PERGAMON MATERIALS SERIES VOLUME CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide PERGAMON MATERIALS SERIES VOLUME CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide PERGAMON MATERIALS SERIES Series Editor: Robert W Cahn FRS Department of Materials Science and Metallurgy, University of Cambridge, UK Vol CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide by N Saunders and A P Miodownik A selection of further titles: Non-equilibrium Processing of Materials edited by C Suryanarayana Phase Transformations in Titanium- and Zirconium-based Alloys by S Banerjee and P Mukhopadhyay Wettability at High Temperatures by N Eustathopoulos, M G Nicholas and B Drevet Ostwald Ripening by S Marsh Nucleation by A L Greer and K F Kelton Underneath the Bragg Peaks: Structural Analysis of Complex Materials by T Egami and S J L Billinge The Coming of Materials Science by R W Calm P E R G A M O N M A T E R I A L S SERIES I-I Calculation of Phase Diagrams A Comprehensive Guide by N Saunders Thermotech Ltd., Guildford, UK and A P Miodownik Professor Emeritus, School of Mechanical and Materials Engineering, University of Surrey, Guildford, UK PERGAMON UK Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 IGB USA Elsevier Science Inc., 655 Avenue of the Americas, New York, NY 10010, USA JAPAN Elsevier Science Japan, 9-15 Higashi-Azabu I-~home, Minato-ku, Tokyo 106, Japan Copyright 1998 Elsevier Science Ltd All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means; electronic, electostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers Library of Congress Cataloging in Publication Data Saunders, N (Nigel) CALPHAD (calculation of phase diagrams) : a comprehensive guide / by N Saunders and A P Miodownik p c m - (Pergamon materials series : v 1) Includes bibliographical references ISBN 0-08-042129-6 (alk paper) Phase diagrams -Data processing Thermochemistry~Data processing I Miodownik, A P (A Peter) II Title III Series QD503.$265 1998 530.4'74 DC21 98-15693 CIP British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-08-0421296 Transferred to digital printing 2005 Contents Series preface Preface Foreword xiv XV xvi CHAPTER INTRODUCTION CHAPTER History of CALPHAD 2.1 Introduction 2.2 The Early Years 2.3 The Intermediate Years 2.4 The Last Decade 2.5 The Current Status of CALPHAD References CHAPTER BASIC THERMODYNAMICS 3.1 Introduction 3.2 The First Law of Thermodynamics 3.2.1 The Definition of Enthalpy and Heat Capacity 3.2.2 Enthalpy of Formation 3.2.3 Hess's Law 3.2.4 Kirchhoff's Law 3.3 The Second Law of Thermodynamics 3.3.1 The Gibbs-Helmholtz Equation 3.3.2 Calculation of Entropy and Gibbs Energy Change from Heat Capacities 3.3.3 The Physical Nature of Entropy 3.4 The Third Law of Thermodynamics 3.5 Thermodynamics and Chemical Equilibrium 3.5.1 The Law of Mass Action and the Equilibrium Constant 7 14 21 24 26 33 33 33 34 36 36 37 38 39 39 40 41 41 41 3.5.2 The Van't Hoff Isotherm 3.6 Solution Phase Thermodynamics 3.6.1 Gibbs Energy of Binary Solutions 3.6.1.1 Ideal Mixing 3.6.1.2 Non-ideal Mixing 3.6.2 Partial Gibbs Energy and Activity in Binary Solutions 3.7 Thermodynamics of Phase Equilibria and Some Simple Calculated Phase Diagrams 3.7.1 Topological Features of Phase Diagrams Calculated Using Regular Solution Theory References CHAPTER EXPERIMENTAL DETERMINATION OF THERMODYNAMIC QUANTITIES AND PHASE DIAGRAMS 4.1 4.2 Introduction Experimental Determination of Thermodynamic Quantities 4.2.1 Calorimetric Methods 4.2.1.1 Measurement of Enthalpy and Heat Capacity 4.2.1.2 Measurement of Enthalpies of Transformation 4.2.2 Gas Phase Equilibria Techniques 4.2.2.1 Static Methods for Measurement of Vapour Pressures 4.2.2.2 The Dew-point and Non-isothermal Isopiestic Methods 4.2.2.3 The Knudsen Effimion and Langmuir Free-Evaporation Methods 4.2.3 Electromotive Force Measurements 4.3 Experimental Determination of Phase Diagrams 4.3.1 Non-isothermal Techniques 4.3.1.1 Thermal Analysis Techniques 4.3.1.2 Chemical Potential Techniques 4.3.1.3 Magnetic Susceptibility Measurements 4.3.1.4 Resistivity Methods 4.3.1.5 Dilatometric Methods 4.3.2 Isothermal Techniques 4.3.2.1 