handbook of heterogenous kinetics (soustelle/handbook of heterogenous kinetics) || front matter

Handbook of Heterogenous Kinetics

Handbook of Heterogenous Kinetics

Michel Soustelle

First published 2006 and 2007 in France by Hermes Science/Lavoisier in 4 volumes entitled: Cinétique hétérogène © LAVOISIER 2006, 2007 First published 2010 in Great Britain and the United States in one volume by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2010 The rights of Michel Soustelle to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data Soustelle, Michel. [Cinétique hétérogène. English] Handbook of heterogenous kinetics / Michel Soustelle. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-100-1 1. Chemical kinetics--Handbooks, manuals, etc. I. Title. QD502.S5613 2010 541'.394--dc22

2009049028 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-100-1

Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

Chapter 1. Definitions and Experimental Approach . . . . . . . . . . . . . . 1 1.1. Thermal transformations of solids . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Classification of transformations . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1. Transformation without formation of a new solid phase . . . . . . 3 1.2.2. Transformation with formation of a new solid phase . . . . . . . . 4

1.3. Speed and rate of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1. Speed of reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.2. Fractional extent and rate of a reaction. . . . . . . . . . . . . . . . . 8 1.3.3. Volumes of the phases and coefficient of expansion of the reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4. Reaction zones of a transformation . . . . . . . . . . . . . . . . . . . . . 10 1.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.2. Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.3. Sizes of a reaction zone . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5. Chemical characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1. Analyses of the gas phases . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5.2. Elementary analyses of the solids . . . . . . . . . . . . . . . . . . . . 13

1.6. Structural characterizations of the solids . . . . . . . . . . . . . . . . . . 13 1.7. Textural characterizations of the solids . . . . . . . . . . . . . . . . . . . 14

1.7.1. The marker method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.7.2. The cavity method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8. Characterization of the evolution of the systems . . . . . . . . . . . . . 17 1.8.1. Curves of evolution: definitions . . . . . . . . . . . . . . . . . . . . . 17 1.8.2. Curves of evolution: experimental obtaining . . . . . . . . . . . . . 17 1.8.3. Curves of evolution: obtained laws . . . . . . . . . . . . . . . . . . . 23

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1.9. Influence of various variables on speed . . . . . . . . . . . . . . . . . . . 26 1.9.1. Influence of temperature . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.9.2. Influence of partial pressures of gases . . . . . . . . . . . . . . . . . 27 1.9.3. Influence of the shapes and sizes of solid particles. . . . . . . . . . 27

Chapter 2. The Real Solid: Structure Elements and Quasi-Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1. Structure elements of a solid . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.1. Definition of a structure element . . . . . . . . . . . . . . . . . . . . 30 2.1.2. Binary solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.3. Symbolic notation of structure elements . . . . . . . . . . . . . . . . 31 2.1.4. Building unit of a solid . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.5. Description and composition of a solid. . . . . . . . . . . . . . . . . 33

2.2. Structure elements of a stoichiometric binary solid . . . . . . . . . . . . 35 2.2.1. Schottky disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2. Frenkel disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.3. Antistructure disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.4. S.A. disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3. Structure elements of a non-stoichiometric binary solid . . . . . . . . . 36 2.3.1. Distance from stoichiometry and structure element . . . . . . . . . 37 2.3.2. The approximation of Wagner of the prevalent defect for ionic solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.3. More complex binary compounds. . . . . . . . . . . . . . . . . . . . 44

2.4. Extension to non-binary compounds. . . . . . . . . . . . . . . . . . . . . 44 2.4.1. The pseudo-binary approximation . . . . . . . . . . . . . . . . . . . 44 2.4.2. Generalization of the approximation of the prevalent defect . . . . 45

2.5. Quasi-chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.1. Definition and characteristics of quasi-chemical reactions . . . . . 46 2.5.2. Homogenous quasi-chemical reactions in the solid . . . . . . . . . 47 2.5.3. The interphase reactions . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5.4. Reactions of solid destruction . . . . . . . . . . . . . . . . . . . . . . 52

