Springer heitjans p krger j (eds) diffusion in condensed matter methods materials models (springer 2005)(ISBN 3540200436)(971s)

Materials now dealt with are, among others, metals and alloys, metallic glasses, semiconductors, oxides, proton-, lithium- and other ion-conductors,nanocrystalline materials, micro- and

Paul Heitjans · Jörg Kärger

Diffusion in

Condensed Matter Methods, Materials, Models

With 448 Figures

ABC

Professor Dr Paul Heitjans

D-04103 Leipzig, Germany Email: kaerger@physik.uni-leipzig.de

Library of Congress Control Number: 2005935206

ISBN -10 3-540-20043-6 Springer Berlin Heidelberg New York

ISBN -13 978-3-540-20043-7 Springer Berlin Heidelberg New York

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Diffusion as the process of migration and mixing due to irregular movement

of particles is one of the basic and ubiquitous phenomena in nature as well

as in society In the latter case the word “particles” may stand for men orideas, and in the former for atoms or galaxies In this sense diffusion is atruly universal and transdisciplinary topic

The present book is confined, of course, to diffusion of atoms and cules As this process shows up in all states of matter over very large time andlength scales, the subject is still very general involving a large variety of nat-ural sciences such as physics, chemistry, biology, geology and their interfacialdisciplines Besides its scientific interest, diffusion is of enormous practicalrelevance for industry and life, ranging from steel making to oxide/carbondioxide exchange in the human lung

mole-It therefore comes as no surprise that the early history of the subject ismarked by scientists from diverse communities, e.g., the botanist R Brown(1828), the chemist T Graham (1833), the physiologist A Fick (1855), themetallurgist W.C Roberts-Austen (1896) and the physicist A Einstein (1905).Today, exactly 150 and 100 years after the seminal publications by Fick andEinstein, respectively, the field is flourishing more than ever with about 10.000scientific papers per year

From the foregoing it is evident that a single volume book on atomic andmolecular diffusion has to be further restricted in its scope As the title says,the book is confined to diffusion in condensed matter systems, so diffusion

in gases is excluded Furthermore, emphasis is on the fundamental aspects ofthe experimental observations and theoretical descriptions, whereas practi-cal considerations and technical applications have largely been omitted The

contents are roughly characterized by the headings Solids, Interfaces,

Liq-uids, and Theoretical Concepts and Models of the four parts under which the

chapters have been grouped

The book consists of 23 chapters written by leading researchers in theirrespective fields Although each chapter is independent and self-contained(using its own notation, listed at the end of the chapter), the editors havetaken the liberty of adding many cross-references to other chapters and sec-tions This has been facilitated by the common classification scheme Further

help to the reader in this respect is provided by an extended common list ofcontents, in addition to the contents overview, as well as an extensive subjectindex.

The book is a greatly enlarged (more than twice) and completely revisededition of a volume first published with Vieweg in 1998 Although the firstedition was very well received (and considered as a “must for students andworkers in the field”), it was felt that, in addition to the broad coverage

of modern methods, materials should also be discussed in greater detail inthe new edition The same applies to theoretical concepts and models This,

in fact, is represented by the new subtitle Methods, Materials, Models of

Diffusion in Condensed Matter.

The experimental Methods include radiotracer and mass spectrometry,

M¨oßbauer spectroscopy and nuclear resonant scattering of synchrotron diation, quasielastic neutron scattering and neutron spin-echo spectroscopy,dynamic light scattering and fluorescence techniques, diffraction and scan-ning tunneling microscopy in surface diffusion, spin relaxation spectroscopy

ra-by nuclear magnetic resonance (NMR) and beta-radiation detected NMR,NMR in a magnetic field gradient, NMR in the presence of an electric field,impedance spectroscopy and other techniques for measuring frequency de-pendent conductivities

Materials now dealt with are, among others, metals and alloys, metallic

glasses, semiconductors, oxides, proton-, lithium- and other ion-conductors,nanocrystalline materials, micro- and mesoporous systems, inorganic glasses,polymers and colloidal systems, biological membranes, fluids and liquid mix-tures The span from simple monoatomic crystals, with defects in thermalequilibrium enabling elementary jumps, to highly complex systems, exem-plarily represented by a biomembrane (cf Fig 12.3), is also indicated on thebook cover

Models in the subtitle stands for theoretical descriptions by, e g., correlation

functions, lattice models treated by (approximate) analytical methods, thetheory of fractals, percolation models, Monte Carlo simulations, molecular dy-namics simulations, phenomenological approaches like the counterion model,the dynamic structure model and the concept of mismatch and relaxation.Despite the large variety of topics and themes the coverage of diffusion incondensed matter is of course not complete and far from being encyclopedic.Inevitably, it reflects to a certain extent also the editors’ main fields of inter-est Nevertheless the chapters are believed to present a balanced selection.The book tries to bridge the transition from the advanced undergradu-ate to the postgraduate and active research stage Accordingly, the variouschapters are in parts tutorial, but they also lead to the forefront of currentresearch without intending to mimic the topicality of proceedings, which nor-mally have a short expiry date It is therefore designed as a textbook or refer-

ence work for graduate and undergraduate students as well as a source bookfor active researchers.

The invaluable technical help of Dr Sylvio Indris (University of nover) in the laborious editing of the chapters, which in some cases includedextensive revision, is highly acknowledged We also thank Jacqueline Lenzand Dr T Schneider from Springer-Verlag for accompanying this project

Han-As ever, the editors have to thank their wives, Maria Heitjans and BirgeK¨arger, for their patience and encouragement

