Preface ...................................................... xiii
Acknowledgments .............................................. xvii
Notation ...................................................... xix
Part I Basic thermodynamics and kinetics of phase
transformations ................................................. 1
1 Introduction ................................................. 3
1.1 What is a phase transition? ............................. 3
1.2 Atoms and materials ..................................... 4
1.3 Pure elements ........................................... 6
1.4 Alloys: unmixing and ordering ........................... 9
1.5 Transitions and transformations ........................ 11
1.6 Brief review of thermodynamics and kinetics ............ 14
Problems .................................................... 18
2 Essentials of T-c phase diagrams ............................ 20
2.1 Overview of the approach ............................... 20
2.2 Intuition and expectations about alloy thermodynamics .. 22
2.3 Free energy curves, solute conservation, and the
lever rule ............................................. 26
2.4 Common tangent construction ............................ 28
2.5 Continuous solid solubility phase diagram .............. 30
2.6 Solid solutions ........................................ 31
2.7 Unmixing phase diagrams ................................ 37
2.8 Eutectic and peritectic phase diagrams ................. 39
2.9 Ternary phase diagrams ................................. 42
2.10 Long-range order in the point approximation ............ 44
2.11 Alloy phase diagrams ................................... 48
Problems .................................................... 49
3 Diffusion ................................................... 52
3.1 The diffusion equation ................................. 52
3.2 Gaussian and error function solutions to the
ID diffusion equation .................................. 56
3.3 Fourier series solutions to the diffusion equation ..... 61
3.4 Bessel functions and other special function solutions
to the diffusion equation .............................. 66
3.5 Kinetic master equation and equilibrium ................ 69
3.6 Linear kinetic response ................................ 71
Problems .................................................... 72
4 Nucleation .................................................. 74
4.1 Terminology and issues ................................. 75
4.2 Critical nucleus ....................................... 76
4.3 Heterogeneous nucleation ............................... 79
4.4 Free energy curves and elastic energy .................. 81
4.5 The nucleation rate .................................... 85
4.6 Time-dependent nucleation .............................. 92
Problems .................................................... 94
5 Effects of diffusion and nucleation on phase
transformations ............................................. 96
5.1 Nonequilibrium processing of materials ................. 96
5.2 Alloy solidification with solute partitioning .......... 99
5.3 Alloy solidification: suppressed diffusion in the
solid phase ........................................... 100
5.4 Alloy solidification: suppressed diffusion in the
solid and liquid ...................................... 105
5.5 Practical issues for alloy solidification and
evaporation ........................................... 106
5.6 Heat flow and kinetics ................................ 109
5.7 Nucleation kinetics ................................... 111
5.8 Glass formation ....................................... 112
5.9 Solid-state amorphization and suppressed diffusion
in a solid phase ...................................... 114
5.10 Reactions at surfaces ................................. 115
5.11 The glass transition .................................. 120
Problems ................................................... 121
Part II The atomic origins of thermodynamics and kinetics ..... 125
6 Energy ..................................................... 127
6.1 Molecular orbital theory of diatomic molecules ........ 127
6.2 Electronic energy bands in solids ..................... 134
6.3 Elastic constants and the interatomic potential ....... 145
6.4 Linear elasticity ..................................... 149
6.5 Misfitting particle ................................... 153
6.6 Surface energy ........................................ 158
Problems ................................................... 161
7 Entropy .................................................... 163
7.1 Static and dynamic sources of entropy ................. 163
7.2 Short-range order and the pair approximation .......... 165
7.3 Local atomic structures described by clusters ......... 168
7.4 Thermodynamics with cluster approximations ............ 170
7.5 Concept of vibrational entropy ........................ 172
7.6 Phonon statistics ..................................... 175
7.7 Lattice dynamics and vibrational entropy .............. 176
7.8 Bond proportion model ................................. 179
7.9 Bond-stiffness-versus-bond-length model ............... 187
Problems ................................................... 190
8 Pressure ................................................... 194
8.1 Materials under pressure at low temperatures .......... 194
8.2 Thermal pressure, a step beyond the harmonic model .... 199
8.3 Free energies and phase boundaries under pressure ..... 200
8.4 Chemical bonding and antibonding under pressure ....... 202
8.5 Two-level system under pressure ....................... 205
8.6 Activation volume ..................................... 208
Problems ................................................... 209
9 Atom movements with the vacancy mechanism .................. 211
9.1 Random walk and correlations .......................... 211
9.2 Phenomena in alloy diffusion .......................... 220
9.3 Diffusion in a potential gradient ..................... 227
9.4 Nonthermodynamic equilibrium in driven systems ........ 232
9.5 Vineyard's theory of diffusion ........................ 235
Problems ................................................... 240
Part III Types of phase transformations ....................... 245
10 Melting .................................................... 247
10.1 Free energy and latent heat ........................... 247
10.2 Chemical trends of melting ............................ 248
10.3 Free energy of a solid ................................ 250
10.4 Entropy of a liquid ................................... 257
10.5 Thermodynamic condition for Tm ........................ 259
10.6 Glass transition ...................................... 261
10.7 Two dimensions ........................................ 264
Problems ................................................... 266
11 Transformations involving precipitates and interfaces ...... 268
11.1 Guinier-Preston zones ................................. 268
11.2 Surface structure and thermodynamics .................. 270
11.3 Surface structure and kinetics ........................ 276
11.4 Chemical energy of a precipitate interface ............ 278
11.5 Elastic energy and shape of growing precipitates ...... 280
11.6 Precipitation at grain boundaries and defects ......... 282
