Diffusion in Solids Field Theory, Solid-State Principles, and Applications

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ISBN: 9780471239727 | 0471239720
Cover: Hardcover
Copyright: 11/30/1999

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This book offers a modern treatment of diffusion in solids, covering such core topics as the transport of mass through the lattice of a crystalline solid. Part I of the book develops basic concepts in diffusion field theory and illustrates them with several applications, while Part II focuses on key solid-state principles needed to apply diffusion theory to real materials.

Preface

xix

Acknowledgments

xxi

PART I FIELD THEORY

(188)

Laws of Diffusion

(14)

Preliminary Definitions

(1)

Diffusion in an Isotropic Medium

(1)

Fick's First Law

(2)

Units

(1)

Vector Form of Fick's First Law

(1)

Mass Conservation

(3)

Fick's Second Law

(1)

Linear Diffusion Equation in Various Coordinate Systems

(1)

Important Diffusion Symmetries

(6)

Exercises

(3)

Bibliography

(2)

Diffusion in Generalized Media

(14)

Diffusivity Tensor

(1)

Principal Directions

(1)

Diffusion in Generalized Media

(3)

Influence of Imposed Symmetry

(9)

Isotropic Materials

(1)

Cubic crystals

(1)

Orthotropic Materials

(1)

Orthorhombic Crystals

(1)

Monoclinic Crystals

(1)

Triclinic Crystals

(1)

Exercises

(3)

Bibliography

(1)

Solutions to the Linear Diffusion Equation

(18)

Linear Diffusion and Transform Methods

(1)

Linear Diffusion into a Semi-Infinite Medium

(4)

Boundary Conditions

(1)

Laplace Transforms

(3)

Behavior of the Concentration Field

(2)

Instantaneous Planar Diffusion Source in an Infinite Medium

(3)

Conservation of Mass for a Planar Source

(2)

Estimating the Error Function

(1)

Thin-Film Configuration

(6)

Exercises

(3)

Bibliography

(2)

Diffusion Couple

(10)

Superposition of Instantaneous Sources

(1)

Diffusion Couples

(3)

Diffusion with a Fixed Boundary Concentration

(1)

Slab Couples

(5)

Exercises

(3)

Bibliography

(1)

Diffusion Point Sources in Higher Dimensions

(16)

Instantaneous Point Source in Three Dimensions

(5)

Transform Solution for a Point Source

(2)

Mass Constraint in Spherical Symmetry

(2)

Source Solutions in One, Two, and Three Dimensions

(1)

Behavior of the Fields Upon Release of a Source

(4)

Choice of Concentration, Time, and Length Scales

(4)

Behaviour of Diffusion Responses at Early Times

(1)

Conservation Integrals for One, Two, and Three Dimensions

(7)

Mass Centroids in Higher Dimensions

(2)

Exercises

(1)

Bibliography

(2)

Generalized Sources

(20)

Generalizing the Point Source in Three Dimensions

(2)

Relationship to Green's Functions

(1)

Instantaneous Line Source Releasing Diffusant in Three Dimensions

(2)

Hadamard's Method of Descent

(1)

Generalization of the Instantaneous Point Source in Two Dimensions

(1)

Instantaneous Line Source in Two Dimensions

(2)

Instantaneous Ring Source

(3)

Continuous Point Sources

(3)

Steady-State Limit of Continuous Point Sources

(1)

Time-Continuous Line Source in Two Dimensions

(8)

Exercises

(6)

Bibliography

(1)

Diffusion-Reaction

(12)

Linear Diffusion--Reaction

(2)

One-Dimensional Diffusion--Reaction

(1)

Boundary Conditions for Diffusion--Reaction

(1)

Imposing Steady-State Conditions

(2)

General Kinetic Limits

(1)

Reaction Limit

(1)

Diffusion Limit

(1)

Mixed Diffusion--Reaction Kinetics

100

(1)

Time-Dependent Reaction-Diffusion

101

(1)

Nonlinear Diffusion--Reaction

102

(5)

Exercises

103

(3)

Bibliography

106

(1)

Linear Flow in Finite Systems

107

(28)

Diffusion in a Single-Phase Composite Slab

107

(6)

No-Flow Boundary Condition

108

(1)

Fictitious Image Sources

109

(4)

Fourier's Method

113

(2)

Application of Fourier Series

115

(4)

Numerical Approximations

119

(4)

Error Function Series Approximation

120

(1)

Fourier Series Approximation

121

(1)

Numerical Comparison

122

(1)

Codastefano's Method

123

(12)

Exercises

126

(6)

Bibliography

132

(3)

Spherical Bodies

135

(26)

Diffusion Around Spheres

136

(4)