August 2005

Part I Solids

1 Diffusion: Introduction and Case Studies in Metals and

Binary Alloys

Helmut Mehrer 3

2 The Elementary Diffusion Step in Metals Studied by the

Interference of Gamma-Rays, X-Rays and Neutrons

Gero Vogl, Bogdan Sepiol 65

3 Diffusion Studies of Solids by Quasielastic Neutron

6 Diffusion in Metallic Glasses and Supercooled Melts

Franz Faupel, Klaus R¨ atzke 249

Part II Interfaces

7 Fluctuations and Growth Phenomena in Surface Diffusion

Michael C Tringides, Myron Hupalo 285

8 Grain Boundary Diffusion in Metals

Christian Herzig, Yuri Mishin 337

9 NMR and β-NMR Studies of Diffusion in

Interface-Dominated and Disordered Solids

Paul Heitjans, Andreas Schirmer, Sylvio Indris 367

10 PFG NMR Studies of Anomalous Diffusion

J¨ org K¨ arger, Frank Stallmach 417

11 Diffusion Measurements by Ultrasonics

Roger Biel, Martin Schubert, Karl Ullrich W¨ urz, Wolfgang Grill 461

12 Diffusion in Membranes

Ilpo Vattulainen, Ole G Mouritsen 471

Part III Liquids

13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems

Dieter Richter 513

14 The Molecular Description of Mutual Diffusion Processes

in Liquid Mixtures

Hermann Weing¨ artner 555

15 Diffusion Measurements in Fluids by Dynamic Light

Scattering

Alfred Leipertz, Andreas P Fr¨ oba 579

16 Diffusion in Colloidal and Polymeric Systems

Gerhard N¨ agele, Jan K G Dhont, Gerhard Meier 619

17 Field-Assisted Diffusion Studied by Electrophoretic NMR

Manfred Holz 717

Part IV Theoretical Concepts and Models

18 Diffusion of Particles on Lattices

Klaus W Kehr, Kiaresch Mussawisade, Gunter M Sch¨ utz, Thomas

Wichmann 745

19 Diffusion on Fractals

Uwe Renner, Gunter M Sch¨ utz, G¨ unter Vojta 793

20 Ionic Transport in Disordered Materials

Armin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer 813

21 Concept of Mismatch and Relaxation for Self-Diffusion

and Conduction in Ionic Materials with Disordered Structure

Klaus Funke, Cornelia Cramer, Dirk Wilmer 857

22 Diffusion and Conduction in Percolation Systems

Armin Bunde, Jan W Kantelhardt 895

23 Statistical Theory and Molecular Dynamics of Diffusion

in Zeolites

Reinhold Haberlandt 915

List of Contributors 949 Index 955

Part I Solids

1 Diffusion: Introduction and Case Studies in Metals and

Binary Alloys

Helmut Mehrer 3

1.1 Introduction 3

1.2 Continuum Description of Diffusion 4

1.2.1 Fick’s Laws for Anisotropic Media 4

1.2.2 Fick’s Second Law for Constant Diffusivity 5

1.2.3 Fick’s Second Law for Concentration-Dependent Diffusivity 6 1.3 The Various Diffusion Coefficients 7

1.3.1 Tracer Diffusion Coefficients 7

1.3.2 Chemical Diffusion (or Interdiffusion) Coefficient 8

1.3.3 Intrinsic Diffusion Coefficients 10

1.4 Experimental Methods 10

1.4.1 Direct Methods 11

1.4.2 Indirect Methods 15

1.5 Dependence of Diffusion on Thermodynamic Variables 17

1.5.1 Temperature Dependence 17

1.5.2 Pressure Dependence 18

1.6 Atomistic Description of Diffusion 19

1.6.1 Einstein-Smoluchowski Relation and Correlation Factor 19

1.6.2 Atomic Jumps and Diffusion 22

1.6.3 Diffusion Mechanisms 23

1.7 Interstitial Diffusion in Metals 27

1.7.1 ‘Normal’ Interstitial Solutes 27

1.7.2 Hydrogen Diffusion 29

1.8 Self-Diffusion in Metals 31

1.8.1 Face-Centered Cubic Metals 32

1.8.2 Body-Centered Cubic Metals 34

1.9 Impurity Diffusion in Metals 35

1.9.1 ‘Normal’ Impurity Diffusion in fcc Metals 36

1.9.2 Slow Diffusion of Transition-Metal Solutes in Aluminium 39 1.9.3 Fast Solute Diffusion in ‘Open’ Metals 40

1.10 Self-Diffusion in Binary Intermetallics 42

1.10.1 Influence of Order-Disorder Transition 43

1.10.2 Coupled Diffusion in B2 Intermetallics 44

1.10.3 The Cu3Au Rule 47

1.11 Interdiffusion in Substitutional Binary Alloys 49

1.11.1 Boltzmann-Matano Method 49

1.11.2 Darken’s Equations 51

1.11.3 Darken-Manning Relations 52

1.12 Multiphase Diffusion in Binary Systems 53

1.13 Conclusion 56

References 60

2 The Elementary Diffusion Step in Metals Studied by the Interference of Gamma-Rays, X-Rays and Neutrons Gero Vogl, Bogdan Sepiol 65

2.1 Introduction 65

2.2 Self-Correlation Function and Quasielastic Methods 66

2.2.1 Quasielastic Methods: M¨oßbauer Spectroscopy and Neutron Scattering 68

2.2.2 Nuclear Resonant Scattering of Synchrotron Radiation 73

2.2.3 Neutron Spin-Echo Spectroscopy 74

2.2.4 Non-Resonant Methods 75

2.3 Experimental Results 77

2.3.1 Pure Metals and Dilute Alloys 77

2.3.2 Ordered Alloys 78

2.4 Conclusion 87

References 89

3 Diffusion Studies of Solids by Quasielastic Neutron Scattering Tasso Springer, Ruep E Lechner 93

3.1 Introduction 93

3.2 The Dynamic Structure Factor 94

3.3 The Rate Equation and the Self-Correlation Function 102

3.4 High Resolution Neutron Spectroscopy 106

3.5 Hydrogen Diffusion in Metals and in Metallic Alloys 115

3.6 Diffusion with Traps 121

3.7 Vacancy Induced Diffusion 124

3.8 Ion Diffusion Related to Ionic Conduction 126

3.9 Proton Diffusion in Solid-State Protonic Conductors 131

3.10 Proton Conduction: Diffusion Mechanism Based on a Chemical Reaction Equilibrium 139

3.11 Two-Dimensional Diffusion 143

3.12 Coherent Quasielastic Scattering 149

3.13 Conclusion 155

References 159

4 Diffusion in Semiconductors

Teh Yu Tan, Ulrich G¨ osele 165

4.1 Introduction 165

4.2 Diffusion Mechanisms and Point Defects in Semiconductors 165

4.3 Diffusion in Silicon 166

4.3.1 Silicon Self-Diffusion 166

4.3.2 Interstitial-Substitutional Diffusion: Au, Pt and Zn in Si 168 4.3.3 Dopant Diffusion 172

4.3.4 Diffusion of Carbon and Other Group IV Elements 177

4.3.5 Diffusion of Si Self-Interstitials and Vacancies 180

4.3.6 Oxygen and Hydrogen Diffusion 182

4.4 Diffusion in Germanium 183

4.5 Diffusion in Gallium Arsenide 184

4.5.1 Native Point Defects and General Aspects 185

4.5.2 Gallium Self-Diffusion and Superlattice Disordering 187

4.5.3 Arsenic Self-Diffusion and Superlattice Disordering 194

4.5.4 Impurity Diffusion in Gallium Arsenide 196

4.5.5 Diffusion in Other III-V Compounds 203

4.6 Conclusion 203

References 205

5 Diffusion in Oxides Manfred Martin 209

5.1 Introduction 209

5.2 Defect Chemistry of Oxides 210

5.2.1 Dominating Cation Disorder 212

5.2.2 Dominating Oxygen Disorder 215

5.3 Self- and Impurity Diffusion in Oxides 216

5.3.1 Diffusion in Oxides with Dominating Cation Disorder 216

5.3.2 Diffusion in Oxides with Dominating Oxygen Disorder 222

5.4 Chemical Diffusion 226

5.5 Diffusion in Oxides Exposed to External Forces 228

5.5.1 Diffusion in an Oxygen Potential Gradient 229

5.5.2 Diffusion in an Electric Potential Gradient 236

5.6 Conclusion 242

5.7 Appendix 243

References 245

6 Diffusion in Metallic Glasses and Supercooled Melts Franz Faupel, Klaus R¨ atzke 249

6.1 Introduction 249

6.2 Characteristics of Diffusion in Crystals 250

6.3 Diffusion in Simple Liquids 251

6.4 General Aspects of Mass Transport and Relaxation in Supercooled Liquids and Glasses 254

6.5 Diffusion in Metallic Glasses 259

6.5.1 Structure and Properties of Metallic Glasses 259

6.5.2 Possible Diffusion Mechanisms 262

6.5.3 Isotope Effect 265

6.5.4 Pressure Dependence 268

6.5.5 Effect of Excess Volume on Diffusion 269

6.6 Diffusion in Supercooled and Equilibrium Melts 270

6.7 Conclusion 276

References 278

Part II Interfaces 7 Fluctuations and Growth Phenomena in Surface Diffusion Michael C Tringides, Myron Hupalo 285

