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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
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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 t h a t which is visible b u t complicated
by t h a t which is invisible b u t simple ...”
J e a n Perrin,
in preface t o Les Atomes (1912)
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a
b
C
Diffusion of an a.datom on a. (110) surface of a fcc crystal by the, exchange
mechanism:
a ) A d a t o m in initial position
t
=
to
b) Saddle point position
t
20
c) Final positmion
t
=
=
to
+
+
6 x
s
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 t o m s .
T h e a t o m coordinates were calculated by a molecular dynamics simulation
using a Lennard-Jones potential with p a r a m e t e r s corresponding t o solid argon
a t 0.4 T,. (see G . d e Lorenzi el al., reference a t end of Ch. VI.)
T h e a u t h o r t h a n k s Drs. Madeleine Meyer and Vassili Pontikis for preparing t h e figure.
Extrait de la publication
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 solidstate 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 t o 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 t o
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 t a learn enough French
to read the “bible,” I also made it possible for them to enjoy much more fruitful
visits t o 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
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Forewo rd
This book was written t o 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
t o 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 t o the more complex cases of mass flow under chemical or thermal
gradients or under electric fields, or diffusion in structures of lower dimensionality (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 t o illustrate important points. The rest of the bibliography
lists references t o 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 t o clarify a number of points. His thanks
go equally t o the secretaries who had to face a difficult stenographic task,
and especially t o 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
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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
Extrait de la publication
TABLE OF CONTENTS
Preface
. . . . . . . . . . . . . .
. . . . . . . . . . . . .
Foreword
Foreword to the English edition .
Translator’s acknowledgments . .
General Bibliography
. . . . . .
Notation . . . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . . .
V
VI1
IX
IX
. . . . . . . . . . . . .
. . . . . . . . . . . . . XIX
. . . . . . . . . . . . XXIII
. . . . . . . . .
1
I. Flux of particles. Fick’s equation . . . . . . . . . . . . . .
II. Time-dependent case . . . . . . . . . . . . . . . . . . .
III. Solutions of the diffusion equation (or Fick’s second law)
. .
111.1 Thin layer or instantaneous source
111.2 Constant surface concentration (diffusion in a
sern-infinite solid)
111.3 Infinite initial distribution
111.4 The Boltzmann transformation
111.5 Concentration-dependent diffusion coefficient
IV. Relation between drift and diffusion. The Nernst-Einstein
. . . . . . . . . . . . . . . . . . . . . . . .
equation
V . The nature of the driving force . . . . . . . . . . . . . . .
VI. A variety of diffusion processes and generalization of Fick’s law
VII. Diffusion with phase change. Multiphase diffusion . . . . . .
. . . .
APPENDIX I: Methods for solving the diffusion equation
APPENDIX II: Diffusion in three dimensions . . . . . . . . . .
APPENDIX III: Conservation at amoving boundary . . . . . . .
1
2
16
22
26
29
30
. . .
33
. . . . . . . .
. . . . . . . . .
33
36
CHAPTER I: DIFFUSION AND DRIFT
CHAPTER II: ATOMIC THEORY OF DIFFUSION
I. A simplified model . . . . . . . . . . . .
II. General theory of random walk
. . . . .
III. Expressions for the mean-square displacement
for the diffusion coefficient . . . . . . . .
IV. Diffusion in the presence of a driving force
V . Explicit form of the function W (X, r ) . .
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13
14
( X ’ ) and
. . . . . . . . .
. . . . . . . . .
45
. . . . . . . . .
46
39
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Atom movements
VI. Variable jump distance . . . . . .
VII. Correlation functions . . . . . . .
VII.l Characterization of the structure
medium
V11.2 Diffusion
VIII. Limitations of Fick’s law . . . . .
APPENDIX: Some definitions . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
in a non-crystalline
. . . . . . . . . . . .
. . . . . . . . . . . .
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. Migration ofpoint defects. Effect of temperature
. . . . . .