2.6. Introduction of foreign elements into a solid . . . . . . . . . . . . . . . . 53 2.6.1. Concepts of impurity and doping agent . . . . . . . . . . . . . . . . 53 2.6.2. The controlled atomic imperfection in stoichiometric solids . . . . 54 2.6.3. The controlled electronic imperfection in non-stoichiometric solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.6.4. Concept of induced valence . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 3. Thermodynamics of Heterogenous Systems . . . . . . . . . . . . 59 3.1. Introduction: aims of thermodynamics . . . . . . . . . . . . . . . . . . . 59 3.2. General survey of thermodynamics of equilibrium . . . . . . . . . . . . 60

3.2.1. Chemical potential of a component in a phase . . . . . . . . . . . . 60

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3.2.2. Variance of a system at equilibrium . . . . . . . . . . . . . . . . . . 64 3.2.3. Associated extensive properties of a transformation, partial molar properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.4. Chemical potential of an ion or a structure element . . . . . . . . . 66 3.2.5. Feasibility of chemical reactions: De Donder inequality . . . . . . 67 3.2.6. Law of mass action for equilibriums . . . . . . . . . . . . . . . . . . 68

3.3. Phenomena leading to solid-gas equilibriums . . . . . . . . . . . . . . . 69 3.3.1. Systems with variance p − 1 . . . . . . . . . . . . . . . . . . . . . . . 70 3.3.2. Systems with variance p . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3.3. Systems with variance p + 1 . . . . . . . . . . . . . . . . . . . . . . . 71

3.4. Thermodynamic approach of solid-gas systems . . . . . . . . . . . . . . 71 3.4.1. Univariant systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.2. Divariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4.3. Trivariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5. Thermodynamics of systems containing solid phases only . . . . . . . 76 3.5.1. Non-variant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5.2. Univariant systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.6. Specific study of quasi-chemical equilibriums. . . . . . . . . . . . . . . 77 3.6.1. Equilibrium between an oxide and oxygen: the Wagner prevalent defect approximation . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.6.2. General equilibrium of an oxide with oxygen in the Brouwer approximation of majority defects . . . . . . . . . . . . . . . . . . . . . . . 79 3.6.3. Doping a solid with foreign elements: quantitative aspect . . . . . 82

3.7. Thermodynamics of systems: water vapor-hydrated salts . . . . . . . . 85 3.7.1. Experimental approach of equilibriums between water vapor and hydrated salts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.7.2. Study of the equilibriums with variance 1 . . . . . . . . . . . . . . . 87 3.7.3. Study of hydrates with variance 2 . . . . . . . . . . . . . . . . . . . . 88

3.8. Sequence of transformations, juxtaposition of stability area . . . . . . 93 3.9. Equilibrium of the formation of a solid from a solution . . . . . . . . . 96

3.9.1. Solubility product and supersaturation . . . . . . . . . . . . . . . . . 96 3.9.2. Extension to formation of a real solid . . . . . . . . . . . . . . . . . 99 3.9.3. Extension to the transformation of a solid into another solid . . . . 99

3.10. Variations in the equilibrium conditions with sizes of solid phases . 100 3.10.1. Variation in equilibrium constant with curvature radii. . . . . . . 100 3.10.2. Influence of curvature radii on tension of vapor . . . . . . . . . . 103 3.10.3. Influence of curvature radii on point defect concentrations . . . . 104

Chapter 4. Elementary Steps in Heterogenous Reactions . . . . . . . . . . . 105 4.1. Nature of elementary steps . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.1.1. The postulate of the activated jump. . . . . . . . . . . . . . . . . . . 107 4.1.2. Voluminal speed of an elementary jump. . . . . . . . . . . . . . . . 110 4.1.3. Total voluminal speed of an elementary step . . . . . . . . . . . . . 114

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4.2. Elementary reactions at solid-solid interfaces . . . . . . . . . . . . . . . 114 4.2.1. The phenomenon of epitaxy . . . . . . . . . . . . . . . . . . . . . . . 115 4.2.2. Creation of an M interstitial cation in MG. . . . . . . . . . . . . . . 115 4.2.3. Creation of a G vacancy anion in MG . . . . . . . . . . . . . . . . . 117 4.2.4. Consumption of a G interstitial anion of MG . . . . . . . . . . . . . 119 4.2.5. Consumption of an M vacancy cation of MG . . . . . . . . . . . . . 120 4.2.6. Creation of the point defects created in the initial solid . . . . . . . 122

