa XK Þ
(iii) Along the NaCl-KCl join when ΔgNaCl and ΔgKCl are extremely
negative, there are only (Na-Cl) and (K-Cl) FNN pairs.
That is, all ni-j and Xi-j are zero except when (i-j) ¼ (Na-Cl)
or (K-Cl).
Along the join,
XNaCl ¼ XNaCl XKCl ¼ XKCl :
nNa ¼ nNaCl nK ¼ nKCl nCl ¼ ðnNaCl + nKCl Þ
XNa ¼ XNaCl =2 XK ¼ XKCl =2 XCl ¼ 0:5
nNaCl ¼ ZnNa ¼ ZnNaCl nKCl ¼ ZnK ¼ ZnKCl
o
o
o
o
o
ΔgNaCl ¼ gNaCl
gNa
gCl
gKo gCl
=Z ΔgKCl ¼ gKCl
=Z
Substitution into the MQM equation in (ii) gives
o
o
+ nKCl gKCl
+ RT ðnNaCl ln XNaCl + nKCl ln XKCl Þ
G ¼ nNaCl gNaCl
+ ðnNaCl + nKCl Þ ln ð2ðZ 2ÞÞ
This is the same as the BW expression in (i) when Z ¼ 2.
The reason for this is the fact that the entropy expression
in the MQM is only exact when Z ¼ 2 as discussed in
Section 16.2.
INDEX
Note: Page numbers followed by f indicate figures and t indicate tables.
A
Absolute enthalpy, 16
Activity
aqueous solutions, 78
component, 46
dilute metallic solutions,
77–78
single ion, 78–79
solution phase, 79–80, 80f
Activity coefficient, 49–50
Henrian, 73–74, 77
Raoultian, 77
Adjoining phase regions, law of,
105–106
Aqueous phase diagram.
See True aqueous phase
diagram
Associate model, 257–260
molten oxide solutions,
325–326
Asymmetrical salt solution, 200,
249–250
solid vs. liquid, 251–252
Asymmetrical solution, 200
Auxiliary functions, 28–29, 31,
118–122
Avogadro’s number
coordination equivalent
fractions, 220–221
ideal Raoultian solution, 48
limiting liquidus slope, 349
regular solution, 50–51
B
Binary compounds, 89–90
Binary eutectic reaction, 63–64,
141–143
Binary temperaturecomposition phase
diagrams
eutectic systems, 63–65
geometric units, 81f
eutectic invariant, 80–81
extension rule, 82
peritectic invariant, 82
grain size, effect of, 83
Henrian ideal solution, 73–76
immiscibility, 67–69
intermediate phases, 70–73
minima and maxima, 59–60
miscibility gaps, 61–63
regular solution theory, 65–67
solid-solid transformation,
56–57, 57f
strain energy, 83
tangent construction, 55
Bivariant region, 36
Boltzmann’s equation, 11, 47
Borate solutions, 328–330
Bragg-Williams models.
See Multiple sublattice
random-mixing; Singlelattice random mixing
C
Catatectics, 81
Cell model, molten
silicates, 327
Ceramic solutions, 246–249
Charge compensation effect,
327–328, 329f
Charge equivalent ionic
fractions, 124f, 129
Charge equivalent site fractions,
249–250
Chemical potential, 12, 29–30
excess, 197
ideal Henrian solution, 75
relative, 44–45
standard state, 77–78
Cluster-site approximation
(CSA), 340
Cluster variation method (CVM),
297, 340
Coherent boundary, 83
Complex anion model, 273–274
Component
activity, 46
activity coefficient, 49–50
choice of variables, 127–128
Composition-dependent
coordination numbers,
269–270, 279
Composition square
order-disorder transitions,
334, 334f
quadruplets, 301, 301f
(Fe,Cr)(C,Va) solution,
240–241, 241f
spinel, 246, 247f
Composition triangle, 85–86
Compound, 72
Compound energy formalism
(CEF), 240
ceramic solutions, 246–249
interstitial solutions, 240–241
molten oxide solutions, 334
nonstoichiometric
compounds, 241–245
Configurational entropy, 47
associate model, 258–259,
259f
ionic liquid model, 293
limiting liquidus slope, 349
modified quasichemical
model, 261, 263f
Raoultian ideal solutions, 194
Reddy-Blander capacity
model, 331–332
two-sublattice MQM, 305–308
377
378
INDEX
Congruently melting compound,
72
Conode. See Tie-line
Constant composition section.
