extrait de la publication…madeleine meyer and vassili pontikis for prepar- ing the figure. extrait...

Extrait de la publication

Atom movements

Diffusion and mass transport

in solids

Jean PHILIBERT Professor of Materials Science

Université de Paris-Sud

Panslated from the French by

Steven J. Rothman Metallurgist, Argonne National Laboratory

PREFACE by

David Lazarus University of Illinois

le3 éditions

Avenue du Hoggar, Zone Industrielle de Courlaboeuf,

B.P. 112, F-91944 Les Ulis Cedex A, France


Tous droits de traduction, d’adaptation et de reproduction par tous procédés, réservés pour tous pays. La Loi du 11 mars 1957 n’autorisant, aux termes des alinéas 2 et 3 de l’article 41, d’une part, que les “copies ou reproductions strictement réservées à l’usage privé du copiste et non destinées à une utilisation collective”, et d’autre part, que les analyses et les courtes citations dans un but d’exemple et d’illustration, “toute représentation intégrale, ou partielle, faite sans le consentement de l’auteur ou de ses ayants droit ou ayants cause est illicite” (alinéa le‘ de l’article 40). Cette représentation ou reproduction, par quelque procédé que ce soit, constituerait donc une contrefaçon sanctionnée par les articles 425 et suivants du code pénal.

@ Les Éditions de Physique 1991

“To explain tha t which is visible b u t complicated by t h a t which is invisible b u t simple ...”

Jean Perrin, in preface t o Les Atomes (1912)


a

b

C

Diffusion of an a.datom on a. (1 10) surface of a fcc crystal by the, exchange mechanism:

a ) Ada tom i n initial position t = t o

b) Saddle point position t = 2 0 + 6 x s

c) Final positmion t = t o + 10 x 10-l2 s

T h e figures show a n “instantaneous” view of two atomic layers, viewed along a direction t h a t makes an angle of 20’ with t h e (110) plane ; each plane contain six <110> strings of eight a toms.

T h e a t o m coordinates were calculated by a molecular dynamics simulation using a Lennard-Jones potential with parameters corresponding t o solid argon a t 0.4 T,. (see G. de Lorenzi el al., reference a t end of Ch. VI.)

T h e au tho r thanks Drs. Madeleine Meyer and Vassili Pontikis for prepar- ing the figure.


Preface

As I write this preface, in January 1989, it is hard for me to believe that a full 23 years have passed since the publication of “LA DIFFUSION DANS LES SOLIDES’’ (Presses Universitaires de France, Paris, 1966). This glorious two-volume work by Yves Adda and Jean Philibert was, until very recently, the basic “bible” for all serious scientists working in the field of diffusion in solids. In 1985 Professor Philibert published a condensed, updated version, suitable as a textbook for advanced students of materials science or solid- state physics : “DIFFUSION E T TRANSPORT DE MATIERE DANS LES SOLIDES’ (Monographies de Physique, les Editions de Physique, Paris, 1985).

Unfortunately, the world includes fewer francophones than persons who wish to, or should, enter into the serious study of the field of solid-state diffusion- an area which is absolutely fundamental to understanding a virtual cornucopia of important phenomena in materials science: nucleation, crystal growth, sintering, hardening, alloying, phase transformations, oxidation, plastic flow, fracture, photography ...... the list is almost endless. Thus, many not raised with a sufficient knowledge of French, (including most of my own graduate students over two decades) have either had to learn enough French to wade slowly and painfully through the Adda-Philibert “bible,” or, far worse, had no access at all to this most important reference.

Finally, a miracle has occured : Dr. S. J . Rothman of Argonne National Laboratory, not only a fluent francophone but also a scientist who himself has made enormous contributions to the field of solid-state diffusion, has made an English-language translation of Professor Philibert’s 1985 text, now entitled “ATOM MOVEMENTS”. Moreover, the new edition has been updated in important ways and includes an extensive set of extremely practical homework exercises to help the serious reader master the field in a professional manner. This, if I may steal a line from Shakespeare, is “ ... a consummation devoutly to be wished.”

