Phase Diagrams & Thermodynamics

Phase Diagrams & Thermodynamics A phase diagram is a graphical representation of the equilibrium state of a system using the intensive variables T and i while p is kept constant. The equilibrium may be calculated from thermodynamic data using G for all relevant phases G Solubility of B in pure A Line Compound p const. T const. Solution (e.g. Liquid) Solution with miscibility gap A B B 1 Modeling

The Tangent Method Graphical evaluation of equilibria from the G() curves G T, p const. Local equilibrium conditions: phase μ A phase μ A μ B μ B μ A μ A : G ' ' G ' ( ) μ B μ B : G '' '' G ( ) '' G ' ' G + (1 ) ( ) and: T T ; p p ' G '' '' G + (1 ) ( ) '' A B 2 Modeling

What do we need for the Calculation? For each phase relevant in the system we need the Gibbs Energy G as a function of the intensive variables p, T, i (analytical epression) 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 p i ) 3 Modeling

Thermodynamic Modeling Literature M. Hillert: Phase Equilibria, Phase Diagrams and Phase Transformations Their Thermodynamic Basis, Cambridge University Press 1998 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 epression 2) Empirical models: eperimental data mathematical epression 4 Modeling

Eample: Simple Empirical Model Consider the representation of G as a power series in T c p But: G G H a + bt + dt 2 +... c p SER 2 G T ( ) 2dT 2 T +... 0 298 Usual course of c p at high temperature T / K This means we need a constant term in c p for a proper description! c p c 2dT... G a + bt + ct lnt + dt 2 +... Representation generally used in SGTE format. Only valid for high temperatures! 5 Modeling

Simple Physical Model: Thermal acancies Consider a pure crystalline solid ( N + N element. The number of possible W arrangements is: N! Nv! )! N: number of atoms N : number of vacancies According to Boltzmann this gives a change in entropy: ΔS k lnw k [( N + N )ln( N + N ) N lnn N lnn ] This may be introduced into the Gibbs Energy: ΔG N N g TΔS g + knt[ln N N + N + N N N ln N + N ] g: energy of formation for one vacancy 6 Modeling

Thermal acancies (2) Regard N as internal variable for a Gibbs energy minimization: D G 0 ( ) T, p, N N g + N kt ln N + N Equilibrium fraction of vacancies (for D 0) y D: thermodynamic driving force eq N N + N ep( g kt ) At equilibrium the internal variable (N ) can now be eliminated: ΔG N N g g + knt[ln(1 y knt ln(1 y + knt[ln N eq ) N + N eq ) + N + N N N ln y RT ln[1 ep( 7 Modeling N ln N + N eq g kt ] )] ]

Solution Phases Thermodynamic properties have to be modeled as a function of composition ΔG solution phase T, p const. line compound e.g. NaCl, GaAs In fact also shows homogeneity range Depends on the scale! line compound AB 2 solution phase Most liquids Solid solutions nonstoichiometric compounds A B B Two component system (binary) 8 Modeling

Ideal Solution G G( p, T, N1, N2,...) G G( p, T, ) (binary system) No difference in the interaction between like and unlike atoms is assumed for the ideal solution : ΔH Δ id id 0 0 A - A A - B B B ΔS id 0 ΔG id 0 ΔS ΔG id id R RT c i 1 c i 1 i ln i i ln i R[ ln + (1 )ln(1 )] RT[ ln + (1 )ln(1 )] As < 1 ln < 0 always stabilizing! 9 Modeling

Ideal Solution (2) 0.5 ln + (1 )ln(1 ) 0.693 0 1 lim a0 Δ S id lim Δ a1 lim a0 S id Δ G id lim Δ a1 G id + + 10 Modeling

Regular Solution [Hildebrand 1929]: Interaction between unlike atoms contributes to ΔH. ΔG ε ( 1 ) + RT[ ln + (1 )ln(1 )] Ecess term Ideal term Define Ecess functions of the form Y Y E + Y IDEAL ε < 0 : Additional stabilization from H E ε 0 : Ideal Solution ε > 0 : Interplay between S (stabilization) and H (destabilization) 11 Modeling

Regular Solution - Eample ε 12.5 kjmol -1 Critical point [Y.A. Chang, University of Wisconsin] 12 Modeling

Regular Solution Eample (2) ε 12.5 kjmol -1 13 Modeling

Regular Solution Eample (3) Resulting phase diagram obtained by the calculation with our regular solution model (ε 12.5 kjmol -1 ) single phase field spinodal curve two phase field 14 Modeling