7.1 Introduction 285

7.2 Surface Diffusion Beyond a Random Walk 286

7.2.1 The Role of Structure and Geometry of the Substrate 286

7.2.2 The Role of Adsorbate-Adsorbate Interactions 288

7.2.3 Diffusion in Equilibrium and Non-Equilibrium Concentration Gradients 290

7.3 Equilibrium Measurements of Surface Diffusion 297

7.3.1 Equilibrium Diffusion Measurements from Diffraction Intensity Fluctuations 297

7.3.2 STM Tunneling Current Fluctuations 306

7.4 Non-Equilibrium Experiments 313

7.4.1 Uniform-Height Pb Islands on Si(111) 313

7.4.2 Measurements of Interlayer Diffusion on Ag/Ag(111) 320

7.5 Conclusion 331

References 333

8 Grain Boundary Diffusion in Metals Christian Herzig, Yuri Mishin 337

8.1 Introduction 337

8.2 Fundamentals of Grain Boundary Diffusion 338

8.2.1 Basic Equations of Grain Boundary Diffusion 338

8.2.2 Surface Conditions 339

8.2.3 Methods of Profile Analysis 340

8.2.4 What Do We Know About Grain Boundary Diffusion? 343

8.3 Classification of Diffusion Kinetics 347

8.3.1 Harrison’s Classification 348

8.3.2 Other Classifications 351

8.4 Grain Boundary Diffusion and Segregation 353

8.4.1 Determination of the Segregation Factor from Grain Boundary Diffusion Data 353

8.4.2 Beyond the Linear Segregation 357

8.5 Conclusion 359

References 364

9 NMR and β-NMR Studies of Diffusion in Interface-Dominated and Disordered Solids Paul Heitjans, Andreas Schirmer, Sylvio Indris 367

9.1 Introduction 367

9.2 Influence of Diffusion on NMR Spin-Lattice Relaxation and Linewidth 369

9.3 Basics of NMR Relaxation Techniques 375

9.4 Method of β-Radiation Detected NMR Relaxation 380

9.5 Intercalation Compounds 384

9.5.1 Lithium Graphite Intercalation Compounds 384

9.5.2 Lithium Titanium Disulfide – Hexagonal Versus Cubic 386

9.6 Nanocrystalline Materials 390

9.6.1 Nanocrystalline Calcium Fluoride 391

9.6.2 Nanocrystalline, Microcrystalline and Amorphous Lithium Niobate 394

9.6.3 Nanocrystalline Lithium Titanium Disulfide 397

9.6.4 Nanocrystalline Composites of Lithium Oxide and Boron Oxide 399

9.7 Glasses 402

9.7.1 Inhomogeneous Spin-Lattice Relaxation in Glasses with Different Short-Range Order 403

9.7.2 Glassy and Crystalline Lithium Aluminosilicates 405

9.8 Conclusion 408

9.9 Appendix 409

References 411

10 PFG NMR Studies of Anomalous Diffusion J¨ org K¨ arger, Frank Stallmach 417

10.1 Introduction 417

10.2 The Origin of Anomalous Diffusion 418

10.3 Fundamentals of PFG NMR 421

10.3.1 The Measuring Principle 421

10.3.2 The Mean Propagator 422

10.3.3 PFG NMR as a Generalized Scattering Experiment 424

10.3.4 Experimental Conditions 425

10.4 PFG NMR Diffusion Studies in Regular Pore Networks 427

10.4.1 The Different Regimes of Diffusion Measurement 428

10.4.2 Intracrystalline Self-Diffusion 430

10.4.3 Correlated Diffusion Anisotropy 431

10.4.4 Transport Diffusion Versus Self-Diffusion 432

10.4.5 Single-File Diffusion 434

10.4.6 Diffusion in Ordered Mesoporous Materials 437

10.5 Anomalous Diffusion by External Confinement 439

10.5.1 Restricted Diffusion in Polystyrene Matrices 440

10.5.2 Diffusion in Porous Polypropylene Membranes 441

10.5.3 Tracing Surface-to-Volume Ratios 444

10.6 Anomalous Diffusion due to Internal Confinement 447

10.6.1 Anomalous Segment Diffusion in Entangled Polymer Melts 448

10.6.2 Diffusion Under the Influence of Hyperstructures in Polymer Solutions 450

10.6.3 Diffusion Under the Influence of Hyperstructures in Polymer Melts 453

10.7 Conclusion 455

References 456

11 Diffusion Measurements by Ultrasonics Roger Biel, Martin Schubert, Karl Ullrich W¨ urz, Wolfgang Grill 461

11.1 Introduction 461

11.2 Diffusion of Hydrogen in Single-Crystalline Tantalum 462

11.3 Observation of Diffusion of Heavy Water in Gels and Living Cells by Scanning Acoustic Microscopy with Phase Contrast 466