The most wonderful aspects of the original Adda-Philibert “bible” are faithfully preserved in Professor Philibert’s French-language 1985 book and again in this English-language edition. This is a work of love by a scientist who understands the field thoroughly and deeply, from its fundamental atomistic aspects to the most practical of its “real-world’’ applications. The selection of topics is superb, and the treatment of each subject is thorough and complete, appropriate iii level for advanced undergraduate or graduate students, as well as active research workers, who demand a thorough grounding in this vital area.

Thus, through the joint efforts of Jean Philibert and Steve Rothman, we finally have available “ATOM MOVEMENTS”, a superb basic text in English,

VI Atom movements

which should be “required reading” for serious students of diffusion throughout the world. My one sadness is that it comes too late for my own graduate students (I have now retired from active research), but then, I can always console myself with the thought that by forcing them ta learn enough French to read the “bible,” I also made it possible for them to enjoy much more fruitful visits to France themselves in their post-student lives!

As a final and personal note, I want to express my own sincere thanks to my old and dear friends and colleagues, Jean Philibert, who wrote the new book, and Yves Adda, who joined with Jean in writing the original “bible,” for all that they have done for the field of solid-state diffusion, in general, and for me and my own research programs over the past decades. Their books, as well as their own vital and basic scientific work in this field, will endure for generations. I am delighted that their work, through this English-language edition, will now be more widely available.

David Lazarus Loomis Laboratory of Physics The University of Illinois Urbana, Illinois, USA


Fore wo r d

This book was written to remedy a deficiency: at this time, an elementary text on diffusion in solids does not exist either in French or in English. On the other hand, literature for specialists at an advanced level is abundant ; during the last fifteen years, a number of colloquia and workshops have resulted in publications, many of which resemble review articles. Still, there is no first book that would prepare a graduate student or beginning researcher to use these review articles or the original literature fruitfully.

The present book is the result of diverse courses on diffusion. It is intended to give as complete an overview as possible of diffusion in solid media, while relating the processes of diffusion to both their physical bases and their applications. In this spirit, certain fundamental aspects of these processes, such as the calculation of correlation factors or the theory of the atomic jump, which require long mathematical derivations, have been considered only on an elementary level, with the important results given without proof. However, when a simple approach was possible, the important relations have been derived, but concentrating more on the physics than on the mathematical formalism.

A series of a real situations is covered in this account, from self-diffusion of radiotracers to the more complex cases of mass flow under chemical or thermal gradients or under electric fields, or diffusion in structures of lower dimension- ality (surfaces and interfaces). In all these analyses, no category of materials was favored ; metals, ionic crystals, oxides, and semiconductors all had their turn. Only polymers were not specifically touched. One chapter is specifically devoted to techniques for studying diffusion, including methods of numerical simulation, and a last and long chapter gives a number of metallurgical phenomena in which diffusion plays a fundamental role.

In the spirit of the book, neither a review of experimental results nor an exhaustive bibliography has been given. Only a few typical results, with their references, are given to illustrate important points. The rest of the bibliography lists references to books and review articles which allow the reader to penetrate the subject more deeply before going to the original literature.

This work is addressed first of all to graduate students, but may serve a larger audience in allowing researchers to refresh their memories on some points of diffusion. They will grasp that the points of view, the approaches, of this apparently classical subject have recently experienced a significant evolution, as shown in the series of colloquia held over the last fifteen years and cited in

VI11 Atom movements

the general bibliography. The background is classical ; the new perspectives open with new materials.

May this small book inspire the reader to futher research and renewal in a field in which several laboratories in our country have long been active.

* * *

The author thanks all those who, by reading a part of the manuscript and by discussion have helped him to clarify a number of points. His thanks go equally to the secretaries who had to face a difficult stenographic task, and especially to Mrs. Marie-Claire Dolou, who took care of a large part of this reproduction, as well as to the publisher, Editions de Physique, who have lavished much care on the production of this volume.