See Isoplethal sections
Constituent diagrams, 139–140
Coordination equivalent
fractions, 199, 219–222
Corresponding extensive
variable, 104
Corresponding phase diagrams,
114–118
Corresponding variable pairs,
130, 130t
Crystallization path
equilibrium and ScheilGulliver solidification,
141, 142t
polythermal liquidus
projection, 88
Curie temperature, 159–160
D
Dalton’s law, 18–19
Darken’s quadratic formalism,
202–203
Debye-Davies equation, 176,
177f
Deoxidation equilibria, steel,
342–345
Dilute Henrian solution, 201
Double ZPF lines, 110
E
Eh-pH diagram
equilibrium constant, 169–170
Henrian ideal solution,
168–169
H2O-Fe system, 167–168, 168f
vs. hydrogen potential, 170,
171f
Nernst equation, 169
vs. oxygen potential, 170, 170f
Ellingham diagram, 24, 24f
End-member, 231, 239
Enthalpy, 13–16
absolute, 16
alkaline-earth silicates,
321–324, 322f
associate model, 258, 258f
Enthalpy change, 26
Enthalpy-composition phase
diagrams, 117–118
Entropy, 11
alkaline-earth silicates,
321–324, 323f
excess, 48
in liquid Al-Sc solutions,
253–255, 256f
of liquid Mg-Sn solutions,
253–255, 255f
third law method, 26–27
Entropy of mixing
configurational, 194
nonconfigurational, 194
Entropy paradox, 258–259
Equilibrium
ideal gas, 18–20
nonideal gas, 20–22
predominance diagrams,
22–24
Equilibrium constant, 19–20
molten oxide solutions, 342
Reddy-Blander capacity
model, 330
Equilibrium cooling, 37
Equilibrium solidification
calculated amounts of phases,
133–134
crystallization paths, 141–143,
142t
invariant reactions, 134–135
quasi-invariant reactions,
135–136
Eutectic phase diagram, 63–65
Eutectic reaction
binary, 63–64
equilibrium solidification,
134–135
Scheil-Gulliver cooling, 139
Eutectic temperature, 63
Eutectoids, 81
Evaporation paths, 165–167
Excess Gibbs energy, 48–49
Excess stability function, 341,
341f
Extension rule
binary phase diagrams, 82
polythermal projections, 131
ternary tie-triangle, 94f, 95
Extensive variable, 104, 105t
F
FactSage, 134, 160, 174
Fe-Cr-C system
full equilibrium phase
diagram, 150, 150f
paraequilibrium phase
diagram, 151, 151f
Ferromagnetic transition,
163–164
First-melting projection, 98
First-nearest-neighbor (FNN)
pair
modified quasichemical
model
single-lattice, 260
two-sublattice, 297
two-sublattice solution,
235–236
First-order transitions, 159
Five-component system
modified quasichemical
model, 282, 283f
single-lattice random mixing,
217, 217f
Fraction lines, phase, 126f, 131
Fugacity, 21, 21f
Fusion reaction, 15
Fusion temperature, 37
G
Galvanic cell, 13, 25
Geometric rules
extensive variable, 104, 105t
law of adjoining phase regions,
105–106
potential variables, 104–105,
105t
Schreinemakers’ rule, 107–108,
122
Gibbs-Duhem equation, 30–31
excess property, 49
ideal Henrian solution, 74
single-lattice random mixing,
199
single-phase region, 121
INDEX
Gibbs energy, 16–18
excess, 48–49
of mixing, 41–42
of pure substance, 27–28
two-sublattice MQM, 303–305
Gibbs energy change, 25
Gibbs-Helmholtz equation, 26,
197–198
Gibbs phase rule, 4–5
applications
Al2SiO5 system, 38
Cu-SO2-O2 system, 39
Fe-Cr-S2-O2 system, 40
Zn-Mg-Al system, 38–39
binary T-X phase diagrams
bivariant region, 36
equilibrium cooling, 37
fusion temperature, 37
three-phase invariants,
