Thermodynamic data
A tutorial course
Session 8: Calculation of phase equilibria from
critically assessed thermodynamic data
Alan Dinsdale
“Thermochemistry of Materials” SRC
Outline
• Describe how phase diagrams can be calculated
from critically assessed thermodynamic data
– What are critically assessed data ?
• How thermodynamic data are modelled
– Temperature
– Composition
– Pressure
• Databases and software
• Applications
2
Why calculate phase diagrams ?
• Provides way to rationalise different but related
sets of experimental properties
• ..... and to extrapolate thermodynamic data from
small systems into higher order systems in such a
way that they allow prediction of
multicomponent phase equilibria
• Now software and data are sufficiently robust for
– monitoring and controlling industrial plant
– choosing materials used in everyday household appliances
– understanding the way in which mankind affects the world and the
environment in which we live
– devising how we can perhaps make the world a safer and better place for
us and our children to live in
3
Principles of Phase
Equilibrium Calculations
Experimental Data
G(T,P,x) Model For Each Phase
Develop Parameters For SMALL Systems
To Reproduce Experimental Data
Industrial Problem
MTDATA
Database
Predictions For
LARGE
Problem Solved
Systems
Validation
4
In a nutshell
Model Gibbs energy over a range of
temperatures and compositions
6
…. in order to
calculate the phase
diagram
7
What are critically assessed data ?
What do we mean by
critically assessed data ?
• Enthalpies of mixing of
liquid Cu-Fe alloys
• Large scatter in
experimental values
• Which data best
represent reality ?
• .. and are these data
consistent
• with …
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…. with the experimental
phase diagram
10
…… and measured activity data
11
What is the aim of a critical
assessment ?
• To generate a set of reliable data or diagrams
which are self consistent and represent all
the available experimental data
• Involves a critical analysis of the
experimental data
• ….. followed by a computer based
optimisation process to reduce the
experimental data into a small number of
model parameters
• ….. using rigorous theoretical basis
underlying thermodynamics
12
Critical analysis of experimental data
• Experimental data: Search and analysis
– Search through standard compilations eg Hultgren, Massalski
– Use a database of references to the literature eg Cheynet
– Carry out a full literature search
• Which properties
– Phase diagram information
• Liquidus / solidus temperatures
• Solubilities
– Thermodynamic information
•
•
•
•
•
•
enthalpies of mixing
vapour pressure data
emf data
heat capacities
Enthalpies of transformation
Ab-initio calculations
13
Obtaining model parameters
• Aim is to determine set of coefficients which gives
best agreement with experimental data
– by least-squares fitting of the thermodynamic functions to selected
set of experimental and ab-initio data
• It is usually carried out with the assistance of a
computer
– Using optimisation software
•
•
•
•
MTDATA optimisation module
PARROT (inside ThermoCalc)
LUKAS program BINGSS
CHEMOPT
14
Calculated enthalpies of mixing
for liquid Fe-Cu alloys
15
Calculated Fe-Cu phase diagram
16
Calculated activities for
Fe-Cu liquid alloys
17
of
phases at fixed
temperature
18
How to model thermodynamic data
need to understand how G changes with T, P and x
of
phases at fixed
temperature
Difference in
Gibbs
energy
between fcc
and liquid Fe
fcc phase is
reference for
both elements
Difference in Gibbs
energy between fcc
and bcc Cu
Change in Gibbs
energy with
composition is
complex
20
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IoM3, London, UK
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Heat capacity of Sn for
different phases
Ttrs
Tfus = 505.078
22
Enthalpy of Sn relative to
298.15 K
ΔfusH
ΔtrsH
23
Entropy of phases of Sn
24
Gibbs energy of phases of Sn relative to BCT
Ttrs
Tfus
25
Mathematical description of Gibbs
energy variation with temperature
• Heat capacity generally represented by 1 or
more expressions of the form:
– Cp = a + b T + c T2 + d T-2
(generally obtained from experiment)
• With enthalpy of formation and entropies
at 298.15 K (or transition enthalpies and
entropies) this leads to expressions for the
Gibbs energy of the form:
– G = A + B T + C T ln(T) + D T2 + ET3 + F T-1
– (relative to some defined reference state)
26
Magnetic materials
• Magnetism has a big influence on the heat
capacity and therefore on the Gibbs energy
Heat capacity of bcc Fe
Gibbs energy of liquid
and bcc phases of Fe
relative to fcc
27
Effect of pressure
G=H–TS+PV
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Effect of composition on
Gibbs energy
Difference
in Gibbs
energy
between
fcc and
liquid Fe
Difference in
Gibbs energy
between fcc
and bcc Cu
Change in
Gibbs
energy with
composition
is complex
 Basic approach is to use a simple theory to model
what we measure and then
 Fit any discrepancies (excess Gibbs energy) to a
power series expression
29
Simplest theory – ideal mixing
• Assumes that the components (elements) mix randomly
without giving off or absorbing any heat and with no
net volume change
• The mixing does result in a change in the entropy and
therefore the Gibbs energy
Gideal = R T [xFe ln(xFe) + xCu ln(xCu)]
30
Real materials – excess Gibbs
energy
• In most cases mixing between components (elements)
is accompanied by a significant heat effect
 which may be simple or complex
 The deviation from ideal behaviour may also vary with
temperature leading to an “excess entropy of mixing”
31
Overall Gibbs energy of mixing
32
Gibbs energy of binary solutions
Pure component Gibbs
energies
G=
xFe GFe + xCu GCu
+ R T [ xFe ln(xFe) + xCu ln(xCu)]
+ xCu xFe [ a + b (xCu-xFe) + c (xCu-xFe)2
+ d (xCu-xFe)3 + …..]
Ideal contribution
to Gibbs energy
Excess Gibbs
energy – in this
case “RedlichKister expression”
where a, b, c, d …. could be temperature dependent
(in practice for Fe-Cu we may need only one or possibly two parameters)
33
Calculated phase diagram
involving two phases
34
From binary to multicomponent
• Multicomponent Gibbs energy given by
G=
Σ xi Gi + R T Σ xi ln(xi) + Gexcess
Various models used to extrapolate excess Gibbs
energy into ternary and higher order systems from data
for binary systems. Extra ternary terms used if required
Kohler
Muggianu
Toop
36
Compound phases
• Stoichiometric phases: variation of Gibbs energy
with T similar to that for phases of elements
• Many important compound phases are stable
over ranges of homogeneity. Crystal structure
indicates sublattices with preferred occupancy.
–
eg: sigma, mu, gamma brass
• Use compound energy formalism to allow mixing
on different sites
–
–
–
(C,Va)1
Laves phases: (Cu,Mg)2 (Cu,Mg)1
Interstitial solution of carbon: (Cr,Fe)1
Spinels: (Fe2+,Fe3+)1 (Fe2+,Fe3+)2 (O2-)4
37
Gibbs energy using compound
energy formalism eg (Cu,Mg) (Cu,Mg)
2
1
• Gibbs energy again has 3 contributions
• Pure compounds with element from each sublattice
Cu:Cu, Cu:Mg, Mg:Cu, Mg:Mg
• Ideal mixing of elements on each sublattice
•
Cu and Mg on first and on second sublattices
• Non-ideal interaction between the elements on each
sublattice but with a specific element on the other
sublattice
– Cu,Mg:Cu Cu,Mg:Mg Cu:Cu,Mg Mg:Cu,Mg
•
38
Thermodynamic databases and
software
Thermodynamic databases
• Two types of databases
– Collections of datasets suitable for potentially diverse
materials eg SGSUB, SGSOL
– Application specific databases: comprehensive set of
data for particular material types
• All databases need to be based upon a set of
standards
– Consistent data for the elements
– Consistent models to represent thermodynamic data
based (where possible) on crystallographic
information
40
Database providers
• Software developers
–
–
–
–
–
MTDATA (NPL)
ThermoCalc (TCSAB)
FactSage (GTT, Ecole Polytechnique Montreal
PANDAT (CompuTherm)
JMatPro (ThermoTech)
• Specialist centres eg SGTE, CEA, Tohoku, Claustal
• International Collaborative Projects
– COST507, COST531, COST MP0602
– Superdata, III-V semiconductor project
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Software packages
General phase diagram calculations
• ThermoCalc: TCSAB, Stockholm, Sweden
• FactSage: GTT, Herzogenrath, Germany
• Pandat: CompuTherm, Madison, USA
• MTDATA: NPL, Teddington, UK
Calculation of materials properties
• JMatPro: Sente Software Ltd, Guildford, UK
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General facilities available
• Single calculation of phase equilibria for defined temperature,
pressure and composition
• Calculations over a range of temperatures for fixed composition
• Mass of phases in equilibrium (and compositions)
• Partial pressures
• Enthalpy changes and heat capacity
• Transformation temperatures
• Phase diagram calculations (binary, ternary, isopleths, liquidus
projections)
• Database management
• Assessment and optimisation capability
• Simple simulations (eg Scheil solidification)
43
Applications
Liquidus projection for
solders
Cu
1
0.9
00
10
0
90
900
900
0
80
0.8
0.7
800
800
700
600
0.4
0.3
700
700

