Diffusion, Osmosis, Active Transport Diffusion Facilitated Diffusion
Diffusion in Condensed Matter - Springer978-3-540-30970-3/1.pdf · Diffusion in Condensed Matter....
Transcript of Diffusion in Condensed Matter - Springer978-3-540-30970-3/1.pdf · Diffusion in Condensed Matter....
Paul Heitjans · Jörg Kärger
Diffusion inCondensed MatterMethods, Materials, Models
With 448 Figures
ABC
Editors
Professor Dr. Paul HeitjansUniversität HannoverInstitut für Physikalische Chemieund ElektrochemieCallinstr. 3–3aD-30167 Hannover, GermanyEmail: [email protected]
Professor Dr. Jörg KärgerUniversität LeipzigInstitut für Experimentelle Physik ILinnéstr. 5D-04103 Leipzig, GermanyEmail: [email protected]
Library of Congress Control Number: 2005935206
ISBN -10 3-540-20043-6 Springer Berlin Heidelberg New YorkISBN -13 978-3-540-20043-7 Springer Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable for prosecution under the German Copyright Law.
Springer is a part of Springer Science+Business Mediaspringeronline.comc© Springer-Verlag Berlin Heidelberg 2005
Printed in The Netherlands
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.
Typesetting: by the authors and S. Indris using a Springer LATEX macro package
Cover design: Cover design: design &production GmbH, Heidelberg
Printed on acid-free paper SPIN: 10816487 56/3141/jl 5 4 3 2 1 0
Preface
Diffusion as the process of migration and mixing due to irregular movementof particles is one of the basic and ubiquitous phenomena in nature as wellas 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 mole-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.
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 diffusionin 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. Thecontents are roughly characterized by the headings Solids, Interfaces, Liq-uids, and Theoretical Concepts and Models of the four parts under which thechapters 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
VIII Preface
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 coverageof 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 ofDiffusion in Condensed Matter.
The experimental Methods include radiotracer and mass spectrometry,Moßbauer spectroscopy and nuclear resonant scattering of synchrotron ra-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 spectroscopyby 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, metallicglasses, 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., correlationfunctions, 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-
Preface IX
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 Han-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.As ever, the editors have to thank their wives, Maria Heitjans and BirgeKarger, for their patience and encouragement.
Hannover, Germany Paul HeitjansLeipzig, Germany Jorg KargerAugust 2005
Contents – Overview
Part I Solids
1 Diffusion: Introduction and Case Studies in Metals andBinary AlloysHelmut Mehrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The Elementary Diffusion Step in Metals Studied by theInterference of Gamma-Rays, X-Rays and NeutronsGero Vogl, Bogdan Sepiol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3 Diffusion Studies of Solids by Quasielastic NeutronScatteringTasso Springer, Ruep E. Lechner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4 Diffusion in SemiconductorsTeh Yu Tan, Ulrich Gosele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5 Diffusion in OxidesManfred Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6 Diffusion in Metallic Glasses and Supercooled MeltsFranz Faupel, Klaus Ratzke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Part II Interfaces
7 Fluctuations and Growth Phenomena in Surface DiffusionMichael C. Tringides, Myron Hupalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8 Grain Boundary Diffusion in MetalsChristian Herzig, Yuri Mishin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
9 NMR and β-NMR Studies of Diffusion in Interface-Dominated and Disordered SolidsPaul Heitjans, Andreas Schirmer, Sylvio Indris . . . . . . . . . . . . . . . . . . . . . 367
XII Contents – Overview
10 PFG NMR Studies of Anomalous DiffusionJorg Karger, Frank Stallmach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
11 Diffusion Measurements by UltrasonicsRoger Biel, Martin Schubert, Karl Ullrich Wurz, Wolfgang Grill . . . . . . 461
12 Diffusion in MembranesIlpo Vattulainen, Ole G. Mouritsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
Part III Liquids
13 Viscoelasticity and Microscopic Motion in Dense PolymerSystemsDieter Richter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
14 The Molecular Description of Mutual Diffusion Processesin Liquid MixturesHermann Weingartner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
15 Diffusion Measurements in Fluids by Dynamic LightScatteringAlfred Leipertz, Andreas P. Froba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