4.3. Elementary reactions at gas-solid interfaces . . . . . . . . . . . . . . . . 122 4.3.1. Consumption of an M interstitial cation of MG. . . . . . . . . . . . 123 4.3.2. Consumption of a G vacancy anion of MG . . . . . . . . . . . . . . 124 4.3.3. Creation of a G interstitial anion in MG . . . . . . . . . . . . . . . . 126 4.3.4. Creation of an M vacancy cation in MG . . . . . . . . . . . . . . . . 128

4.4. The apparent energies of activation of interface reactions . . . . . . . . 130 4.5. The areal speed of an interface reaction. . . . . . . . . . . . . . . . . . . 130

Chapter 5. Chemical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1. Introduction: nature of diffusing particles in a solid . . . . . . . . . . . 131

5.1.1. Origin of the diffusion in a solid . . . . . . . . . . . . . . . . . . . . 131 5.1.2. Mechanisms of diffusion in a solid . . . . . . . . . . . . . . . . . . . 132

5.2. Flux of diffusion and velocity of diffusing particles . . . . . . . . . . . 135 5.3. The laws of Fick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3.1. First law of Fick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.2. Second law of Fick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3.3. Expression of the laws of Fick in various axes systems . . . . . . . 142 5.3.4. Solutions of the laws of Fick . . . . . . . . . . . . . . . . . . . . . . . 144 5.3.5. Self-diffusion and diffusion of the associated defect . . . . . . . . 148

5.4. Steady state obstructed diffusion . . . . . . . . . . . . . . . . . . . . . . . 150 5.5. Diffusion under electric field . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.5.1. Expression of flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.5.2. Electric conductivity and diffusion . . . . . . . . . . . . . . . . . . . 155 5.5.3. Diffusion in a semiconductor with electronic conduction under null current and without accumulation . . . . . . . . . . . . . . . . . 157

5.6. Diffusion in two mediums separated by a mobile interface . . . . . . . 161 5.6.1. Danckwerts solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.6.2. Example of application . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.6.3. Wagner pseudo-steady state approximation . . . . . . . . . . . . . . 166

Chapter 6. Chemical Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1. Definitions: physical adsorption and chemical adsorption. . . . . . . . 169 6.2. Adsorption thermodynamics and chemisorption equilibrium . . . . . . 170

6.2.1. Experimental results on adsorption equilibrium . . . . . . . . . . . 170 6.2.2. The Langmuir model of chemisorption equilibrium . . . . . . . . . 171

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6.2.3. Dissociative adsorption and the Langmuir model . . . . . . . . . . 173 6.2.4. Chemisorption of gas mixtures in the Langmuir model . . . . . . . 175 6.2.5. Adsorption isotherms that do not follow the Langmuir model . . . 176

6.3. Kinetics of chemisorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.1. Velocity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.3.2. Role of temperature on the kinetics of adsorption . . . . . . . . . . 180

6.4. Chemisorption and structure elements. . . . . . . . . . . . . . . . . . . . 181 6.4.1. Ways of chemisorption modeling . . . . . . . . . . . . . . . . . . . . 182 6.4.2. The concepts used in the quasi-chemical description of adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.4.3. Modes of adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.4.4. Modifications of the properties of adsorption of a solid. . . . . . . 191

Chapter 7. Mechanisms and Kinetics of a Process . . . . . . . . . . . . . . . 195 7.1. Speeds and reactivities of reactions taking place in only a single zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.1.1. Voluminal speed in a zone . . . . . . . . . . . . . . . . . . . . . . . . 195 7.1.2. Reactivity of a transformation in a given zone . . . . . . . . . . . . 197

7.2. Transformations with several zones . . . . . . . . . . . . . . . . . . . . . 201 7.2.1. Postulate of the decomposition of a reaction in elementary steps . 201 7.2.2. Reaction mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.2.3. Material balance in a reaction zone . . . . . . . . . . . . . . . . . . . 203 7.2.4. Setting in the equation of mechanism – example . . . . . . . . . . . 205

7.3. Linear reaction mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.3.1. Definition and classification of the linear mechanisms . . . . . . . 210 7.3.2. Multiplying coefficients of a linear mechanism . . . . . . . . . . . 211

