T KINETICS OF MATERIALS Robert W. Balluffi Samuel M. Allen W. Craig Carter With Editorial Assistance from Rachel A. Kemper Department of Materials Science and Engineering Massachusetts Institute of Tech nology Cambridge, Massachusetts WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Copyright @ 2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or com- pleteness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situ- ation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Depart- ment with the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Balluffi, Robert W., 1924- Kinetics of Materials / Robert W. Balluffi, Samuel M. Allen, W. Craig Cart,er; edited by Rachel A. Kemper; p. cm. Includes bibliographical references and index. ISBN 13 978-0-471-24689-3 ISBN-10 0-471-24689-1 1.Materials-Mechanical Properties. 2. Materials science I. Allen, Samuel M. 11. Carter, W. Craig. 111. Kemper, Rachel A. IV. Title. TA404.8.B35 2005 620.1 ' 1292-dc22 2005047793 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 CONTENTS Preface Acknowledgments Notation S ymbols-Roman Symbols-Greek 1 Introduction 1.1 Thermodynamics and Kinetics 1.1.1 1.1.2 Averaging Classical Thermodynamics and Constructions of Kinetic Theories 1.2 Irreversible Thermodynamics and Kinetics 1.3 Mathematical Background 1.3.1 Fields 1.3.2 Variations 1.3.3 1.3.4 Fluxes 1.3.5 Accumulation 1.3.6 Conserved and Nonconserved Quantities 1.3.7 Matrices, Tensors, and the Eigensystem Continuum Limits and Coarse Graining Bibliography Exercises xvii xix xx xxi xxv 1 2 2 4 5 7 7 7 8 10 11 12 13 16 16 V Vi CONTENTS PART I MOTION OF ATOMS AND MOLECULES BY DIFFUSION 2 Irreversible Thermodynamics: Coupled Forces and Fluxes 2.1 Entropy and Entropy Production 2.1.1 Entropy Production 2.1.2 Conjugate Forces and Fluxes 2.1.3 2.2 Linear Irreversible Thermodynamics 2.2.1 2.2.2 2.2.3 2.2.4 Onsager’s Symmetry Principle Basic Postulate of Irreversible Thermodynamics General Coupling between Forces and Fluxes Force-Flux Relations when Extensive Quantities are Constrained Introduction of the Diffusion Potential Bibliography Exercises 3 Driving Forces and Fluxes for Diffusion 3.1 Concentration Gradients and Diffusion 3.1.1 3.1.2 Self-Diffusion: Diffusion in the Absence of Chemical Effects Self-Diffusion of Component i in a Chemically Homogeneous Binary Solution Diffusion of Substitutional Particles in a Chemical Concentration Gradient Diffusion of Interstitial Particles in a Chemical Concentration Gradient On the Algebraic Signs of Diffusivities 3.1.3 3.1.4 3.1.5 3.1.6 Summary of Diffusivities Electrical Potential Gradients and Diffusion 3.2.1 3.2.2 Electromigration in Metals 3.3 Thermal Gradients and Diffusion 3.4 Capillarity and Diffusion 3.2 Charged Ions in Ionic Conductors 3.4.1 3.4.2 Boundary Conditions 3.5.1 3.5.2 Stress as a Driving Force for Diffusion: Formation of 3.5.3 3.5.4 Summary of Diffusion Potentials The Flux Equation and Diffusion Equation 3.5 Stress and Diffusion Effect of Stress on Mobilities Solute-Atom Atmosphere around Dislocations Influence of Stress on the Boundary Conditions for Diffusion: Diffusional Creep Bibliography 23 23 25 27 27 28 28 30 32 33 35 36 41 41 42 44 44 52 53 53 54 55 55 56 57 58 61 61 61 62 64 66 67 CONTENTS vii Exercises 68 4 The Diffusion Equation 4.1 Fick’s Second Law 4.1.1 4.1.2 4.1.3 Linearization of the Diffusion Equation Relation of Fick’s Second Law to the Heat Equation Variational Interpretation of the Diffusion Equation Geometrical Interpretation of the Diffusion Equation when Diffusivity is Constant Scaling of the Diffusion Equation 4.2 Constant Diffusivity 4.2.1 4.2.2 4.2.3 Superposition Diffusivity as a Function of Concentration Diffusivity as a Function of Time Diffusivity as a Function of Direction 4.3 4.4 4.5 Bibliography Exercises 5 Solutions to the Diffusion Equation 5.1 