J. Philibert


Foreword to the English Edition

The good reception given this work in the scientific community and the urging of several colleagues have encouraged the author to prepare an English edition.

The title chosen for this edition, evoking that of the AÇM seminar published in 1952, is intended to indicate the aim of this book: to understand the processes encountered in Materials Science which are governed by the movement of atoms. As for the text itself, it has been revised, expanded, and corrected, and, last but not least, a set of exercices of various levels of difficulty has been added.

The author wishes to thank all those who have made suggestions about the book, and especially the translator ; his many suggestions have considerably improved the original text, so that it may be of even better service to its readers.

J . Philibert, October 1988

Translator’s Acknowledgments

Dr. Charles Wiley and Prof. Jean Philibert read the translation manuscript ; I thank them for their many excellent suggestions, which helped greatly to improve the clarity of the translation. I especially thank Prof. Philibert for his constant friendly encouragement. I am grateful to Dr. David Price for reading and correcting the parts on neutron diffraction, and to Drs. Alex McKale and Nestor Zalucec for assistance with word processing. I thank my wife, Ms. Barbara Rothman, for her frequent suggestions of the right word or the correct grammar, and for her patience and support during the course of this work.

S. J . Rothman, October 1988

TABLE OF CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . V Foreword . . . . . . . . . . . . . . . . . . . . . . . . . VI1 Foreword to the English edition . . . . . . . . . . . . . . IX Translator’s acknowledgments . . . . . . . . . . . . . . . IX General Bibliography . . . . . . . . . . . . . . . . . . . XIX Notation . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII

CHAPTER I: DIFFUSION AND DRIFT . . . . . . . . . 1

I . Flux of particles. Fick’s equation . . . . . . . . . . . . . . 1 II . Time-dependent case . . . . . . . . . . . . . . . . . . . 2

III . Solutions of the diffusion equation (or Fick’s second law) . . 5 111.1 Thin layer or instantaneous source 111.2 Constant surface concentration (diffusion in a

111.3 Infinite initial distribution 111.4 The Boltzmann transformation 111.5 Concentration-dependent diffusion coefficient

sern-infinite solid)

IV. Relation between drift and diffusion. The Nernst-Einstein equation . . . . . . . . . . . . . . . . . . . . . . . . 13

VII. Diffusion with phase change. Multiphase diffusion . . . . . . 22

V. The nature of the driving force . . . . . . . . . . . . . . . 14 VI. A variety of diffusion processes and generalization of Fick’s law 16

APPENDIX I: Methods for solving the diffusion equation . . . . 26 APPENDIX II: Diffusion in three dimensions . . . . . . . . . . 29 APPENDIX III: Conservation at amoving boundary . . . . . . . 30

CHAPTER II: ATOMIC THEORY OF DIFFUSION . . . 33

I. A simplified model . . . . . . . . . . . . . . . . . . . . 33 II. General theory of random walk . . . . . . . . . . . . . . 36

for the diffusion coefficient . . . . . . . . . . . . . . . . . 39 45 46

III. Expressions for the mean-square displacement ( X ’ ) and

IV. Diffusion in the presence of a driving force V . Explicit form of the function W (X, r )

. . . . . . . . . . . . . . . . . . . .


XII Atom movements

VI. Variable jump distance . . . . . . . . . . . . . . . . . . VII. Correlation functions . . . . . . . . . . . . . . . . . . .

VII.l Characterization of the structure in a non-crystalline

V11.2 Diffusion medium

VIII. Limitations of Fick’s law APPENDIX: Some definitions . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

CHAPTER III: DIFFUSION MECHANISMS AND CORRELATION EFFECTS . . . . . . . . . . . .

I. Mechanisms of diffusion . . . . . . . . . . . . . . . . . . 1.1 Direct interchange 1.2 Mechanisms involving point defects

II. Definition of the correlation factor . . . . . . . . . . . . . III. The encounter model . . . . . . . . . . . . . . . . . . . IV. A simple simulation of self-diffusion and electromigration . . . V. Methods of calculating the correlation factor . . . . . . . . .