37–38
tie-line, 35–36
univariant region, 36
components, 33–34
composition variables, 34
degrees of freedom, 34
number of phases, 34
H
Helmholtz energy, 28
Henrian activity coefficient,
73–74, 77
Reddy-Blander capacity
model, 331
Henrian ideal solution, 73–76
Eh-pH diagrams, 168–169
Henry’s law, 74, 76, 348
Homogeneity range, 70–72
I
Ideal gas equation, 18–19
Ideal Raoultian solution, 46–48,
194–195
Immiscibility, 67–69
Incongruently melting
compound, 72–73
Intermediate phases, 70–73
Interstitial solutions, 240–241
Invariant reaction, 64–65,
134–135
Ionic liquid model, 292–293
Isobaric temperaturecomposition phase
diagram, 105–106, 105f
Isoplethal phase diagram, 3, 5f,
144–147, 146–147f
Isoplethal sections, 95–97
Isothermal phase diagram, 3, 4f
Scheil-Gulliver solidification,
144, 146f
solid and liquid phases, 60, 60f
J
€necke coordinates, 116–117
Ja
K
Kohler model, 208f, 209–210,
210f
L
Lattice stabilities, 75–76, 201
Law of adjoining constituent
regions, 147
Law of adjoining phase regions
(LAPR), 105–106, 149
Lens-shaped phase diagrams,
53–55
Lever rule, 36, 38
binary eutectic system, 65
oxide phase diagrams, 124
ternary isothermal sections,
91–93
Limiting liquidus slope, 346–349
Liquidus isotherms, 88
Liquidus projection
LiF-NaF-KF system, 213–214,
213f
quaternary phase diagrams,
99–100
reciprocal ternary system
(Li,K)(F,Br), 232, 233f
(Na,K)(F,Cl), 232, 234f
Scheil-Gulliver constituent
diagrams, 140–141, 140f,
143–144, 143f
ternary phase diagrams, 88–91
two-sublattice MQM
(Li,Na)(F,Cl) system,
315–316f, 317
379
(Na,Ca)(F,Cl) system,
316–317f, 317
Long range ordering (LRO),
162–163, 229, 333
M
Magnetic contributions, 346
Magnetic ordering.
See Ferromagnetic
transition
Majority defects, 241
Mechanical mixing energy,
231–232
Metastable phase boundaries,
190
Minimum Gibbs energy phase
diagrams
Fe-Ni-Cr system, 157, 157f
Ni-Cr system, 155–157, 156f
Miscibility gap, 61–63
Bragg-Williams vs. MQM, 290f
critical points, 114–116
liquid-liquid, 67
liquid-liquid island, 232
Modified quasichemical model
(MQM)
and Bragg-Williams
expression, 264–268
vs. Bragg-Williams model,
288–291
closed explicit form, 286–287
composition-dependent
coordination numbers,
269–270
enthalpy of mixing, 262, 263f
equilibrium distribution, 262
first-nearest-neighbor, 260
Mg-Sn system optimization,
270–272
molten oxide solutions,
324–325
multicomponent solution
composition-dependent
coordination numbers,
279
database, 287–288
interpolation formulae,
280–286
pair-exchange reaction, 278
380
INDEX
Modified quasichemical model
(MQM) (Continued)
standard Gibbs energy, 280
pair-exchange reaction, 260
polynomial expansion, 264,
268–269
Molar enthalpy, 44
liquid Al-Ca solutions,
253–255, 254f
liquid Al-Sc solutions,
253–255, 255f
liquid Mg-Sn solutions,
253–255, 254f
Molar Gibbs energy of mixing,
41, 42f
Mole fraction, 41
ideal Raoultian solution, 46
integral and partial, 42, 42f
Molten oxide solutions
alkaline-earth silicates
enthalpy of mixing, 321–324,
322f
entropy of mixing, 321–324,
323f
associate model, 325–326
boron, network forming,
328–330
CaO-SiO2 system
phase diagram, 321–324,
323f
structure, 319–321, 320f
cell model, 327
charge compensation effect,
327–328, 329f
modified quasichemical