11.4 Conclusion 468

References 469

12 Diffusion in Membranes Ilpo Vattulainen, Ole G Mouritsen 471

12.1 Introduction 471

12.2 Short Overview of Biological Membranes 473

12.3 Lateral Diffusion of Single Molecules 477

12.3.1 Lateral Tracer Diffusion Coefficient 477

12.3.2 Methods to Examine Lateral Tracer Diffusion 479

12.3.3 Lateral Diffusion of Lipids and Proteins 482

12.4 Rotational Diffusion of Single Molecules 491

12.5 Lateral Collective Diffusion of Molecules in Membranes 493

12.5.1 Fick’s Laws 493

12.5.2 Decay of Density Fluctuations 494

12.5.3 Relation Between Tracer and Collective Diffusion 495

12.5.4 Methods to Examine Lateral Collective Diffusion 497

12.5.5 Lateral Collective Diffusion in Model Membranes 498

12.6 Diffusive Transport Through Membranes 500

12.7 Conclusion 503

References 505

Part III Liquids

13 Viscoelasticity and Microscopic Motion in Dense Polymer Systems

Dieter Richter 513

13.1 Introduction 513

13.2 The Neutron Scattering Method 514

13.2.1 The Neutron Spin-Echo Technique Versus Conventional Scattering 516

13.2.2 Neutron Spin Manipulations with Magnetic Fields 516

13.2.3 The Spin-Echo Principle 518

13.3 Local Chain Dynamics and the Glass Transition 519

13.3.1 Dynamic Structure Factor 521

13.3.2 Self-Correlation Function 527

13.4 Entropic Forces – The Rouse Model 529

13.4.1 Neutron Spin-Echo Results in PDMS Melts 531

13.4.2 Computer Simulations 534

13.5 Long-Chains Reptation 537

13.5.1 Theoretical Concepts 537

13.5.2 Experimental Observations of Chain Confinement 538

13.6 Intermediate Scale Dynamics 540

13.7 The Crossover from Rouse to Reptation Dynamics 543

13.8 Conclusion 550

References 552

14 The Molecular Description of Mutual Diffusion Processes in Liquid Mixtures Hermann Weing¨ artner 555

14.1 Introduction 555

14.2 Experimental Background 558

14.3 Phenomenological Description of Mutual Diffusion 559

14.4 Thermodynamics of Mutual Diffusion 564

14.5 Linear Response Theory and Time Correlation Functions 567

14.6 The Time Correlation Function for Mutual Diffusion 569

14.7 Properties of Distinct-Diffusion Coefficients 571

14.8 Information on Intermolecular Interactions Deduced from Diffusion Data 573

14.9 Conclusion 576

References 577

15 Diffusion Measurements in Fluids by Dynamic Light Scattering Alfred Leipertz, Andreas P Fr¨ oba 579

15.1 Introduction 579

15.2 Basic Principles 580

15.2.1 Spectrum of Scattered Light 580

15.2.2 Correlation Technique 582

15.2.3 Homodyne and Heterodyne Techniques 587

15.3 The Dynamic Light Scattering Experiment 589

15.3.1 Setup 589

15.3.2 Signal Statistics and Data Evaluation 594

15.4 Thermophysical Properties of Fluids Measured by Dynamic Light Scattering 597

15.4.1 Thermal Diffusivity 597

15.4.2 Mutual Diffusion Coefficient 600

15.4.3 Dynamic Viscosity 601

15.4.4 Sound Velocity and Sound Attenuation 604

15.4.5 Landau-Placzek Ratio 606

15.4.6 Soret Coefficient 606

15.4.7 Derivable Properties 607

15.5 Related Techniques 608

15.5.1 Surface Light Scattering 608

15.5.2 Forced Rayleigh Scattering 613

15.6 Conclusion 615

References 617

16 Diffusion in Colloidal and Polymeric Systems Gerhard N¨ agele, Jan K G Dhont, Gerhard Meier 619

16.1 Introduction 619

16.2 Principles of Quasielastic Light Scattering 620

16.2.1 The Scattered Electric Field Strength 620

16.2.2 Dynamic Light Scattering 624

16.2.3 Dynamic Structure Factors 626

16.3 Heuristic Considerations on Diffusion Processes 628

16.3.1 Very Dilute Colloidal Systems 629

16.3.2 Diffusion Mechanisms in Concentrated Colloidal Systems 636 16.4 Fluorescence Techniques for Long-Time Self-Diffusion of Non-Spherical Particles 660

16.4.1 Fluorescence Recovery After Photobleaching 661

16.4.2 Fluorescence Correlation Spectroscopy 669

16.5 Theoretical and Experimental Results on Diffusion of Colloidal Spheres and Polymers 675

16.5.1 Colloidal Spheres 676

16.5.2 Polymer Blends and Random Phase Approximation 697

16.6 Conclusion 709

References 712

17 Field-Assisted Diffusion Studied by Electrophoretic NMR

Manfred Holz 717

17.1 Introduction 717

17.2 Principles of Electrophoretic NMR 719

17.2.1 Electrophoresis 719

17.2.2 Pulsed Field Gradient NMR for the Study of Drift Velocities 720

17.3 NMR in Presence of an Electric Direct Current Technical Requirements, Problems and Solutions 725

17.4 ENMR Sample Cells 727

17.5 ENMR Experiments (1D, 2D and 3D) and Application Examples 728 17.5.1 1D ENMR Applications 729

17.5.2 2D and 3D Experiments 734

17.5.3 Mobility and Velocity Distributions Polydispersity and Electro-Osmotic Flow 737

17.6 Conclusion 738

References 741

Part IV Theoretical Concepts and Models 18 Diffusion of Particles on Lattices Klaus W Kehr, Kiaresch Mussawisade, Gunter M Sch¨ utz, Thomas Wichmann 745

18.1 Introduction 745

18.2 One Particle on Uniform Lattices 748

18.2.1 The Master Equation 748

18.2.2 Solution of the Master Equation 749

18.2.3 Diffusion Coefficient 751

18.2.4 Extensions 752

18.3 One Particle on Disordered Lattices 753

18.3.1 Models of Disorder 753

18.3.2 Exact Expression for the Diffusion Coefficient in d = 1 755

18.3.3 Applications of the Exact Result 757

18.3.4 Frequency Dependence in d = 1: Effective-Medium Approximation 758

18.3.5 Higher-Dimensional Lattices: Approximations 762

18.3.6 Higher-Dimensional Lattices: Applications 766

18.3.7 Remarks on Other Models 769

18.4 Many Particles on Uniform Lattices 771

18.4.1 Lattice Gas (Site Exclusion) Model 771

18.4.2 Collective Diffusion 772

18.4.3 Tracer Diffusion for d > 1 773

18.4.4 Tagged-Particle Diffusion on a Linear Chain 774

18.5 Many Particles on Disordered Lattices 778

18.5.1 Models with Symmetric Rates 778

18.5.2 Selected Results for the Coefficient of Collective Diffusion in the Random Site-Energy Model 780

18.6 Conclusion 783

18.7 Appendix 784

18.7.1 Derivation of the Result for the Diffusion Coefficient for Arbitrarily Disordered Transition Rates 784

18.7.2 Derivation of the Self-Consistency Condition for the Effective-Medium Approximation 787

18.7.3 Relation Between the Relative Displacement and the Density Change 789

References 790

19 Diffusion on Fractals Uwe Renner, Gunter M Sch¨ utz, G¨ unter Vojta 793

19.1 Introduction: What a Fractal is 793

19.2 Anomalous Diffusion: Phenomenology 797

19.3 Stochastic Theory of Diffusion on Fractals 802

19.4 Anomalous Diffusion: Dynamical Dimensions 803

19.5 Anomalous Diffusion and Chemical Kinetics 806

19.6 Conclusion 809

References 810

20 Ionic Transport in Disordered Materials Armin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer 813

20.1 Introduction 813

20.2 Basic Quantities 816

20.2.1 Tracer Diffusion 816

20.2.2 Dynamic Conductivity 817

20.2.3 Probability Distribution and Incoherent Neutron Scattering 817

20.2.4 Spin-Lattice Relaxation 818

20.3 Ion-Conducting Glasses: Models and Numerical Technique 819

20.4 Dispersive Transport 822

20.5 Non-Arrhenius Behavior 832

20.6 Counterion Model and the “Nearly Constant Dielectric Loss” Response 835

20.7 Compositional Anomalies 839

20.8 Ion-Conducting Polymers 843

20.8.1 Lattice Model of Polymer Electrolytes 843

20.8.2 Diffusion through a Polymer Network: Dynamic Percolation Approach 846

20.8.3 Diffusion in Stretched Polymers 849

20.9 Conclusion 850

References 852

21 Concept of Mismatch and Relaxation for Self-Diffusion

and Conduction in Ionic Materials with Disordered Structure

Klaus Funke, Cornelia Cramer, Dirk Wilmer 857

21.1 Introduction 85721.2 Conductivity Spectra of Ion Conducting Materials 86121.3 Relevant Functions and Some Model Concepts for Ion Transport

in Disordered Systems 86421.4 CMR Equations and Model Conductivity Spectra 86721.5 Scaling Properties of Model Conductivity Spectra 87121.6 Physical Concept of the CMR 87421.7 Complete Conductivity Spectra of Solid Ion Conductors 87721.8 Ion Dynamics in a Fragile Supercooled Melt 88021.9 Conductivities of Glassy and Crystalline Electrolytes Below

10 MHz 88321.10 Localised Motion at Low Temperatures 88721.11 Conclusion 891References 892

22 Diffusion and Conduction in Percolation Systems

Armin Bunde, Jan W Kantelhardt 895

22.1 Introduction 89522.2 The (Site-)Percolation Model 89522.3 The Fractal Structure of Percolation Clusters near pc 89722.4 Further Percolation Systems 90122.5 Diffusion on Regular Lattices 90322.6 Diffusion on Percolation Clusters 90422.7 Conductivity of Percolation Clusters 90522.8 Further Electrical Properties 90622.9 Application of the Percolation Concept: Heterogeneous Ionic

