Diffusion in Solids »

Diffusion is a vital topic in solid-state physics and chemistry, physical metallurgy and materials science. Diffusion processes are ubiquitous in solids at elevated temperatures. A thorough understanding of diffusion in materials is crucial for materials development and engineering. This book first gives an account of the central aspects of diffusion in solids, for which the necessary background is a course in solid state physics. It then provides easy access to important information about diffusion in metals, alloys, semiconductors, ion-conducting materials, glasses and nanomaterials. Several diffusion-controlled phenomena, including ionic conduction, grain-boundary and dislocation pipe diffusion, are considered as well.

Graduate students in solid-state physics, physical metallurgy, materials science, physical and inorganic chemistry or geophysics will benefit from this book as will physicists, chemists, metallurgists, materials engineers in academic and industrial research laboratories.

1 History and Bibliography of Diffusion 1

1.1 Pioneers and Landmarks of Diffusion 2

References 16

1.2 Bibliography of Solid-State Diffusion 18

Part I Fundamentals of Diffusion

2 Continuum Theory of Diffusion 27

2.1 Fick's Laws in Isotropic Media 27

2.1.1 Fick's First Law 28

2.1.2 Equation of Continuity 29

2.1.3 Fick's Second Law - the 'Diffusion Equation' 30

2.2 Diffusion Equation in Various Coordinates 31

2.3 Fick's Laws in Anisotropic Media 33

References 35

3 Solutions of the Diffusion Equation 37

3.1 Steady-State Diffusion 37

3.2 Non-Steady-State Diffusion in one Dimension 39

3.2.1 Thin-Film Solution 39

3.2.2 Extended Initial Distribution and Constant Surface Concentration 41

3.2.3 Method of Laplace Transformation 45

3.2.4 Diffusion in a Plane Sheet - Separation of Variables 47

3.2.5 Radial Diffusion in a Cylinder 50

3.2.6 Radial Diffusion in a Sphere 51

3.3 Point Source in one, two, and three Dimensions 52

References 53

4 Random Walk Theory and Atomic Jump Process 55

4.1 Random Walk and Diffusion 56

4.1.1 A Simplified Model 56

4.1.2 Einstein-Smoluchowski Relation 58

4.1.3 Random Walk on a Lattice 60

4.1.4 Correlation Factor 62

4.2 Atomic Jump Process 64

References 66

5 Point Defects in Crystals 69

5.1 Pure Metals 70

5.1.1 Vacancies 70

5.1.2 Divacancies 72

5.1.3 Determination of Vacancy Properties 74

5.1.4 Self-Interstitials 79

5.2 Substitutional Binary Alloys 80

5.2.1 Vacancies in Dilute Alloys 81

5.2.2 Vacancies in Concentrated Alloys 82

5.3 Ionic Compounds 83

5.3.1 Frenkel Disorder 84

5.3.2 Schottky Disorder 85

5.4 Intermetallics 86

5.5 Semiconductors 88

References 91

6 Diffusion Mechanisms95

6.1 Interstitial Mechanism 95

6.2 Collective Mechanisms 97

6.3 Vacancy Mechanism 98

6.4 Divacancy Mechanism 100

6.5 Interstitialcy Mechanism 100

6.6 Interstitial-Substitutional Exchange Mechanisms 102

References 103

7 Correlation in Solid-State Diffusion 105

7.1 Interstitial Mechanism 107

7.2 Interstitialcy Mechanism 107

7.3 Vacancy Mechanism of Self-diffusion 108

7.3.1 A 'Rule of Thumb' 108

7.3.2 Vacancy-tracer Encounters 109

7.3.3 Spatial and Temporal Correlation 112

7.3.4 Calculation of Correlation Factors 112

7.4 Correlation Factors of Self-diffusion 115

7.5 Vacancy-mediated Solute Diffusion 116