Metallography 4.3.2.2 X-rays 4.3.2.3 Quantitative Determination of Phase Compositions in Multi-Phase Fields 4.3.2.4 Sampling/l/quilibriation Methods 4.3.2.5 Diffusion Couples References vi 43 44 45 45 46 47 50 55 57 61 61 61 61 62 64 67 68 68 68 69 72 72 73 75 77 78 78 80 80 81 83 83 84 85 CHAPTER THERMODYNAMIC MODELS FOR SOLUTION AND COMPOUND PHASES 5.1 Introduction 5.2 Stoichiometrie Compounds 5.3 Random Substitutional Models 5.3.1 Simple Mixtures 5.3.1.1 Dilute Solutions 5.3.1.2 Ideal Solutions 5.3.1.3 Non-Ideal Solutions: Regular and Non-Regular Solution Models 5.3.1.4 The Extrapolation of the Gibbs Excess Energy to Multi-Component Systems 5.4 Sublattiee Models 5.4.1 Introduction 5.4.2 The Generalised Multiple Sublattiee Model 5.4.2.1 Definition of Site Fractions 5.4.2.2 Ideal Entropy of Mixing 5.4.2.3 Gibbs Energy Reference State 5.4.2.4 Gibbs Excess Energy of Mixing 5.4.3 Applications of the Sublattice Model 5.4.3.1 Line Compounds 5.4.3.2 Interstitial Phases 5.4.3.3 Complex lntermetaUie Compounds with Significant Variation in Stoiehiometry 5.4.3.4 Order-Disorder Transformations 5.5 Ionic Liquid Models 5.5.1 The Cellular Model 5.5.2 Modified Quasichemieal Models 5.5.3 Sublattice Models 5.5.4 Associated Solution Models 5.6 Aqueous Solutions References CHAPTER PHASE STABILITIES 91 91 92 92 93 93 94 95 97 99 99 100 100 100 101 102 103 103 104 105 106 110 1I0 112 114 117 120 124 129 6.1 Introduction 6.2 Thermochemieal Estimations 6.2.1 General Procedure for Allotropie Elements 6.2.2 General Procedure for Non-Allotropie Elements 6.2.2.1 The Van Laar Technique for Estimating Melting Points 6.2.2.2 The Estimation of Metastable Entropies of Melting 6.2.2.3 Determination of Transformation Enthalpies in Binary Systems vii 129 129 129 132 134 135 139 6.2.2.4 Utilisation of Stacking Fault Energies 6.2.3 Summary of the Current Status of Thermochemical Estimates 6.3 Ab Initio Electron Energy Calculations 6.3.1 Comparison Between FP and TC Lattice Stabilities 6.3.2 Reconciliation of the Difference Between FP and TC Lattice Stabilities for Some of the Transition Metals 6.4 The Behaviour of Magnetic Elements 6.4.1 Fe 6.4.2 Co 6.4.3 Ni 6.4.4 Mn 6.5 The Effect of Pressure 6.5.1 Basic Addition of a PA V Term 6.5.2 Making the Volume a Function of T and P 6.5.3 Effect of Competing States 6.6 Determination of Interaction Coefficients for Alloys and Stability of Counter-Phases 6.6.1 The Prediction of Liquid and Solid Solution Parameters 6.6.1.1 Empirical and Semi-Empirical Approaches 6.6.1.2 Ab Initio Electron Energy Calculations 6.6.2 The Prediction of Thermodynamic Properties for Compounds 6.6.2.1 The Concept of Counter-Phases 6.6.2.2 Structure Maps 6.6.2.3 The Miedema Model and Other Semi-Empirical Methods 6.6.2.4 Ab lnitio Electron Energy Calculations 6.7 Summary References CHAPTER ORDERING MODELS 141 141 142 144 148 153 153 158 159 159 160 160 161 162 165 166 166 168 168 168 170 170 171 172 173 181 7.1 Introduction 7.1.1 Definition of Long-Range Order 7.1.2 Definition of Short-Range Order 7.1.3 Magnetic Ordering vs Structural Ordering 7.1.4 Continuous vs Discontinuous Ordering 7.2 General Principles of Ordering Models 7.2.1 Interaction Parameters 7.2.2 Hierarchy of Ordering Models 7.3 Features of Various Ordering Models 7.3.1 The Monte Carlo Method 7.3.2 The BWG Approximation 7.3.2.1 BWG Enthalpies 7.3.2.2 Approximate Derivation of 7~r 7.3.2.3 Magnetic Interactions in the BWG Treatment viii 181 181 182 183 183 184 184 184 188 188 189 189 190 191 7.3.2.4 BWG and Anti-Phase Boundary Energies 7.3.3 The Cluster Variation Method (CVM) 7.3.3.1 Site-Occupation Parameters 7.3.3.2 Effective Cluster Interactions 7.3.3.3 Effective Pair Interaction Parameters 7.3.3.4 Use of the General Perturbation Method 7.3.3.5 General form of the CVM Enthalpy 7.3.3.6 Relation of BWG, Pair and CVM Enthalpies 7.3.4 CVM Entropy 7.3.4.1 Criteria for Judging CVM Approximations 7.3.4.2 Entropy on the Pair Interaction Model 7.3.4.3 Entropy on the Tetrahedron Approximation 7.3.4.4 Implementation of CVM 7.3.5 The Cluster Site Approximation (CSA) 7.3.6 Simulation of CVM in the Framework of a Sub-Lattice Model 7.4 