Diffusion in Solids: Field Theory, Solid-State Principles, and Applications

481 Pages·1999·13.44 MB·English

by  Martin Eden Glicksman

DIFFUSION
IN SOLIDS
FIELD THEORY,
SOLID-STATE PRINCIPLES,
AND APPLICATIONS
MARTIN EDEN GLICKSMAN
DIFFUSI. ON IN SOLl1DS
Field Theory, Solid-State Principles,
and Applications
Martin Eden Glicksman
Rensselaer Polytechnic Institute
90
A Wu uv IN111R11C'1HNCB P11111.1cATtoN
JOUN WILEY & SONS, INC.
r , 1w Vc>tk • Chlch, I ,r • Welnhelm • Brisban• • Singapore • Toror
DIFFUSI. ON IN SOLl1DS
Field Theory, Solid-State Principles,
and Applications
Martin Eden Glicksman
Rensselaer Polytechnic Institute
90
A Wu uv IN111R11C'1HNCB P11111.1cATtoN
JOUN WILEY & SONS, INC.
r , 1w Vc>tk • Chlch, I ,r • Welnhelm • Brisban• • Singapore • Toror
ThiK book is printed on acid-free paper. @)
Copyright © 2000 by John Wiley & Sons, Inc. All rights reserved.
Published simultaneously in Canada.
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l.lbrary of Congress Cataloging•in-Publication Data: ,
GlickHman, M. E.
DlffuHion in solids : field theory, solid-state principles, and
opplic11tlons / Martin E. Glicksman.
p. cm.
lncludcH hibliographicul references und inde)(.
ISRN 0-471-23972-0 (alk. paper)
I, Kirkendall effect. I. Title.
QC176.8.DSO57 2000 99-21381
SJ0,4' 13 ·dc2I
or
Prlntad In th, l lnlted s111oa Amerlc.i1.
20 IV IN 17 lti
To Nat Henry, my father

CONTENTS
PREFACE
ACKNOWLEDGMENTS
PART I FIELD THEORY
Chapter 1 Laws of Diffusion
I . I Preliminary Definitions
l .2 Diffusion in an Isotropic Medium
1.3 Fick's First Law
IA Units
1.5 Vector Form of Fick's First Law
1.6 Mass Conservation
1.7 Fick's Second Law
1.8 Linear Diffusion Equation in Various Coordinate Systems
1.9 Important Diffusion Symmetries
Exercises
Bibliography
Chapter 2 Diffusion In Generalized Media
2, I l)lffu1lvlty Tensor
2.2 Prlnclpul DlrectlonN
:2, ~ DlffuNlon In Ocmernll1,c.,d Modl11
viii CONTENTS
2.4 Influence of Imposed Symmetry 22
2.4.1 Isotropic Materials 24
2.4.2 Cubic crystals 24
2.4.3 Orthotropic Materials 25
2.4.4 Orthorhombic Crystals 26
2.4.5 Monoclinic Crystals 26
2.4.6 Triclinic Crystals 26
Exercises 27
Bibliography 30
Chapter 3 Solutions to the Linear Diffusion Equation 31
3.1 Linear Diffusion and 1ransforrn Methods 31
3.2 Linear Diffusion into a Semi-Infinite Medium 32
3.2.1 Boundary Conditions 33
3.2.2 Laplace Transfonns 33
3.3 Behavior of the Concentration Field 36
3.4 Instantaneous Planar Diffusion Source in an Infinite Medium 38
3.5 Conservation of Mass for a Planar Source 41
3.5.1 Estimating the Error Function 42
3.6 Thin-Film Configuration 43
Exercises 44
Bibliography 4 7
Chapter 4 Diffusion Couple 49
4.1 Superposition of Instantaneous Sources 49
4.2 Diffusion Couples .. 50
4.3 Diffusion with a Fixed Boundary Concentration 53
4.4 Slab Couples 54
Exercises 55
Bibliography 58
Chapter es Diffusion Point Sources In Higher Dlmen1lon1 59
,.1 lnNtantaneou11 Point Source in Three Dimcm•lonN 59
,.1.1 Tran•form Solution for II Point Sm111·, 60
,. 1.2 Mnu C'on•tr1ln1 In Spherknl , \ • uu u t12
CONTENTS
5.2 Source Solutions in One, Two, and Three Dimensions
5.3 Behavior of the Fields Upon Release of a Source
5.3.1 Choice of Concentration, Time, and Length Scales
5.3.2 Behavior of Diffusion Responses at Early Times
5.4 Conservation Integrals for One, Two, and Three Dimensions
5.4.1 Mass Centroids in Higher Dimensions
Exercises
Bibliography
Chapter 6 Generalized Sources
6.1 Generalizing the Point Source in Three Dimensions
6.1.1 Relationship to Green's Functions
6.2 Instantaneous Line Source Releasing Diffusant in Three
Dimensions
6.2.1 Hadamard's Method of Descent
6.3 Generalization of the Instantaneous Point Source in Two
Dimensions
6.4 Instantaneous Line Source in Two Dimensions
6.5 Instantaneous Ring Source
6.6 Continuous Point Sources
6.6.1 Steady-State Limit of Continuous Point Sources
6.7 Time-Continuous Line Source in Two Dimensions
Exercises
Bibliography
Chapter 7 Diffusion-Reaction
7.1 Linear Diffusion-Reaction
7.1.1 One-Dimensional Diffusion-Reaction
7 .1.2 Boundary Conditions for Diffusion-Reaction
7.2 Imposing Steady-State Conditions
7.3 General Kinetic Limits
7 .3.1 Reaction Limit
7.3,2 Dlffuilion Limit
7,4 Mlx0LI Dlll\111lon Rt1ncllon Kinetic•
X CONTENTS
7.5 Time-Dependent Reaction-Diffusion 101
7.6 Nonlinear Diffusion-Reaction 102
Exercises 103
Bibliography 106
Chapter 8 Linear Flow in Finite Systems 107
8.1 Diffusion in a Single-Phase Composite Slab 107
8.1.1 No-Flow Boundary Condition 108
8.1.2 Fictitious Image Sources 109
8.2 Fourier's Method 113
8.3 Application of Fourier Series 115
8.4 Numerical Approximations 119
8.4.1 Error Function Series Approximation 120
8.4.2 Fourier Series Approximation 121
8.4.3 Numerical Comparison 122
8.5 Codastefano's Method 123
Exercises 126
Bibliography 132
Chapter 9 Spherical Bodies 135
9.1 Diffusion Around Spheres 136
9.1.1 Boundary Conditions 136
9.1.2 Spherical Diffusion Solution 138
9.2 Behavior of the Spherical Concentration Field 140
9.3 Interfacial Fluxes 141
"'
9.4 Quasi-Static Growth and Dissolution 142
9.5 Moving Boundaries 145
9.6 Evolution of a Spherical Particle 145
9.6.1 Conditions on the Phases 145
9.6.2 Conditions at the a-/J Interface 146
9.6.3 Moving Boundary Solution 146
9.7 The Characteristic Equation 150