Mats Hillert Department of Materials Science and Engineering KTH, Stockholm Phase Equilibria, Phase Diagrams and Phase Transformations Their Thermodynamic Basis CAMBRIDGE UNIVERSITY PRESS
Contents Preface xiii Basic concepts of thermodynamics 1 1.1 External State variables 1 1.2 Internal State variables 4 1.3 The first law of thermodynamics 6 1.4 Freezing-in conditions 10 1.5 Reversible and irreversible processes 12 1.6 The second law of thermodynamics 15 1.7 Condition of internal equilibrium 21 1.8 Driving force 23 1.9 The combined first and second law 25 1.10 General conditions of equilibrium 28 1.11 Characteristic State functions 29 1.12 Entropy 33 V4 Manipulation of thermodynamic quantities 37 2.1 Evaluation of one characteristic State function from another 37 2.2 Internal variables at equilibrium 38 2.3 Equations of State 42 2.4 Experimental conditions 43 2.5 Notation for partial derivatives 47
VIII Contents 2.6 Use ofvarious derivatives 48 2.7 Comparison between C v and C P 52 2.8 Change of independent variables 54 2.9 Maxwell relations 56 Systems with variable composition 59 3.1 Chemical potential 59 3.2 Molar quantities 61 3.3 More about characteristic State functions 64 3.4 Various forms of the combined law 66 3.5 Partial quantities 70 3.6 Relations for partial quantities 73 3.7 Alternative variables for composition 75 3.8 The lever rule 79 3.9 The tie-line rule 81 3.10 Different sets of components 84 3.11 Constitution and constituents 86 3.12 Chemical potentials in a phase with sublattices 88 Evaluation and use of driving force 93 4.1 Irreversible thermodynamics 93 4.2 Calculation of equilibrium 93 4.3 Evaluation of the driving force 95 4.4 Evaluation of integrated driving force as function of T or P 97 4.5 Driving force for molecular reactions 99 4.6 Onsager's reciprocity relations 101 4.7 Driving force and entropy production in diffusion 103 4.8 Effective driving force 106 Stability 109 5.1 Introduction 109 5.2 Some necessary conditions of stability 111 5.3 Sufficient conditions of stability 113 5.4 Summary of stability conditions 116 5.5 Limit of stability 118 5.6 Limit of stability ofalloys 121 5.7 Chemical capacitance 125 5.8 Limit of stability in phases with sublattices 126 5.9 Le Chatelier's principle 129 Applications of molar Gibbs energy diagrams 134 6.1 Molar Gibbs energy diagrams for binary Systems 134 6.2 Instability of binary Solutions 139
Contents 6.3 Illustration of the Gibbs-Duhem relation 140 6.4 Two-phase equilibria in binary Systems 143 6.5 Allotropic phase boundaries 146 6.6 Effect of a pressure difference on a two-phase equilibrium 148 6.7 Driving force for the formation of a new phase 151 6.8 Partitionless transformation under local equilibrium 155 6.9 Activation energy for a fluctuation 158 6.10 Ternary Systems 160 6.11 Solubility product 163 Phase equilibria and potential phase diagrams 167 o Q 7.1 Gibbs'phase rule 167 7.2 Fundamental property diagram 170 7.3 Topology of potential phase diagrams 176 7.4 Potential phase diagrams in binary and multinary Systems 182 7.5 Sections of potential phase diagrams 185 7.6 Binary Systems 186 7.7 Ternary Systems 190 7.8 Direction of phase fields in potential phase diagrams 195 7.9 Extremum in temperature and pressure 199 Molar phase diagrams 203 8.1 Molar axes 203 8.2 Sets ofconjugatepairscontaining molar variables 208 8.3 Phase boundaries 212 8.4 Sections of molar phase diagrams 214 8.5 Schreinemakers'rule 216 8.6 Topology ofsectioned molar diagrams 222 Projected and mixed phase diagrams 226 9.1 Schreinemakers'projection of potential phase diagrams 226 9.2 The phase field rule and projected diagrams 229 9.3 Relation between molar diagrams and Schreinemakers' projected diagrams 234 9.4 Coincidence of projected surfaces 236 9.5 Projection of higher-order invariant equilibria 241 9.6 The phase field rule and mixed diagrams 244 9.7 Selection of axes in mixed diagrams 248 9.8 Konovalov's rule 251 9.9 General rule for singular equilibria 255 Direction of phase boundaries 258 10.1 Useof distribution coefficient 258