Empirical Routes 7.4.1 Specific Heat (Cp) Approximation 7.4.2 General Polynomial Approximation 7.5 Role of Lattice Vibrations 7.5.1 Interaction of Ordering and Vibrational Entropy 7.5.2 Kinetic Development of Ordered States 7.6 Integration of Ordering into Phase Diagram Calculations 7.6.1 Predictions Restricted to Phases of Related Symmetry 7.6.2 Predictions Using Only First-Principles Plus CVM 7.6.3 Methods Which Maximise the First-Principles Input 7.6.4 The Mixed CVM-CALPHAD Approach 7.6.5 Applications of FP-CVM Calculations to Higher-Order Metallic Alloys 7.6.6 Applications to More Complex Structures 7.7 Comments on the use of ordering treatments in CALPHAD calculations 7.7.1 General Comments 7.7.2 The Prediction of Ordering Temperatures References CHAPTER THE ROLE OF MAGNETIC GIBBS ENERGY 8.1 Introduction 8.1.1 Polynomial Representation of Magnetic Gibbs Energy 8.1.2 Consideration of the Best Reference State 8.1.3 Magnitude of the Short-Range Magnetic Order Component 8.2 Derivation of the Magnetic Entropy 8.2.1 Theoretical Value for the Maximum Magnetic Entropy 8.2.2 Empirical Value for the Maximum Magnetic Entropy 8.2.3 Explicit Variation in Entropy with Magnetic Spin Number and Temperature 192 193 194 196 199 199 200 200 200 201 201 202 203 203 205 206 206 208 208 208 209 210 211 211 211 214 215 217 220 220 221 222 229 229 230 232 232 233 233 234 234 This Page Intentionally Left Blank Author index ,~gren, J 17 Ansara, I xiii K6ster, W 25 Kubaschewski, O 14 Barin, I 14 Brewer, L 12, 140 van Laar, J J Machlin, E S 16 Meijering, J L Miedema, A R 16, 19, 167, 170, 172, 418 Miodownik, A P xiii Cahn, R W xii, xiii Colinet, C xiii Engel, N 12 Nishizawa, T Hillert, M Hume-Rothery, W 10, 11 Pelton, A D 15 Kattner, U xiii Kaufman, L xiii, xiv, 8, 9, 10, 11, 12, 14, 21,130, 168, 447 Saunders, N xiii Small, C xiii Sundman, B xiii, 17 467 This Page Intentionally Left Blank Subject index Ab-initio approach, 21, 142-153, 168, 171, 464 (see also First Principles Calculations) Accident analysis in nuclear reactors, 395-398 Activities, 44, 47-50, 61, 63-72, 275,286, 383 partial Gibbs energy and activity, 47-50 activity coefficient, 50, 61,384 measurement of, 63-72 EMF and activity, 69-70 Allotropy Co, 158-159 enthalpy differences, 142 Fe, 153-157 metastable, 21, 134, 142, 162 Mn, 159-160 Ti, 131,314 Alloy design, 368-380 of magnetic materials, 368-372 of RS metal matrix composites, 372-376 of duplex stainless steels, 376-378 of high strength Co-Ni steels, 378-380 ALLOYDATA, 17 Anti-ferromagnetism, 229, 240 Anti-phase boundary (APB) energies, 25, 192-193 Associated Solution Model, 117-119 Basic Thermodynamics, 33-57 BINGSS, 290 Bohr magneton, definition, 229 Bragg-Williams-Gorsky (BWG) treatment, 99, 184, 187, 189-194, 200, 208, 211,235-237 BWG and anti-phase boundary energies, 192-193 BWG enthalpies, 189-190 BWG in the sublattice model, 109, 205 magnetic interactions in the BWG treatment, 191-192, 235-236 BWG ordering temperature, 187, 190191,194, 221-222 Brillouin-Langevin formalism, 237 Bronsted-Guggenheim equation, 160 Calculation of phase equilibria (see Computational Methods) Calorimetric methods, 61-67, 74 adiabiatic calorimeter, 62, 64, 66 combustion bomb calorimeter, 65 differential Scanning Calorimetry (DSC), 64, 74 differential Thermal Analysis (DTA), 64, 65, 290 direct reaction calorimetry, 65, 66 levitation calorimetry, 62 isoperibol calorimeter, 62 isothermal calorimeter, 62 solute-solvent drop calorimetry, 67 CALPHAD Inc., 15 CALPHAD journal, 1, 16, 299 20 year index, 299 Central Atom Model, 110 Chemical Activities, (see Activities) Chemical Vapour Deposition (CVD), 389391 CVD diagram, 389 ChemOpt, 285 ChemSage, 25, 276, 393, 465 Cladding failure in oxide fuel pins of nuclear reactors, 395 Clapeyron equation, 160 Cluster site approximation (CSA), 187, 469 470 Subject index 197, 203-205 Cluster Variation Method (CVM), 3, 24, 25, 184, 187, 193-206, 21 0-222, 237 applications to complex structures, 217-220 comparison of CVM, BWG and Pair interaction enthalpies, 200 criteria for judging CVM approximations, 201 CVM enthalpy, 200 CVM entropy, 200-203 effective cluster