Theory of solidification

Stephen H. Davis0521650801, 9780521650809, 9780511019241

The processes of freezing and melting were present at the beginnings of the Earth and continue to dominate the natural and industrial worlds. The solidification of a liquid or the melting of a solid involves a complex interplay of many physical effects. This book systematically presents the field of continuum solidification theory based on instability phenomena. An understanding of the physics is developed by using examples of increasing complexity with the object of creating a deep physical insight applicable to more complex problems. Applied mathematicians, engineers, physicists, and materials scientists will all find this volume of interest.

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Table of contents :
Abstract ……Page 1
Series title ……Page 3
Title ……Page 5
Date-line ……Page 6
Dedication ……Page 7
Contents ……Page 9
Preface ……Page 13
1 Introduction ……Page 15
2.1.1 Mathematical model ……Page 21
2.1.2 One-dimensional freezing from a cold boundary ……Page 23
2.1.3 One-dimensional freezing from a cold boundary: Small undercooling ……Page 27
2.1.4 One-dimensional freezing into an undercooled melt ……Page 29
2.1.5 One-dimensional freezing into an undercooled melt: Effect of kinetic undercooling ……Page 32
2.2.1 Boundary conditions ……Page 35
2.2.2 Growth of a nucleus in an undercooled melt ……Page 40
2.2.3 Linearized instability of growing nucleus ……Page 46
2.2.4 Linearized instability of a plane front growing into an undercooled melt ……Page 49
2.2.5 Remarks ……Page 53
3.1 Mathematical model ……Page 56
3.2 Directional solidification ……Page 59
3.3 Basic state and approximate models ……Page 60
3.4 Linearized instability of a moving front in directional solidification ……Page 62
3.5 Mechanism of morphological instability ……Page 70
3.6 More general models ……Page 71
3.7 Remarks ……Page 73
4.1.1 Two-dimensional theory ……Page 76
4.1.2 Two-dimensional theory for wave number selection ……Page 80
4.1.3 Three-dimensional theory ……Page 86
4.2 Long-scale theories ……Page 90
4.2.1 Small segregation coefficient ……Page 91
4.2.2 Small segregation coefficient and large surface energy ……Page 92
4.2.3 Near absolute stability ……Page 94
4.3 Remarks ……Page 96
5.1 Surface energy and kinetics ……Page 100
5.2 Directional solidification with “small” anisotropy ……Page 105
5.3 Directional solidification with “small” anisotropy: Stepwise growth ……Page 111
5.4 Unconstrained growth with “small” anisotropy ……Page 119
5.4.1 Two-dimensional crystal and one-dimensional front ……Page 124
5.4.2 Three-dimensional crystal and two-dimensional front ……Page 125
5.5 Unconstrained growth with “large” anisotropy – One-dimensional interfaces ……Page 135
5.6 Unconstrained growth with “large” anisotropy – Two-dimensional interfaces ……Page 149
5.7 Faceting with constant driving force ……Page 153
5.8 Coarsening ……Page 166
5.9 Remarks ……Page 170
6 Disequilibrium ……Page 176
6.1 Model of rapid solidification ……Page 178
6.2 Basic state and linear stability theory ……Page 181
6.3 Thermal effects ……Page 185
6.4 Linear-stability theory with thermal effects ……Page 186
6.4.2 Oscillatory mode ……Page 187
6.4.3 The two modes ……Page 191
6.5 Cellular modes in the FTA: Two-dimensional bifurcation theory ……Page 195
6.6 Oscillatory modes in the FTA: Two-dimensional bifurcation theory ……Page 197
6.7 Strongly nonlinear pulsations ……Page 203
6.7.1 Small $beta$ ……Page 204
6.7.2 Large $beta$ ……Page 212
6.7.3 Numerical simulation ……Page 217
6.8.1 Pulsatile-cellular interactions ……Page 218
6.8.2 Oscillatory-cellular interactions ……Page 219
6.8.3 Oscillatory-pulsatile interactions ……Page 220
6.9 Phenomenological models ……Page 222
6.10 Remarks ……Page 225
7 Dendrites ……Page 229
7.1 Isolated needle crystals ……Page 231
7.2 Approximate selection arguments ……Page 235
7.3 Selection theories ……Page 243
7.4 Arrays of needles ……Page 251
7.5 Remarks ……Page 265
8 Eutectics ……Page 269
8.1 Formulation ……Page 270
8.2 Approximate theories for steady growth and selection ……Page 275
8.3 Instabilities ……Page 281
8.4 Remarks ……Page 284
9 Microscale Fluid Flow ……Page 288
9.1 Formulation ……Page 290
9.2.1 Free convection ……Page 293
9.2.2 Benard convection ……Page 294
9.3 Directional solidification and volume-change convection ……Page 297
9.4 Directional solidification and buoyancy-driven convection ……Page 301
9.5 Directional solidification and forced flows ……Page 306
9.6 Directional solidification with imposed cellular convection ……Page 318
9.7 Flows over Ivantsov needles ……Page 325
9.8 Remarks ……Page 333
10 Mesoscale Fluid Flow ……Page 338
10.1 Formulation ……Page 339
10.2 Planar solidification between horizontal planes ……Page 340
10.3 Mushy-zone models ……Page 345
10.4 Mushy zones with volume-change convection ……Page 350
10.5 Mushy zones with buoyancy-driven convective instability ……Page 355
10.6 An oscillatory mode of convective instability ……Page 363
10.7 Weakly nonlinear convection ……Page 370
10.8 Chimneys ……Page 371
10.9 Remarks ……Page 377
11 Phase-Field Models ……Page 380
11.1 Pure materials – A model system ……Page 381
11.2 Pure materials – A deduced system ……Page 386
11.3 Pure materials – Computations ……Page 388
11.4 Remarks ……Page 390
Index ……Page 393