model, 324–325
nonoxide anions, 330–333
PbO-SiO2 melts, Qi species,
321, 322f
reciprocal ionic liquid model,
326–327
Reddy-Blander capacity
model, 330–333
SiO2 system
network modification,
321f
structure, 319, 320f
Molten salt solution
liquid-liquid mixing, 272–273
multiple sublattice randommixing, 252
single-lattice random mixing,
200
single-lattice SRO, 274, 275f
two-sublattice modified
quasichemical model,
300–301
Monotectic reaction, 68
Monotectics, 81
MQM. See Modified
quasichemical model
(MQM)
Muggianu model, 208f, 209–212
Muggianu-Redlich-Kister model,
211–212
Multicomponent solution, 52,
189
Multiple sublattice randommixing
compound energy formalism,
240
ceramic solutions, 246–249
interstitial solutions,
240–241
nonstoichiometric
compounds, 241–245
end-member activities, 239
Temkin model, 249–252
three-sublattice model,
230–231
two-sublattice solution, 231
polynomial expansion, 232
reciprocal excess term,
235–239
(Li,K)(F,Br) reciprocal
ternary system, 232, 233f
(Na,K)(F,Cl) reciprocal
ternary system, 232, 234f
Multisublattice model, 200
N
Nernst equation, 25, 169
Network modifiers, 321
Network formers, 321
Neumann-Kopp rule, 272
Node, 107–108
Nonconfigurational entropy of
mixing, 194
Nonideal gas, equilibrium, 20–22
Nonstoichiometric compound,
72, 241–245
O
One-component system, 106f,
119
One-sublattice Bragg-Williams
model, 272
Order-disorder transition,
162–163, 162f
Au-Cu phase diagram, 337,
337f
composition square, 334, 334f
compound energy formalism,
334
equilibrium value, 336, 336f
equimolar line, 335–336, 335f
first-order, 338, 339f
long range ordering, 333
second-order, 337f
three ordered phases, 340
Orthogonal Legendre
polynomials, 198
Oxide phase diagrams
FeO-SiO2 system, 124f, 125
Fe2O3-SiO2 system, 123f, 124
Fe3O4-SiO2 system, 122–123,
123f
phase fraction lines, 126f, 127
P
Pair-exchange reaction, 260, 274
Pandat, 137, 190
Paraequilibrium, 8
Paraequilibrium phase diagram
Fe-Cr-C-N system, 154, 155f
Fe-Cr-C system
molar ratio vs. molar ratio,
152–153, 153f
molar ratio vs. temperature,
151, 151f, 154f
law of adjoining phase regions,
150
Paraequilibrium state, 149
Partial excess Gibbs energies,
197
Partial molar enthalpy, 348
Partial molar Gibbs energy, 43
INDEX
Partial molar properties, 43–44
Peritectic reaction, 70
equilibrium solidification,
134–135
Scheil-Gulliver cooling, 139
Peritectics, 82
Peritectoids, 82
Phase diagrams
Al2SiO5 system, 3, 7f
compilations, 183–184
definition, 3
Fe-Cr-S2-O2 system, 3, 7f
Fe-Cr-V-C system, 3, 6f
Fe-Ni-O2 system, 3, 6f
general rules, 6–8
isoplethal, 5f
isothermal, 4f
single-valued, 110–114
temperature composition, 4f
Phase Diagrams for Ceramists,
184
Phase fraction lines, 126f, 127,
131
Physical vapor deposition (PVD),
155
Point defects, 241
Polynomial expansion, 197
Polythermal liquidus
projections, 133–134
Al-Sc-Mg system, 288, 289f
extension rule, 131
Potential variable, 104–105, 105t,
153–154
Pourbaix diagram. See Eh-pH
diagram
Predominance phase diagrams,
22–24
Pressure-composition (P-X)
phase diagram, 58–59
Proeutectic/primary
constituent, 63–64
Q
Quasi-binary phase, 96–97
Quasichemical model, 314
Quasi-invariant reactions,
135–136
Quasi-peritectic reaction, 90,
133–135
Quasi sublattices, 229–230