interactions, 184, 196198 effective pair interaction parameters, 199 CVM and FP calculations, 211-215, 221 mixed CVM-CALPHAD approach, 21 4-215, 220-221 natural iteration method, 203 simulation, in sub-lattice framework model, 205-206 site occupation parameters, 194-196 Coherent potential approximation (CPA), 171 Colinet extrapolation method, 98 Complex precipitation sequences, 349-355 in 7XXX Al-alloys, 349-352 in (Ni,Fe)-based superalloys, 352-354 in microa-lloyed steels, 354-355 Computational methods, 3, 50-57, 261294, 463-465 Gaussian least-squares method, 290, 292, 293 Lagrange's method, 275, 276 Marquardt modification, 293 Newton-Raphson, 52, 203, 265, 276, 283 related to ordering, 24-25, 106-109, 181-222, 464 robustness of code, 281-284 simplex code, 276 speed of, 281-284 thermodynamic models for solutions, 440-46, 91-124 Computer programmes (Consolidated List) ALLOYDATA, 17 BINGSS, 290 ChemOpt, 285 ChemSage, 25, 276, 393, 465 DICTRA, 20, 25, 422, 423, 433-440, 450, 464 F*A*C*T, 15, 18, 276 FITBIN, 285 Lukas programme, 17, 262, 285, 286, 290-293, 465 MANLABS suite of progranmaes, 17 MTDATA, 17, 276 PARROT, 17, 262, 285, 286, 289, 292293 PMLFKT, 276, 465 ProCAST, 465 SOLGAS, 276 SOLGASMIX, 15, 18, 19, 276, 308 TERGSS, 290 Thermo-Calc, 17, 25, 243, 276, 292, 293, 309, 422, 433, 434, 464 Connolly method, 198, 214 Continuous vs discontinuous ordering, 183 Counter phases, 168-169 Coupling of thermodynamics and kinetics, 20, 25, 422-458 Crystal structure determination of, 82 and sub-lattice model, 103-109 Curie temperature, 154-159, 230-244, 250-253, 370 Databases assessed data-bases, 312-313 for Al-alloys, 310, 312 for III-V compounds, 19 for the prediction of mineral solubilities, 124 for Iron alloys 309 FeDATA, 309, 312, 333, 336, 360 TCFe, 309, 336 for Ni-alloys, 310, 312, 447 for oxides, 114, 382, 398 SGTE solution database, 309 for Ti-alloys, 310, 312 for TiAI alloys, 366 Subject index substance databases, 17, 24, 308, 310, 389, 392 multi-component, 31 0-312 incompatibility between, 464 vision of extensive database, Debye equation, 133 Debye temperature, 130, 138-139, 142, 152, 159, 209 DICTRA, 20, 25,422, 423,433-440, 450, 464 Effective cluster interactions, 184, 196198 Effective pair interaction parameters, 199 Einstein equation, 133 Elasticity elastic constants, 150-152, 165, 209 modulus, 20, 372-376 Electromotive force (EMF), 61, 69-72, 75, 80, 291 measurement of, 69-72 Electron-atom ratio, xvi, 16 Electron concentration, 170 Electron density, xvi, 167 Electron energy calculations, (see First Principle calculations) Electron microprobe analysis (EPMA), 80, 84 Electronegativity, 170 Electronic specific heat, 130 Ellingham diagrams, 44 Enthalpy, 34-36, 50 and heat capacity, 34 of formation, 36, 38, 61, 66, 67, 167, 168, 170-172, 189, 264 of mixing, 96, 117, 118, 148, 192, 211, 289 of vaporisation, 166 partial, 61, 69, 264 Entropy, 39-41, 50, 182 additional contributions, 150, 152, 153 anomalies, 21,152 configurational, 112, 114 excess entropy of mixing, 96 ideal entropy of mixing, 45-46, 94, 100-101, 105 471 magnetic, 154, 233-234 of fusion, 20-21, 134, 135-138, 149, 168, 264, 454 of transformation, 134, 136, 138, 142 partial, 69, 264 quasichemical, 112 statistical description, 40 structural, 136, 181 vibrational, 139, 157, 219, 464 Equilibrium constant, 41, 42, 26 l quasi-chemical equilibrium constant, I 13 Estimation of, effective cluster interactions, 198 interaction coefficients for solution phases, 166-168 interaction parameters, 165-172, 196200 magnetic parameters, 244-246 metastable entropies of melting, 135139 ordering temperatures, 221,222 stability of counter phases, 168-172 van Laar technique for estimating melting points, 134, 152 Excess Gibbs Energy of mixing 46-47, 53-55, 91, 96, 102-103, 104, 105, 107, l l6, 118, 120, 122 partial excess Gibbs Energy, 49-50, 93, 120 for multicomponent systems method of Colinet, 98 Kohler's equation, 97-99 Muggianu's equation, 97-99 Toop's equation, 98-99 Experimental methods for the determination of phase diagrams, 72-85, 301-303 atom probe, 83 cooling curves, 74 dilatometry, 78-80 electrical resistance, 78 electron