7.5.1 Face-Centered Cubic Solvents 117

7.5.2 Body-Centered Cubic Solvents 120

7.5.3 Diamond Structure Solvents 121

7.6 Concluding Remarks 122

References 124

8 Dependence of Diffusion on Temperature and Pressure 127

8.1 Temperature Dependence 127

8.1.1 The Arrhenius Relation 127

8.1.2 Activation parameters - Examples 130

8.2 Pressure Dependence 132

8.2.1 Activation Volumes of Self-diffusion 135

8.2.2 Activation Volumes of Solute Diffusion 139

8.2.3 Activation Volumes of Ionic Crystals 140

8.3 Correlations between Diffusion and Bulk Properties 141

8.3.1 Melting Properties and Diffusion 141

8.3.2 Activation Parameters and Elastic Constants 146

8.3.3 Use of Correlations 147

References 147

9 Isotope Effect of Diffusion 151

9.1 Single-jump Mechanisms 151

9.2 Collective Mechanisms 155

9.3 Isotope Effect Experiments 155

References 159

10 Interdiffusion and Kirkendall Effect 161

10.1 Interdiffusion 161

10.1.1 Boltzmann Transformation 162

10.1.2 Boltzamann-Matano Method 163

10.1.3 Sauer-Freise Method 166

10.2 Intrinsic Diffusion and Kirkendall Effect 168

10.3 Darken Equations 170

10.4 Darken-Manning Equations 172

10.5 Microstructural Stability of the Kirkendall Plane 173

References 176

11 Diffusion and External Driving Forces 179

11.1 Overview 179

11.2 Fick's Equations with Drift 181

11.3 Nernst-Einstein Relation 182

11.4 Nernst-Einstein Relation for Ionic Conductors and Haven Ratio 184

11.5 Nernst-Planck Equation - Interdiffusion in Ionic Crystals 186

11.6 Nernst-Planck Equation versus Darken Equation 188

References 189

12 Irreversible Thermodynamics and Diffusion 191

12.1 General Remarks 191

12.2 Phenomenological Equations of Isothermal Diffusion 193

12.2.1 Tracer Self-Diffusion in Element Crystals 193

12.2.2 Diffusion in Binary Alloys 195

12.3 The Phenomenological Coefficients 199

12.3.1 Phenomenological Coefficients, Tracer Diffusivities, and Jump Models 202

12.3.2 Sum Rules - Relations between Phenomenological Coefficients 204

References 205

Part II Experimental Methods

13 Direct Diffusion Studies 209

13.1 Direct versus Indirect Methods 209

13.2 The Various Diffusion Coefficients 212

13.2.1 Tracer Diffusion Coefficients 212

13.2.2 Interdiffusion and Intrinsic Diffusion Coefficients 214

13.3 Tracer Diffusion Experiments 215

13.3.1 Profile Analysis by Serial Sectioning 217

13.3.2 Residual Activity Method 222

13.4 Isotopically Controlled Heterostructures 223

13.5 Secondary Ion Mass Spectrometry (SIMS) 224

13.6 Electron Microprobe Analysis (EMPA) 227

13.7 Auger-Electron Spectroscopy (AES) 230

13.8 Ion-beam Analysis: RBS and NRA 231

References 234

14 Mechanical Spectroscopy 237

14.1 General Remarks 237

14.2 Anelasticity and Internal Friction 239

14.3 Techniques of Mechanical Spectroscopy 242

14.4 Examples of Diffusion-related Anelasticty 244

14.4.1 Snoek Effect (Snoek Relaxation) 244

14.4.2 Zener Effect (Zener Relaxation) 247

14.4.3 Gorski Effect (Gorski Relaxation) 248

14.4.4 Mechanical Loss in Ion-conducting Glasses 249

14.5 Magnetic Relaxation 250

References 251

15 Nuclear Methods 253

15.1 General Remarks 253

15.2 Nuclear Magnetic Relaxation (NMR) 253

15.2.1 Fundamentals of NMR 254

15.2.2 Direct Diffusion Measurement by Field-Gradient NMR 256

15.2.3 NMR Relaxation Methods 258

15.3 Mossbauer Spectroscopy (MBS) 264