VI. Types of correlation factors . . . . . . . . . . . . . . . . VJ.1 Dynamic correlations VI.2 Physical correlation VI.3 Meaning of the physical correlation factor VI.4 Compounds with a high concentration of defects

VII. 1 The potential-barrier model VII.2 More refined models VII.3 The isotope effect VII.4 Numerical simulation VII.5 Some simple applications of the potential-barrier model

APPENDIX II: Percolation . . . . . . . . . . . . . . . . . .

VII. Migration ofpoint defects. Effect of temperature . . . . . .

APPENDIX I: Calculation of (cos O ) . . . . . . . . . . . . . .

CHAPTER IV: SELF-DIFFUSION . . . . . . . . . . .

I . The self-diffusion coefficient . . . . . . . . . . . . . . . . II . Variation of the diffusion coefficient with temperature . . . . .

11.1 Vacancy mechanism 11.2 Divacancy mechanism 11.3 Interstitial mechanism 11.4 Several mechanisms operating simultaneously

ITT. Anisotropy of diffusion . . . . . . . . . . . . . . . . . . IV. Deviations from the Arrhenius law . . . . . . . . . . . . . V. The isotope effect . . . . . . . . . . . . . . . . . . . .

48 49

55 56

61

61

67 69 73 76 77

83

91 92

97

97 97

102 103 106


ï‘able of coiiteiits XII1

VI. Effect of pressure . . . . . . . . . . . . . . . . . . . . . VII. Empirical correlations . . . . . . . . . . . . . . . . . . VIII. Self-diffusion in metals . . . . . . . . . . . . . . . . . . IX. Self-diffusion in semiconductors . . . . . . . . . . . . . .

IX.l Ionization of the point defects IX.2 Compound semiconductors

X.l Alkali halides X.2 Silver halides X.3 The fluorite structure X.4 Oxides

X. Self-diffusion in ionic crystals . . . . . . . . . . . . . . .

XI. Molecular crystals . . . . . . . . . . . . . . . . . . . .

CHAPTER V: SOLUTE DIFFUSION IN PURE MATERIALS. DIFFUSION IN ALLOYS . . . .

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . II. Solute diffusion at infinite dilution

11.1 The five-frequency model (FCC) 11.2 Models for the BCC structure 11.3 Comparison of self- and solute diffusion 11.4 Application to metals II .5 Ultra-fast diffusers

111.1 The solutes C, N , and O 111.2 Hydrogen and its isotopes (D, T )

IV.l Diffusion of homovalent solutes IV.2 Diffusion of heterovalent solutes

V. l Substitutional solutes V.2 Interstitial impurities

VI.l Effect of the solute concentration V1.2 Determination of the jump frequency ratios VI.3 The effect of substitutional impurities on the diffusion

. . . . . . . . . . . . .

I I I . Interstitial solid solutions . . . . . . . . . . . . . . . . .

IV. Ionic crystals . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . V . Semiconductors

VI. Dilute alloys . . . . . . . . . . . . . . . . . . . . . . .

of interstitials

VII. 1 Disordered alloys VII.2 Ordered alloys

VII. Diffusion in homogeneous concentrated alloys

VIII. Superionic conductors . . . . . . . . . . . . . . . . . . IX. Amorphous materials . . . . . . . . . . . . . . . . . . .

. . . . . . .

IX.l Amorphous metals (or metallic glasses) IX.2 Oxide glasses

110 112 114 121

125

143

149

149 150

164

172

173

179

184

191 196


XIV A tom movements

C H A P T E R VI: D I F F U S I O N A N D DRIFT I N ALLOYS A N D C O M P O U N D S . . . . . . . . . . . . . . . . . 203

I. Intrinsic diffusion coefficients . . . . . . . . . . . . . . . 203 1.1 Interdiffusion of two metals A/B 1.2 Interdiffusion of two ionic crystals AX/BX

11.1 Darken’s equations 11.2 Experimental verification and the Kirkendall effect 11.3 Marker movement. The Kirkendall interface 11.4 Sources and sinks for vacancies. Kirkendall porosity