16 Diffusion in Colloidal and Polymeric SystemsGerhard Nagele, Jan K.G. Dhont, Gerhard Meier . . . . . . . . . . . . . . . . . . . 619
17 Field-Assisted Diffusion Studied by Electrophoretic NMRManfred Holz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
Part IV Theoretical Concepts and Models
18 Diffusion of Particles on LatticesKlaus W. Kehr, Kiaresch Mussawisade, Gunter M. Schutz, ThomasWichmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
19 Diffusion on FractalsUwe Renner, Gunter M. Schutz, Gunter Vojta . . . . . . . . . . . . . . . . . . . . . . 793
20 Ionic Transport in Disordered MaterialsArmin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer . . . . . 813
21 Concept of Mismatch and Relaxation for Self-Diffusionand Conduction in Ionic Materials with Disordered StructureKlaus Funke, Cornelia Cramer, Dirk Wilmer . . . . . . . . . . . . . . . . . . . . . . . 857
Contents – Overview XIII
22 Diffusion and Conduction in Percolation SystemsArmin Bunde, Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895
23 Statistical Theory and Molecular Dynamics of Diffusionin ZeolitesReinhold Haberlandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955
Contents – In Detail
Part I Solids
1 Diffusion: Introduction and Case Studies in Metals andBinary AlloysHelmut Mehrer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Continuum Description of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Fick’s Laws for Anisotropic Media . . . . . . . . . . . . . . . . . . . . . 41.2.2 Fick’s Second Law for Constant Diffusivity . . . . . . . . . . . . . 51.2.3 Fick’s Second Law for Concentration-Dependent Diffusivity 6
1.3 The Various Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Tracer Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Chemical Diffusion (or Interdiffusion) Coefficient . . . . . . . . 81.3.3 Intrinsic Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 Indirect Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Dependence of Diffusion on Thermodynamic Variables . . . . . . . . . 171.5.1 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5.2 Pressure Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Atomistic Description of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.6.1 Einstein-Smoluchowski Relation and Correlation Factor . . 191.6.2 Atomic Jumps and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 221.6.3 Diffusion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7 Interstitial Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.1 ‘Normal’ Interstitial Solutes . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.2 Hydrogen Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8 Self-Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8.1 Face-Centered Cubic Metals . . . . . . . . . . . . . . . . . . . . . . . . . . 321.8.2 Body-Centered Cubic Metals . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.9 Impurity Diffusion in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.9.1 ‘Normal’ Impurity Diffusion in fcc Metals . . . . . . . . . . . . . . 361.9.2 Slow Diffusion of Transition-Metal Solutes in Aluminium . 391.9.3 Fast Solute Diffusion in ‘Open’ Metals . . . . . . . . . . . . . . . . . 40
1.10 Self-Diffusion in Binary Intermetallics . . . . . . . . . . . . . . . . . . . . . . . . 42
XVI Contents – In Detail
1.10.1 Influence of Order-Disorder Transition . . . . . . . . . . . . . . . . . 431.10.2 Coupled Diffusion in B2 Intermetallics . . . . . . . . . . . . . . . . . 441.10.3 The Cu3Au Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.11 Interdiffusion in Substitutional Binary Alloys . . . . . . . . . . . . . . . . . 491.11.1 Boltzmann-Matano Method . . . . . . . . . . . . . . . . . . . . . . . . . . 491.11.2 Darken’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.11.3 Darken-Manning Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.12 Multiphase Diffusion in Binary Systems . . . . . . . . . . . . . . . . . . . . . . 531.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2 The Elementary Diffusion Step in Metals Studied by theInterference of Gamma-Rays, X-Rays and NeutronsGero Vogl, Bogdan Sepiol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.2 Self-Correlation Function and Quasielastic Methods . . . . . . . . . . . . 66