15.4 Quasielastic Neutron Scattering (QENS) 269

15.4.1 Examples of QENS studies 278

15.4.2 Advantages and Limitations of MBS and QENS 279

References 281

16 Electrical Methods 285

16.1 Impedance Spectroscopy 285

16.2 Spreading Resistance Profiling 290

References 293

Part III Diffusion in Metallic Materials

17 Self-diffusion in Metals 297

17.1 General Remarks 297

17.2 Cubic Metals 299

17.2.1 FCC Metals - Empirical Facts 299

17.2.2 BCC Metals - Empirical Facts 301

17.2.3 Monovacancy Interpretation 302

17.2.4 Mono-and Divacancy Interpretation 303

17.3 Hexagonal Close-Packed and Tetragonal Metals 306

17.4 Metals with Phase Transitions 308

References 311

18 Diffusion of Interstitial Solutes in Metals 313

18.1 'Heavy' Interstitial Solutes C, N, and O 313

18.1.1 General Remarks 313

18.1.2 Experimental Methods 314

18.1.3 Interstitial Diffusion in Dilute Interstial Alloys 316

18.2 Hydrogen Diffusion in Metals 317

18.2.1 General Remarks 317

18.2.2 Experimental Methods 318

18.2.3 Examples of Hydrogen Diffusion 320

18.2.4 Non-Classical Isotope Effects 323

References 324

19 Diffusion in Dilute Substitutional Alloys 327

19.1 Diffusion of Impurities 327

19.1.1 'Normal' Impurity Diffusion 327

19.1.2 Impurity Diffusion in Al 332

19.2 Impurity Diffusion in 'Open' Metals - Dissociative Mechanism 333

19.3 Solute Diffusion and Solvent Diffusion in Alloys 336

References 338

20 Diffusion in Binary Intermetallics 341

20.1 General Remarks 341

20.2 Influence of Order-Disorder Transitions 344

20.3 B2 Intermetallics 346

20.3.1 Diffusion Mechanisms in B2 Phases 347

20.3.2 Example B2 NiAl 351

20.3.3 Example B2 Fe-Al 353

20.4 Li2 Intermetallics 355

20.5 D03 Intermetallics 357

20.6 Uniaxial Intermetallics 360

20.6.1 L10 Intermetallics 360

20.6.2 Molybdenum Disilicide (C11b structure) 362

20.7 Laves Phases 364

20.8 The Cu3Au Rule 366

References 367

21 Diffusion in Quasicrystalline Alloys 371

21.1 General Remarks on Quasicrystals 371

21.2 Diffusion Properties of Quasicrystals 373

21.2.1 Icosahedral Quasicrystals 374

21.2.2 Decagonal Quasicrystals 379

References 381

Part IV Diffusion in Semiconductors

22 General Remarks on Semiconductors 385

22.1 'Semiconductor Age' and Diffusion 386

22.2 Specific Features of Semiconductor Diffusion 389

References 392

23 Self-diffusion in Elemental Semiconductors 395

23.1 Intrinsic Point Defects and Diffusion 396

23.2 Germanium 398

23.3 Silicon 402

References 406

24 Foreign-Atom Diffusion in Silicon and Germanium 409

24.1 Solubility and Site Occupancy 409

24.2 Diffusivities and Diffusion Modes 412

24.2.1 Interstitial Diffusion 414

24.2.2 Dopant Diffusion 416

24.2.3 Diffusion of Hybrid Foreign Elements 420

24.3 Self-and Foreign Atom Diffusion - a Summary 421

References 422

25 Interstitial-Substitutional Diffusion 425

25.1 Combined Dissociative and Kick-out Diffusion 425

25.1.1 Diffusion Limited by the Flow of Intrinsic Defects 427

25.1.2 Diffusion Limited by the Flow of Interstitial Solutes 429

25.1.3 Numerical Analysis of an Intermediate Case 430

25.2 Kick-out Mechanism 431

25.2.1 Basic Equations and two Solutions 431

25.2.2 Examples of Kick-Out Diffusion 434

25.3 Dissociative Mechanism 439

25.3.1 Basic Equations 439

25.3.2 Examples of Dissociative Diffusion 440

References 445