11.7 The eutectoid reaction and ferrous metallurgy ......... 285
11.8 The Kolmogorov-Johnson-Mehl-Avrami growth equation .... 291
11.9 Coarsening ............................................ 293
Problems ................................................... 296
12 Spinodal decomposition ..................................... 298
12.1 Concentration fluctuations and the free energy of
solution .............................................. 298
12.2 Adding a square gradient term to the free energy
F(c) .................................................. 300
12.3 Constrained minimization of the free energy ........... 304
12.4 The diffusion equation ................................ 309
12.5 Effects of elastic energy ............................. 311
Problems ................................................... 313
13 Phase field theory ......................................... 315
13.1 Spatial distribution of phases and interfaces ......... 315
13.2 Solidification ........................................ 318
13.3 Ginzburg-Landau equation and order parameter .......... 319
13.4 Interfaces between domains ............................ 322
Problems ................................................... 330
14 Method of concentration waves and chemical ordering ........ 332
14.1 Structure in real space and reciprocal space .......... 332
14.2 Symmetry and the star ................................. 338
14.3 The free energy in &-space with concentration waves ... 341
14.4 Symmetry invariance of free energy and Landau-
Lifshitz rule for second-order phase transitions ...... 344
14.5 Thermodynamics of ordering in the mean field
approximation with long-range interactions ............ 349
Problems ................................................... 353
15 Diffusionless transformations .............................. 355
15.1 Dislocations and mechanisms ........................... 356
15.2 Twinning .............................................. 360
15.3 Martensite ............................................ 362
15.4 The crystallographic theory of martensite ............. 368
15.5 Landau theory of displacive phase transitions ......... 370
15.6 First-order Landau theory ............................. 374
15.7 Crystal instabilities and phonons ..................... 377
Problems ................................................... 381
16 Thermodynamics of nanomaterials ............................ 383
16.1 Grain boundary structure .............................. 384
16.2 Grain boundary energy ................................. 386
16.3 Gibbs-Thomson effect .................................. 387
16.4 Energies of free electrons confined to
nanostructures ........................................ 390
16.5 Configurational entropy of nanomaterials .............. 392
16.6 Vibrational entropy ................................... 394
16.7 Gas adsorption ........................................ 397
16.8 Characteristics of magnetic nanoparticles ............. 399
16.9 Elastic energy of anisotropic microstructures ......... 401
Problems ................................................... 402
17 Magnetic and electronic phase transitions .................. 404
17.1 Overview of magnetic and electronic phase
transitions ........................................... 405
17.2 Exchange interactions ................................. 410
17.3 Correlated electrons .................................. 414
17.4 Thermodynamics of ferromagnetism ...................... 417
17.5 Spin waves ............................................ 420
17.6 Thermodynamics of antiferromagnetism .................. 423
17.7 Ferroelectric transition .............................. 426
17.8 Domains ............................................... 428
Problems ................................................... 430
18 Phase transitions in quantum materials ..................... 432
18.1 Bose-Einstein condensation ............................ 432
18.2 Superfluidity ......................................... 435
18.3 Condensate wavefunction ............................... 437
18.4 Superconductivity ..................................... 440
18.5 Quantum critical behavior ............................. 448
Problems ................................................... 452
Part IV Advanced topics ....................................... 453
19 Low-temperature analysis of phase boundaries ............... 455
19.1 Ground-state analysis for T = 0 ....................... 456
19.2 Richards, Allen, Cahn ground-state maps ............... 457
19.3 Low but finite temperatures ........................... 458
19.4 Analysis of equiatomic bcc alloys ..................... 463
19.5 High-temperature expansion of the partition function .. 465
Problems ................................................... 466
20 Cooperative behavior near a critical temperature ........... 468
20.1 Critical exponents .................................... 468
20.2 Critical slowing down ................................. 469
20.3 The Rushbrooke inequality ............................. 471
20.4 Scaling theory ........................................ 472
20.5 Scaling and decimation ................................ 474
20.6 Partition function for one-dimensional chain .......... 475
20.7 Partition function for two-dimensional lattice ........ 479
Problems ................................................... 482
21 Elastic energy of solid precipitates ....................... 483
21.1 Transformation strains and elastic energy ............. 483
21.2 Real space approach ................................... 485
21.3 k-space approach ...................................... 488
Problems ................................................... 491
22 Statistical kinetics of ordering transformations ........... 492
22.1 Ordering transformations with vacancies ............... 493
22.2 B2 ordering with vacancies in the point
approximation ......................................... 495
22.3 Vacancy ordering ...................................... 499
22.4 Kinetic paths ......................................... 500
Problems ................................................... 506
23 Diffusion, dissipation, and inelastic scattering ........... 508
23.1 Atomic processes and diffusion ........................ 508
23.2 Dissipation and fluctuations .......................... 512
23.3 Inelastic scattering .................................. 515
23.4 Phonons and quantum mechanics ......................... 518
Problems ................................................... 521
24 Vibrational thermodynamics of materials at high
temperatures ............................................... 522
24.1 Lattice dynamics ...................................... 522
24.2 Harmonic thermodynamics ............................... 527
24.3 Quasiharmonic thermodynamics .......................... 528
24.4 Thermal effects beyond quasiharmonic theory ........... 531
24.5 Anharmonicity and phonon-phonon interactions .......... 532
24.6 Electron-phonon interactions and temperature .......... 536
Problems ................................................... 538
Further reading ............................................ 540
References ................................................. 545
Index ......................................................... 554
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