2.2.1 Quasielastic Methods: Moßbauer Spectroscopy andNeutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2.2 Nuclear Resonant Scattering of Synchrotron Radiation . . . 732.2.3 Neutron Spin-Echo Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 742.2.4 Non-Resonant Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.3.1 Pure Metals and Dilute Alloys . . . . . . . . . . . . . . . . . . . . . . . . 772.3.2 Ordered Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3 Diffusion Studies of Solids by Quasielastic NeutronScatteringTasso Springer, Ruep E. Lechner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2 The Dynamic Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3 The Rate Equation and the Self-Correlation Function . . . . . . . . . . 1023.4 High Resolution Neutron Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 1063.5 Hydrogen Diffusion in Metals and in Metallic Alloys . . . . . . . . . . . 1153.6 Diffusion with Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.7 Vacancy Induced Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.8 Ion Diffusion Related to Ionic Conduction . . . . . . . . . . . . . . . . . . . . 1263.9 Proton Diffusion in Solid-State Protonic Conductors . . . . . . . . . . . 1313.10 Proton Conduction: Diffusion Mechanism Based on a Chemical
Reaction Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393.11 Two-Dimensional Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.12 Coherent Quasielastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Contents – In Detail XVII
4 Diffusion in SemiconductorsTeh Yu Tan, Ulrich Gosele . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.2 Diffusion Mechanisms and Point Defects in Semiconductors . . . . . 1654.3 Diffusion in Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.3.1 Silicon Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.3.2 Interstitial-Substitutional Diffusion: Au, Pt and Zn in Si . 1684.3.3 Dopant Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724.3.4 Diffusion of Carbon and Other Group IV Elements . . . . . . 1774.3.5 Diffusion of Si Self-Interstitials and Vacancies . . . . . . . . . . . 1804.3.6 Oxygen and Hydrogen Diffusion . . . . . . . . . . . . . . . . . . . . . . 182
4.4 Diffusion in Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.5 Diffusion in Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.5.1 Native Point Defects and General Aspects . . . . . . . . . . . . . . 1854.5.2 Gallium Self-Diffusion and Superlattice Disordering . . . . . 1874.5.3 Arsenic Self-Diffusion and Superlattice Disordering . . . . . . 1944.5.4 Impurity Diffusion in Gallium Arsenide . . . . . . . . . . . . . . . . 1964.5.5 Diffusion in Other III-V Compounds . . . . . . . . . . . . . . . . . . 203
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5 Diffusion in OxidesManfred Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.2 Defect Chemistry of Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
5.2.1 Dominating Cation Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 2125.2.2 Dominating Oxygen Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.3 Self- and Impurity Diffusion in Oxides . . . . . . . . . . . . . . . . . . . . . . . . 2165.3.1 Diffusion in Oxides with Dominating Cation Disorder . . . . 2165.3.2 Diffusion in Oxides with Dominating Oxygen Disorder . . . 222
5.4 Chemical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.5 Diffusion in Oxides Exposed to External Forces . . . . . . . . . . . . . . . 228
5.5.1 Diffusion in an Oxygen Potential Gradient . . . . . . . . . . . . . 2295.5.2 Diffusion in an Electric Potential Gradient . . . . . . . . . . . . . 236
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2425.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
6 Diffusion in Metallic Glasses and Supercooled MeltsFranz Faupel, Klaus Ratzke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.2 Characteristics of Diffusion in Crystals . . . . . . . . . . . . . . . . . . . . . . . 2506.3 Diffusion in Simple Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516.4 General Aspects of Mass Transport and Relaxation in
Supercooled Liquids and Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
XVIII Contents – In Detail
6.5 Diffusion in Metallic Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596.5.1 Structure and Properties of Metallic Glasses . . . . . . . . . . . . 2596.5.2 Possible Diffusion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 2626.5.3 Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2656.5.4 Pressure Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2686.5.5 Effect of Excess Volume on Diffusion . . . . . . . . . . . . . . . . . . 269
6.6 Diffusion in Supercooled and Equilibrium Melts . . . . . . . . . . . . . . . 2706.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Part II Interfaces
7 Fluctuations and Growth Phenomena in Surface DiffusionMichael C. Tringides, Myron Hupalo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2857.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2857.2 Surface Diffusion Beyond a Random Walk . . . . . . . . . . . . . . . . . . . . 286
7.2.1 The Role of Structure and Geometry of the Substrate . . . 2867.2.2 The Role of Adsorbate-Adsorbate Interactions . . . . . . . . . . 2887.2.3 Diffusion in Equilibrium and Non-Equilibrium
Concentration Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2907.3 Equilibrium Measurements of Surface Diffusion . . . . . . . . . . . . . . . . 297