Boundary Conditions

136

(2)

Spherical Diffusion Solution

138

(2)

Behavior of the Spherical Concentration Field

140

(1)

Interfacial Fluxes

141

(1)

Quasi-Static Growth and Dissolution

142

(3)

Moving Boundaries

145

(1)

Evolution of a Spherical Particle

145

(5)

Conditions on the Phases

145

(1)

Conditions at the &alfa;--β Interface

146

(1)

Moving Boundary Solution

146

(4)

The Characteristic Equation

150

(4)

Diffusion Boundary Layers

154

(7)

Massive Transformations

155

(1)

Exercise

156

(2)

Bibliography

158

(3)

Steady-State Diffusion

161

(14)

Permeation in Linear Flow

161

(2)

Molecular Dissociation at Surfaces

162

(1)

Steady-State Permeation

162

(1)

Permeability Units

163

(1)

Steady Flow with Variable D

163

(5)

Boundary Conditions

164

(4)

Steady-State Diffusion in Cylinders: Constant D

168

(2)

Steady-State Diffusion in Spheres

170

(5)

Exercise

172

(2)

Bibliography

174

(1)

Inverse Methods

175

(14)

Time-Dependent Diffusion with Variable Diffusivity

176

(1)

Boltzmann's Transformation

176

(2)

Matano's Geometry

178

(2)

Matano Interface

178

(2)

Application of the Boltzmann-Matano Method

180

(2)

Application of the Method of Sauer, Freise, and den Broeder

182

(7)

Exercise

186

(1)

Bibliography

187

(2)

PART II SOLID-STATE PRINCIPLES

189

(230)

Random Walks and Diffusion

191

(22)

Background

191

(1)

Diffusive Motions

192

(1)

Random Walks

193

(1)

Random Numbers

193

(1)

One-Dimensional Random Walks

194

(3)

Combinatoric Formulation

197

(2)

Random Walks in Higher Dimensions

199

(4)

Diffusion Viewed as a Stochastic Process

203

(2)

Diffusivity: A Microscopic Transport Property

205

(8)

Time Reversal

206

(1)

Einstein's Formula

207

(1)

Exercises

208

(4)

Bibliography

212

(1)

Structure and Diffusion

213

(22)

Random Walks in Crystals

213

(3)

Primitive (Simple) Cubic

214

(1)

Body-Centered Cubic

214

(1)

Face-Centered Cubic

215

(1)

Constraints Imposed by the Structure

216

(2)

Interpreting Diffusivities in Cubic Crystals

218

(2)

Diffusion Mechanisms

220

(7)

Interstitial Diffusion

220

(3)

Ring Diffusion

223

(1)

Vacancy-Assisted Diffusion

224

(1)

Interstitialcy Diffusion

225

(2)

Diffusion in FCC Crystals

227

(1)

Lattice Vacancies

228

(2)

Divacancies

230

(5)

Exercises

232

(1)

Bibliography

233

(2)

Correlation Effects in Diffusion

235

(10)

Markov Sequences and Correlated Walks

235

(2)

Estimates of the Correlation Factor: Pure Materials

237

(5)

Jumping Probability Estimate

238

(4)

Computation of Correlation Factors in Pure Crystals

242

(3)

Exercises

243

(1)

Bibliography

244

(1)

Vacancy-Assisted Diffusion

245

(20)

Dynamical Theory

245

(3)

Flynn's Model

246

(2)

Quasi-Chemical Theory

248

(5)

Activated Diffusion Complexes

248

(1)

Energy-Momentum Phase Space

248

(4)

Self-Diffusion in Crystals

252

(1)

Empirical Relationships

253

(3)

Measures of Lattice Stability

253

(1)

Activation Energy for Self-Diffusion

253

(2)

Corresponding States

255

(1)

Correlation with Diffuser Size

256

(1)

Arrhenius Correlation

256

(2)

Dependence of D on Composition

258

(1)

Le Claire's Correlation

258

(1)

Correlations of D and Q with Composition

259

(1)

Empirical, Observations

260

(5)

Exercises

261

(2)

Bibliography

263

(2)

Diffusion in Dilute Alloys

265

(18)

Five-Frequency Exchange Model

266

(5)

Isotope Effect

271

(2)

Diffusion of H+ and D+ with Trapping

273

(4)

Fick's Law with Trapping

275

(2)

Saturable Diffusion Traps

277

(1)

Irreversible Traps

278

(5)

Exercises

279

(2)

Bibliography

281

(2)

Kirkendall Effect

283

(26)

Nonreciprocal Diffusion

284

(1)

Kirkendall Effect

285

(3)

Kirkendall's Diffusion Couple

285

(1)