Conductors 90822.9.1 Interfacial Percolation and the Liang Effect 90822.9.2 Composite Micro- and Nanocrystalline Conductors 91022.10 Conclusion 912References 913

23 Statistical Theory and Molecular Dynamics of Diffusion

in Zeolites

Reinhold Haberlandt 915

23.1 Introduction 91523.2 Some Notions and Relations of Statistical Physics 91623.2.1 Statistical Thermodynamics 91623.2.2 Statistical Theory of Irreversible Processes 91923.3 Molecular Dynamics 92223.3.1 General Remarks 92223.3.2 Procedure of an MD Simulation 923

23.3.3 Methodical Hints 92523.4 Simulation of Diffusion in Zeolites 92523.4.1 Introduction 92523.4.2 Simulations 92623.4.3 Results 92823.5 Conclusion 942References 944

List of Contributors 949 Index 955

Solids

Metals and Binary Alloys

Helmut Mehrer

1.1 Introduction

Diffusion in solids is an important topic of physical metallurgy and materialsscience Diffusion processes play a key role in the kinetics of many microstruc-tural changes that occur during processing of metals, alloys, ceramics, semi-conductors, glasses, and polymers Typical examples are nucleation of newphases, diffusive phase transformations, precipitation and dissolution of asecond phase, recrystallization, high-temperature creep, and thermal oxida-tion Direct technological applications concern, e.g., diffusion doping duringfabrication of microelectronic devices, solid electrolytes for battery and fuelcells, surface hardening of steel through carburization or nitridation, diffusionbonding, and sintering

The atomic mechanisms of diffusion in crystalline materials are closelyconnected with defects Point defects such as vacancies or interstitials are thesimplest defects and often mediate diffusion Dislocations, grain boundaries,phase boundaries, and free surfaces are other types of defects of crystallinesolids They can act as diffusion short circuits, because the mobility of atomsalong such defects is usually much higher than in the lattice

This chapter will concentrate on bulk diffusion in solid metals and alloys.Most of the solid elements are metals Furthermore, diffusion properties andatomic mechanisms of diffusion have most thoroughly been investigated inmetallic solids On the other hand, many of the physical concepts, which havebeen developed for metals, apply to diffusion in all crystalline solids Thoseeffects, which are unique to non-metallic systems such as charge effects inionic crystals and semiconductors, are treated in Chaps 4 and 5

For a comprehensive treatment of diffusion in solid matter the reader isreferred to the textbooks of Shewmon [1], Philibert [2], Heumann [3], Allnattand Lidiard [4], and Glicksman [5] A critical collection of data for diffusion inmetals and alloys was edited in 1990 by Mehrer [6] Recent developments can

be found in the proceedings of a series of international conferences on sion in Materials’ [7–9] The field of grain- and interphase-boundary diffusion

‘Diffu-is described in Chap 8 and in the book of Kaur, M‘Diffu-ishin and Gust [10] Thebook of Crank [11] provides an excellent introduction to the mathematics ofdiffusion

1.2 Continuum Description of Diffusion

1.2.1 Fick’s Laws for Anisotropic Media

The diffusion of atoms through a solid can be described by Fick’s equations

The first equation relates the diffusion flux j (number of atoms crossing a

unit area per second) to the gradient of the concentration c (number of atoms

per unit volume) via

The quantity D is denoted as diffusion coefficient tensor or as diffusivity

tensor The dimensions of its components are length2 time−1 Its SI units

are [m2s−1] Equation (1.1) implies thatD varies with direction In general

the diffusion flux and the concentration gradient are not always antiparallel.They are antiparallel for an isotropic medium

For anisotropic media and non-cubic crystalline solidsD is a symmetric

tensor of rank 2 [12] Each symmetric second rank tensor can be reduced todiagonal form The diffusion flux is antiparallel to the concentration gradient

only for diffusion along the orthogonal principal directions If x1, x2, x3denote

these directions and j1, j2, j3the pertaining components of the diffusion flux,(1.1) can be written as

where D I , D II , D III denote the three principal diffusivities The diffusion

coefficient for a direction (α1, α2, α3) is obtained from

D(α1, α2, α3) = α2D I + α2D II + α2D III , (1.3)

where α i denote the direction cosine of the diffusion flux with axis i Equation

(1.3) shows that anisotropic diffusion is completely described by the threeprincipal diffusion coefficients

For uniaxial (hexagonal, tetragonal, trigonal) crystals and decagonal

qua-sicrystals with the unique axis parallel to the x3 axis we have D I = D II =

D III and (1.3) reduces to

where Θ denotes the angle between diffusion direction and crystal axis For

isotropic media such as amorphous metals and inorganic glasses, cubic tals and icosahedral quasicrystals

Then the diffusion coefficient tensor is reduced to a scalar quantity

Steady state methods for measuring diffusion coefficients, like the ation method [3], are directly based on Fick’s first law In non-steady state

perme-situations diffusion flux and concentration vary with time t and position x.

In addition to Fick’s first law a balance equation is necessary For particleswhich undergo no reactions (no chemical reaction, no reactions between dif-ferent types of sites in a crystal, etc.) this is the continuity equation

1.2.2 Fick’s Second Law for Constant Diffusivity

In diffusion studies with trace elements very tiny amounts of the diffusingspecies can be applied Then the composition of the sample during the inves-

tigation does practically not change (see also Sect 1.3) and D is independent

of the tracer concentration Also diffusion in ideal solutions is described by aconcentration-independent diffusion coefficient Then for diffusion in a certain

the determination of D from measurements of the concentration distribution

as a function of position and time We consider two simple examples which,however, are often relevant for the analysis of experiments

Thin-Film Solution

If a thin layer of the diffusing species (M atoms per unit area) is concentrated

at x = 0 of a semi-infinite sample, the concentration after time t is described

Dt, called diffusion length, is a characteristic distance for

diffusion problems The experimental determination of diffusion coefficients

by the tracer method discussed in Sect 1.4 is based on (1.9) It is applicable

if√

Dt is much larger than the initial layer thickness.

Error Function Solution

Suppose that at t = 0 the concentration of the diffusing species is c(x, 0) = 0 for x > 0 Then, if for t > 0 the concentration at x = 0 is maintained at

c(0, t) = c s, the appropriate solution of (1.8) is

c(x, t) = c serfc

2√ Dt

dif-c s of that species at the surface It is, e.g., applicable to the in-diffusion of

a volatile solute into a non-volatile solvent, to the carburization of a metal

in a carbon containing ambient, and to in-diffusion of a solute from an

inex-haustible diffusion source into a solvent with solubility c s

1.2.3 Fick’s Second Law for Concentration-Dependent Diffusivity

Let us consider a case of great practical importance, in which the chemicalcomposition during diffusion varies over a certain concentration range Dif-fusing particles will experience different chemical environments and hencedifferent diffusion coefficients This situation is denoted as interdiffusion or

as chemical diffusion We use the symbol ˜D to indicate that the diffusion

co-efficient is concentration dependent ˜D is denoted as interdiffusion coefficient

or as chemical diffusion coefficient Fick’s second law (1.7) for diffusion in a

certain direction x then reads

depen-concentration-depth profiles will be considered in Sect 1.11 of this chapter

1.3 The Various Diffusion Coefficients

Diffusion in materials is characterized by several diffusion coefficients Inthis section we describe various experimental situations which entail differ-ent diffusion coefficients We will, however, concentrate on bulk diffusion inunary and binary systems Diffusion in ternary systems produces mathemat-ical complexities which are beyond the scope of this chapter We will focus

on bulk diffusion since diffusion along grain boundaries and along surfaces

is treated in Chaps 7 and 8 of this book In this section we will distinguishthe various diffusion coefficients by lower and upper indices We will drop theindices in the following sections again, whenever it is clear which diffusioncoefficient is meant