7.3.1 Equilibrium Diffusion Measurements from DiffractionIntensity Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7.3.2 STM Tunneling Current Fluctuations . . . . . . . . . . . . . . . . . . 3067.4 Non-Equilibrium Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
7.4.1 Uniform-Height Pb Islands on Si(111) . . . . . . . . . . . . . . . . . 3137.4.2 Measurements of Interlayer Diffusion on Ag/Ag(111) . . . . 320
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
8 Grain Boundary Diffusion in MetalsChristian Herzig, Yuri Mishin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3378.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3378.2 Fundamentals of Grain Boundary Diffusion . . . . . . . . . . . . . . . . . . . 338
8.2.1 Basic Equations of Grain Boundary Diffusion . . . . . . . . . . . 3388.2.2 Surface Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3398.2.3 Methods of Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3408.2.4 What Do We Know About Grain Boundary Diffusion? . . . 343
8.3 Classification of Diffusion Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3478.3.1 Harrison’s Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3488.3.2 Other Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
8.4 Grain Boundary Diffusion and Segregation . . . . . . . . . . . . . . . . . . . . 3538.4.1 Determination of the Segregation Factor from Grain
Boundary Diffusion Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Contents – In Detail XIX
8.4.2 Beyond the Linear Segregation . . . . . . . . . . . . . . . . . . . . . . . . 3578.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
9 NMR and β-NMR Studies of Diffusion in Interface-Dominated and Disordered SolidsPaul Heitjans, Andreas Schirmer, Sylvio Indris . . . . . . . . . . . . . . . . . . . . . 3679.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3679.2 Influence of Diffusion on NMR Spin-Lattice Relaxation and
Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3699.3 Basics of NMR Relaxation Techniques . . . . . . . . . . . . . . . . . . . . . . . . 3759.4 Method of β-Radiation Detected NMR Relaxation . . . . . . . . . . . . . 3809.5 Intercalation Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.5.1 Lithium Graphite Intercalation Compounds . . . . . . . . . . . . 3849.5.2 Lithium Titanium Disulfide – Hexagonal Versus Cubic . . . 386
9.6 Nanocrystalline Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3909.6.1 Nanocrystalline Calcium Fluoride . . . . . . . . . . . . . . . . . . . . . 3919.6.2 Nanocrystalline, Microcrystalline and Amorphous
Lithium Niobate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3949.6.3 Nanocrystalline Lithium Titanium Disulfide . . . . . . . . . . . . 3979.6.4 Nanocrystalline Composites of Lithium Oxide and Boron
Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3999.7 Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
9.7.1 Inhomogeneous Spin-Lattice Relaxation in Glasses withDifferent Short-Range Order . . . . . . . . . . . . . . . . . . . . . . . . . . 403
9.7.2 Glassy and Crystalline Lithium Aluminosilicates . . . . . . . . 4059.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4089.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
10 PFG NMR Studies of Anomalous DiffusionJorg Karger, Frank Stallmach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41710.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41710.2 The Origin of Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 41810.3 Fundamentals of PFG NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
10.3.1 The Measuring Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42110.3.2 The Mean Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42210.3.3 PFG NMR as a Generalized Scattering Experiment . . . . . 42410.3.4 Experimental Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
10.4 PFG NMR Diffusion Studies in Regular Pore Networks . . . . . . . . . 42710.4.1 The Different Regimes of Diffusion Measurement . . . . . . . . 42810.4.2 Intracrystalline Self-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 43010.4.3 Correlated Diffusion Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 43110.4.4 Transport Diffusion Versus Self-Diffusion . . . . . . . . . . . . . . . 43210.4.5 Single-File Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
XX Contents – In Detail
10.4.6 Diffusion in Ordered Mesoporous Materials . . . . . . . . . . . . . 43710.5 Anomalous Diffusion by External Confinement . . . . . . . . . . . . . . . . 439
10.5.1 Restricted Diffusion in Polystyrene Matrices . . . . . . . . . . . . 44010.5.2 Diffusion in Porous Polypropylene Membranes . . . . . . . . . . 44110.5.3 Tracing Surface-to-Volume Ratios . . . . . . . . . . . . . . . . . . . . . 444
10.6 Anomalous Diffusion due to Internal Confinement . . . . . . . . . . . . . 44710.6.1 Anomalous Segment Diffusion in Entangled Polymer
Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44810.6.2 Diffusion Under the Influence of Hyperstructures in
Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45010.6.3 Diffusion Under the Influence of Hyperstructures in
Polymer Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45310.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
11 Diffusion Measurements by UltrasonicsRoger Biel, Martin Schubert, Karl Ullrich Wurz, Wolfgang Grill . . . . . . 46111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46111.2 Diffusion of Hydrogen in Single-Crystalline Tantalum . . . . . . . . . . 46211.3 Observation of Diffusion of Heavy Water in Gels and Living
Cells by Scanning Acoustic Microscopy with Phase Contrast . . . . 46611.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
12 Diffusion in MembranesIlpo Vattulainen, Ole G. Mouritsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47112.2 Short Overview of Biological Membranes . . . . . . . . . . . . . . . . . . . . . 47312.3 Lateral Diffusion of Single Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 477
12.3.1 Lateral Tracer Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . 47712.3.2 Methods to Examine Lateral Tracer Diffusion . . . . . . . . . . . 47912.3.3 Lateral Diffusion of Lipids and Proteins . . . . . . . . . . . . . . . . 482
12.4 Rotational Diffusion of Single Molecules . . . . . . . . . . . . . . . . . . . . . . 49112.5 Lateral Collective Diffusion of Molecules in Membranes . . . . . . . . . 493
12.5.1 Fick’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49312.5.2 Decay of Density Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 49412.5.3 Relation Between Tracer and Collective Diffusion . . . . . . . 49512.5.4 Methods to Examine Lateral Collective Diffusion . . . . . . . . 49712.5.5 Lateral Collective Diffusion in Model Membranes . . . . . . . 498
12.6 Diffusive Transport Through Membranes . . . . . . . . . . . . . . . . . . . . . 50012.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
Contents – In Detail XXI
Part III Liquids
13 Viscoelasticity and Microscopic Motion in Dense PolymerSystemsDieter Richter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51313.2 The Neutron Scattering Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514
13.2.1 The Neutron Spin-Echo Technique Versus ConventionalScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
13.2.2 Neutron Spin Manipulations with Magnetic Fields . . . . . . 51613.2.3 The Spin-Echo Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
13.3 Local Chain Dynamics and the Glass Transition . . . . . . . . . . . . . . . 51913.3.1 Dynamic Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52113.3.2 Self-Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
13.4 Entropic Forces – The Rouse Model . . . . . . . . . . . . . . . . . . . . . . . . . . 52913.4.1 Neutron Spin-Echo Results in PDMS Melts . . . . . . . . . . . . 53113.4.2 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534
13.5 Long-Chains Reptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53713.5.1 Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53713.5.2 Experimental Observations of Chain Confinement . . . . . . . 538
13.6 Intermediate Scale Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54013.7 The Crossover from Rouse to Reptation Dynamics . . . . . . . . . . . . . 54313.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
14 The Molecular Description of Mutual Diffusion Processesin Liquid MixturesHermann Weingartner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55514.2 Experimental Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55814.3 Phenomenological Description of Mutual Diffusion . . . . . . . . . . . . . 55914.4 Thermodynamics of Mutual Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 56414.5 Linear Response Theory and Time Correlation Functions . . . . . . . 56714.6 The Time Correlation Function for Mutual Diffusion . . . . . . . . . . . 56914.7 Properties of Distinct-Diffusion Coefficients . . . . . . . . . . . . . . . . . . . 57114.8 Information on Intermolecular Interactions Deduced from
Diffusion Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57314.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
15 Diffusion Measurements in Fluids by Dynamic LightScatteringAlfred Leipertz, Andreas P. Froba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
XXII Contents – In Detail
15.2 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58015.2.1 Spectrum of Scattered Light . . . . . . . . . . . . . . . . . . . . . . . . . . 58015.2.2 Correlation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58215.2.3 Homodyne and Heterodyne Techniques . . . . . . . . . . . . . . . . 587
15.3 The Dynamic Light Scattering Experiment . . . . . . . . . . . . . . . . . . . 58915.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58915.3.2 Signal Statistics and Data Evaluation . . . . . . . . . . . . . . . . . . 594
