Phase Diagrams & Thermodynamics - Universität Wien Extension to higher order Systems 1) Solutions:...

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



phases

Optimisation of model parameters

(by error minimisation procedures)




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.)




phases






25 Modeling



phases






G(P,T,x1,...,xi,)

Data selection and input

Weight assignment

Parameter evaluation by non-linear least squares regression


26 Modeling



phases






Comparison with input and derived data

Compatibility with similar and higher order systems


27 Modeling



phases






Database implementation

Extrapolation to higher order

Materials simulation

etc.


28 Modeling

optimized A-B

extrapolated A-B-C

optimized A-C

optimized B-C

optimized A-B-C

“a few” key data


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


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


33 Modeling

λ

βα

ε


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


45 Modeling

Vapor pressure data


46 Modeling

Pressures over Ni2In from EMF and Knudsen sources


47 Modeling

Enthalpies from EMF and Calorimetric sources


48 Modeling

Fit with phase diagram data


49 Modeling

Phase Diagrams & Thermodynamics - Universität Wien Extension to higher order Systems 1) Solutions:...

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