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 I: Calculation of (cos O ) . . . . . . . . . . . . . .
APPENDIX II: Percolation . . . . . . . . . . . . . . . . . .
CHAPTER IV: SELF-DIFFUSION
48
49
. . . . . . . . . . .
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
. . . . . . . . . . . . . . . .
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. . . . .
55
56
61
61
67
69
73
76
77
83
91
92
97
. . . . .
97
97
. . . . .
. . . . .
. . . .
102
103
106
ï‘able of coiiteiits
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. Self-diffusion in ionic crystals
X.l Alkali halides
X.2 Silver halides
X.3 The fluorite structure
X.4 Oxides
XI. Molecular crystals . . . . . . . . . .
XII1
. . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
110
112
114
121
. . . . . . . . . . .
125
. . . . . . . . . .
143
. . . . . . . . . . .
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
III. Interstitial solid solutions . . . . . . . . . . . . . . . . .
111.1 The solutes C, N , and O
111.2 Hydrogen and its isotopes (D, T )
IV. Ionic crystals
. . . . . . . . . . . . . . . . . . . . . .
IV.l Diffusion of homovalent solutes
IV.2 Diffusion of heterovalent solutes
. . . . . . . . . . . . . . . . . . . . .
V . Semiconductors
V . l Substitutional solutes
V.2 Interstitial impurities
VI. Dilute alloys . . . . . . . . . . . . . . . . . . . . . . .
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
of interstitials
VII. Diffusion in homogeneous concentrated alloys
. . . . . . .
VII. 1 Disordered alloys
VII.2 Ordered alloys
VIII. Superionic conductors . . . . . . . . . . . . . . . . . .
IX. Amorphous materials . . . . . . . . . . . . . . . . . . .
IX.l Amorphous metals (or metallic glasses)
IX.2 Oxide glasses
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149
150
164
172
173
179
184
191
196
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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 A L L O Y S
. . . . . . . . . . . . . . . . .
AND COMPOUNDS
I. Intrinsic diffusion coefficients
. . . . . . . . . . . . .
1.1 Interdiffusion of two metals A/B
1.2 Interdiffusion of two ionic crystals AX/BX
II. The interdiffusion coefficient . . . . . . . . . . . . . .
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. Chemical diffusion in compounds . . . . . . . . . . . .
III. 1 Chemical diffusion coefficient
111.2 Ambipolar diffusion and the Nernst electric field
111.3 Application to the oxidation of a pure metal
IV. The effective diffusion coefficient . . . . . . . . . . . .
APPENDIX I: Variable molar volume. Problem of the frame
of reference . . . . . . . . . . . . . . . .
. . . . . . . . . . .
APPENDIX II: Kroger-Vink notation
APPENDIX III: Deviation from stoichiometry in a binary oxide
APPENDIX IV: Ambipolar diffusion in a binary oxide . . . .
203
. .
203
. .
207
. .
221
. .
229
. .
. .
233
241
242
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
DIMENSIONALITY . . . . . . . . . . . . . . . . . .
249
I.
II.
III.
IV.
V.
. .
.
.
Part 1. - Internal short- circuits
(dislocations, interfaces)
Phenomenology
. . . . . . . . . . . . . . . . . . . . .
1.1 Fisher’s model
1.2 Regimes of diffusion
Analytical solutions
. . . . . . . . . . . . . . . . . . .
11.1 Grain boundaries
II.2 Subgrain boundaries
11.3 Interfaces between dissimilar phases
II .4 Dislocations
11.5 Solute diffusion
A tom’c models . . . . . . . . . . . . . . . . . . . . . .
Effect of temperature . . . . . . . . . . . . . . . . . . .
Experimental methods and results . . . . . . . . . . . . .
V . l Experimental methods
251
255
266
269
270
xv
Table of contents
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
III. Experimental methods and results . . . . . . . . . .
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
APPENDIX I: Grain-boundary diffusion . . . . . . . . .
