Kinetics, Transport, and Structure in Hard and Soft Materials

Table of Contents......Page 7
Part I: Tools: Elements of Diffusional Transport......Page 14
1.1 Introduction......Page 16
1.2.1 Average Thermodynamic Properties......Page 18
1.2.2 Maxwell-Boltzmann Velocity Distributions......Page 23
1.2.2.1 Distribution of Component Velocities......Page 24
1.2.2.2 Distribution of Speeds......Page 27
1.2.3 Diffusional Transport of Noninteracting Particles......Page 28
1.2.3.1 Flux of Maxwellian Particles......Page 29
1.2.3.2 The Diffusion Coefficient and Fick’s 1st Law......Page 30
1.2.3.3 Collision Probabilities and the Mean Free Path......Page 31
1.3.1 Fick’s 1st Law: Additional Comments......Page 33
1.3.2 Fick’s 2nd Law......Page 35
1.3.2.1 Fick’s 2nd Law in Cylindrical Coordinates......Page 37
1.4.1 Flow through a Planar Layer......Page 38
1.4.2 Steady State Flow through Nonplanar Surfaces: Cylinder......Page 39
1.4.3 Steady State Flow through a Spherical Interface......Page 40
1.5.1 Solution to Fick’s 2nd Law Using Fourier Integral Transforms......Page 41
1.5.2 Solution to Fick’s 2nd Law in Three Dimensions Using Laplace Transforms......Page 44
1.6.1 Diffusion from a Semi-Infinite Source......Page 49
1.6.2 Diffusion from a Finite Source of Thickness 2h......Page 50
1.6.3 Desporption/Absorption of a Species from a Sample of Finite Dimensions......Page 51
1.6.4 Permeation Experiments......Page 52
1.6.5 Time-Dependent Fluxes: Weight Gain Experiments......Page 53
1.7 Concluding Remarks......Page 54
1.8 Problems......Page 55
1.10.2 Fourier Integral Transforms of Derivatives......Page 61
2.1 Introduction......Page 64
2.2.1 Binomial Distribution Function......Page 65
2.2.3 The Gaussian Distribution Function......Page 67
2.3 Correlation Functions......Page 69
2.3.1 Pair Correlation Functions and the Static Structure Factor......Page 72
2.3.3 Pair Distribution Function......Page 73
2.4 Langevin Analysis......Page 76
2.4.2 Mean Square Velocity......Page 77
2.4.3 Mean Square Displacement......Page 78
2.4.5 Nernst-Einstein Equation......Page 79
2.5 Light Scattering: Measurement of Diffusion......Page 80
2.5.1 The Scattered Field......Page 81
2.5.2 Scattering from a Dilute Collection of Molecules......Page 82
2.5.3 Measurement of Diffusion......Page 83
2.6 Problems for Chapter 2......Page 85
2.8 References......Page 88
Part II: Diffusion in Crystalline Materials......Page 90
3.1 Introduction......Page 92
3.2.1 Bravais Lattices......Page 94
3.2.2 Unit Cells, Crystal Directions, and Crystal Planes......Page 95
3.2.3 Atomic Defects in Crystals......Page 99
3.3.1 Random Walk in 3-D......Page 102
3.3.1.1 The Jump Frequency, Γ......Page 104
3.3.2 Debye Frequency......Page 106
3.3.2.1 An Expression for the Tracer Diffusion Coefficient......Page 107
3.4 Atomic Transport in Crystals via a Single Vacancy Mechanism......Page 108
3.4.1 Self-Diffusion and Tracer Diffusion via a Vacancy Mechanism......Page 109
3.5 The Equilibrium Vacancy Concentration......Page 111
3.5.1 Vacancy Concentration in Crystals: Experiment versus Theory......Page 112
3.6 Divacancies and Their Effect on Diffusion......Page 116
3.7 Diffusion of Interstitials in Crystals......Page 119
3.8 Ring Mechanism of Atomic Diffusion......Page 120
3.10 Diffusion in the Presence of Impurities......Page 121
3.10.2.1 Concentration of Vacancies and Impurities in a Dilute Alloy......Page 122
