Phase Diagrams & ThermodynamicsA phase diagram is a graphical representation of the equilibrium state of a system using the intensive variables T and xi while p is kept constant.
The equilibrium may be calculated from thermodynamic data using G for all relevant phases
xB
G p = const.T = const.
Solubility of Bin pure A
A B
Solution(e.g. Liquid)
„Line Compound“
Solution withmiscibility gap
1 Modeling
The Tangent Method
Graphical evaluation of equilibria from the G(x) curves
A Bx
GT, p = const.
x‘ x‘‘
μA‘ =μA‘‘ μB‘ =
μB‘‘
Local equilibrium conditions:
μA’ = μA’’:
μB’ = μB’’:
and:T’ = T’’; p’ = p’’
''''''''' )()(xGxG
xGxG
∂∂
−=∂∂
−
''''''''' )()1()()1(xGxG
xGxG
∂∂
−+=∂∂
−+
phase ‘phase ‘‘
2 Modeling
What do we need for the Calculation?
For each phase relevant in the system we need the Gibbs Energy Gas a function of the intensive variables p, T, xi (analytical expression)
The combination of these Gibbs energies defines our thermodynamic model.
The minimum of G for the system, and thus the phase equilibria, can be calculated by minimization procedures.
Phases:• pure condensed substances (elements, compounds)• solutions (liquid and solid solutions)• nonstoichiometric compounds• gas phase (consisting of different gas species i with partial
pressure pi)
3 Modeling
Thermodynamic Modeling
M Hillert:“By modeling we shall understand the selection of some assumptions
from which it is possible to calculate the properties of a system”
1) Physical models: hypothesis ⇒ mathematical expression
2) Empirical models: experimental data ⇒ mathematical expression
4 Modeling
LiteratureM. Hillert: „Phase Equilibria, Phase Diagrams and Phase Transformations – Their Thermodynamic Basis“, Cambridge University Press 1998
Example: Simple Empirical Model
Consider the representation of G as a power series in T
...2 +++=−= dTbTaHGG SER
...2)( 2
2+−=
∂
∂−= dT
TGTcp⇒
cp
T / K0 298
Usual course of cpat high temperature
But:
This means we need a constant term in cp for a proper description!
...ln
....22 ++++=
−−−=
dTTcTbTaG
dTccp Representation generally used in SGTE format.Only valid for high temperatures !
5 Modeling
Simple Physical Model: Thermal Vacancies
N: number of atomsNV: number of vacancies
Consider a pure crystalline solid element. The number of possible arrangements is: !!
)!(vV
NNNNW +
=
]lnln)ln()[(ln VVVV NNNNNNNNkWkS −−++⋅==Δ
According to Boltzmann this gives a change in entropy:
This may be introduced into the Gibbs Energy:
]ln[lnV
VVV
V
V
NNN
NN
NNNkNTgN
STgNG
++
++⋅=
Δ−⋅=Δg: energy of formation
for one vacancy
6 Modeling
Thermal Vacancies (2)
Regard NV as internal variable for a Gibbs energy minimization:
VV
NpTV NN
NkTgNGD
++=
∂∂
==− ln)(0 ,,
⇒ Equilibrium fraction of vacancies (for D = 0)
D: thermodynamic driving force
)exp(kTg
NNNyV
VeqV −=
+=
At equilibrium the internal variable (NV) can now be eliminated:
)]exp(1ln[)1ln(
]ln)1[ln(
]ln[ln
kTgRTykNT
yN
NykNTgN
NNN
NN
NNNkNTgNG
eqV
eqV
VeqVV
VVV
VV
−−=−=
=+−+⋅=
=+
++
+⋅=Δ
7 Modeling
Solution PhasesThermodynamic properties have to be modeled as a function of composition
A BxB
ΔG
Two component system (binary)
T, p = const.
„line compound“AB2
„solution phase““line compound”e.g. NaCl, GaAsIn fact also shows homogeneity range⇒ Depends on the scale!
“solution phase”⇒ Most liquids⇒ Solid solutions⇒ nonstoichiometriccompounds
8 Modeling
Ideal Solution
No difference in the interaction between like and unlike atoms is assumed for the ideal solution :
A - A = A - B = B – B
),,(,...),,,( 21 xTpGGNNTpGG =⇒= (binary system)
00
0
0
≠Δ⇒≠Δ
=Δ
=Δ
idid
id
id
GS
V
H
∑ −−+−=−=Δ=
c
iii
id xxxxRxxRS1
)]1ln()1(ln[ln
∑ −−+==Δ=
c
iii
id xxxxRTxxRTG1
)]1ln()1(ln[ln
As x < 1 ⇒ lnx < 0 ⇒ always stabilizing!