microscopy, 83 electron microprobe anaylsis (EPMA), 84-85 EMF methods, 75-77 magnetic susceptibility, 77-78 472 Subject index metallography, 80-81 sampling/equilibriation techniques, 8384 thermal analysis, 73-75 x-rays, 81-82 F*A*C*T, 15, 18, 276 Faraday constant, 69 Fick's law, 435 First law of thermodynamics, 33 First-principle (FP) calculations, 10, 12, 13, 15, 21, 22, 25, 141, 142-153, 159, 162, 168, 171-173, 190, 193, 211-217, 244, 248, 463-464 band structure calculations, 10, 12 lattice stabilities, (see Phase Stabilities) historical aspects, 10, 12, 13, 15, 21, 22, 25 in relation to magnetism, 13, 244, 248 in relation to ordering, 211-217 future application, 463-464 FITBIN, 285 Gaussian least-squares method, 290, 292, 293 General perturbation method (GPM), 199, 2OO Geological systems, 19, 20, 24 Gibbs-Duhem equation, 70, 94 Gibbs energy curves/diagrams, 46, 47, 50-55, 130, 142, 153, 160, 184, 417422 Gibbs energy minimisation, 50-55, 182, 261, 262, 265-276, 281-284, 299, 464 Gibbs energy models (see Solution phase models) Gibbs-Thompson effect, 457 Glass formation from the liquid state, 416-417 glass forming ability (GFA), 20, 423, 451-454 glass forming range (GFR), 416-417, 418 liquid-glass transition, 133, 142, 153, 417, 419, 451 TTT diagrams, 451-452 from the solid state, 417-420 Heat capacity (Cp), 14, 35-38, 39, 61, 62, 64, 93, 132, 133 at either constant pressure or volume, 36 calculation of entropy and Gibbs energy from, 39 of a calorimeter, 62 Debye equation, 133 direct measurement, 62 Einstein equation, 133 and enthalpy, 34 measurements, 41, 62 Helmholtz energy, 39, 162 Henry's law, 93, 383, 384 Hess's law, 36-37 Hildebrand's solubility parameters, 166 History of CALPHAD, 7-26 Hot salt corrosion in gas turbines, 392 Ideal solution model, 9, 11, 45, 94-95 Interaction parameters, estimation of, 165-172, 196-200 Ionic Liquid Models, 110-119 Ionic two-sublattice model, 115-117, 286 Ising model, 183, 230 Isothermal martensite, 255 Johnson-Mehl-Avrami kinetic formalism, 451 Kirchoff's law, 37-38 Kohler's equation, 97-99 Lagrange's method, 275, 276 Lattice stabilities (see Phase Stabilities) definition, 129 LennardJones pair potentials, 217 Lindemann equation, 138 Line Compounds, 103-I 04 Liquid )glass transition, 133, 142, 153, 417, 419, 451 Lukas programme, 17, 262, 285, 286, 290-293, 465 Subject index Magnetic Gibbs Energy effects, 9, 12-13, 153-160, 207, 229-256, 369-373, 380 algorithms for magnetic Gibbs energy, 156-157, 237-240 anti-ferromagnetism, 229, 240 definition, 229 in specific elements Co, 158-159, 233, 237, 244, 245, 246 Cr, 244, 245, 246 Fe, 9, 12-13, 153-157, 229-232, 237, 240, 245, 247 Mn, 159-160, 245, 246 Ni, 159, 232-233,237, 240, 244, 245 Cp models, 238-240 model of Hillert and Jarl, 239 model of Inden, 238-239 model of Chuang et al., 239 effect of solute additions 240-244, 248-252, 369-373, 380 estimation of magnetic parameters, 244-246 ferromagnetism, 229, 238-240 definition, 229 ferri-magnetism, 229, 240 definition, 229 interaction with external magnetic fields, 253-256 Nishizawa Horn, 249, 370-373 metastable phases, 250-252 magnetic entropy, 154, 233-234 magnetic enthalpy, 154, 234-237 magnetic moments, tabulated values, 245 thermochemical moments, 244 variation with temperature, 246 magnetic reference state, 232 magnetic short-range order, 183, 230, 232-233, 237, 239 magnetic vs structural ordering, 183 magnetic susceptibility measurements, 77 magnetism and stacking fault energy (SFE), 159, 252-253 multiple magnetic states, 12, 157, 240, 246-248 two-gamma state concept, 12, 157, 246 473 Schottky model, 246-247 polynomial representations of magnetic Gibbs energy, 9, 230-232 MANLABS suite of programmes, 17 Martensite, 22, 155, 156, 157, 232, 248, 251,255, 363, 374, 375, 380, 424 deformation induced, 380 Mass action, law of, 41 Matte-slag-gas reactions in Cu-Fe-Ni sulphide ores, 381-382 Mechanical alloying, 418 Mechanical instability, 21, 139, 150-153, 172 Mendeleev number, 170 Metallic Glasses, (see Glass formation) Miedema model, 16, 19, 167-168, 170, 418 Minimisation of Gibbs energy, 50-55, 182, 261, 262, 265-276, 281-284, 299, 