Steady-State Solutions 5.1.1 One Dimension 5.1.2 Cylindrical Shell 5.1.3 Spherical Shell 5.1.4 Variable Diffusivity 5.2 Non-Steady- St ate Diffusion 5.2.1 5.2.2 5.2.3 Method of Superposition 5.2.4 5.2.5 Method of Laplace Transforms 5.2.6 Instantaneous Localized Sources in Infinite Media Solutions Involving the Error Function Method of Separation of Variables: Diffusion on a Finite Domain Estimating the Diffusion Depth and Time to Approach Steady State Bibliography Exercises 6 Diffusion in Multicomponent Systems 6.1 General Formulation 6.2 Solving the Diffusion Equations 6.2.1 Constant Diffusivities 6.2.2 Concentration-Dependent Diffusivities 77 77 78 79 80 81 81 81 83 85 87 88 91 91 99 100 100 101 102 102 103 103 105 107 107 110 113 114 114 131 131 134 135 139 viii CONTENTS 6.3 Measurement of Diffusivities Bibliography Exercises 7 Atomic Models for Diffusion 7.1 Thermally Activated Atomic Jumping 7.1.1 7.1.2 7.1.3 Many-Body Model Diffusion as a Series of Discrete Jumps 7.2.1 One-Particle Model with Square Potential-Energy Wells One-Particle Model with Parabolic Potential-Energy Wells 7.2 Relation of Diffusivity to the Mean-Square Particle Displacement 7.2.2 Diffusion and Random Walks 7.2.3 Diffusion with Correlated Jumps Bibliography Exercises 8 Diffusion in Crystals 8.1 Atomic Mechanisms 8.1.1 Ring Mechanism 8.1.2 Vacancy Mechanism 8.1.3 Interstitialcy Mechanism 8.1.4 Interstitial Mechanism 8.1.5 8.2.1 Metals 8.2.2 Ionic Solids 8.3.1 8.3.2 Determination of Diffusivities Diffusion Mechanisms in Various Materials 8.2 Atomic Models for Diffusivities 8.3 Diffusional Anelasticity (Internal Friction) Anelasticity due to Reorientation of Anisotropic Point Defects Bibliography Exercises 9 Diffusion along Crystal Imperfections 9.1 The Diffusion Spectrum 9.2 Grain Boundaries Regimes of Grain-Boundary Short-circuit Diffusion in a Polycryst a1 Analysis of Diffusion in the A, B, and C Regimes Mechanism of Fast Grain-Boundary Diffusion 9.2.1 9.2.2 9.2.3 141 141 141 145 145 146 148 149 154 155 156 158 158 159 163 163 164 164 165 167 167 169 169 177 183 183 189 189 190 209 209 214 214 218 221 CONTENTS ix 9.3 Dislocations 9.4 Free Surfaces Bibliography Exercises 222 223 224 225 10 Diffusion in Noncrystalline Materials 229 10.1 Free-Volume Model for Liquids 229 10.2 Diffusion in Amorphous Metals 232 10.2.1 Self-Diffusion 232 10.2.2 Diffusion of Small Interstitial Solute Atoms 234 10.3 Small Atoms (Molecules) in Glassy Polymers 239 10.4 Alkali Ions in Network Oxide Glasses 240 10.5 Diffusion of Polymer Chains 241 10.5.1 Structure of Polymer Chains 241 10.5.2 Diffusion of Isolated Polymer Chains in Dilute Solutions 243 10.5.3 Diffusion of Densely Entangled Polymer Chains by Reptation 245 Bibliography PART I1 MOTION OF DISLOCATIONS AND INTERFACES 11 Motion of Dislocations 11.1 Glide and Climb 11.2 Driving Forces on Dislocations 11.2.1 Mechanical Force 11.2.2 Osmotic Force 11.2.3 Curvature Force 11.2.4 Total Driving Force on a Dislocation 11.3.1 Glide in Perfect Single Crystals 11.3.2 Glide in Imperfect Crystals Containing Various Obstacles 11.3.3 Some Experimental Observations 11.3.4 Supersonic Glide Motion 11.3.5 Contributions of Dislocation Motion to Anelastic Behavior 11.3 Dislocation Glide 11.4 Dislocation Climb 11.4.1 Diffusion-Limited vs. Source-Limited Climb Kinetics 11.4.2 Experimental Observations 11.4.3 Analyses of Two Climb Problems Bibliography Exercises 12 Motion of Crystalline Surfaces 12.1 Thermodynamics of Interface Motion 247 253 253 255 255 256 257 258 258 258 263 264 265 266 266 267 269 269 274 275 285 285 X CONTENTS 12.2 Motion of Crystal/Vapor Interfaces 12.2.1 Structure of Crystal/Vapor Surfaces 12.2.2 Crystal Growth from a Supersaturated Vapor 12.2.3 Surfaces as Sinks for Supersaturated Lattice Vacancies 12.3.1 Structure of Crystal/Liquid Interfaces 12.3.2 Crystal Growth