Quaternary phase diagrams
liquidus projections, 99–100
solidus projections, 100–102
R
Raoultian activity coefficient, 77
Raoult’s law, 77, 241
Reciprocal salt system, 128–129
Reddy-Blander capacity model,
330–333
Redlich-Kister polynomials, 198,
346
Regular solution theory, 50–52
binary phase diagrams, 65–67
single-lattice random mixing,
195–196
Relative chemical potential,
44–45
Reversible work, 12–13, 17
S
Saddle points, 90, 136
Saturation limit, 165
Scheil-Gulliver constituent
diagrams, 137
Al-Li system, 137–140
Au-Bi-Sb system
isoplethal phase, 144–147,
146–147f
isothermal phase, 144, 146f
liquidus projection,
140–141, 140f, 143–144,
143f
microstructural
constituents, 144, 144t
geometrical properties,
147–148
Scheil-Gulliver solidification, 8
crystallization paths, 141–143,
142t
nonequilibrium, 136–137
Schreinemakers’ rule, 95, 107,
122
Scientific Group Thermodata
Europe (SGTE), 183
Second-nearest-neighbor (SNN)
pairs, 237–238, 308–309
Second-nearest-neighbor SRO
381
complex anion model,
273–274
enthalpy of mixing, 272–273,
273f
in molten KCl-CoCl2 solutions,
274, 275f
Second-order transition
definition, 160
heat capacity of α-Fe, 160, 161f
order-disorder, 162–163, 162f
T-composition phase diagram,
160, 161f
Short-range ordering (SRO)
first-nearest-neighbor, 297,
314
second-nearest-neighbor,
308–309
complex anion model,
273–274
enthalpy of mixing, 272–273,
273f
in molten KCl-CoCl2
solutions, 274, 275f
Single ion activity, 78–79
Single-lattice random mixing
coordination equivalent
fractions, 219–222
Darken’s quadratic formalism,
202–203
ideal Raoultian solution,
194–195
interaction parameter
formalism
e23, 224–225
molar ratio representation,
225
unified, 224
Wagner’s, 222–223
lattice stabilities, 201
phase diagram optimization
algorithms, 206
concept of, 203
NaNO3-KNO3 system, 204,
204–205f
virtual data, use of, 206–207
polynomial expansion, excess
Gibbs energy, 196
caveat for optimizations,
199–200
382
INDEX
Single-lattice random mixing
(Continued)
equivalent fractions,
198–199
Gibbs-Helmholz equation,
197–198
Redlich-Kister polynomial,
198
quaternary solution
binary terms, 216–218
partial excess Gibbs energy,
219
ternary terms, 218–219
regular solution theory,
195–196
ternary solution
geometric models, 207–209,
208f
Kohler vs. Muggianu model,
212–213
liquidus projection, 213–214
polynomial expansion, 207,
209
terms, 214–216
thermal vacancies, 226
Single-sublattice modified
quasichemical model
associate models, 257–260
and Bragg-Williams
expression, 264–268
characteristic of solutions,
253–256
closed explicit form, 286–287
composition-dependent
coordination numbers,
269–270
enthalpy of mixing, 262, 263f
equilibrium distribution, 262
first-nearest-neighbor, 260
vs. ideal random mixing,
274–277
Mg-Sn system optimization,
270–272
pair-exchange reaction, 260
polynomial expansion, 264,
268–269, 277–278
second-nearest-neighbor
complex anion model,
273–274
enthalpy of mixing, 272–273,
273f
in molten KCl-CoCl2
solutions, 274, 275f
Single-valued phase diagrams,
110–114
Site exchange energy, 247
Solid ionic salt solution, 229
Solidus projections
quaternary phase diagram,
100–102
ternary phase diagram, 97–99
Solvus lines, 63
3D space model, ternary,