APPENDIX II: Evolution of the profile of a surface by material
transport . . . . . . . . . . . . . . . .
. . .
. . .
276
279
. . .
284
. . .
29 1
. . .
293
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 . . . . . . . . . .
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303
.
.
303
306
309
.
337
344
.
.
.
.
350
35 1
353
358
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CHAPTER IX: TECHNIQUES FOR THE STUDY OF
DIFFUSION . . . . . . . . . . . . . . . . . . . .
. .
361
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
III. Indirect 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
IV. Data processing . . . . . . . . . . . . . . . . . . .
1 V . i Concentration profiles
IV.2 Variation of D with temperature
IV.3 The interdiffusion coefficient
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. Nuclear methods . . . . . . . . . . . . . . . . . . . .
II. 1 Incoherent neutron scattering
11.2 Nuclear magnetic resonance
11.3 Mossbauer effect
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
. . . . . . . . . .
Desorption of a gas by detrapping . . .
APPENDIX II: The Snoek effect . . . . . . . . . . .
. . .
Calculation of the relaxation time
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364
371
377
382
390
. . . . .
. . . . .
404
404
. . . . .
41 1
413
413
414
. . . . .
. . . . .
. . . . .
Table of contents
CHAPTER X: THE STUDY OF SOME DIFFUSIONCONTROLLED 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
. .
II. Oxidation . . . . . . . . . . . . . . . . . . . .
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. Precipitation and Aging . . . . . . . . . . . . . . , . .
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
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. Plastic deformation a t high temperature
VII. 1 Diffusional creep
V11.2 Growth of voids at the grain boundaries during high
temperature plastic deformation
APPENDIX I: Rate constant for oxidation . . . . . . . . . .
APPENDIX II: Effective diffusion coefficient for coalescence
. .
APPENDIX III: Growth of voids at grain boundaries during
high-temperature deformation . . . . . . . . .
EXERCISES . . . . . . . . . . . . . . . . . . . . . . .
INDEX
. . . . . . . . . . . . . . . . . . . . . . . .
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421
432
447
462
487
492
500
512
514
516
521
569
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General Bibliography
I. - B O O K S
A t o m Movements, J . H. Hollomon ed., ASM, Cleveland (1951).
SHEWMON
P. G., Diffusion i n Solids (McGraw-Hill, New York) 1963.
Diffusion in BCC Metals, ASM, Cleveland (1965)
ADDAY. 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 i n Crystals (Van Nostrand,
Princeton) 1968.
Atomic Transport i n 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).
FLYNNC. P., Point Defects and Diffusion (Clarendon Press, Oxford) 1972.
Atomic Diffusion i n Semiconductors, D. Shaw ed. (Plenum, New York) 1973.
TUCKB., 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 i n Solids, J. H. Crawford and L. M. Slifkin eds. (Plenum Press,
New York)
Vol. 1, General and Ionic Crystals (1972)
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552
Atom movements
curve defined in the second question.)
4”) Can an activation energy for grain-boundary diffusion be defined?
Compare it t o the activation energies for volume diffusion.
5”) Study the graph of log E u s . z2. Does the “diffusion tail” corresponding
to grain-boundary diffusion appear linear? To what precision? What can be
concluded from this?
(1) A. ATKINSONand R. I. TAYLOR,
Philos. Mag. A43 (1981) 979-998.
51
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DENUDED ZONE NEAR GRAIN BOUNDARIES
A denuded zone, i.e., a zone in which precipitates do not appear, is classically 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 supersaturation is thus insufficient t o produce precipitation near the grain boundaries
(see figure).
0
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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 t o
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 t o carry out a series of "point" analyses
in a direction perpendicular t o 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)
distance to the
boundary, (nm)
Li (at. %)
2,
O
38 100 145
205
250
300
O - 0.45 0.62 1.4 2.85 2.2, 3.15 2.85 3.5, 4.9
460 500 560 750 760 810 850 910
350
405
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 t o 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 t o the precipitates (Figure).
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