# Phase Diagrams & Thermodynamics

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1 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

2 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

3 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

4 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

5 Eample: Simple Empirical Model Consider the representation of G as a power series in T c p But: G G H a + bt + dt c p SER 2 G T ( ) 2dT 2 T 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 Representation generally used in SGTE format. Only valid for high temperatures! 5 Modeling

6 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

7 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 ] )] ]

8 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

9 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

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

11 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

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

13 Regular Solution Eample (2) ε 12.5 kjmol Modeling

14 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

15 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:... ) ( ) ( 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

16 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

17 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

18 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

19 Kohler Model A B(AB) C(AC) 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) A(AB) C B Symmetric Etrapolation B(BC) C(BC) 19 Modeling

20 Muggianu Model A B(AB) C(AC) 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) A(AB) C B Symmetric Etrapolation B(BC) C(BC) 20 Modeling

21 Toop Model A(AC) C C(AC) A Asymmetric Component 0.8 B(AB) 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

22 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

23 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

24 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

25 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

26 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

27 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

28 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

29 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

30 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

31 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

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

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

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

35 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

36 1 - Liquidus and Solidus Curves 1000 ε l ε s T(K) 900 ε l s ε 5kJ Solid Liquid 850 ε l s ε 15kJ Mole Fraction of B 36 Modeling

37 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 / mol 700 Solid 10kJ / mol Mole Fraction of B ε l s ε 37 Modeling

38 3 - Peritectic Phase Diagram ε l ε s 15 kj Liquid 900 T(K) Solid ε l s ε 12kJ / mol Solid+Solid Mole fraction of B 38 Modeling

39 4 - Eutectic Phase Diagram ε l s ε 0 15kJ / mol Liquid T(K) Solid ε l s ε 20kJ / mol 25kJ / mol ε l s ε 0 10kJ / mol Solid+Solid Mole fraction of B 39 Modeling

40 5 - Monotectic Phase Diagram ε l s ε 17.5kJ 13.5kJ / mol / mol Liquid 1000 Liq+liq T(K) Liquid+Solid Solid Solid+Solid Mole fraction of B 40 Modeling

41 6 - Syntectic Phase Diagram ε l s ε 25kJ / mol 10kJ / mol Liquid 1200 Liquid+liquid T(K) 900 Liquid+Solid Solid 600 Solid+Solid Mole fraction of B 41 Modeling

42 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 Mole fraction of B 42 Modeling

43 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

44 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

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

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

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

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

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

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