Redlich-Kister Polynoms Common standard model for solution modeling. Etension of the regular solution model for the modeling of all kinds of asymmetric shapes. I I G B A E Δ ) 1 ( General epression for the binary Redlich-Kister:... ) ( ) ( 2 2 1 0 + + + B A B A L L L I Δ n k k B A k B A E L G 0 ) ( L is modeled as a function of T e.g.: or higher powers of T bt a L k + 15 Modeling

Sublattice Models Nonstoichiometric 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: acancies Interstitials Substitutions Occur on different sublattices! Crystal structure and defect mechanisms must be known! X-ray diffraction investigations Spectroscopy Diffusion studies, etc 16 Modeling

Eample: TiO 2- Sublattice Model Rutile structure type Tetragonal P4 2 /mnm Ti 4+ : 2a (0,0,0) O 2-, a 2-4f (0.3,0.3,0) Sublattice notation: (Ti 4+ ) 1 (O 2-,a 2- ) 2 ΔG + y 0 2 y y O 2 a G 2 0 TiO n 2 k 0 k + y a 2 L( T )( y G O 0 Tia 2 2 y + a 2RT ( y 2 ) k O 2 ln y O 2 + y a 2 ln y a 2 ) (y Site fraction) [Waldner and Eriksson, CALPHAD 1999] 17 Modeling

Etension to higher order Systems 1) Solutions: The thermodynamic properties of the solution are etrapolated from the thermodynamic properties of the subsystems using different geometrical models. e.g.: - Kohler Model (symmetric) - Muggianu Model (symmetric) - Toop Model (asymmetric) Etrapolation with or without additional interaction parameters 2) Compounds: Up to now it is not possible to predict compound formation Eperiments necessary! Higher order compounds are modeled as line compounds (only temperature dependence) or with suitable sublattice models according to the crystal structure. 18 Modeling

Kohler Model A 0.0 1.0 0.1 0.9 B(AB) 0.2 0.8 C(AC) 0.3 0.7 0.4 0.6 0.5 0.5 0.6 0.4 0.7 0.3 ΔG + + E ABC A A( AB) A A( AC) B B( BC) B B( AB) C C( AC) C G C( BC) E AB G G E AC E BC A(AC) 0.8 0.2 A(AB) C 0.9 0.1 1.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 B Symmetric Etrapolation B(BC) C(BC) 19 Modeling

Muggianu Model A 0.0 1.0 0.1 0.9 B(AB) 0.2 0.8 C(AC) 0.3 0.7 0.4 0.6 0.5 0.5 0.6 0.4 0.7 0.3 ΔG + + E ABC A A( AB) A A( AC) B B( BC) B B( AB) C C( AC) C G C( BC) E AB G G E AC E BC A(AC) 0.8 0.2 A(AB) C 0.9 0.1 1.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 B Symmetric Etrapolation B(BC) C(BC) 20 Modeling

Toop Model A(AC) C 1.0 0.9 0.8 0.7 C(AC) 0.6 0.5 0.4 0.3 0.2 0.1 0.0 A 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 0.9 Asymmetric Component 0.8 B(AB) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 A(AB) B ΔG + + E ABC A A( AB) A A( AC) B B( BC) B B( AB) C C( AC) C G C( BC) E AB G G E AC E BC Asymmetric Etrapolation B(BC) C(BC) 21 Modeling

The CALPHAD Method CALPHAD 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, i ) for all phases in the system. 3) Optimization of model parameters for best representation of the eperimental data interconsistency 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 of input data Thermodynamic modeling of the phases The CALPHAD approach [G. Cacciamani, Genova University] 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.) 23 Modeling

The CALPHAD Approach (1) Evaluation and selection of input data Optimisation of model parameters (by error minimisation procedures) Thermodynamic modeling of the phases Calculation (phase diagrams, property diagrams, etc.) and Comparison (to the input data) Stoichiometric compounds Ordered solutions Disordered solid solutions Liquids etc. Applications (databases, predictions, simulations, etc.) 24 Modeling

The CALPHAD Approach (2) Evaluation and selection of input data Optimisation of model parameters (by error minimisation procedures) Thermodynamic modeling of the phases Calculation (phase diagrams, property diagrams, etc.) and Comparison (to the input data) Eperiments (DTA, DSC, calorimetry, EMF, vapor pressure, LOM, SEM, X-ray diffraction, etc.) Estimates (periodic properties, chemical criteria, etc.) Theory (ab-initio, semi-empirical, etc.) Applications (databases, predictions, simulations, etc.) 25 Modeling

The CALPHAD Approach (3) Evaluation and selection of input 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, 1,..., i,) Data selection and input Weight assignment Parameter evaluation by non-linear least squares regression 26 Modeling

The CALPHAD Approach (4) Evaluation and selection of input 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 27 Modeling