15.4 Thermophysical Properties of Fluids Measured by DynamicLight Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59715.4.1 Thermal Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59715.4.2 Mutual Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 60015.4.3 Dynamic Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60115.4.4 Sound Velocity and Sound Attenuation . . . . . . . . . . . . . . . . 60415.4.5 Landau-Placzek Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60615.4.6 Soret Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60615.4.7 Derivable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
15.5 Related Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60815.5.1 Surface Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60815.5.2 Forced Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
15.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
16 Diffusion in Colloidal and Polymeric SystemsGerhard Nagele, Jan K.G. Dhont, Gerhard Meier . . . . . . . . . . . . . . . . . . . 61916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61916.2 Principles of Quasielastic Light Scattering . . . . . . . . . . . . . . . . . . . . 620
16.2.1 The Scattered Electric Field Strength . . . . . . . . . . . . . . . . . . 62016.2.2 Dynamic Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62416.2.3 Dynamic Structure Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
16.3 Heuristic Considerations on Diffusion Processes . . . . . . . . . . . . . . . 62816.3.1 Very Dilute Colloidal Systems . . . . . . . . . . . . . . . . . . . . . . . . 62916.3.2 Diffusion Mechanisms in Concentrated Colloidal Systems . 636
16.4 Fluorescence Techniques for Long-Time Self-Diffusion ofNon-Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66016.4.1 Fluorescence Recovery After Photobleaching . . . . . . . . . . . . 66116.4.2 Fluorescence Correlation Spectroscopy . . . . . . . . . . . . . . . . . 669
16.5 Theoretical and Experimental Results on Diffusion of ColloidalSpheres and Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67516.5.1 Colloidal Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67616.5.2 Polymer Blends and Random Phase Approximation . . . . . 697
16.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
Contents – In Detail XXIII
17 Field-Assisted Diffusion Studied by Electrophoretic NMRManfred Holz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71717.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71717.2 Principles of Electrophoretic NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
17.2.1 Electrophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71917.2.2 Pulsed Field Gradient NMR for the Study of Drift
Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72017.3 NMR in Presence of an Electric Direct Current. Technical
Requirements, Problems and Solutions . . . . . . . . . . . . . . . . . . . . . . . 72517.4 ENMR Sample Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72717.5 ENMR Experiments (1D, 2D and 3D) and Application Examples 728
17.5.1 1D ENMR Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72917.5.2 2D and 3D Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73417.5.3 Mobility and Velocity Distributions. Polydispersity and
Electro-Osmotic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73717.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
Part IV Theoretical Concepts and Models
18 Diffusion of Particles on LatticesKlaus W. Kehr, Kiaresch Mussawisade, Gunter M. Schutz, ThomasWichmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74518.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74518.2 One Particle on Uniform Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
18.2.1 The Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74818.2.2 Solution of the Master Equation . . . . . . . . . . . . . . . . . . . . . . 74918.2.3 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75118.2.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752
18.3 One Particle on Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 75318.3.1 Models of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75318.3.2 Exact Expression for the Diffusion Coefficient in d = 1 . . . 75518.3.3 Applications of the Exact Result . . . . . . . . . . . . . . . . . . . . . . 75718.3.4 Frequency Dependence in d = 1: Effective-Medium
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75818.3.5 Higher-Dimensional Lattices: Approximations . . . . . . . . . . . 76218.3.6 Higher-Dimensional Lattices: Applications . . . . . . . . . . . . . . 76618.3.7 Remarks on Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