0.6
Temperature
/C
Temperature
/C
0.5
x(C
u)
x(Cu)
(Cu)
Cu3Sn()
U1
600
600
U2
U3
500
500
U4
500
400
400
0.2
0.2
(Ag)
U5
400

0
0.0
0
0.0
Ag

Ag3Sn()
300
300
300
200
200
0
0.0
0.2
0.2
0.4
0.4
x(Sn) x(Sn)
0.6
0.6
0.8
0.8
(Sn)
1
1.0
E1
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
x(Ag) x(Ag)
Sn
45
Isopleths: Mixing electrician’s solder
with lead free solder
46
Plot of mass of phases with
variation of temperature
47
Calculated heat capacity and
volume change
48
Use of thermodynamic data to
predict thermophysical properties
Calculation of surface
tension:
equilibrium between bulk and surface
50
Oxide liquid modelled as nonideal mixture of species
51
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Calculated viscosity
52
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Engineering toolkits
Virtual Measurement Systems
• Use MTDATA to calculate properties such as liquidus and solidus
temperatures, enthalpy, heat capacity and density
• Develop and use thermodynamics as basis for modelling other
thermophysical properties
• Simple interface - user is shielded from complexity of models
• Easy to export data to Excel and other software
54
Amalgam Toolkit for
compact fluorescent lamp
design
•
Provide an easy to use
tool for lamp design
engineers
 Amalgam thermodynamic
database + MTDATA api
 Automatic selection of
elements and
compositions for
amalgams to optimise light
output
 Light output calculations
derived from calculated
materials chemistry
55
Relative Light Output results
56
Final thoughts
• Now possible to calculate phase equilibria for a wide
range of materials
• Rely on good quality critically assessed data compiled
into comprehensive databases
• A number of organisations are involved in developing
database
• Other sorts of properties can be modelled from a
thermodynamic basis
• Engineering toolkits offer potential to bring phase
diagram calculations to the non specialist
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