3.10.2.2 Substitutional Impurity-Vacancy Pair Diffusion......Page 124
3.12 Effects of Pressure on Diffusion......Page 127
3.13 Diffusion Near Dislocations and Grain Boundaries......Page 128
3.15 Problems for Chapter 3......Page 130
3.16 References and Additional Reading......Page 134
4.1 Introduction......Page 136
4.2 Defects in Ionic Crystals......Page 137
4.3 Frenkel Defect Concentration......Page 138
4.4 Schottky Defect Concentration......Page 139
4.5 Diffusional Transport of Cationic and Ionic Defects......Page 140
4.6 Diffusivity of Frenkel Defects......Page 142
4.7 Diffusion of Schottky Defects......Page 143
4.8 The Effect of Multivalent Impurities on Conductivity......Page 144
4.9 Comments on Transport in Alkali Halide Crystals: Transport Coefficients......Page 146
4.10 Problems for Chapter 4......Page 148
4.11 References and Additional Reading......Page 150
5.1 Introduction......Page 152
5.3 Self-Diffusion in Silicon and Germanium......Page 153
5.4 Diffusion of Dopants......Page 156
5.4.1 Mechanisms of Atomic Transport......Page 157
5.4.2 Examples......Page 158
5.5 Concluding Remarks......Page 159
5.7 References......Page 160
Part III: Diffusional Transport in Systems That Lack Long-Range Structural Order......Page 162
6.1 Introduction and Context......Page 164
6.2 Classification of Polymers......Page 166
6.3.1 Freely Jointed Chain Model......Page 168
6.3.2 Freely Rotating Chain Model......Page 169
6.3.3 Hindered Rotation Chain Model......Page 171
6.3.3.1 Persistence Length......Page 172
6.3.4 Single Chain Statistics: Excluded Volume Effects......Page 173
6.4 Phenomenology of the Viscoelastic Behavior of Polymers......Page 175
6.4.1 Maxwell and Voigt Phenomenological Models......Page 177
6.4.2 The Viscosity: Experimental Observations......Page 181
6.4.2.1 Temperature Dependence of the Viscosity......Page 182
6.4.3 Time-Temperature-Superposition and Shift Factors......Page 184
6.4.4 Oscillatory Shear Measurements......Page 186
6.4.5 Connections between G(t) and Frequency Domain Experiments......Page 188
6.5 Microscopic Model for Diffusion and Viscoelasticity in Polymer Melts......Page 192
6.5.1 Rouse Model: Unentangled Chains......Page 193
6.5.2 Reptation: Dynamics of Entangled Chains......Page 195
6.5.3 The Stress Relaxation Modulus, the Viscosity, and the Steady State Compliance......Page 200
6.5.4 The Entanglement, the Molecular Weight, and the Critical Molecular Weight......Page 203
6.5.5 The Viscosity of Polymers......Page 205
6.5.6 The Diffusion Coefficient of Entangled Chains......Page 206
6.5.7 Temperature Dependence of Diffusion......Page 209
6.5.8 Tube Length Fluctuations......Page 211
6.5.9 Constraint Release Mechanism......Page 212
6.5.10 Dynamic Moduli G\'(w) and G\"(w)......Page 215
6.6 Concluding Remarks......Page 216
6.7 Problems for Chapter 6......Page 217
6.8 References......Page 221
7.1 Introduction......Page 224
7.2 The Structure of Inorganic Network Glass Formers: An Introduction......Page 226
7.3 Bulk Transport Processes Inorganic Network Glass Formers......Page 229
7.3.1 Temperature Dependence of the Viscosity: The VTF Equation......Page 230
7.3.1.1 Comments Regarding the Glass Transition......Page 231
7.3.2 Temperature Dependence of the Viscosity: Adam-Gibbs Model......Page 233
7.4 Connection between Kinetic and Thermodynamic Fragility......Page 235