9 Modeling
Ideal Solution (2)
0 1x
+∞=∂Δ∂
xSid
xlim
0a
−∞=∂Δ∂
xSid
xlim
1a
−∞=∂Δ∂
xGid
xlim
0a
+∞=∂Δ∂
xGid
xlim
1a
693.0)1ln()1(ln −=−−+ xxxxx = 0.5
10 Modeling
Regular Solution[Hildebrand 1929]: Interaction between unlike atoms contributes to ΔH.
)]1ln()1(ln[)1( xxxxRTxxG −−++−⋅=Δ ε
Excess term Ideal term
⇒ Define Excess functions of the form Y = YE + YIDEAL
ε < 0 :
Additional stabilization
from HE
ε = 0 :
Ideal Solution
ε > 0 :
Interplay between S (stabilization)
and H (destabilization)
11 Modeling
Regular Solution - Example
[Y.A. Chang, University of Wisconsin]
ε = 12.5 kJmol-1
Critical point
12 Modeling
Regular Solution – Example (2)
ε = 12.5 kJmol-1
13 Modeling
Regular Solution – Example (3)
Resulting “phase diagram” obtained by the calculation with our regular solution model (ε = 12.5 kJmol-1)
single phase field
two phase field
spinodal curve
14 Modeling
Redlich-Kister PolynomsCommon standard model for solution modeling. Extension of the regular solution model for the modeling of all kinds of asymmetric shapes.
IxxIxxG BAE ⋅−=⋅=Δ )1( General expression for the binary
Redlich-Kister:
...)()( 2210 +−+−+= BABA xxLxxLLI
∑ −⋅=Δ=
n
k
kBA
kBA
E xxLxxG0
)(
L is modeled as a function of T
e.g.: or higher powers of TbTaLk +=
15 Modeling
Sublattice ModelsNonstoichiometric compounds require composition dependent modeling. Usually they have more than one sublattice.
⇒ No adequate representation by conventional Redlich-Kister models!
Usual case for crystalline phases:
• Vacancies• Interstitials Occur on different sublattices!• Substitutions
⇒ Crystal structure and defect mechanisms must be known!
• X-ray diffraction investigations• Spectroscopy• Diffusion studies, etc…
16 Modeling
Example: TiO2-x Sublattice Model
Rutile structure typeTetragonal P42/mnm
Ti4+: 2a (0,0,0) O2-, Va2- 4f (0.3,0.3,0)
Sublattice notation: (Ti4+)1(O2-,Va2-)2
∑ −+
+++=Δ
=−−−−
−−−−−−
n
k
kVaO
kVa
VaVaOOTiVaVaTiOO
yyTLyy
yyyyRTGyGyG
00
00
))((
)lnln(2
2222
22222222
(y…Site fraction) [Waldner and Eriksson, CALPHAD 1999]
17 Modeling
Extension to higher order Systems1) Solutions:
The thermodynamic properties of the solution are extrapolated from the thermodynamic properties of the subsystems using different geometrical models.e.g.: - Kohler Model (symmetric)
- Muggianu Model (symmetric)- Toop Model (asymmetric)
Extrapolation with or without additional interaction parameters
2) Compounds:Up to now it is not possible to predict compound formation ⇒Experiments necessary!
Higher order compounds are modeled as line compounds (only temperature dependence) or with suitable sublattice modelsaccording to the crystal structure.
18 Modeling
Kohler Model
B0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
xA(AC)
xB(AB)
xC(BC)
EBC
BCCBCB
CB
EAC
ACCACA
CA
EAB
ABBABA
BA
EABC
Gxxxx
Gxxxx
Gxxxx
G
)()(
)()(
)()(
+
+
=Δ
xC(AC)
xA(AB)
xB(BC)
Symmetric Extrapolation
19 Modeling
Muggianu Model
B0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
xA(AC)
xC(BC)
xB(AB)
Symmetric Extrapolation
EBC
BCCBCB
CB
EAC
ACCACA
CA
EAB
ABBABA
BA
EABC
Gxxxx
Gxxxx
Gxxxx
G
)()(
)()(
)()(
+
+
=Δ
xC(AC)
xA(AB)
xB(BC)
20 Modeling
Toop Model
B0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
A
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
xA(AC)xA(AB)
xC(BC)
Asymmetric Extrapolation
EBC
BCCBCB
CB
EAC
ACCACA
CA
EAB
ABBABA
BA
EABC
Gxxxx
Gxxxx
Gxxxx
G
)()(
)()(
)()(
+
+
=Δ
xC(AC)
xB(AB)
xB(BC)
Asymmetric Component
21 Modeling
The CALPHAD MethodCALPHAD = Calculation of Phase Diagrams
⇒ Critical assessment and thermodynamic optimization of binary and higher order systems
1) Literature Assessment:evaluation of all available literature sources
2) Modeling of the Gibbs energies G(p,T,xi) for all phases in the system.