18.4 Many Particles on Uniform Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 77118.4.1 Lattice Gas (Site Exclusion) Model . . . . . . . . . . . . . . . . . . . . 77118.4.2 Collective Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77218.4.3 Tracer Diffusion for d > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77318.4.4 Tagged-Particle Diffusion on a Linear Chain . . . . . . . . . . . . 774
18.5 Many Particles on Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . 778
XXIV Contents – In Detail
18.5.1 Models with Symmetric Rates . . . . . . . . . . . . . . . . . . . . . . . . 77818.5.2 Selected Results for the Coefficient of Collective
Diffusion in the Random Site-Energy Model . . . . . . . . . . . . 78018.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78318.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
18.7.1 Derivation of the Result for the Diffusion Coefficient forArbitrarily Disordered Transition Rates . . . . . . . . . . . . . . . . 784
18.7.2 Derivation of the Self-Consistency Condition for theEffective-Medium Approximation . . . . . . . . . . . . . . . . . . . . . 787
18.7.3 Relation Between the Relative Displacement and theDensity Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790
19 Diffusion on FractalsUwe Renner, Gunter M. Schutz, Gunter Vojta . . . . . . . . . . . . . . . . . . . . . . 79319.1 Introduction: What a Fractal is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79319.2 Anomalous Diffusion: Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . 79719.3 Stochastic Theory of Diffusion on Fractals . . . . . . . . . . . . . . . . . . . . 80219.4 Anomalous Diffusion: Dynamical Dimensions . . . . . . . . . . . . . . . . . 80319.5 Anomalous Diffusion and Chemical Kinetics . . . . . . . . . . . . . . . . . . 80619.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810
20 Ionic Transport in Disordered MaterialsArmin Bunde, Wolfgang Dieterich, Philipp Maass, Martin Meyer . . . . . 81320.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81320.2 Basic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816
20.2.1 Tracer Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81620.2.2 Dynamic Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81720.2.3 Probability Distribution and Incoherent Neutron
Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81720.2.4 Spin-Lattice Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818
20.3 Ion-Conducting Glasses: Models and Numerical Technique . . . . . . 81920.4 Dispersive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82220.5 Non-Arrhenius Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83220.6 Counterion Model and the “Nearly Constant Dielectric Loss”
Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83520.7 Compositional Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83920.8 Ion-Conducting Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843
20.8.1 Lattice Model of Polymer Electrolytes . . . . . . . . . . . . . . . . . 84320.8.2 Diffusion through a Polymer Network: Dynamic
Percolation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84620.8.3 Diffusion in Stretched Polymers . . . . . . . . . . . . . . . . . . . . . . . 849
20.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 850References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852
Contents – In Detail XXV
21 Concept of Mismatch and Relaxation for Self-Diffusionand Conduction in Ionic Materials with Disordered StructureKlaus Funke, Cornelia Cramer, Dirk Wilmer . . . . . . . . . . . . . . . . . . . . . . . 85721.1 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
10MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88321.10 Localised Motion at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . 88721.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892
22 Diffusion and Conduction in Percolation SystemsArmin Bunde, Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89522.1 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 . . . . . . 910
22.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913
23 Statistical Theory and Molecular Dynamics of Diffusionin ZeolitesReinhold Haberlandt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91523.2 Some Notions and Relations of Statistical Physics . . . . . . . . . . . . . 916
23.2.1 Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 91623.2.2 Statistical Theory of Irreversible Processes . . . . . . . . . . . . . 919
23.3 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92223.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92223.3.2 Procedure of an MD Simulation . . . . . . . . . . . . . . . . . . . . . . . 923
XXVI Contents – In Detail
23.3.3 Methodical Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92523.4 Simulation of Diffusion in Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
23.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92523.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92623.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928
23.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955