7.5 “Strong” versus “Fragile” Network Glass Melts, a Structural Connection......Page 236
7.5.1 Influence of Alkali Content on Heat Capacity and Activation Energy for Flow......Page 237
7.5.2 The Viscosity of Mixed Alkali Glass Melts......Page 239
7.5.3 Effect of Alkali Composition on Tg......Page 240
7.6 Relaxation Functions......Page 241
7.7.1 Primary Relaxations......Page 243
7.7.2 Secondary Mechanical Relaxations (T < Tg)......Page 245
7.9 Ionic Conductivity and Diffusion......Page 248
7.9.3 Comments Regarding Ionic Conductivity in Network Glasses......Page 250
7.9.3.1 The Electrical Modulus Representation......Page 253
7.10 Secondary Relaxations in ECR and MR Experiments......Page 254
7.11.1.1 Option I......Page 256
7.11.1.2 Option II......Page 257
7.11.2 Mixed Alkali Glasses......Page 258
7.13 Problems for Chapter 7......Page 259
7.14 References......Page 262
7.15 Appendix......Page 265
8.1 Introduction......Page 268
8.2 Temperature Dependencies of Relaxations......Page 269
8.2.1 Dispersive Dynamics Associated with Disorder......Page 270
8.3 Comments on Dynamics in the Supercooled State......Page 273
8.4 Comments on the Stokes-Einstein Relationship......Page 274
8.6 References......Page 275
Part IV: Instabilities and Pattern Formation in Materials......Page 278
9.1 Introduction......Page 280
9.2 Free Energy of Mixing of a Binary Polymer-Polymer Mixture......Page 282
9.2.1 Phase Diagram of a Simple Binary Mixture......Page 284
9.3.1 Linearized Theory for the Early Stages of Spinodal Decomposition......Page 286
9.4 An Example Involving a Polymer-Polymer Mixture......Page 291
9.5 Remarks Regarding Spinodal Decomposition......Page 293
9.6 Nucleation......Page 294
9.6.2 Elements of the Classical Theory of Nucleation......Page 295
9.6.3 Steady State Growth Rate......Page 297
9.7 Heterogeneous Nucleation......Page 299
9.9 Problems for Chapter 9......Page 300
9.10 References for Spinodal Decomposition......Page 302
9.11 References for Nucleation and Growth......Page 303
10.1 Introduction......Page 306
10.2.1 Onsager Analysis......Page 309
10.2.2 The Darken Equation......Page 311
I0.2.3 Marker Velocity......Page 313
10.3 The Hartley-Crank Equation......Page 314
10.4 Interdiffusion in Polymers......Page 315
10.5 Measurements of Interdiffusion......Page 318
10.5.1 Marker Experiments......Page 319
10.7 Problems for Chapter 10......Page 322
10.8 References......Page 323
11.1 Introduction......Page 326
11.2.1 Elementary Concepts of Classical Capillarity......Page 328
11.2.1.1 Effect of Curvature on the Properties of Small Systems......Page 331
11.3 Moving Front in a Supercooled Melt......Page 333
11.3.1 Stationary Solutions (planar interface, k = 0)......Page 337
11.3.2 Linear Stability Analysis......Page 338
11.4 Instabilities at an Interface in a Supersaturated Environment......Page 341
11.5 Brief Comments on Microstructure......Page 343
11.6 Problems for Chapter 11......Page 344
11.7 References and Further Reading......Page 347
12.1 Introduction......Page 350
12.2.1 Instabilities in Macroscopically Thick Films......Page 351
12.3.1 Pattern Formation in Nanometer-Thick Films......Page 353
12.3.2 Fingering in Ultrathin Films......Page 355
12.4 Instabilities Involving Macroscopic or Bulk Flows......Page 356
12.4.1 Rayleigh-Bénard Instability......Page 357
12.4.2 Rayleigh Instability......Page 358
12.6 References......Page 359
12.7 Further Reading......Page 360