3) Optimization of model parameters for best representation of the experimental data ⇒ inter-consistency of data!
Data Sources:
• Thermodynamics (Calorimetry, EMF, vapor pressure)
• Phase Diagram Studies (DTA/DSC, X-ray diffraction, optical microscopy, SEM/EPMA, …)
• Other Methods (Diffusion studies, magnetic investigations, …)
22 Modeling
Evaluation and selection ofinput data
Thermodynamic modeling of the
phases
Optimization of model parameters
(by error minimization procedures)
Calculation(phase diagrams, property diagrams, etc.)
and Comparison (to the input data)
Applications(databases, predictions, simulations, etc.)
TheCALPHADapproach
[G. Cacciamani, Genova University]
23 Modeling
Evaluation and selection ofinput data
Thermodynamic modeling of the
phases
Optimisation of model parameters
(by error minimisation procedures)
Calculation(phase diagrams, property diagrams, etc.)
and Comparison (to the input data)
Applications(databases, predictions, simulations, etc.)
Stoichiometric compounds
Ordered solutions
Disordered solid solutions
Liquids
etc.
The CALPHAD Approach (1)
24 Modeling
Experiments (DTA, DSC, calorimetry, EMF, vapor pressure, LOM, SEM, X-ray diffraction, etc.)
Estimates (periodic properties, chemical criteria, etc.)
Theory (ab-initio, semi-empirical, etc.)
The CALPHAD Approach (2)
Evaluation and selection ofinput data
Thermodynamic modeling of the
phases
Optimisation of model parameters
(by error minimisation procedures)
Calculation(phase diagrams, property diagrams, etc.)
and Comparison (to the input data)
Applications(databases, predictions, simulations, etc.)
25 Modeling
Evaluation and selection ofinput data
Thermodynamic modeling of the
phases
Optimisation of model parameters
(by error minimisation procedures)
Calculation(phase diagrams, property diagrams, etc.)
and Comparison (to the input data)
Applications(databases, predictions, simulations, etc.)
G(P,T,x1,...,xi,)
Data selection and input
Weight assignment
Parameter evaluation by non-linear least squares regression
The CALPHAD Approach (3)
26 Modeling
Evaluation and selection ofinput data
Thermodynamic modeling of the
phases
Optimisation of model parameters
(by error minimisation procedures)
Calculation(phase diagrams, property diagrams, etc.)
and Comparison (to the input data)
Applications(databases, predictions, simulations, etc.)
Comparison with input and derived data
Compatibility with similar and higher order systems
The CALPHAD Approach (4)
27 Modeling
Evaluation and selection ofinput data
Thermodynamic modeling of the
phases
Optimisation of model parameters
(by error minimisation procedures)
Calculation(phase diagrams, property diagrams, etc.)
and Comparison (to the input data)
Applications(databases, predictions, simulations, etc.)
Database implementation
Extrapolation to higher order
Materials simulation
etc.
The CALPHAD Approach (5)
28 Modeling
optimized A-B
extrapolated A-B-C
optimized A-C
optimized B-C
optimized A-B-C
“a few” key data
The CALPHAD Approach (6)
29 Modeling
optimized A-B-C
extrapolated A-B-C-D
optimized A-B-D
optimized A-C-D
optimized A-B-C-D
“a few” key data
optimized B-C-D
The CALPHAD Approach (7)
30 Modeling
Example: Hypothetical Binaries, Ideal Solution
SGTE parameter representation for pure elements (stable and meta-stable phases) in combination with the ideal solution model.
⇒Calculation of hypothetical binary phase diagrams.