464 Miscibility Gaps, 51-53, 57, 110, 286, 299, 354 355 in binary systems, 51-53 in higher order systems, 354-355 Mixed CVM-CALPHAD approach, 214215, 220-22 l Monte Carlo calculations, 25, 109, 185, 187, 188-189, 194, 201, 210, 212, 220, 22 l, 222, 313 MTDATA, 17, 276 Muggianu's equation, 9799 Multi-component calculations general Background, 309-312 step-by Step examples 313-332 quantitative verification of calculated equilibria, 332-344 selected examples, 344-402 Multiple electronic states, 130 Natural iteration method, 203 Nearest-neighbour interactions, 95, 170, 182, 184, 189, 190, 193, 198, 199, 209 Neel temperature, 133, 219, 229, 241, 243, 246 Newton-Raphson, 52, 203, 265, 276, 283 474 Subject index Nishizawa Horn, 249, 370-373 Ordering models, 181-222, (see also Bragg-Willtams-Gorsky, Cluster Variation Method and MonteCarlo) continuous vs discontinuous ordering, 183-184 empirical methods, 206-208 general principles, 184-188 hierarchy of ordering models, 184-188 kinetic development of ordered states, 209-210 integration of ordering in phase diagram calculations, 210-221 interaction parameters, 184 long-range order, definition, 181-182 magnetic vs structural ordering, 183 short-range order, definition, 182-183 vibrational energy effects, 208-210 Oxide inclusions, calculation of liquidus and solidus, 386-389 PARROT, 17, 262, 285, 286, 289, 292293 Path-probability method, 210 Pettifor-Hasegawa model, 157 PHACOMP, 16, 344, 347, 359 Phase stabilities, 129-173 comparison of FP and TC values, 10, 11-12, 21 144-153, 172-173 Brewer-Engel approach, 140-141 Counter phases, concept of, 168-169 Counter phases, estimation of stability, 168-172 Miedema model, 170 Machlin model, 170 ab-initio calculations, 171-172 estimation of metastable entropies of melting, 135-139 first principle (FP) calculations, 21, 142-153, 168, 171-172, 173, 463-464 interaction coefficients, estimation for solution phases, 166-168 model of Kaufman and Bernstein, 166-167 Miedema model, 167 ab-initio calculations, 168 lattice stabilities, 9, 11, 21, 129-162, 172, 213, 214, 304 of magnetic elements, 153-160 pressure, the effect of, 160-165 pressure, competing states, 162-165 stacking fault energies, relation to 141 structure maps, 170 thermochemical (TC), 9,11, 21, 129141, 144-153, 172 van Laar technique for estimating melting points, 134, 152 Pitting Resistance Equivalent (PRE), 327, 378 PMLFKT, 276, 465 Polymerisation Models, 110 Predominant nucleation maps, 454 456 Pressure, 34-35, 42 equilibrium constant, 42 effect on phase stabilities, 160-165 competing states, 162-165 use of a PAV term, 160-161 effect of T and P on volume, 161-162 ProCAST, 465 Production of Si in an electric arc furnace, 393-394 Pseudopotentials, 145, 171 Quasichemical model of Guggenheim, 187 modified quasi-chemical model, 112114 Radiation effects on Silicide precipitation in Ni-alloys, 398-401 Rapid Solidification, 20, 372-376, 412, 416-417-421,423, 451 458 Redlich-Kister equation, 96 Reference state, 43, 53, 91, 95, 101-102, 116, 132, 232 standard state of a species, 43 Gibbs energy reference state, 53, 95 in the sublattice model, 101-102, 116 Subject index standard element reference state (SER), 92, 132 Regular Solution model, 11, 46, 95-96 application, 55-57 Rhenium, effect on TCP formation in Nibase superalloys, 345-348 Richard's rule, 135 Robustness of software codes, 281-284 Rose equation of state, 162 Salts, 112, 114, 392 Schottky model, 246-247 Schrodinger equation, 143, 171 Scientific Group Thermodata Europe (SGTE), 18, 245, 301,308 Second law thermodynamics, 38-39 Semiconductors, 22, 212, 217, 286-289 Sensitivity factor analysis, 356-360 in heat treatment of duplex stainless steels, 356-358 for cr phase formation Ni-based superalloys, 359-360 for liquid phase sintering of M2 steels, 36O Sigma formation in Ni-based superalloys, 344-345, 359-360 Site occupation parameters, 194-196 Slags, 24, 110-119, 299, 380-389 SOLGAS, 276 SOLGASMIX, 15, 18, 19, 276, 308 Solidification, 440-450 (see also Rapid Solidification) of Al-alloys, 444-445 of cast irons, 445-446 lever rule conditions, 442 modelling of solidification under 