III. 1 Chemical diffusion coefficient 111.2 Ambipolar diffusion and the Nernst electric field 111.3 Application to the oxidation of a pure metal

II. The interdiffusion coefficient . . . . . . . . . . . . . . . . 207

III. Chemical diffusion in compounds . . . . . . . . . . . . . . 221

IV. The effective diffusion coefficient . . . . . . . . . . . . . . 229 APPENDIX I: Variable molar volume. Problem of the frame

of reference . . . . . . . . . . . . . . . . . . 233 APPENDIX II: Kroger-Vink notation . . . . . . . . . . . . . 241 APPENDIX III: Deviation from stoichiometry in a binary oxide . . 242 APPENDIX IV: Ambipolar diffusion in a binary oxide . . . . . . 244

C H A P T E R VII: D I F F U S I O N I N M E D I A O F L O W E R D I M E N S I O N A L I T Y . . . . . . . . . . . . . . . . . . 249

Part 1. - Internal short-circui ts (dislocations, interfaces)

I. Phenomenology . . . . . . . . . . . . . . . . . . . . . 251 1.1 Fisher’s model 1.2 Regimes of diffusion

11.1 Grain boundaries II. 2 Subgrain boundaries 11.3 Interfaces between dissimilar phases II .4 Dislocations 11.5 Solute diffusion

II. Analytical solutions . . . . . . . . . . . . . . . . . . . 255

III. A tom’c models . . . . . . . . . . . . . . . . . . . . . . 266 IV. Effect of temperature . . . . . . . . . . . . . . . . . . . 269 V. Experimental methods and results . . . . . . . . . . . . . 270

V . l Experimental methods

Table of contents xv

V.2 Experimental results VI. Diffusion-induced grain-boundary migration (DIGM) . . . . .

Part. 2. - Surface diffusion I. The structure of surfaces . . . . . . . . . . . . . . . . .

II. Mechanisms of diffusion . . . . . . . . . . . . . . . . . . 11.1 Self-diffusion 11.2 Solute diffusion

111.1 Field-ion microscopy 111.2 Diffusion of radiotracers 111.3 Topographic methods 111.4 Laser-induced thermal desorption (LITD) 111.5 Other methods

III. Experimental methods and results . . . . . . . . . . . . .

APPENDIX I: Grain-boundary diffusion . . . . . . . . . . . .

transport . . . . . . . . . . . . . . . . . . . APPENDIX II: Evolution of the profile of a surface by material

CHAPTER VIII: PHENOMENOLOGICAL THEORY OF DIFFUSION . . . . . . . . . . . . . . . . . . . . . .

I . Review of the Thermodynamics of Irreversible Processes (T.I.P.) . . . . . . . . . . . . . . . . . . . . . . . . .

II. The application of T.I.P. to diffusion in solids . . . . . . . . III. Applications of the phenomenological equations . . . . . . .

111.1 Diffusion of a radioactive tracer in a pure material 111.2 Interdiffusion of A and B 111.3 Flux of material arising from a flux of point defects:

segregation induced by quenching or irradiation 111.4 Electromigration in a substitutional binary alloy III .5 Thermomigr at ion 111.6 Problems connected with non-conserved species

IV. Ternary systems . . . . . . . . . . . . . . . . . . . . . V. Heterogeneous solid solutions: effect of composition gradients .

V.l Expression for the Gibbs free energy V.2 Interdiffusion V.3 Evolution of a modulation of composition

APPENDIX I: Chemical potential of vacancies . . . . . . . . . APPENDIX II: Diffusion in anisotropic media . . . . . . . . . . APPENDIX III: The frame of reference APPENDIX IV: The square root diffusivity . . . . . . . . . . .

. . . . . . . . . . . . .

274

276 279

284

29 1

293

303

303 306 309

337 344

350 35 1 353 358


XVI A tom movements

CHAPTER IX: TECHNIQUES FOR THE STUDY OF DIFFUSION . . . . . . . . . . . . . . . . . . . . . .