X Contents 10.2 Calculation of allotropic phase boundaries 260 10.3 Variation of a chemical potential in a two-phase field 263 10.4 Direction of phase boundaries 265 10.5 Congruent melting points 269 10.6 Vertical phase boundaries 276 10.7 Slope of phase boundaries in isothermal sections 277 10.8 The effect of a pressure difference between two phases 281 Sharp and gradual phase transformations 283 11.1 Experimental conditions 283 11.2 Characterization of phase transformations 285 11.3 Microstructural character 290 11.4 Phase transformations in alloys 292 11.5 Classification of sharp phase transformations 294 11.6 Applications of Schreinemakers' projection 298 11.7 Scheil's reaction diagram 303 11.8 Gradual phase transformations at fixed composition 304 11.9 Phase transformations controlled by a chemical potential 308 Transformations at constant composition 312 12.1 The phase field rule at constant composition 312 12.2 Reaction coefficients in sharp transformations for p = c + 1 314 12.3 Graphical evaluation of reaction coefficients 317 12.4 Reaction coefficients in gradual transformations for p = c 319 12.5 Driving force for sharp phase transformations 321 12.6 Driving force under constant chemical potential 325 12.7 Reaction coefficients at a constant chemical potential 328 12.8 Compositional degeneracies for p = c 330 12.9 Effect to two or more compositional degeneracies for p = c 1 337 12.10 Compositional degeneracies and the phase field rule 340 12.11 Overlapping transformations 343 Partitionless transformations 348 13.1 Deviation from local equilibrium 348 13.2 Adiabatic phase transformation 349 13.3 Quasi-adiabatic phase transformation 351 13.4 Partitionless transformations in binary System 354 13.5 Partial chemical equilibrium 358 13.6 Transformations in steel under quasi-paraequilibrium 362 13.7 Transformations in steel under partitioning of alloying elements 366 Limit of stability, critical phenomena and interfaces 368 14.1 Transformations and transitions 368
Contents XI ^ Q 14.2 Order-disorder transitions 372 14.3 Miscibility gaps 376 14.4 Spinodal decomposition 381 14.5 Tri-critical points 385 14.6 Interfaces 390 14.7 Nucleation 395 Methods of modelling 401 15.1 General principles 401 15.2 Choice of characteristic State function 402 15.3 Reference states 404 15.4 Representation ofgibbs energy offormation 407 15.5 Use of power series in T 409 15.6 Representation of pressure dependence 411 15.7 Application of physical modeis 414 15.8 Ideal gas 415 15.9 Real gases 417 15.10 Mixtures of gas species 421 15.11 Black-body radiation 423 15.12 Electron gas 425 Modelling of disorder 427 16.1 Introduction 427 16.2 Thermal vacancies in a crystal 428 16.3 Topological disorder 432 16.4 Heat capacity due to thermal vibrations 434 16.5 Relation between vibrational energy and elastic properties 439 16.6 Magnetic contribution to thermodynamic properties 440 16.7 A simple physical model for the magnetic contribution 443 16.8 The physical background ofmagnetism 446 16.9 Random mixture ofatoms 449 16.10 Restricted random mixture 452 16.11 Crystals with stoichiometric vacancies 453 16.12 Interstitial Solutions 455 Mathematical modelling of Solution phases 458 17.1 Ideal Solution 458 17.2 Mixing quantities 460 17.3 Excess quantities 461 17.4 Empirical approach to substitutional Solutions 462 17.5 Real Solutions 468 17.6 Applications of the Gibbs-Duhem relation 472 17.7 Dilute Solution approximations 476
XII Contents 17.8 Predictions for Solutions in higher-order Systems 480 17.9 Numerical methods of predictions for higher-order Systems 482 Solution phases with sublattices 487 18.1 Sublattice Solution phases 487 18.2 Interstitial Solutions 489 18.3 Reciprocal Solution phases 491 18.4 Combination of interstitial and substitutional Solution 495 18.5 Phases with variable order 496 18.6 Ionic solid Solutions 499 Physical Solution modeis 504 19.1 Concept of nearest-neighbour bond energies 504 19.2 Random mixing model for a substitutional Solution 506 19.3 Deviation from random distribution 508 19.4 Short-range order 510 19.5 Long-range order 512 19.6 Long-and short-range order 515 19.7 The Compound energy model with short-range order 517 19.8 Interstitial ordering 520 19.9 Composition dependence of physical effects 522 References 527 Index 529