Program: FazDiaGr by G. Garzeł
Data base: 4d and 5d elements of Group 4-8:Zr, Hf, Nb, Ta, Mo, W, Re, Rh, Ir, Pd, Pt, Ru, Os
Phases: Liquid (λ), fcc (α), bcc (β), hcp (ε)
31 Modeling
λ
βα
ε
Hypothetical Binaries, Ideal Solution (1)
32 Modeling
λ
βα
ε
Hypothetical Binaries, Ideal Solution (2)
33 Modeling
λ
βα
ε
Hypothetical Binaries, Ideal Solution (3)
34 Modeling
Binary Phase diagram features modeled with use of the regular solution model
• Cigar shape of the solid/liquid phase boundaries• Maximum congruent melting of the solid phase• Minimum congruent melting of the solid phase• Peritectic phase diagram• Eutectic phase diagram
KmolJSSKTKT
mB
mA
mB
mA
⋅==
==
/201000 ,800
Example: Modeling using Regular Solutions
Regular solution parametersεl and εs are varied
35 Modeling
800
850
900
950
1000
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)
0== sεε l
800
850
900
950
1000
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)
kJs 5== εε l
800
850
900
950
1000
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)
kJs 15−== εε l
1 - Liquidus and Solidus Curves
Liquid
Solid
36 Modeling
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)Liquid
Solid
00
==
sεε l
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)
molkJs /100−=
=εε l
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)
0/10
=−=
smolkJ
εε l
2 - Maximum and Minimum
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
Mole Fraction of B
T(K
)
0/12− molkJ
==
sεε l
37 Modeling
400
500
600
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
Solid
Solid+Solid
Liquid
T(K
)
molkJs /12== εε l
Mole fraction of B
3 - Peritectic Phase Diagram
400
500
600
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
kJs 15== εε l
38 Modeling
400
500
600
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
Liquid
molkJs /100
==
εε l
Solid+Solid
Solid
T(K
)
Mole fraction of B
400
500
600
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
molkJs /150
==
εε l
400
500
600
700
800
900
1000
1100
0 0.2 0.4 0.6 0.8 1
molkJmolkJ
s /25/20
=−=
εε l
4 - Eutectic Phase Diagram
39 Modeling
5 - Monotectic Phase Diagram
40 Modeling
500
600
700
800
900
1000
1100
1200
0 0.2 0.4 0.6 0.8 1
Liquid
Liq+liq
Solid
Solid+Solid
Liquid+Solid
molkJmolkJ
s /5.13/5.17
==
εε l
Mole fraction of B
T(K
)
6 - Syntectic Phase Diagram
41 Modeling
300
600
900
1200
1500
1800
0 0.2 0.4 0.6 0.8 1
Liquid
Liquid+liquid
Solid
Solid+Solid
Liquid+Solid
Mole fraction of B
T(K
)
molkJmolkJ
s /10/25
==
εε l
7 – Monotectic + Peritectic Phase Diagram
42 Modeling
300
600
900
1200
1500
0 0.2 0.4 0.6 0.8 1
Liquid
Liquid+liquid
Solid
Solid+Solid
Liquid+Solid
Mole fraction of B
T(K
)
molkJmolkJ
s /20/20
==
εε l
Solid
Example: Modeling of binary In-NiCALPHAD assessment by Waldner and Ipser
[Massalski’s Phase Diagram Compilation]
Models:
L, (Ni):Redlich-Kister solution models
δ:(Ni,Va)1(In,Ni)1sublattice model
ξ, ξ’: (Ni,Va)1(Ni)1(In,Ni)1sublattice models
Ni3In, Ni2In, NiIn, Ni2In3, Ni3In7: Stoichiometric
43 Modeling
Modeling of binary In-Ni (1)
Data Sources:
• Phase diagram: mainly 2 papers ( + older literature)• Vapor pressure data: 3 papers• EMF measurements: 5 papers• Calorimetry: 3 papers • Crystal structure: various literature on defect mechanisms of the
structure type
Used as input for the optimization procedure
44 Modeling
Calorimetric data
Modeling of binary In-Ni (2)
45 Modeling
Vapor pressure data
Modeling of binary In-Ni (3)
46 Modeling
Pressures over Ni2In from EMF and Knudsen sources
Modeling of binary In-Ni (4)
47 Modeling
Enthalpies from EMF and Calorimetric sources
Modeling of binary In-Ni (5)
48 Modeling
Fit with phase diagram data
Modeling of binary In-Ni (6)
49 Modeling
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