The CALPHAD Approach (5) Evaluation and selection of input 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 Etrapolation to higher order Materials simulation etc. 28 Modeling

The CALPHAD Approach (6) optimized A-B optimized A-C optimized B-C etrapolated A-B-C a few key data optimized A-B-C 29 Modeling

The CALPHAD Approach (7) optimized A-B-C optimized A-B-D optimized A-C-D optimized B-C-D etrapolated A-B-C-D a few key data optimized A-B-C-D 30 Modeling

Eample: Hypothetical Binaries, Ideal Solution SGTE parameter representation for pure elements (stable and metastable 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

Hypothetical Binaries, Ideal Solution (1) λ β ε α 32 Modeling

Hypothetical Binaries, Ideal Solution (2) λ β ε α 33 Modeling

Hypothetical Binaries, Ideal Solution (3) λ β ε α 34 Modeling

Eample: Modeling using Regular Solutions Binary Phase diagram features modeled with use of the regular solution model Cigar shape of the solid/liquid phase boundaries Maimum congruent melting of the solid phase Minimum congruent melting of the solid phase Peritectic phase diagram Eutectic phase diagram T S m A m A 800 S m B K, 20 T m B J / 1000 K mol K Regular solution parameters ε l and ε s are varied 35 Modeling

1 - Liquidus and Solidus Curves 1000 ε l ε s 0 950 T(K) 900 ε l s ε 5kJ Solid Liquid 850 ε l s ε 15kJ 800 0 0.2 0.4 0.6 0.8 1 Mole Fraction of B 36 Modeling

2 - Maimum and Minimum 1100 ε l s ε 0 10kJ / mol Liquid 1000 T(K) 900 ε l s ε 0 0 ε l s ε 12kJ 0 800 / mol 700 Solid 10kJ / mol 0 0 0.2 0.4 0.6 0.8 1 Mole Fraction of B ε l s ε 37 Modeling

3 - Peritectic Phase Diagram 1100 1000 ε l ε s 15 kj Liquid 900 T(K) 800 700 Solid ε l s ε 12kJ / mol 600 500 Solid+Solid 400 0 0.2 0.4 0.6 0.8 1 Mole fraction of B 38 Modeling

4 - Eutectic Phase Diagram 1100 1000 900 ε l s ε 0 15kJ / mol Liquid T(K) 800 700 600 Solid ε l s ε 20kJ / mol 25kJ / mol ε l s ε 0 10kJ / mol 500 400 Solid+Solid 0 0.2 0.4 0.6 0.8 1 Mole fraction of B 39 Modeling

5 - Monotectic Phase Diagram 1200 1100 ε l s ε 17.5kJ 13.5kJ / mol / mol Liquid 1000 Liq+liq T(K) 900 800 Liquid+Solid Solid 700 600 Solid+Solid 500 0 0.2 0.4 0.6 0.8 1 Mole fraction of B 40 Modeling

6 - Syntectic Phase Diagram 1800 1500 ε l s ε 25kJ / mol 10kJ / mol Liquid 1200 Liquid+liquid T(K) 900 Liquid+Solid Solid 600 Solid+Solid 300 0 0.2 0.4 0.6 0.8 1 Mole fraction of B 41 Modeling

7 Monotectic + Peritectic Phase Diagram 1500 ε l s ε 20kJ 20kJ / mol / mol Liquid 1200 Liquid+liquid T(K) 900 Liquid+Solid Solid 600 Solid Solid+Solid 300 0 0.2 0.4 0.6 0.8 1 Mole fraction of B 42 Modeling

Models: Eample: Modeling of binary In-Ni CALPHAD assessment by Waldner and Ipser L, (Ni): Redlich-Kister solution models δ: (Ni,a) 1 (In,Ni) 1 sublattice model ξ, ξ : (Ni,a) 1 (Ni) 1 (In,Ni) 1 sublattice models Ni 3 In, Ni 2 In, NiIn, Ni 2 In 3, Ni 3 In 7 : Stoichiometric [Massalski s Phase Diagram Compilation] 43 Modeling

Modeling of binary In-Ni (1) Data Sources: Phase diagram: mainly 2 papers ( + older literature) apor 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

Modeling of binary In-Ni (2) Calorimetric data 45 Modeling

Modeling of binary In-Ni (3) apor pressure data 46 Modeling

Modeling of binary In-Ni (4) Pressures over Ni 2 In from EMF and Knudsen sources 47 Modeling

Modeling of binary In-Ni (5) Enthalpies from EMF and Calorimetric sources 48 Modeling

Modeling of binary In-Ni (6) Fit with phase diagram data 49 Modeling