Part 1. - Diffusion over a long distance I. Alethodology of the measurements. Sample preparation . .

. . . . II. Determination of the diffusion profile c(z, y , z , t ) 11.1 Non-destructive methods 11.2 Destructive methods

111.1 Radiotracers: decrease of surface activity 111.2 Gas-solid diffusion couples 111.3 Micrographic methods 111.4 Autoradiography 111.5 Synthetic modulated structures (interdiffusion) 111.6 Transmission electron microscopy 111.7 Electrical resistivity

1V. i Concentration profiles IV.2 Variation of D with temperature IV.3 The interdiffusion coefficient

III. Indirect methods . . . . . . . . . . . . . . . . . . .

IV. Data processing . . . . . . . . . . . . . . . . . . .

Part 2. - Methods based on the measurement of jump frequencies

I. Relaxation induced by an external stimulus . . . . . . . . 1.1 Mechanical relaxation 1.2 Magnetic relaxation 1.3 Dielectric relaxation

II. 1 Incoherent neutron scattering 11.2 Nuclear magnetic resonance 11.3 Mossbauer effect

II. Nuclear methods . . . . . . . . . . . . . . . . . . . .

Part 3. - Computer simulation I. Statistical calculations . . . . . . . . . . . . . . . . . .

II. Defect characteristics and diffusion mechanisms . . . . . . . 11.1 The goals of simulation 11.2 Models 11.3 Methods

APPENDIX I: Diffusion of gases . . . . . . . . . . . . . . .

APPENDIX II: The Snoek effect Desorption of a gas by detrapping

Calculation of the relaxation time

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

361

36 1 364

371

377

382

390

404 404

41 1 413 413 414


Table of contents

CHAPTER X: THE STUDY OF SOME DIFFUSION- CONTROLLED PROCESSES . . . . . . . . . . . .

I . Diffusion in multi-phase systems and formation of intermediate compounds . . . . . . . . . . . . . . . . . . . 1.1 Nature of the phases formed by interdiffusion 1.2 Experimental studies of multiphase diffusion 1.3 The kinetics of phase growth 1.4 Problems connected with nucleation 1.5 Ternary systems

11.1 Oxidation of a pure metal 11.2 Oxidation of a binary alloy AB

III. Sintering . . . . . . . . . . . . . . . . . . . . 111.1 Stage 1 of sintering identical spherical particles 111.2 Stage 3 of sintering

IV.l Growth of a precipitate IV.2 Dissolution of a precipitate IV.3 Coalescence IV.4 Elimination of vacancies IV.5 Segregation to dislocations

II. Oxidation . . . . . . . . . . . . . . . . . . . .

IV. Precipitation and Aging . . . . . . . . . . . . . .

V . The solidification of an alloy . . . . . . . . . . . . VI. Diffusion under irradiation . . . . . . . . . . . .

VI.l Defect concentrations. Balance equations VI.2 Steady state VI.3 Tracer self-diffusion in the steady state VI.4 Cascade mixing

VII. 1 Diffusional creep VII. Plastic deformation a t high temperature . . . . .

. .

. .

. .

, . .

. . .

. . .

. . .

V11.2 Growth of voids at the grain boundaries during high

APPENDIX I: Rate constant for oxidation . . . . . . . . . .

high-temperature deformation . . . . . . . . .

temperature plastic deformation

APPENDIX II: Effective diffusion coefficient for coalescence . . APPENDIX III: Growth of voids at grain boundaries during

EXERCISES . . . . . . . . . . . . . . . . . . . . . . . INDEX . . . . . . . . . . . . . . . . . . . . . . . .

XVII

421

421

432

447

462

487 492

500

512 514

516

521 569


General Bibliography

I. - BOOKS

Atom Movements, J . H. Hollomon ed., ASM, Cleveland (1951).

SHEWMON P. G., Diffusion in Solids (McGraw-Hill, New York) 1963.

