Kinetics of First Order Phase Transitions

Kinetics of First-order Phase Transitions......Page 6
Contents......Page 8
Foreword......Page 14
Preface......Page 16
1 Introduction......Page 18
2.1 Introduction......Page 24
2.2 Basic Kinetic Equations......Page 26
2.3 Ratio of the Coefficients of Absorption and Emission of Particles......Page 27
2.3.1 Traditional Approach......Page 28
2.3.2 A New Method of Determination of the Coefficients of Emission......Page 33
2.4 Generalization to Multicomponent Systems......Page 39
2.4.1 Traditional Approach......Page 40
2.4.2 New Approach......Page 41
2.4.3 Applications......Page 42
2.5 Generalization to Arbitrary Boundary Conditions......Page 43
2.6 Initial Conditions for the Cluster-Size Distribution Function......Page 45
2.7.1 Motivation......Page 47
2.7.2 Effective Diffusion Coefficients......Page 48
2.7.3 Evolution of the Cluster-Size Distribution Functions......Page 53
2.8 Conclusions......Page 54
3.1 Introduction......Page 56
3.2 Basic Kinetic Equations......Page 58
3.3 Nonsteady-State Effects in the Initial Stage of Nucleation......Page 63
3.3.1 Approximative Solution in the Range 1 (<~) n (<~) n(c) – δn(c)......Page 64
3.3.2 Time Scale of Establishment of Steady-State Cluster-Size Distributions in the Range 1 (<~) n (<~) n(c) – δn(c)......Page 67
3.3.4 Steady-State Nucleation Rate and Steady-State Cluster-Size Distribution in the Range 1 (<~) n (<~) n(c) + δn(c)......Page 68
3.4 Flux and Cluster Distributions in the Range of Supercritical Cluster Sizes......Page 71
3.4.1 Results in the Range n(c) (<~) n (<~) 8n(c)......Page 72
3.4.2 Results in the Range n (<~) 8n(c)......Page 74
3.5 Time Interval for Steady-State Nucleation......Page 82
3.5.1 Kinetically Limited Growth......Page 83
3.5.3 Nonsteady-State Time Lag and the Time Scale of Steady-State Nucleation......Page 85
3.6.1 Number of Clusters Formed by Nucleation......Page 86
3.6.2 Average Size of the Clusters......Page 87
3.6.3 Time Interval of Independent Growth......Page 88
3.7 Time of Steady-State Nucleation and Induction Time......Page 90
3.8.2 Basic Equations......Page 93
3.8.3 Applications......Page 98
3.9.1 Results for the Range of Cluster Sizes n (<~) n(c)......Page 103
3.9.2 Results for the Range of Cluster Sizes n (<~) n(c)......Page 104
3.9.3 Integral Characteristics of the Nucleation–Growth Process......Page 106
3.10 Conclusions......Page 108
4.1.1 Introduction: Formulation of the Problem......Page 110
4.1.2 Asymptotic Behavior of the Critical Cluster Size......Page 113
4.1.3 Asymptotic Behavior of the Distribution Function......Page 117
4.1.4 Boundary Effects and Theory of Sintering......Page 122
4.1.5 Diffusive Decomposition Involving Different Mass-transfer Mechanisms......Page 126
4.1.6 Effects of Competition of Several Mass-Transfer Mechanisms......Page 130
4.1.7 Asymptotic Stability of Solid Solutions......Page 136
4.2.2 Canonical Form of the Basic System of Equations......Page 142
4.2.3 Coarsening in the Case of Power-Dependent Initial Cluster Size Distributions......Page 148
4.2.4 Coarsening in the Case of Exponentially Decaying Initial Cluster-Size Distributions......Page 152
4.2.5 Generalizations......Page 158
4.3.1 Introduction......Page 160
4.3.2 Basic Equations and Their Solution......Page 161
4.3.3 Regions of Phase Coexistence in Composition Space......Page 169
4.3.4 Competition of Different Phases in Coarsening......Page 173
4.3.5 Formation of Precipitates of Nonstoichiometric Composition......Page 178
4.3.6 Comparison with Experimental Data......Page 180
4.3.7 Conclusions......Page 182
5.1 Introduction......Page 188
5.2 Analysis of Statistical Approaches: “Equilibrium Distribution” of Classical Nucleation Theory, Fisher’s Droplet, and Similar Models......Page 189
5.3 Thermodynamic Approach: On the Possibility of Evolution of Monodisperse Cluster-Size Distributions......Page 192
5.4.1 Basic Kinetic Equations: General Expression......Page 195
5.4.2 Determination of the Coefficients of Emission......Page 196
5.4.4 Description of Growth Processes of Clusters......Page 198
5.4.5 Application to the Description of Nucleation......Page 201
5.4.6 Basic Kinetic Equations for Different Important Growth Mechanisms......Page 202
5.5.1 Precipitation in a Perfect Solution......Page 204