Part V Diffusion and Conduction in Ionic Materials

26 Ionic Crystals 449

26.1 General Remarks 449

26.2 Point Defects in Ionic Crystals 451

26.2.1 Intrinsic Defects 452

26.2.2 Extrinsic Defects 454

26.3 Methods for the Study of Defect and Transport Properties 456

26.4 Alkali Halides 458

26.4.1 Defect Motion, Tracer Self-diffusion, and Ionic Conduction 458

26.4.2 Example NaCl 462

26.4.3 Common Features of Alkali Halides 467

26.5 Silver Halides AgCl and AgBr 468

26.5.1 Self-diffusion and Ionic Conduction 469

26.5.2 Doping Effects 471

References 473

27 Fast Ion Conductors 475

27.1 Fast Silver-Ion Conductors 477

27.1.1 AgI and related Simple Anion Structures 477

27.1.2 RbAg4I5 and related Compounds 479

27.2 PbF2 and other Halide Ion Conductors 480

27.3 Stabilised Zirconia and related Oxide Ion Conductors 481

27.4 Perovskite Oxide Ion Conductors 482

27.5 Sodium B-Alumina and related Materials 482

27.6 Lithium Ion Conductors 484

27.7 Polymer Electrolytes 485

References 488

Part VI Diffusion in Glasses

28 The Glassy State 493

28.1 What is a Glass? 493

28.2 Volume-Temperature Diagram 494

28.3 Temperature-Time-Transformation Diagram 496

28.4 Glass Families 498

References 501

29 Diffusion in Metallic Glasses 503

29.1 General Remarks 503

29.2 Structural Relaxation and Diffusion 506

29.3 Diffusion Properties of Metallic Glasses 509

29.4 Diffusion and Viscosity in Class-forming Alloys 517

References 518

30 Diffusion and Ionic Conduction in Oxide Glasses 521

30.1 General Remarks 521

30.2 Experimental Methods 526

30.3 Gas Permeation 529

30.4 Examples of Diffusion and Ionic Conduction 530

References 542

Part VII Diffusion along High-Diffusivity Paths and in Nanomaterials

31 High-diffusivity Paths in Metals 547

31.1 General Remarks 547

31.2 Diffusion Spectrum 548

31.3 Empirical Rules for Grain-Boundary Diffusion 549

31.4 Lattice Diffusion and Microstructural Defects 551

References 552

32 Grain-Boundary Diffusion 553

32.1 General Remarks 553

32.2 Grain Boundaries 554

32.2.1 Low-and High-Angle Grain Boundaries 555

32.2.2 Special High-Angle Boundaries 557

32.3 Diffusion along an Isolated Boundary (Fisher Model) 559

32.4 Diffusion Kinetics in Polycrystals 568

32.4.1 Type A Kinetics Regime 568

32.4.2 Type B Kinetics Regime 570

32.4.3 Type C Kinetics Regime 574

32.5 Grain Boundary Diffusion and Segregation 576

32.6 Atomic Mechanisms of Grain-Boundary Diffusion 579

References 580

33 Dislocation Pipe Diffusion 583

33.1 Disloction Pipe Model 584

33.2 Solutions for Mean Thin Layer Concentrations 586

References 591

34 Diffusion in Nanocrystalline Materials 593

34.1 General Remarks 593

34.2 Synthesis of Nancrystalline Materials 594

34.2.1 Powder Processing 594

34.2.2 Heavy Plastic Deformation 596

34.2.3 Chemical and Related Synthesis Methods 598

34.2.4 Devitrification of Amorphous Precursors 598

34.3 Diffusion in Poly - and Nanocrystals 599

34.3.1 Grain Size and Diffusion Regimes 599

34.3.2 Effective Diffusivities in Poly - and Nanocrystals 604

34.4 Diffusion in Nanocrystalline Metals 606

34.4.1 General Remarks 606

34.4.2 Structural Relaxation and Grain Growth 607

34.4.3 Nanomaterials with Bimodal Grain Structure 608

34.4.4 Grain Boundary Triple Junctions 612

34.5 Diffusion and Ionic Conduction in Nanocrystalline Ceramics 612

References 618

Index 639