3.012 Fundamentals of Materials Science
Fall 2003
Lecture 15: 10.29.03 Phase changes and phase diagrams of single-component
materials
Today:
LAST TIME .............................................................................................................................................................................................. 2
BEHAVIOR OF THE CHEMICAL POTENTIAL/MOLAR FREE ENERGY IN SINGLE-COMPONENT MATERIALS .................................................... 3
The free energy at phase transitions .................................................................................................................................................. 3
PHASES AND PHASE DIAGRAMS SINGLE-COMPONENT MATERIALS ........................................................................................................... 5
Phases of single-component materials ............................................................................................................................................... 5
Phase diagrams of single-component materials ................................................................................................................................ 6
The Gibbs Phase Rule ........................................................................................................................................................................ 6
Example single-component phase diagrams ...................................................................................................................................... 8
Constraints on the shape of coexistence curves: The Clausius-Clapeyron equation ....................................................................... 11
REFERENCES ......................................................................................................................................................................................... 12
Reading:
Supplementary Reading:
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Lecture 15 – Single-component phase diagrams and phase transitions
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Last time
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Behavior of the chemical potential/molar free energy in single-component materials
The free energy at phase transitions

Earlier in the term, we examined the behavior of the heat capacity and enthalpy as heat was added to singlecomponent materials near a phase transition. What happens to the free energy at a phase transition? This can
be analyzed by plotting the free energy vs. temperature. Let’s consider the case of a solid that is melting to
become a liquid at constant pressure:
o
We can plot the free energy as a function of temperature for the liquid, and for the solid. The free energy
curves of each phase are unique- even though both phases are composed of the same atoms in singlecomponent systems, they have unique interactions in the solid and liquid states. Thus the free energy of
the liquid can be determined for very low temperatures where the liquid is not stable. Similarly, the free
energy for the solid can be determined at high temperature where it is not the stable phase:
(© W.C. Carter1 p. 105)
o

So given two free energy curves, how do we identify the stable phase at a given temperature? The Gibbs
statement of equilibrium says the stable phase has the lowest free energy. Below Tm, the solid has the
lowest free energy- above Tm, the liquid has the lowest free energy. At T=Tm, the two phases are in
equilibrium- Gliquid = Gsolid.
We showed earlier in the term that the enthalpy has a discontinuous jump at the melting point- the transformation
to the liquid state absorbs thermal energy and ‘stores’ it in the liquid. The enthalpy of melting Hm is absorbed as
the phase fraction of liquid moves from 0 to 1:
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(© W.C. Carter p. 105)
(Eqn 1)
o
If we plot the Gibbs free energy vs. the heat absorbed by the material (remember, the heat absorbed dq =
dH at constant pressure), we get the lower diagram. During the transformation, the free energy of the
solid and liquid are equal and constant.
o
If the free energy is constant and the enthalpy of the system is steadily increasing, what must the
behavior of the entropy during the transformation? Using the definition of the Gibbs free energy:
G  H  TS


(Eqn 2)
Let’s write an expression for the total change in free energy that occurs during the phase
transformation:
Gsolidliquid  0  H m  Tm Sm


(Eqn 3)
Hm  TmSm
Thus we find:
or
Tm 
H m
Sm
…the product of the melting temperature and the entropy exactly matches the changing enthalpy during the phase
transition, to maintain a constant free energy- consistent with our earlier analysis of the heat stored in phase transitions.
This equation
shows us how to measure the entropy of melting, if we know the melting temperature and the enthalpy of


melting.
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Phases and phase diagrams single-component materials
Phases of single-component materials

Day-to-day experience tells us that many materials exist in multiple forms as a function of the environmental
conditions- particularly temperature. Solids melt, liquids boil, and solids can sublimate directly to vapor.
Stability of polymorphs

In addition to these fundamental states of aggregation, liquids and solids may have multiple structural forms that
are stable under different conditions of temperature and pressure. In solids, this is the stability of different crystal
structures. Such structural variants (that have the same composition in single-component materials) are known
as polymorphs or allotropes (an allotrope typically refers to different structural forms of pure elements). Many
compounds have 4 or 5 different stable crystal structures, depending on the temperature!

Our discussion above outlined that the stable phase of a material is that with the lowest free energy- thus, the
structural rearrangements in polymorphs must give materials with different free energies. Since G = H –TS, we
may expect that these differences may reside in differences in enthalpy or differences in entropy. In general, both
the enthalpy and entropy may differ. Consider the molar entropy of iron as a function of temperature, from a plot
we showed earlier:
(Gaskell2)

Fe has several allotropes- the low temperature  phase (BCC), then the  phase (also BCC- this was once
believed to be a phase, but was shown to be a magnetic phase transition and no longer considered a separate
phase from  3),  phase (FCC), and finally  phase (BCC), before melting to the liquid state. In accordance with
the requirement for stability- that the entropy increase with increasing temperature, the entropy is larger in each
subsequent phase.
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Because G = H – TS, high entropy phases are stable at high temperature, while low enthalpy
phases are stable at low temperature.
Phase diagrams of single-component materials

Phase diagrams are maps of the phases present in a system at equilibrium as a function of 2 or more
thermodynamic variables. They are extremely useful as practical indicators of the equilibrium phase behavior of
even complex systems.

