The Mathematics of Diffusion

J. Crank0-19-853344-6, 9780198533443

Table of contents :
Contents……Page 7
1.1. The diffusion process……Page 8
1.3. Differential equation of diffusion……Page 9
1.4. Anisotropic media……Page 12
2.2. Method of reflection and superposition……Page 18
2.3. Method of separation of variables……Page 24
2.4. Method of the Laplace transform……Page 26
2.5. Solutions in two and three dimensions……Page 31
2.6. Other solutions……Page 33
3.2. Instantaneous sources……Page 35
3.3. Semi-infinite media……Page 39
3.4. The infinite composite medium……Page 45
3.5. The semi-infinite composite medium……Page 48
3.6. Weber’s disc……Page 49
4.2. Steady state……Page 51
4.3. Non-steady state……Page 54
4.4. Edge effects in membranes……Page 71
4.5. Approximate two-dimensional solutions……Page 72
5.2. Steady state……Page 76
5.3. Non-steady state: solid cylinder……Page 78
5.4. The hollow cylinder……Page 89
5.5. The region bounded internally by the cylinder r = a……Page 94
6.2. Steady state……Page 96
6.3. Non-steady state……Page 97
6.4. Hollow sphere……Page 105
6.5. The region bounded internally by the sphere r = a……Page 109
7.1. Time-dependent diffusion coefficients……Page 111
7.2. Concentration-dependent diffusion: infinite and semi-infinite media……Page 112
7.4. Methods of moments……Page 136
7.5. Approximation by orthogonal functions……Page 142
8.1. Numerical solutions……Page 144
8.2. Non-dimensional variables……Page 145
8.3. Physical derivation of a numerical solution……Page 146
8.4. Finite-difference solution: explicit method……Page 148
8.5. Crank-Nicolson implicit method……Page 151
8.6. Other boundary conditions……Page 153
8.7. Finite-difference formulae for the cylinder and sphere……Page 155
8.8. Composite media……Page 156
8.9. Two- and three-dimensional diffusion……Page 157
8.10. Singularities: local solutions……Page 159
8.11. Compatibility, convergence, stability……Page 164
8.12. Steady-state problems……Page 165
9.1. Steady state……Page 167
9.2. Non-steady-state conditions……Page 175
9.3. Concentration-distance curves……Page 176
9.4. Sorption- and desorption-time curves……Page 186
9.5. Diffusion-controlled evaporation……Page 198
9.6. Effect of a surface skin……Page 202
10.2. A frame of reference when the total volume of the system remains constant……Page 210
10.3. Alternative frames of reference……Page 212
10.4. Intrinsic diffusion coefficients……Page 216
10.5. ‘Self’-diffusion coefficients……Page 219
10.6. Methods of measurement……Page 221
10.7. Weighted-mean diffusion coefficients……Page 257
10.8. Radiotracer methods……Page 258
11.1. Glassy polymers……Page 261
11.2. Characteristic features……Page 262
11.3. Mathematical models……Page 264
11.4. Conclusion……Page 271
12.2. Laminates……Page 273
12.3. Disperse phase in a continuum……Page 276
12.4. The mathematical problem of a two-phase system……Page 277
13.1. Introduction……Page 293
13.2. Discontinuous diffusion coefficients……Page 294
13.4. General problem of the moving boundary……Page 305
13.5. Radially symmetric phase growth……Page 315
13.6. Approximate analytic solutions……Page 317
13.7. Finite-difference methods……Page 320
13.8. Other methods……Page 332
14.2. Instantaneous reaction……Page 333
14.3. Irreversible reaction……Page 336
14.4. Reversible reaction……Page 344
14.5. A bimolecular reaction……Page 356
14.6. Reduced sorption curves……Page 357
14.7. Constant reaction rate……Page 358
15.2. Uptake of water by a textile fibre……Page 359
15.3. Two possible equilibrium conditions……Page 360
15.4. Propagation of two disturbances……Page 361
15.5. Equations for diffusion of heat and moisture……Page 362
15.6. Solution of the equations……Page 364
15.7. Surface temperature changes accompanying the sorption of vapours……Page 374
15.8…….Page 381
Tables……Page 382
References……Page 406
Author index……Page 414
Subject index……Page 418