from an Undercooled Liquid 12.3 Interface Motion during Solidification Bibliography Exercises 286 287 288 291 292 292 292 294 295 13 Motion of Crystalline Interfaces 303 13.1 Thermodynamics of Crystalline Interface Motion 13.2 Conservative and Nonconservative Motion 13.3 Conservative Motion 13.3.1 Glissile Motion of Sharp Interfaces by Interfacial Dislocation Glide 13.3.2 Thermally Activated Motion of Sharp Interfaces by Glide and Climb of Interfacial Dislocations 13.3.3 Thermally Activated Motion of Sharp Interfaces by Atom Shuffling 13.3.4 Thermally Activated Motion of Diffuse Interfaces by Self-Diffusion 13.3.5 Impediments to Conservative Interface Motion 13.3.6 Observations of Thermally Activated Grain-Boundary Motion 13.4.1 Source Action of Sharp Interfaces 13.4.2 Diffusion-Limited Vs. Source-Limited Kinetics 13.4 Nonconservative Motion Bibliography Exercises 303 304 305 305 308 311 312 312 315 317 317 321 324 325 PART Ill MORPHOLOGICAL EVOLUTION DUE TO CAPILLARY AND APPLIED MECHANICAL FORCES 14 Surface Evolution due to Capillary Forces 337 14.1 Isotropic Surfaces 338 14.1.1 Flattening of Free Surfaces by Surface Diffusion 338 14.1.3 Evolution of Perturbed Cylinder by Vapor Transport 345 14.1.4 Evolution of Perturbed Cylinder by Surface Diffusion 345 14.1.2 Surface Evolution by Vapor Transport 341 14.1.5 Thermodynamic and Kinetic Morphological Wavelengths 346 14.2 Anisotropic Surfaces 346 CONTENTS xi 14.2.1 Some Geometrical Aspects of Anisotropic Surfaces 346 14.2.2 Rate of Morphological Interface Evolution 350 Exercises 354 Bibliography 353 15 Coarsening due to Capillary Forces 15.1 Coarsening of Particle Distributions 15.1.1 Classical Mean-Field Theory of Coarsening 15.1.2 Beyond the Classical Mean-Field Theory of Coarsening 15.2.1 Grain Growth in Two Dimensions 15.2.2 Grain Growth in Three Dimensions 15.2 Grain Growth Bibliography Exercises 16 Morphological Evolution: Diffusional Creep, and Sintering 16.1 Morphological Evolution for Simple Geometries 16.1.1 Evolution of Bamboo Wire via Grain-Boundary Diffusion 16.1.2 Evolution of a Bundle of Parallel Wires via Grain-Boundary Diffusion 16.1.3 Evolution of Bamboo Wire by Bulk Diffusion 16.1.4 Neck Growth between Two Spherical Particles via Surface Diffusion 16.2.1 Diffusional Creep of Two-Dimensional Polycrystals 16.2.2 Diffusional Creep of Three-Dimensional Polycrystals 16.3.1 Sintering Mechanisms 16.3.2 Sintering Microstructures 16.3.3 Model Sintering Experiments 16.3.4 Scaling Laws for Sintering 16.3.5 Sintering Mechanisms Maps 16.2 Diffusional Creep 16.3 Sintering Bibliography Exercises 363 363 363 371 373 373 379 382 384 387 388 389 39 1 392 394 395 395 398 400 400 40 1 403 403 405 406 408 PART IV PHASE TRANSFORMATIONS 17 General Features of Phase Transformations 419 17.1 Order Parameters 420 17.1.1 One-Component or Fixed Stoichiometry Systems 420 17.1.2 Two-Component Systems 423 17.2 Molar Free-Energy Changes 428 [...]... Bibliography Exercises 19 Nucleation 19 .1 Homogeneous Nucleation 19 .1. 1 Classical Theory of Nucleation in a One-Component System without Strain Energy 19 .1. 2 Classical Theory of Nucleation in a Two-Component System without Strain Energy 19 .1. 3 Effect of Elastic Strain Energy 19 .1. 4 Nucleus Shape of Minimum Energy 19 .1. 5 More Complete Expressions for the Classical Nucleation Rate 19 .1. 