1.3.1 Tracer Diffusion Coefficients

In diffusion studies with trace elements (tagged by their radioactivity or

by their isotopic mass) tiny amounts of the diffusing species can be used.Although there will be a gradient in the concentration of the trace element,its total concentration can be kept so small that the overall composition ofthe sample during the investigation does practically not change1 In suchcases a constant tracer diffusion coefficient is appropriate for the analysis ofthe experiments

Self-Diffusion Coefficient

If the diffusion of A-atoms in a solid element A is studied, one speaks ofself-diffusion Studies of self-diffusion utilize a tracer isotope A∗ of the same

element A typical initial configuration for a tracer self-diffusion experiment

is illustrated in Fig 1.1a If the applied tracer layer is very thin as compared

to the average diffusion length, the tracer self-diffusion coefficient DA∗

A isobtained from such an experiment

The connection between the macroscopically defined tracer self-diffusioncoefficient and the atomistic picture of diffusion is the famous Einstein-Smoluchowski relation discussed in detail in Sect 1.6 In simple cases it reads

DA∗

A = f DE with DE = l

2

where l denotes the jump length and τ the mean residence time of an atom

on a certain site of the crystal2 The quantity f is the correlation factor For self-diffusion in cubic crystals f is a numeric factor Its value is characteristic

of the lattice geometry and the diffusion mechanism (see Sect 1.6) In some

textbooks the quantity DE is denoted as the Einstein diffusion coefficient3

In a homogeneous binary AXB1−X alloy or compound two tracer diffusion

coefficients for both, A∗ and B∗ tracer atoms, can be measured A typical

experimental starting configuration is displayed in Fig 1.1b We denote the

tracer diffusion coefficients by DA∗

Impurity Diffusion Coefficient

When the diffusion of a trace solute C∗ in a monoatomic solvent A or in

a homogeneous binary solvent AXB1−X (Fig 1.1) is measured, the tracer

1.3.2 Chemical Diffusion (or Interdiffusion) Coefficient

So far we have considered in this section cases where the concentration dient is the only cause for the flow of matter We have seen that such situ-ations can be studied using tiny amounts of trace elements in an otherwisehomogeneous material However, from a general viewpoint a diffusion flux isproportional to the gradient of the chemical potential

gra-The chemical potential of a species i in a binary alloy is given by (cf.

In (1.16) G denotes Gibbs free energy, n i the number of moles of species i,

T the temperature, and p the hydrostatic pressure The chemical potential

3This notation is a bit misleading, since the original Einstein-Smoluchowski lation relates the total macroscopic mean square displacement of atoms to thediffusion coefficient (see Sect 1.6), in which correlation effects are included

re-Fig 1.1.Initial configurations for diffusion experiments:

a) Thin layer of A∗on A: tracer diffusion in pure elements

b) Thin layer of A∗or B∗ on homogeneous alloy: tracer diffusion of alloy nents

compo-c) Thin layer of C∗on element A or homogeneous alloy: Impurity diffusion.d) Diffusion couple between metal-hydrogen alloy and a pure metal

e) Diffusion couple between pure end-members

f) Diffusion couple between two homogeneous alloys

depends on the alloy composition For ideal solutions the chemical potentialsare

i depend on T and p only In this case the gradient of the chemical

potential is directly proportional to the logarithmic gradient of the tration In non-ideal solutions the gradient of the chemical potential givesrise to an ‘internal’ driving force As a consequence the interdiffusion diffu-sion coefficient is concentration-dependent and Fick’s equation in the form

Pd-Interdiffusion results in a composition gradient in the diffusion zone diffusion profiles are analysed by the Boltzmann-Matano method or relatedprocedures This method will be described in Sect 1.11 It permits to deducethe concentration dependence of the interdiffusion coefficient

from the experimental diffusion profile

1.3.3 Intrinsic Diffusion Coefficients

The intrinsic diffusion coefficients (sometimes also component diffusion

coeffi-cients) DAand DBof a binary A-B alloy describe diffusion of the components

A and B relative to the lattice planes The diffusion rates of A and B atomsare usually not equal Therefore, in an interdiffusion experiment a net flux ofatoms across any lattice plane exists The shift of lattice planes with respect

to a sample fixed axis is denoted as Kirkendall effect, which is illustrated

in Fig 1.27 in Sect 1.11 The Kirkendall shift can be observed by rating inert markers at the initial interface of a diffusion couple This shiftwas for the first time observed for Cu/Cu-Zn diffusion couples by Kirkendalland coworkers [13] In the following decades work on many different alloysystems and a variety of markers demonstrated that the Kirkendall effect is

incorpo-a widespreincorpo-ad phenomenon of interdiffusion

The intrinsic diffusion coefficients DA and DB of a substitutional binaryA-B alloy are related to the interdiffusion coefficient ˜D and the marker veloc-

ity vK (Kirkendall velocity) These relations were deduced for the first time

by Darken [14] and refined later on by Manning [15] They will be discussed

in Sect 1.11 If the quantities ˜D and vK are known from experiment theintrinsic diffusion coefficients can be deduced

We emphasize that the intrinsic diffusion coefficients and the tracer

diffu-sion coefficients are different DAand DBpertain to diffusion in a composition

where Φ denotes the so-called thermodynamic factor (see Sect 1.11) In a

metal-hydrogen system ususally only the H atoms are mobile Then the trinsic diffusion coefficient and the chemical diffusion coefficient of hydrogenare identical

in-1.4 Experimental Methods

Methods for the measurement of diffusion coefficients can be grouped into twomajor categories: Direct methods are based on Fick’s laws and the phenom-enological definition of the diffusion coefficients given in Sect 1.3 Indirectmethods are not based directly on Fick’s laws Their interpretation requires amicroscopic model of the atomic jump processes and then uses the Einstein-Smoluchowski relation to deduce a diffusion coefficient

1.4.1 Direct Methods

Tracer Method

The tracer method is the most direct and most accurate technique for thedetermination of diffusion coefficients in solids Since very tiny amounts oftrace isotopes can be applied in tracer experiments, the chemical composi-tion of the sample is practically not influenced by the tracer In this wayself-diffusion and impurity diffusion can be studied in a material which ishomogeneous – apart from the tracer gradient

As indicated schematically in Fig 1.2 the tracer is usually deposited onto

a polished, flat surface of the diffusion sample Normally a radioactive isotope

of the investigated atomic species is used as tracer Enriched stable isotopeshave also been used in a few cases Evaporation, dripping of a liquid solution,and electrodeposition of the tracer onto the surface are common depositiontechniques Sometimes the tracer is ion-implanted as a thin layer below thesample surface in order to overcome disturbing surface oxide hold-up and sol-ubility problems [16] The sample is usually encapsulated in quartz ampoulesunder vacuum or inert (e.g Ar) atmosphere and an isothermal diffusion an-

neal is performed at temperature T for some diffusion time t For

tempera-tures below 1500 K quartz ampoules and resistance furnaces can be used Forhigher temperatures more sophisticated heating techniques are necessary.The best way to determine the resulting concentration-depth profile isserial sectioning of the sample and subsequent determination of the amount

of tracer per section

For average diffusion lengths of at least several ten micrometers ical sectioning techniques are applicable (for a review see, e.g., [17]) Lathesand microtomes are appropriate for ductile, grinding devices for brittle sam-

mechan-ples For extended diffusion anneals and diffusivities D > 10 −15m2s−1 lathe

sectioning is sufficient whereas diffusivities D > 10 −17m2s−1 are accessible

by microtome sectioning In favourable cases, grinder sectioning can be used

of the tracer, diffusion anneal, serial sectioning, and evaluation of the penetrationprofile – are indicated

Diffusion studies at lower temperatures require measurements of verysmall diffusivities Measurements of diffusion profiles with diffusion lengths

in the micrometer or nanometer range are possible using sputter ing techniques Devices for serial sectioning of radioactive diffusion samples

section-by ion-beam-sputtering are described in [18, 19] Such devices permit serialsectioning of shallow diffusion zones, which correspond to average diffusion

lengths between several nm and a few µm This implies that for

anneal-ing times of about 106 s a diffusivity range between D ≈ 10 −24m2s−1 and

D ≈ 10 −16m2s−1 can be examined.