5.5.2 Effect of Nonlinear Inhibition of Cluster Growth on the Shape of the Cluster-Size Distributions......Page 209
5.5.3 Application of Fisher’s Expression for the Work of Cluster Formation......Page 213
5.6 Selected Applications and Conclusions......Page 215
5.7 Discussion......Page 218
6.1 Introduction......Page 220
6.2.1 Models of Elastic Stress in Cluster Growth and Coarsening......Page 222
6.2.2 Theoretical Description of Coarsening at a Nonlinear Increase of the Energy of Elastic Deformations with Cluster Volume: A First Approach......Page 223
6.3.1 Mathematical Formulation of the Problem and General Solution......Page 225
6.3.2 Approximations and Numerical Results......Page 228
6.4 Coarsening in a System of Weak Pores......Page 233
6.5.1 A First Approximation......Page 236
6.5.2 General Approach: Description of the Method......Page 238
6.5.3 Results......Page 240
6.6 Influence of Stochastic Effects on Coarsening in Porous Materials......Page 241
6.7 Discussion......Page 242
7.1 Introduction......Page 244
7.2.1 Preliminary Estimates......Page 245
7.2.2 Basic Kinetic Equations......Page 247
7.2.3 Results of the Numerical Solution of the Kinetic Equations......Page 249
7.2.4 Discussion......Page 252
7.3.2 Basic Equations......Page 254
7.3.3 Damped Sources......Page 256
7.3.4 Undamped Sources......Page 260
7.4.1 Introduction......Page 264
7.4.2 Diffusion Mechanism of Radiation-Induced Shrinkage of the Precipitates......Page 265
7.4.3 Effect of the Precipitate Incoherence and the Solute Atom Transition into the Interstitial Sites and Back in the Lattice Sites......Page 268
7.4.4 The Case of Incoherent Precipitation......Page 272
7.4.5 Conclusion......Page 273
8.1 Introduction......Page 274
8.2 Basic Set of Equations......Page 276
8.3 The Stage of Nucleation of Clusters of the Newly Evolving Phase......Page 281
8.4 The Transient Stage......Page 289
8.5 Kinetic Equations and Thermodynamic Relationships Accounting for Solute–Solute Interactions......Page 292
8.6 Rate of Change of the Number of Structural Elements of an Aggregate of the New Phase......Page 297
8.7 The Coefficient of Components Mass Transfer......Page 299
8.8 Steady-State Nucleation Rate......Page 302
8.9 Influence of Interaction of the Solute Components on Coarsening Processes......Page 305
8.10 Discussion and Conclusion......Page 306
9.1 Introduction......Page 308
9.2.1 Reduced Equations Describing the Process of Bubble Nucleation......Page 309
9.2.2 Time of Establishment of Steady-State Nucleation......Page 313
9.2.3 Quasistationary Distribution of Subcritical Bubbles......Page 316
9.2.4 Distribution Function of Bubbles in the Range N(c) < N < N(~)......Page 317
9.2.5 Distribution Function of Bubbles in the Range N < N(~)......Page 319
9.3 The Intermediate Stage......Page 324
9.4 The Late Stage......Page 331
9.5 Results of Numerical Computations......Page 339
9.6 Conclusions......Page 342
9.A.1 Some Mathematical Transformations......Page 343
9.A.2 Estimation of the Conditions when Merging of Colliding Bubbles can be Neglected......Page 344
10.1 Introduction......Page 346
10.2 Homogeneous Nucleation in Mixtures: Theory......Page 348
10.3.1 Spin Echoes in Restricted Geometry and Cluster Sizes......Page 351
10.3.2 Experimental Details......Page 352
10.3.3 Results and Discussion......Page 354
10.4.1 Experimental Results......Page 356
10.4.2 Discussion......Page 357
10.4.3 Conclusion......Page 362
10.5 Influence of the Degree of Supercooling on the Kinetics of Phase Separation in Solid Mixtures of (4)He in (3)He......Page 363
10.6 Comparison between Experiments and Conclusions......Page 366
11.1 Introduction......Page 370
11.2.1 The Cahn–Hilliard–Cook Equation......Page 372
11.2.2 Thermodynamic Aspects......Page 374
11.2.3 Results of Numerical Calculations......Page 376
11.2.4 Theoretical Interpretation......Page 379
11.2.5 Discussion......Page 381
11.3 Generalized Cluster Model Approach to the Description of Phase Separation: The Model System......Page 382
11.4.1 Thermodynamic Analysis......Page 385
11.4.2 Kinetics versus Thermodynamics in Phase Separation......Page 390
11.5.1 Thermodynamic Analysis......Page 393
11.5.2 Kinetics......Page 401
11.5.3 Transition from Independent Cluster Growth to Coarsening......Page 409
11.6 Results and Discussion......Page 412
References......Page 416
Index......Page 430