Reading the phase diagram:
o

Single-component phase diagrams have phase boundaries (or coexistence curves) and phase fields.
Within a phase field, the denoted form of the material is the equilibrium state. At the P,T values making
up the phase boundaries, the phases on either side of the boundary are in equilibrium- they coexist at
these conditions.
Phase diagrams obey the laws of thermodynamics! Thus there are constraints on the structure of phase
diagrams. To understand these constraints and discuss single-component phase diagrams further, we will first
introduce the Gibbs phase rule.
The Gibbs Phase Rule

Recall from the last lecture that the condition for equilibrium at constant temperature and pressure for closed
systems is given by a set of equations for the chemical potentials of each component:
1  1  1  ...  1P
(Eqn 4)
2  2  2  ...  2P
…

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C  C  C  ...  CP

Since each = sign in the above set provides one independent equation, we have in total C(P-1) equations.
However, we have an additional set of relationships between the variables of the system at equilibrium, via the
Gibbs
Duhem equation. Recall Gibbs-Duhem is:
C
0  V dP  S dT   N pj d pj
p
(Eqn 5)
p
j1
o

o
…which can be written for each of the P phases present in a system. Thus in total we have C(P-1) + P
equations. The number of degrees of freedom in the system, D, is the number of equations the system
must satisfy at equilibrium minus the total number of variables describing the system.
Here, we have (CP + 2) total variables (chemical potential of each component in each phase,
temperature, and pressure). Thus we have for D:
D  (CP  2) [C(P 1)  P]
D P  C  2
(Eqn 6)
(Eqn 7)



GIBBS PHASE RULE
(Eqn 7) is called the Gibbs phase rule. This is a very useful equation, as it specifies the number
of phases that can co-exist for a given condition of a system.
Application of the phase rule

Let’s return to our model single-component phase diagram above, and apply the phase rule to it. Consider first a
single-phase region:
D=C+2–P=1+2–1=2
(Eqn 8)
o

There are two degrees of freedom in a single-phase region of a single-component system- thus 2
parameters of the system can vary and that single phase will remain in equilibrium. This is shown in the
diagram by the range of T and P values that remain in the solid phase, for example.
What about the boundaries between two phases? If two phases are in equilibrium at the boundary, then we have:
D=C+2–P=1+2–2=1
(Eqn 9)
o
(Eqn 10)

Thus only one variable can vary independently along the coexistence curve- the other variable must be
coupled to it. Mathematically, for a P-T diagram:
dP
 f (P,T)
dT
Lastly, let’s consider the vertex of the solid-liquid, liquid-gas, and solid-gas coexistence curves:
(Eqn 11)
D = C
+2–P=1+-3=0
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In other words, there is one unique value of pressure and temperature that can allow the 3 phases to coexist; any change in the variables causes the equilibrium to shift to one between only 1 or 2 phases.
Example single-component phase diagrams

The following examples of single-component phase diagrams were prepared by Angela Tong (MIT) for Prof. W.C.
Carter:
Lecture 15 – Single-component phase diagrams and phase transitions
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Lecture 15 – Single-component phase diagrams and phase transitions
Fall 2003
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Other examples:
(Bergeron and Risbud4)
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Constraints on the shape of coexistence curves: The Clausius-Clapeyron equation

Just as the relationship between thermodynamic variables dictates the shape of free energy curves, equilibrium
relationships have something to say about the shape of curves on a phase diagram. An important relationship
known as the Clausius-Clapeyron equation provides a link between the conditions for 2-phase equilibrium and the
slope of the 2-phase line on the P vs. T diagram:
For 2 phases in equilibrium (let’s use the example of solid and liquid in equilibrium at the melting
temperature):
o
S  L
(Eqn 12)

To find a nearby condition of T and P where the two phases are still in equilibrium, we must
have:5

(Eqn 13)
 S  d S   L  d L
(Eqn 14)
0  VdP  SdT   N j d j
d S  d L
which implies:
C


GIBBS-DUHEM
j1
EQUATION

Gibbs-Duhem must hold for both phases:

0  V L dP  S L dT  N L d L
(Eqn 15)
0  V S dP  S S dT  N S d S
(Eqn 16)

(Eqn 17)


S S dT  V S dP  d S
or:
The two chemical potential differentials are equal at equilibrium, so:

V
(Eqn 18)
S L dT  V L dP  d L
S L dT  V L dP  S S dT  VS dP


or:

L
V S dP  S L  S S dT
Finally, we re-arrange to get:
dP
dT


equilibrium
Sm
H m

Vm Tm Vm
CLAUSIUS-CLAPEYRON EQUATION
The Clausius-Clapeyron equation dictates the slope of the two-phase co-existence curve for single-component
materials. We already know that the enthalpy change on melting is typically positive, therefore the sign of the
change in volume on melting will usually dictate whether the slope of P vs. T is positive or negative.

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References
1.
2.
3.
4.
5.
Carter, W. C. (2002).
Gaskell, D. R. Introduction to Metallurgical Thermodynamics (Hemisphere, New York, 1981).
Reed-Hill, R. E. & Abbaschian, R. Physical Metallurgy Principles (PWS Publishing, Boston, 1994).
Bergeron, C. G. & Risbud, S. H. Introduction to Phase Equilibria in Ceramics (American Ceramic Society,
Westerville, OH, 1984).
Denbigh, K. The Principles of Chemical Equilibrium (Cambridge University Press, New York, 1997).
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