'Scheil' conditions, 382, 443-447 models incorporating back diffusion, 447-450 treatment of Brodie and Flemings, 448 treatment of Clyne and Kurz, 448, 449 treatment of Ohnaka, 448 treatment of Chen et al., 448, 449 using DICTRA, 450 using finite difference methods, 450 475 of Ni-based superalloys, 446-447 Scheil equation, 443 Solidification, rapid, 20, 372-376, 412, 416-417, 421,423, 451-458 high-pressure gas atomisation (HPGA), 372, 454-456 glass forming ability (GFA), 20, 423, 451-454 glass forming range (GFR), 416-417, 418 melt spinning, 372 predominant nucleation maps, 454 456 solute drag effect, 457 partitionless transformation, 457 To criterion, 416-417 Solid state amorphisation, 417-420 'Solute-drag' effects, 457 Solution phase models, 44-46, 91-124 Aqueous solutions, 91, 97, 120-124 Bronsted-Guggenheim equation, 121 Davies equation, 121, 122 Debye-Huckel limiting law, 120 extended Debye-Huckel expression, 121 Pitzer models, 122, 124 extension of Bromley and Zematis, 121 molarity, definition of, 120 molality, definition of, 120 cellular model, 110-112 complex intermetallic phases, 105-106 associated solution model, 117-119 central atom model, 110 'complex' model, 110 dilute solutions, 93-94, 309, 426, 428, 430 ideal solutions, 45, 94-95 interstitial solutions, 97, 100, 104-105 ionic liquid models, 110-119 ionic two-sublattice model, 115-117, 286 line compounds, 103-104 modified quasichemical model, 112-114 polymerisation models, 110 regular solution model, 11, 46, 95-96 use of, 55-57 476 Subject index Redlich-Kister equation, 96 sublattice model, 17, 99-109, 114-117, 205-206, 286, 306 application of regular solution model in, 102 application of sub-regular solution model in, 102 two-sublattice order-disorder (2SLOD) model, 106-109 Stacking fault energies, 141, 159, 165, 221,246, 252-253 Stirling's approximation, 46, 94, 202 Stress Corrosion Cracking (SCC), 395 Structure Maps, 170 Sub-lattice model, 17, 99-109, 114-117, 205-206, 286, 306 incorporation of regular solution behaviour, 102 incorporation of sub-regular solution behaviour, 102 simulation of ordering in a twosublattice model, 106-109 Sub-regular solution model, 11, 96 Sulphide capacities of multi-component Taylor's approximation, 276, 426 Temperature/pressure diagrams, 162, 164 TERGSS, 290 Thermo-Calc, 17, 25, 243, 276, 292, 293, 309, 422, 433, 434, 464 Thermodynamic optimisation, 284-294 Third law thermodynamics, 41 Time-Temperature-Transformation diagrams, 423-432, 435, 451-452 for glass formation, 451-452 continuous cooling diagram, 435 treatment of Kirkaldy et al., 424-426 treatment of Bhadeshia, 426-428 treatment of Kirkaldy and Venugopolan, 428-432 treatment of Enomoto, 432-433 Toop's equation, 98-99 Topological features of binary diagrams, 9, 25, 55-57 Undercooling, 440-458 (see also Rapid Solidification) Van Laar technique for estimating melting points, 134, 152 Van't Hoff isotherm, 43-44 Vapour deposition, 20, 420-421 Vapour pressure methods for measuring of activity, 67-69 dew-point, 68 static and quasi-static, 68 Knudsen effusion, 68 Langmuir free-evaporation, 69 Vibrational Entropy, 139, 157, 219, 464 Wagner activity coefficients, 426 Warren-Cowley coefficients, 183 Widmanstitten ferrite, 424, 428 Wigner-Seitz atomic cells, 167 Zener-Hillert expression, 424 Zener's magnetic model, 241 ALLOY INDEX Al-alloys, 84, 321-327, 349-352, 423, 444 445 IXXX, 445 2XXX, 349 3XXX, 321-327, 445 6XXX, 445 7XXX, 349-352, 444, 445 AI-Si casting alloys, 423, 443, 445 Copper Alloys, 93, 457 Cu- 13%Ag, 457 Fe-alioys, 93, 100, 104-105, 235, 249, 277, 309, 327-332, 333, 335-336, 338, 354-355, 356-357, 360, 369370, 372-380, 386, 424, 426, 428, 430, 436, 445-446, 449, 450 AFI410, 378 cast irons, 445-446 duplex stainless steels, 277, 327-332, 335-336, 356-357, 376 high-speed steels, 338, 360 high strength Co-Ni steels, 378-380 high-strength low-alloy (HLSA) steels, 309, 354 low-alloy steels, 424, 426, 428, 430 medium-alloy steels, 422, 428, 430 Subject index micro-alloyed steels, 354-355 SAF2205, 327-332, 378 Zeronl00, 277, 356-357 lnvar alloys, 12, 243, 246 NiAl-alloys, 362-365 Ni-alloys, 24, 83, 217, 221, 335, 338, 