Theory of solidification

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface......Page 15
1 Introduction......Page 17
References......Page 22
2.1.1 Mathematical Model......Page 23
2.1.2 One-Dimensional Freezing from a Cold Boundary......Page 25
Notes......Page 28
2.1.3 One-Dimensional Freezing from a Cold Boundary: Small Undercooling......Page 29
2.1.4 One-Dimensional Freezing into an Undercooled Melt......Page 31
2.1.5 One-Dimensional Freezing into an Undercooled Melt: Effect of Kinetic Undercooling......Page 34
2.2.1 Boundary Conditions......Page 37
2.2.2 Growth of a Nucleus in an Undercooled Melt......Page 42
2.2.3 Linearized Instability of Growing Nucleus......Page 48
2.2.4 Linearized Instability of a Plane Front Growing into an Undercooled Melt......Page 51
2.2.5 Remarks......Page 55
References......Page 56
3.1 Mathematical Model......Page 58
3.2 Directional Solidification......Page 61
3.3 Basic State and Approximate Models......Page 62
3.4 Linearized Instability of a Moving Front in Directional Solidification......Page 64
3.5 Mechanism of Morphological Instability......Page 72
3.6 More General Models......Page 73
3.7 Remarks......Page 75
References......Page 76
4.1.1 Two-Dimensional Theory......Page 78
4.1.2 Two-Dimensional Theory for Wave Number Selection......Page 82
4.1.3 Three-Dimensional Theory......Page 88
4.2 Long-Scale Theories......Page 92
4.2.1 Small Segregation Coefficient......Page 93
4.2.2 Small Segregation Coefficient and Large Surface Energy......Page 94
4.2.3 Near Absolute Stability......Page 96
4.3 Remarks......Page 98
References......Page 99
5.1 Surface Energy and Kinetics......Page 102
5.2 Directional Solidification with "Small" Anisotropy......Page 107
5.3 Directional Solidification with "Small" Anisotropy: Stepwise Growth......Page 113
5.4 Unconstrained Growth with "Small" Anisotropy......Page 121
5.4.1 Two-Dimensional Crystal and One-Dimensional Front......Page 126
5.4.2 Three-Dimensional Crystal and Two-Dimensional Front......Page 127
5.5 Unconstrained Growth with "Large" Anisotropy – One-Dimensional Interfaces......Page 137
5.6 Unconstrained Growth with "Large" Anisotropy – Two-Dimensional Interfaces......Page 151
5.7 Faceting with Constant Driving Force......Page 155
5.8 Coarsening......Page 168
5.9 Remarks......Page 172
References......Page 173
6 Disequilibrium......Page 178
6.1 Model of Rapid Solidification......Page 180
6.2 Basic State and Linear Stability Theory......Page 183
6.3 Thermal Effects......Page 187
6.4 Linear-Stability Theory with Thermal Effects......Page 188
6.4.2 Oscillatory Mode......Page 189
6.4.3 The Two Modes......Page 193
6.5 Cellular Modes in the FTA: Two-Dimensional Bifurcation Theory......Page 197
6.6 Oscillatory Modes in the FTA: Two-Dimensional Bifurcation Theory......Page 199
6.7 Strongly Nonlinear Pulsations......Page 205
6.7.1 Small beta......Page 206
6.7.2 Large beta......Page 214
6.7.3 Numerical Simulation......Page 219
6.8.1 Pulsatile–Cellular Interactions......Page 220
6.8.2 Oscillatory–Cellular Interactions......Page 221
6.8.3 Oscillatory–pulsatile interactions......Page 222
6.9 Phenomenological Models......Page 224
6.10 Remarks......Page 227
References......Page 228
7 Dendrites......Page 231
7.1 Isolated Needle Crystals......Page 233
7.2 Approximate Selection Arguments......Page 237
7.3 Selection Theories......Page 245
7.4 Arrays of Needles......Page 253
7.5 Remarks......Page 267
References......Page 268
8 Eutectics......Page 271
8.1 Formulation......Page 272
8.2 Approximate Theories for Steady Growth and Selection......Page 277
8.3 Instabilities......Page 283
8.4 Remarks......Page 286
References......Page 289
9 Microscale fluid flow......Page 290
9.1 Formulation......Page 292
9.2.1 Free Convection......Page 295
9.2.2 Bénard Convection......Page 296
9.3 Directional Solidification and Volume-Change Convection......Page 299
9.4 Directional Solidification and Buoyancy-Driven Convection......Page 303
9.5 Directional Solidification and Forced Flows......Page 308
9.6 Directional Solidification with Imposed Cellular Convection......Page 320
9.7 Flows over Ivantsov Needles......Page 327
References......Page 335
10 Mesoscale fluid flow......Page 340
10.1 Formulation......Page 341
10.2 Planar Solidification between Horizontal Planes......Page 342
10.3 Mushy-Zone Models......Page 347
10.4 Mushy Zones with Volume-Change Convection......Page 352
10.5 Mushy Zones with Buoyancy-Driven Convective Instability......Page 357
10.6 An Oscillatory Mode of Convective Instability......Page 365
10.7 Weakly Nonlinear Convection......Page 372
10.8 Chimneys......Page 373
References......Page 379
11 Phase-field models......Page 382
11.1 Pure Materials – A Model System......Page 383
11.2 Pure Materials – A Deduced System......Page 388
11.3 Pure Materials – Computations......Page 390
11.4 Remarks......Page 392
References......Page 393
Index......Page 395