86–87
Spinel solution, 229, 246, 247f
Spinodal curve, 62–63
Spinodal points, 62–63
Stability criterion, 120
Stable diagonal, 232–235
Stable pair, 232–235
Standard Gibbs energy, 18, 280,
330–331
Standard molar enthalpy, 14–16,
14f
Standard state, 45
chemical potential, 77–78
of gas, 21
Stirling’s approximation, 47,
349
Stoichiometric compound, 72
Straight tie-lines, 129
Strain energy, 83
Subregular solution, 196
Sulfide capacity model, 332–333,
333f
Symmetrical salt solutions,
233–234f, 249–250
Syntectics, 82
T
Tangent construction, 43
Taylor expansions, 236–237,
277–278
Temkin model
multiple sublattice randommixing, 249–252
single-lattice random mixing,
200
Temperature-composition
phase diagram, 3, 4f,
117–118
bivariant region, 36
equilibrium cooling, 37
fusion temperature, 37
K2CO3-Na2CO3 system,
60, 61f
second-order transition, 160,
161f
three-phase invariants, 37–38
tie-line, 35–36
univariant region, 36
Ternary temperaturecomposition phase
diagrams
composition triangle, 85–86
isopleth, 95–97
isothermal section
Bi-Sn-Cd system, 91, 92f
lever rule, 91–93
topology, 93–95
polythermal liquidus
projection, 89–90, 89f
crystallization path, 88
liquidus isotherms, 88
peritectic reaction, 91
quasi-peritectic reaction, 90
Zn-Mg-Al system, 91, 92f
solidus projections/firstmelting, 97–99
space model, 86–87
Ternary terms, 214–216
Thermal vacancies, 226
Thermo-Calc, 137, 190
Thermodynamics, 4–5
auxiliary functions, 28–29
first law, 10
fundamental equation of,
12–13
nomenclature, 10
real gases, 21
second law, 11–12
third law, 26–27
Thermodynamic variable, 3
Third-order transition, 160
Three-phase invariants, 37–38
Tie-line, 35–36
isoplethal section, 96
INDEX
paraequilibrium phase,
152–153
reciprocal salt system, 128f,
129
straight, 129
ternary isothermal section,
91–93
Tie-triangle, 90–91, 94f
Toop approximation, 208f,
210–211
Transitions
ferromagnetic, 163–164
first-order, 159
order-disorder, 162–163, 162f
second-order, 160
third and fourth-order, 160
Tricritical point, 160
Triple point, 38–39
True aqueous phase diagram
Debye-Davies equation, 176,
177f
Eh-pH coordinates, replotting
of, 180–181, 181–182f
equilibrium composition, 174,
175t, 176
H2O-O2-CO2-Cu-HCl system,
180, 180f
H2O-O2-CO2-Cu system,
178–180, 179f
H2O-O2-HCl-Cu-Au system,
178f
H2O-O2-HCl-Cu system, 178,
179f
H2O-O2- HCl-NaOH-Cu
system, 172, 173f
hydrogen potential, 177f
Pourbaix diagram, Cu-H2O
system, 172, 173f
system size, 172
ZPF lines, 172
Two-dimensional phase
diagram, 104
Two-sublattice modified
quasichemical model,
296
complexes/molecules,
300–302
configurational entropy,
305–308
coordination equivalent, 298
FNN pairs, 297, 314
Gibbs energy equation,
303–305
liquidus projection
(Li,Na)(F,Cl) system,
315–316f, 317
(Na,Ca)(F,Cl) system,
316–317f, 317
383
pair positions, 296–297
quadruplets, 297, 297f,
299–300
salt solution, 296
second-nearest-neighbor
pairs, 308–314
U
Unified interaction parameter
formalism, 224
Univariant region, 36, 87
V
Vacanconium, 245
Virtual data, 206–207
W
Wagner’s interaction parameter
formalism, 222–223
Z
Zero constituent fraction (ZCF)
lines, 148
Zero-phase-fraction (ZPF) lines,
108–110, 131
binary T-composition phase
diagram, 109f
double, 174–176
true phase diagrams, 172