6 Nonclassical Models... volume J m-3 s-l K-' 7- Characteristic time S @i Diffusion potential for species i J @ Electrical potential Site fraction J C-l z P P X ~ c v-lm-l s-l - 0, Ri, (R) Atomic volume, atomic volume of component i, average atomic volume m3 W Angular frequency S -1 = Jm-3 CHAPTER 1 INTRODUCTI0N Kinetics of Materials is the study of the rates at which various processes occur in materials- knowledge of which is... tensor) m-2 M,M Mobility, mobility tensor various M ,O Atomic or molecular weight of species i kg N;' m Mass Number $7 s i, j , I f,$ N N Total number of atoms or molecules in subsystem Nc Number of components in a solution N O Avogadro's number ,-3 S s -1 -1 s-l s-l S K -1 -' N-' kg - - 6.023 x SYMBOLS-ROMAN xxiii SYMBOLS-ROMAN Symbol Definition Units n Number per unit volume (concentration) m-3 a Unit... Cahn-Hilliard Equation 18 .3.2 Allen-Cahn Equation 18 .3.3 Numerical Simulation and the Phase-Field Method 18 .4 Decomposition and Order-Disorder: Initial Stages 18 .4 .1 Cahn-Hilliard: Critical and Kinetic Wavelengths 18 .4.2 Allen-Cahn: Critical Wavelength 18 .5 Coherency-Strain Effects 18 .5 .1 Generalizations of the Cahn-Hilliard and Allen-Cahn Equations 18 .5.2 Diffraction and the Cahn-Hilliard Equation Bibliography... Index Topic Index 6 01 60 1 603 605 605 608 608 610 611 611 612 615 617 620 623 639 PREFACE This textbook has evolved from part of the first-year graduate curriculum in the Department of Materials Science and Engineering at the Massachusetts Institute of Technology (MIT) This curriculum includes four required semester-long subjects Materials at Equilibrium,” “Mechanical Properties of Materials, ” “Electrical,... xiii 5 01 502 502 504 508 510 512 512 514 515 516 518 518 519 524 526 533 533 534 537 538 540 540 543 543 543 546 547 547 548 549 550 550 555 xiv CONTENTS 23 .1 23.2 23.3 23.4 General Features Nucleus Morphology and Energy Coherency Loss during Growth Two Example Systems 23.4 .1 Cu-Co System 23.4.2 A1-Cu System Bibliography 24 Martensitic Transformations 24 .1 General Features 24.2 Crystallography 24.2 .1 Lattice... assumptions of local equilibrium states This approach is consistent 4 CHAPTER 1 INTRODUCTION , - _-_ - t u - ( 9 moles of B/(total moles A+B) (a) - tangent (x> moles of B/(total moles A+B) (b) Figure 1. 1: (a) Curves of the mininium free energy of homogeneous cy and / phases a 3 s a furictioii of average (overall) composition, ( X ) ,at constant P and T G is the free energy of 1 mole of solution... J, J mol-l, J m-3 7f, H , h Enthalpy, enthalpy per mole (or particle), enthalpy density J, Jmol-l, Jm-3 h Planck constant 6.626 x 1 0-3 4 J s 14 ,Ii Current of electrical charge, current of species i c s -1 , Unit vectors parallel to Cartesian coordinates 2,y, z m-2 J Flux, flux of species i Nucleation rate K Thermal conductivity J K Rate constant various k Boltzmann constant 1. 38 x ~ o - ~ ~ J K - ~ Lap... Stability of a l p Interface during Diffusion-Limited Particle Growth 20.3.3 Stability of Liquid/Solid Interface during Binary Alloy Solidification 20.3.4 Analyses of Interfacial Stability Bibliography Exercises 21 Concurrent Nucleation and Growth 21. 1 Overall Rate of Discontinuous Transformation 21. 1 .1 Time-Cone Analysis of Concurrent Nucleation and Growth 21. 1.2 Transformations near the Edge of a Thin... CONTENTS 17 .3 Continuous and Discontinuous Transformations Bibliography 18 Spinodal and Order-Disorder Transformations 18 .1 Interdiffusivity a t Unstable Compositions 18 .2 Diffuse Interface Theory 18 .2 .1 Free Energy of an Inhomogeneous System 18 .2.2 Structure and Energy of Diffuse Interfaces 18 .2.3 Diffusion Potential for Transformation 18 .3 Evolution Equations for Order Parameters 18 .3 .1 Cahn-Hilliard . Crystal/Vapor Interfaces 12 .2 .1 Structure of Crystal/Vapor Surfaces 12 .2.2 Crystal Growth from a Supersaturated Vapor 12 .2.3 Surfaces as Sinks for Supersaturated Lattice Vacancies 12 .3 .1 Structure of. Features of Phase Transformations 419 17 .1 Order Parameters 420 17 .1. 1 One-Component or Fixed Stoichiometry Systems 420 17 .1. 2 Two-Component Systems 423 17 .2 Molar Free-Energy Changes 428. 88 91 91 99 10 0 10 0 10 1 10 2 10 2 10 3 10 3 10 5 10 7 10 7 11 0 11 3 11 4 11 4 13 1 13 1 13 4 13 5 13 9 viii CONTENTS 6.3 Measurement of Diffusivities Bibliography Exercises 7 Atomic