Kirkendall Experiment

286

(2)

Significance of kirkendall Marker Motion

288

(1)

Nonreciprocal Atomic Fluxes

289

(3)

Intrinsic Diffusivities: Vacancy Wind

292

(4)

Diffusion--Advection

296

(4)

Mass Conservation with Diffusion-Advection

297

(2)

Steady-State Diffusion-Advection

299

(1)

One-Dimensional Diffusion-Advection

300

(1)

Darken's Analysis of the Kirkendall Effect

300

(5)

Strain Effects

305

(4)

Exercise

306

(1)

Bibliography

307

(2)

Influence of Solution Ideality

309

(14)

Generalized Forces

309

(1)

Analogy of Electrical Resistance

310

(1)

Atom Mobilities

310

(3)

Constrained Equilibrium

311

(1)

Diffusion Forces

312

(1)

Chemically Inhomogeneous Systems

313

(2)

Limiting Cases

315

(1)

Tracer Diffusion

316

(7)

Exercise

319

(3)

Bibliography

322

(1)

Diffusional Anelasticity

323

(20)

Background

323

(1)

Standard Linear Solid

324

(4)

Cyclic Stress and Strain

325

(3)

Internal Friction

328

(4)

Diffusional Anelasticity

332

(1)

Internal Friction in BCC Materials

332

(1)

Geometry of Interstitial Sites in BCC

332

(1)

Interstitial Subpopulations in BCC

333

(3)

Reorientation of Atom Pairs in FCC Materials

336

(2)

Internal Friction Methods

338

(1)

Summary of Diffusional Anelastic Effects

339

(4)

Exercises

340

(2)

Bibliography

342

(1)

Field-Assisted Diffusion

343

(16)

Background

344

(1)

Diffusion Currents in Homogeneous Ionic Solids

345

(1)

Measurement of Ionic Conductivity in Solids

346

(1)

Defects in Ionic Solids

347

(2)

Equilibria Among Charged Point Defects

347

(2)

Experiments in Ionic Conductors

349

(1)

Irreversible Thermodynamics and Diffusion

350

(2)

Isothermal Binary Diffusion

352

(1)

Vacancies in Thermal Equilibrium

353

(1)

Net Vacancy Flux

354

(5)

Exercise

355

(2)

Bibliography

357

(2)

Multiparticle Diffusion: Capillary Effects

359

(18)

Length Scales in Microstructures

360

(2)

Interfacial Free Energy

361

(1)

Role of Diffusional Transport

361

(1)

Kelvin's Equation

362

(6)

Influence of Curvature on the Phase Diagram

365

(1)

Influence of Curvature on Solubility

366

(2)

Dimensionless Diffusion Potential

368

(1)

Mean-Field Kinetic Equation

368

(3)

Experimental Observations

371

(6)

Exercise

373

(2)

Bibliography

375

(2)

Population Dynamics

377

(14)

Continuity Equation

377

(2)

Scale-Factor Population Dynamics

379

(3)

Particle Distribution Function

382

(2)

Volume Fraction Constraint

384

(2)

Coarsening Kinetics

386

(5)

Experimental Observations

387

(1)

Exercise

388

(2)

Bibliography

390

(1)

Multicomponent Diffusion

391

(14)

Fick's First Low for Multicomponent Diffusion

392

(4)

Multicomponent Fick's First Law in One Dimension

393

(2)

Multicomponent Intrinsic Diffusivities and the Interdiffusion Coefficients

395

(1)

Fick's Second Law for Multicomponent Diffusion

396

(2)

Multicomponent Diffusion Couples

398

(1)

Square-Root Diffusivity Method

399

(1)

Approximation Using Effective Binary Diffusion Coefficients

400

(5)

Exercises

400

(4)

Bibliography

404

(1)

Multicomponent Diffusion: Profiler Program

405

(14)

Background

406

(1)

Profiler

406

(2)

Application to a Ternary Alloy

408

(2)

Extrema in the Composition Profile

409

(1)

Composition Directions, Eigenvalues, and Eigenvectors

410

(3)

Zero-Flux Planes

411

(1)

Properties of the [r] Matrix

411

(2)

Multicomponent Diffusion Paths

413

(2)

Multicomponent Concentration Profiles

415

(4)

Exercise

415

(2)

Bibliography

417

(2)

Appendix A Ten-Place Error Function Values

419

(6)

Table A.1

419

(3)

Table A.2

422

(3)

Appendix B Instruction Manual for Profiler

425

(16)

Index

441

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Diffusion in Solids Field Theory, Solid-State Principles, and Applications

Summary

Author Biography

Table of Contents

Supplemental Materials