Diffusion in BCC Metals, ASM, Cleveland (1965)

ADDA Y. and PHILIBERT J., La Diffusion dans les Solides, 2 vols. (P.U.F., Paris) 1966.

QuÉRÉ Y., Défauts Ponctuels dans les Métaux (Masson et Cie, Paris) 1967.

MANNING J. R., DiDusion Kinetics f o r Atoms in Crystals (Van Nostrand, Princeton) 1968.

Atomic Transport in Solids and Liquids, A. Lodding and T. Lagerwall eds., Zeits. Naturforsch., Tübingen (1970).

DiDusion Processes, J . N. Sherwood, A. V. Chadwick, W. M. Muir and F . L. Switon eds., 2 vols (Gordon and Breach, London) 1971.

Difluszon ASM Seminar, ASM, Cleveland (1972).

FLYNN C. P., Point Defects and Diffusion (Clarendon Press, Oxford) 1972.

Atomic Diffusion in Semiconductors, D. Shaw ed. (Plenum, New York) 1973.

TUCK B., Introduction to Diffusion in Semiconductors (IEE Monograph Series 16, Inst. Electr. Eng.) 1974.

Di'usion in Solids, Receni Developments, A. S. Nowick and J. J. Burton eds., (Academic Press, New York) 1975.

Point Defects in Solids, J. H. Crawford and L. M. Slifkin eds. (Plenum Press, New York)

Vol. 1, General and Ionic Crystals (1972)


552 Atom movements

curve defined in the second question.) 4”) Can an activation energy for grain-boundary diffusion be defined?

Compare it to the activation energies for volume diffusion. 5”) Study the graph of log E us . z2. Does the “diffusion tail” corresponding

to grain-boundary diffusion appear linear? To what precision? What can be concluded from this?

(1) A. ATKINSON and R. I. TAYLOR, Philos. Mag. A43 (1981) 979-998.

51 - DENUDED ZONE NEAR GRAIN BOUNDARIES

A denuded zone, i.e., a zone in which precipitates do not appear, is clas- sically observed around grain boundaries in light metals after quenching and annealing. This is attributed to the “pumping” of the solute element by the grain boundaries, where it forms fine intergranular precipitates. The supersat- uration is thus insufficient to produce precipitation near the grain boundaries (see figure).

0

0

0

I

0

#

0

0

I 0

0

I

I

0

#

4

e

I

I

#

0

0

0

Such a study was carried out on an AI-Li alloy (2.5 wt.% Li). The alloy was annealed a t 500°C and quenched, then aged for different times at 2OOOC to cause precipitation of the metastable phase 6‘ - AIBLi.

1”) The width of the denuded zone, lo, was measured as a function of the annealing time:

Exercises 553

Can a value of the diffusion coefficient of lithium at 200°C be deduced from these data?

2") An analytical transmission electron microscope with electron beam about 20 nm in diameter was used to carry out a series of "point" analyses in a direction perpendicular to the boundary on a thin foil. The results for a treatment of 48 h at 200°C are given in the table below.

Table of Analyses (1)

2, distance to the boundary, (nm) O 38 100 145 205 250 300

Li (at. %) O - 0.45 0.62 1.4 2.85 2.2, 3.15 2.85 3.5, 4.9

350 405 460 500 560 750 760 810 850 910 3.62 3.15, 4.4 5.05 4.9 6.3 6.15 7.4 7.25 7.9 7.4

What is the theoretical profile that these points should fit? Deduce the value of the diffusion coefficient, and compare it to the value obtained in the first question. Plot the experimental points on a graph with the theoretical profile obtained from the best value of D.

(1) SAINFORT P., Thesis, University of Grenoble, 1985.

52 - ZONE IMPOVERISHED BY INTERGRANULAR PRECIPI- TATION

According to the classical model of intergranular precipitation, the precipitates grow by the diffusion of solute toward the grain boundary, the solute being drained off along the boundary to the precipitates (Figure).


extrait de la publication…madeleine meyer and vassili pontikis for prepar- ing the figure. extrait...

Documents