344-348, 352-354, 359, 415, 433, 438, 446-447, 450 CMSX4, 347-348 IN625, 352-354 IN718, 447 1N939, 338 Ni- 19Cr- 1.5AI- 14Co-4.3Mo-3Ti, 438 Nimonie 263, 415 Rene N4, 447 SRR99, 338 U720, 345-346, 348, 359 Waspaloy, 338 Ti-alloys, 162, 311-312, 314-321,333, 338, 415 IMI 834, 312, 333 Ti-6AI-4V, 311-312, 314-321,338 Ti-10V-2A1-3Fe, 333, 338 SP700, 338 TiAI-alloys, 83, 310, 366-368 T i-4 AI-2Nb- 1Mn-0.5W-0.5 Mo0.2Si, 368 Ti-48AI-2Cr-2Nb, 366 Ti-48AI-2Mn-2Nb, 366 Ti-48AI-2Nb-2Cr- 1W, 368 Zircaloy, 395 SYSTEMS INDEX Binary Systems Ag-Bi, 55 Ag-Pb, 55 Ag-Pt, 57 Ag-Si, 452 AI-Co, 212 AI-Cr, 456 AI-Fe, 212, 321 AI-Li, 78 AI-Mn, 321 AI-Si, 321 477 AI-Sn, 75-76 AI-Ti, 314, 366 AI-Zr, 456 Au-Cu, 212 Au-La, 418 Au-Pt, 57 Au-Si, 55, 417, 452 Au-V, 148, 152 CaCOrMgCO3, 218 CaO-MgO, 219 Cd-Pb, 55 Cd-Zn, 55 Co-Cu, 57 Co-Ga, 212 Co-Ti, 300 Co-V, 252 Co-Zn, 250 Co-Zr, 419 Cr-Rh, Cu-Ag, 279 Cu-AI, 135 Cu-Au, 109, 194, 205, 211 Cu-Mg, 280 Cu-Pb, 57 Cu-Rh, 57 Cu-S, 117 Cu-Sn, 73 Cu-TI, 57 Cu-Zn, 135, 193, 207, 212 Cu-Zr, 455 CulnSe2-Zn2Se2, 217 Fe-C, 156, 160, 411-412 Fe-Co, 156, 193, 21 l, 212, 237, 246, 255 Fe-Cr, 156, 10, 211,255, 304, 306 Fe-Nb, 77 Fe-Ni, 46, 156, 241,246, 421 Fe-Si, 252 Fe-V, 301-302 Hf-C, 82 GaSb-InSb, 217 Ga-Te, 286 ln-Pb, 74 InP-InSb, 217 Mg-Fe, 57 Mg-Mn, 57 478 Subject index Mg-Zn, 349 MnO-NiO, 71 Mo-C, 300 Mo-Re, 81 Na2SO4-Na2CrO4, 392 Nb-AI, 208, 213 Ni-AI, 213-214, 215, 304 Ni-Au, 194 Ni-Cr, 139-140, 152, 422, 440 Ni-Cu, 53-55, 266-272 Ni-Ga, 212 Ni-Nb, 420 Ni-Si, 400 Ni-Ti, 213, 412, 417 Ni-Zn, 241 Pd-Si, 455 Pd-P, 455 Pd-V, 188 Ru-Nb, 211 Ti-Al, 314, 366 Ti-Be, 452 Ti-O, 301,316 Ti-Rh, 211 Ti-Si, 415 Ti-V, 314, Ti-W, 57 Tl-In, 160 SiO2-CaO, 110, 300 U-O, 394 W-C, 305 Zr-Be, 455 Ternary Systems AI-Cu-Mg, 281 AI-Fe-Mn, 321 AI-Fe-Si, 324-325 AI-Mn-Si, 324-325 AI-Ni-Co, 370 CaO-FeO-SiO2, 110-112, 387 CaO-MgO-AI203, 301 CaO-MgO-SiO2, 387 Cr-Fe-W, 302 Cu-In-Se, 212 Cu-Ni-Ag, 280 Cu-Zn-Mn, 208 Fe-AI-Ga, 212 Fe-Co-AI, 211,212 Fe-Co-Cr, 370 Fe-Co-Zn, 249 Fe-Cr-N, 281 Fe-Cr-B, 103, 374 Fe-Cr-C, 104-105, 306, 433 Fe-Cr-Mo, 329 Fe-Cr-N, 281 Fe-Cr-Ni, 327, 400, 440, 449 Fe-Cr-Si, 330 Fe-Mn-C, 433 Fe-Mn-S, 117 Fe-Mo-B, 168 Fe-Mo-C, 306 Fe-Ni-Mn, 241 Fe-O-SiO2, 119 Fe-Si-AI, 212 Fe-Si-Co, 212 Ga-As-Sb, 212 Ga-In-P, 212 Hf-Ti-Be, 417 Hf-Zr-Be, 417 In-P-Pb, 212 Ni-AI-Cr, 217 Ni-AI-Fe, 78, 84, 362, 370 Ni-AI-Ti, 212, 217, 303 Ni-Cr-Cu, Ni-Mo-Re, 301 Ru-Nb-Zr, 217 Ti-AI-Mo, 212 Ti-AI-Nb, 212, 217, 301 Ti-AI-O, 316 Ti-AI-Ta, 84 Ti-AI-V, 311, 316 Ti-AI-W, 212 Ti-Ta-Nb, 313 Ti-V-O, 316-318 Ti-Zr-B, 417, 454 U-Pt-O, 394 Zn-In-Pb, 69 Higher-order Systems AI-Cu-Mg-Zn, 349 AI-Fe-Mn-Si, 321 C-H-N-O, 308 Cu-Fe-Ni-S-O, 381 Subject index Fe-Co-Ni-Cr-Mo-W-C, 378-380 Fe-Cr-Ni-Mo-Mn-Si-C-N, 376 Fe-Cr-Mo-B, 374 Fe-Cr-Mo-Ni-B, 372 Fe-Cr-Ni-B, 374 Fe-Cr-Ni-C, 301 H, H2, H20, N, N2, NH, NO, O, 02 OH, 275 H20, Li, Na, Mg, NH4, Mg, Ca, Sr, Ba, F, CI, Br, I, OH, CNS, NO3, C103, CIO4, S, SO3, CH3COO, HCOO and (COO)2, 124 (Li, Na, K)(F, Cl, OH, CO3, SO4), 115 H20, Na, K, Mg, Ca, H, Cl, SO4, OH, HCO3, CO3, CO2, 124 NaCI-NaOH-Na2CrO4-Na2SO4, 392 Ni-AI-Ta-Cr, 109 Ni-AI-Ti-Cr, 109 Ni-Co-Mo-W, 106 479 SiC-ZrO2-A1203-SIO2, 308 SiO2-Al203-CaO-MgO-MnO, 386387 SiO2-AI203-CaO-MgO-MnO-FeONa20-K20-TiO2-Ti203-ZrO2-S, ll4 SiO2-AI203-TiO2-CaO-MgO-MnOFeO-S, 383-386 Ti-AI-V-O-C-N-Fe, 31 l TiCI4-SiH4-H2-Ar, 391 U-Cs-I-Ba-Zr-Mo, 308 UO2-ZrO2-SiO2-CaO-MgO-AI203, 398 UO2-ZrO2-SiO2-CaO-MgO-AI203Sr-BaO-La203, 398 WCI4-SiCI2H2-H2-Ar, 390 WCI4-SiH4-H2-Ar, 389 WF6-SiH4-H2-Ar, 389 Ya-B-Cu-O, 22, 219 This Page Intentionally Left Blank ...PERGAMON MATERIALS SERIES VOLUME CALPHAD (Calculation of Phase Diagrams): A Comprehensive Guide PERGAMON MATERIALS SERIES Series Editor: Robert W Cahn FRS Department of Materials Science and Metallurgy,... Library of Congress Cataloging in Publication Data Saunders, N (Nigel) CALPHAD (calculation of phase diagrams) : a comprehensive guide / by N Saunders and A P Miodownik p c m - (Pergamon materials... of the CALPHAD journal, which both acted as a cumulative record of progress in making calculations and as an invaluable depository of validated parameters The aim of the journal was not primarily