Provided that the experimental conditions were chosen in such a waythat the deposited layer is thin compared with the mean diffusion length,the distribution after the diffusion anneal is described by (1.9) If radioactivetracers are used, the specific activity per section (count rate divided by thesection mass) is proportional to the tracer concentration The count rate

is conveniently determined by nuclear counting facilities (γ- or β-counting,

depending on the isotope) According to (1.9) a plot of the logarithm ofthe specific activity versus the penetration distance squared is linear, if bulkdiffusion is the dominating diffusion process Its slope equals −(4D ∗ t) −1.

From the slope and the diffusion time the tracer diffusivity D ∗ is obtained.

An obvious advantage of the tracer method is that a determination of theabsolute tracer concentration is not necessary

Fig 1.3 shows a penetration profile of the radioisotope59Fe in the metallic phase Fe3Si obtained by grinder sectioning [20] Gaussian behaviour

inter-as stated by (1.9) is observed over several orders of magnitude in tion An example for a penetration profile of59Fe in the intermetallic phase

concentra-Fe3Al obtained with the sputtering device described in [18] is displayed inFig 1.4 according to [21] From diffusion profiles of the quality of Figs 1.3and 1.4 diffusion coefficients can be determined with an accuracy of a fewpercent

Deviations from the Gaussian behaviour in experimentally determinedpenetration profiles may occur for many reasons We mention two frequentones:

– Grain boundaries in a polycrystal often act as diffusion short-circuits withenhanced mobility of atoms Grain boundaries usually cause a ‘grain-boundary tail’ in the deeper penetrating part of the profile In this ‘tail’region the concentration of the diffuser is enhanced with respect to merebulk diffusion

– Evaporation losses of the tracer itself and/or of the diffusion sample willcause deviations from Gaussian behaviour in the near-surface region.For a more detailed discussion of implications and pitfalls of the tracermethod the reader is referred to [17] The grain-boundary tails mentionedabove can be used for a systematic study of grain-boundary diffusion in bi-

or polycrystals as described in Chap 8 and in [10]

ra-dioisotope 59Fe in Fe3Si obtained by

grinder sectioning The solid line

rep-resents a fit of the thin-film solution of

Fick’s second law to the data points

ra-dioisotope 59Fe in Fe3Al obtained bysputter sectioning The solid line rep-resents a fit of the thin-film solution ofFick’s second law to the data points

In some cases several tracer isotopes of the same element are available.Differences between the isotopic masses lead to isotope effects in diffusion.Isotope effects are interesting phenomena although the differences betweendiffusivities of two isotopes of the same element are usually a few percentonly An exception is hydrogen with its three isotopes H, D, and T, whichhave significantly different masses (see Sect 1.7.2) Isotope effects of self- andsolute-diffusion in metals can contribute useful information about the atomicmechanisms of diffusion For a detailed discussion the reader is referred to [22–24] (see also Chap 6, Sect 6.5.3)

Other Profiling and Detection Methods

Several other profiling and detection methods can be used to measureconcentration-depth profiles We mention the more important ones:

1 Secondary Ion Mass Spectrometry (SIMS)

As already mentioned foreign elements or stable isotopes of the matrixcan be used as tracers in combination with SIMS for depth profiling SIMS

is mainly appropriate for the diffusion of foreign elements Contrary toself-diffusion studies by radiotracer experiments, in the case of stabletracers the natural abundance of the stable isotope in the matrix limitsthe concentration range of the diffusion profile Highly enriched isotopesshould be used An example of this technique can be found in a recent

Fig 1.5. fusion profile of a

Interdif-Fe70Al30–Fe50Al50

couple measured byEMPA

study of Ni self-diffusion in the intermetallic compound Ni3Al in whichthe highly enriched stable64Ni isotope was used [25] Average diffusion

lengths between several nm and several µm are accessible.

2 Electron Microprobe Analysis (EMPA)

In EMPA an electron beam of several tens of keV with a diameter ofabout one micrometer stimulates X-ray emission in the diffusion zone

of the sample The diffusion profile can be obtained by analysing theintensity of the characteristic radiation of the elements in a line scan alongthe diffusion direction The detection limit is about 10−3 to 10−4 mole

fractions depending on the element Light elements cannot be analysed

Because of the finite size of the excited volume (several µm3) only fairlylarge diffusion coefficients ≥ 10 −15m2s−1 can be measured EMPA is

mainly appropriate for interdiffusion studies An example of a phase interdiffusion profile resulting from a Fe70Al30–Fe50Al50 couple isshown in Fig 1.5 according to [26] The analysis of interdiffusion profiles

single-is dsingle-iscussed in Sect 1.11

3 Auger Electron Spectroscopy (AES)

AES in combination with sputter profiling can be used to measure

dif-fusion profiles in the range of several nm to several µm It is, however,

only applicable to diffusion of foreign atoms since AES only discriminatesbetween different elements

4 Rutherford Backscattering Spectrometry (RBS)

In RBS experiments a high-energy beam of monoenergetic α-particles is

used These particles are preferentially scattered by heavy nuclei in the

sample and the energy spectrum of scattered α-particles can be used to

determine the concentration-depth distribution of scattering nuclei Thistechnique is mainly suitable for detecting heavy elements in a matrix of

substantially lower atomic weight Due to the energy straggling of the

incident beam the profile depth is limited to less than a few µm.

5 Nuclear Reaction Analysis (NRA)

High energy particles can also be used to study diffusion of light elements,

if the nuclei undergo a suitable resonant nuclear reaction An example isdiffusion of boron in an alloy During irradiation with high energy pro-

tons α-particles are emitted from the nuclear reaction11B + p→8B + α.

The concentration profile of11B can be determined from the number and

energy of emitted α-particles as a function of the incident proton energy.

Like in RBS energy straggling limits the depth resolution of NRA.RBS and NRA methods need a depth calibration which is based on notalways very accurate data for the stopping power in the matrix for thoseparticles emitted by the nuclear reaction Also the depth resolution is usu-ally inferior to that achievable in careful radiotracer and SIMS profilingstudies

6 Field Gradient Nuclear Magnetic Resonance (FG NMR, PFG NMR)

Nuclear magnetic resonance (NMR) measurements in a magnetic fieldgradient (FG) or in a pulsed field gradient (PFG, see Chap 10) provide

a direct macroscopic method for diffusion studies In a magnetic fieldgradient the Larmor frequency of a nuclear moment depends on its po-sition FG NMR and PFG NMR utilize the fact that nuclear spins thatdiffuse in a magnetic field gradient experience an irreversible phase shift,which leads to a decrease in transversal magnetization This decrease can

be observed in so-called spin-echo experiments [27, 28] A measurement

of the diffusion-related part of the spin echo provides the diffusion efficient without any further hypothesis In contrast to tracer diffusion,

co-FG NMR and Pco-FG NMR techniques permit diffusion measurements inisotopically pure systems These techniques are applicable for relatively

large diffusion coefficients D
10−13m2s−1 [29].

1.4.2 Indirect Methods

Indirect methods are based on phenomena which are influenced by the fusive jumps of atoms Some of these methods are often sensitive to one or

dif-a few dif-atomic jumps only Qudif-antities like reldif-axdif-ation times, reldif-axdif-ation rdif-ates

or linewidths are measured Using a microscopic model of the jump processthe mean residence time of the diffusing species is determined and then viathe Einstein-Smoluchowski relation (see Sect 1.6) the diffusivity is deduced.Indirect methods can be grouped into two categories – relaxation methods(mechanical and magnetic) and nuclear methods

Relaxation Methods (Mechanical and Magnetic)

Mechanical relaxation methods make use of the fact that atomic motion in

a material can be induced by external influences such as the application

of constant or oscillating mechanical stress In ferromagnetic materials theinteraction between the magnetic moments and local order can give rise tovarious relaxation phenomena similar to those observed in anelasticity Agreat variety of experimental devices have been used for such studies Theirdescription is, however, beyond the scope of this chapter

Some of the more important relaxation phenomena related to diffusionare the following [30–32]:

The Snoek effect is observed in bcc metals which contain interstitial

solutes such as C, N, or O These solutes occupy octahedral or tetrahedralinterstitial sites These sites have tetragonal symmetry, which is lower thanthe cubic symmetry of the matrix Therefore the lattice distortions caused byinterstitial solutes give rise to elastic dipoles Under the influence of externalstress these dipoles can reorient (para-elasticity) The reorientation of solutesgives rise to a strain relaxation or an internal friction peak The relaxationtime or the (frequency or temperature) position of the internal friction peakcan be used to deduce information about the mean residence time of a solute

A Snoek effect of interstitial solutes in fcc metals cannot be observed, becausethe interstitial sites have cubic symmetry

The Gorski effect is due to solutes in a solvent which produce a lattice

dilatation In a macroscopic strain gradient solutes redistribute by diffusion.This redistribution gives rise to an anelastic relaxation The Gorski effect

is detectable if the diffusion coefficient of the solute is high enough Gorskieffect measurements have been widely used for studies of hydrogen diffusion

in metals [30]

In substitutional A-B alloys the reorientation of solute-solvent pairs under

the influence of stress can give rise to an anelastic relaxation called Zener

effect.

Nuclear Methods

Examples of nuclear methods are NMR, M¨oßbauer spectroscopy (MBS), andquasielastic neutron scattering (QENS) Since MBS, QENS, NMR and PFGNMR are the subjects of the Chaps 2, 3, 9 and 10 and QENS also of a recenttextbook [33] we confine ourselves here to a few remarks:

The width of the resonance line and the spin-lattice relaxation rate T −1

1

in NMR have contributions which are due to the thermally activated jumps

of atoms Measurements of the ’diffusional narrowing’ of the linewidth or

of T −1

1 as a function of temperature permit a determination of the mean

residence time τ of the atoms NMR methods are mainly appropriate for

self-diffusion measurements on solid or liquid metals In favourable cases (e.g

Li and Na) self-diffusion coefficients between 10−18 and 10−10 m2s−1 are

accessible (see [6]) In the case of foreign atom diffusion, NMR studies sufferfrom the fact that a signal from nuclear spins of the minority component must

be detected Nevertheless, detailed studies were conducted, e g., in the case

of Li-based Li-Mg and Li-Ag alloys via the spin-lattice relaxation of polarizedradioactive8Li nuclei [34]

The linewidth in MBS and QENS both have a contribution ∆Γ which is

due to the diffusion jumps of atoms This diffusion broadening can be

ob-served only in systems with fairly high diffusivities since ∆Γ must at least

be comparable with the natural linewidth in MBS experiments or with theenergy resolution of the neutron spectrometer in QENS experiments Appro-priate probes for MBS must be available The usual working horse in MBS

is the isotope 57Fe although there are a few other M¨oßbauer isotopes able (e.g.119Sn,115Eu,161Dy) MBS has been mainly used to study fast Fediffusion QENS experiments are suitable for fast diffusing elements with asizable incoherent scattering cross section for neutrons Examples are hydro-gen diffusion in metals or hydrides and Na self-diffusion (see Chap 3).Neither MBS nor QENS are routine methods for diffusion measurements.The most interesting aspect is that these methods can provide microscopicinformation about the elementary jump process of atoms For single crystals

avail-∆Γ depends on the crystal orientation This orientation dependence can be

used to deduce information about the jump direction and about the jumplength (see Chaps 2 and 3), which is not accessible by conventional diffusionstudies

For a more comprehensive discussion of experimental methods for thedetermination of diffusion coefficients we refer the reader to the already men-tioned textbooks on diffusion [1–3] and to Chap 1 in [6] as well as to a recentarticle [29] where also an overview of the accessible windows for the mean

residence time τ are given.

1.5 Dependence of Diffusion on Thermodynamic

Variables

So far we have said nothing about the dependence of diffusion processes upon

thermodynamic variables, i.e upon temperature T and hydrostatic pressure

p In binary systems also variations of the diffusivity with the variable

’compo-sition’ are of interest These variations can range from very slight to striking.They will be not considered in this section since they depend very much onthe system Examples can be found in Sect 1.10

1.5.1 Temperature Dependence

It is well known that diffusion coefficients in solids generally depend ratherstrongly on temperature, being low at low temperatures but appreciable at

high temperatures Empirically, measurements of diffusion coefficients over acertain temperature range may be often, but by no means always, described

by an Arrhenius relation4

D = D0exp

− ∆H RT

where ∆S is the diffusion entropy and D 

0 contains geometric factors, thecorrelation factor, the lattice parameter squared, and an attempt frequency

of the order of the Debye frequency

In an Arrhenius diagram the logarithm of the diffusivity is plotted versus

the reciprocal temperature T −1 For a diffusion process with a

temperature-independent activation enthalpy ∆H the Arrhenius diagram is a straight line

with slope −∆H/R From its intercept for T −1 → 0 the pre-exponential

factor D0 can be deduced Such simple Arrhenius behaviour should, ever, not be considered to be universal Departures from it may arise formany reasons, ranging from fundamental aspects of the atomic mechanism,temperature dependent activation parameters6, effects associated with im-purities or microstructural features such as grain boundaries Nevertheless,(1.20) provides a very useful standard

how-1.5.2 Pressure Dependence

The variation of diffusivity with hydrostatic pressure p is far less pronounced

than with temperature Usually the diffusivity decreases as the pressure is

4Equation (1.20) is often also written as

D = D0exp − ∆H

kBT

.

If the first version of the Arrhenius equation is used the unit of ∆H is kJ mol −1

If the second version is used the appropriate unit of ∆H is eV per atom Note

that 1 eV per atom = 96.472 kJ mol−1 The gas constant R and the Boltzmann constant kBare related via R = NAkB = 8.314 ×10 −3kJ mol−1K−1 , where N

A

denotes the Avogadro constant

5The symbol Q for the activation enthalpy is also widely used in the literature.