7.4. Linear mechanisms in pseudo-steady state modes . . . . . . . . . . . . 213 7.4.1. Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.4.2. Theorem of “the equality of rates” of a linear mechanism in pseudo-steady state modes . . . . . . . . . . . . . . . . . . . . . . . . . . 214 7.4.3. Relations between various forms of the rates (speed) of reactions with a linear mechanism in pseudo-steady state modes . . . 217 7.4.4. Volumes of the phases and coefficient of expansion of a reaction with a linear mechanism in pseudo-steady state modes . . . 219 7.4.5. Setting in equation of a linear mechanism in pseudo-steady state modes . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.5. Pure modes or modes with a rate-determining step . . . . . . . . . . . . 220 7.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.5.2. Theorem of the concentrations in pure mode . . . . . . . . . . . . . 221 7.5.3. Reactivity of the rate-determining step in pure mode . . . . . . . . 225 7.5.4. Application of the method of the pure modes . . . . . . . . . . . . . 226 7.5.5. Rate of the reaction in pure modes . . . . . . . . . . . . . . . . . . . 227

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7.5.6. Examples of resolutions of pure modes . . . . . . . . . . . . . . . . 228 7.5.7. Pure modes far from equilibrium . . . . . . . . . . . . . . . . . . . . 230

7.6. Mixed modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.6.1. Definition: pseudo-steady state mixed modes. . . . . . . . . . . . . 234 7.6.2. Solving a pseudo-steady state mixed mode . . . . . . . . . . . . . . 234

7.7. Generalization, rate of a linear mechanism in pseudo-steady state mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.8. Mixed non-pseudo-steady state modes . . . . . . . . . . . . . . . . . . . 242 7.9. Equivalent reaction of a linear subset in local pseudo-steady state mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.9.1. Local pseudo-steady state modes . . . . . . . . . . . . . . . . . . . . 245 7.9.2. Application to the “elementary” steps . . . . . . . . . . . . . . . . . 248

7.10. Reactions with separable rates . . . . . . . . . . . . . . . . . . . . . . . 248 7.11. Influence of intensive variables on the kinetic laws . . . . . . . . . . . 250

7.11.1. The first kind of changes of laws . . . . . . . . . . . . . . . . . . . 251 7.11.2. The second kind changes of laws . . . . . . . . . . . . . . . . . . . 252 7.11.3. The third kind changes of laws. . . . . . . . . . . . . . . . . . . . . 252

7.12. Distance from equilibrium for a reaction . . . . . . . . . . . . . . . . . 252 7.12.1. Distance of an elementary step from equilibrium. . . . . . . . . . 253 7.12.2. Pseudo-steady state mode with a rate-determining step . . . . . . 254

7.13. Processes concerned in a heterogenous reaction . . . . . . . . . . . . . 255

Chapter 8. Nucleation of a New Solid Phase . . . . . . . . . . . . . . . . . . . 257 8.1. Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 8.2. Examples of nucleation diagram . . . . . . . . . . . . . . . . . . . . . . . 258 8.3. Interfacial energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

8.3.1. Definition of interfacial energy . . . . . . . . . . . . . . . . . . . . . 260 8.3.2. Microscopic interpretation . . . . . . . . . . . . . . . . . . . . . . . . 261 8.3.3. Effective interfacial energy. . . . . . . . . . . . . . . . . . . . . . . . 268 8.3.4. Relation between energy and the interfacial area. . . . . . . . . . . 271

8.4. Formation molar Gibbs energy of clusters . . . . . . . . . . . . . . . . . 272 8.4.1. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 8.4.2. Homogenous nucleation within a liquid phase: Volmer approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 8.4.3. Homogenous nucleation within a solid phase . . . . . . . . . . . . . 277 8.4.4. Heterogenous primary nucleation starting from a fluid phase . . . 277 8.4.5. Heterogenous primary nucleation starting from a solid phase . . . 282

8.5. Kinetics of nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 8.5.1. Reaction pathway and localization of the phenomena . . . . . . . . 285 8.5.2. Rate and frequency of nucleation . . . . . . . . . . . . . . . . . . . . 289 8.5.3. Various considered modes . . . . . . . . . . . . . . . . . . . . . . . . 290 8.5.4. Kinetics of pseudo-steady state modes of condensation. . . . . . . 291

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8.5.5. Kinetics of pseudo-steady state modes of condensation on potential nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.5.6. Intervention of diffusion in the process of nucleation . . . . . . . . 306

Chapter 9. Growth of a Solid Phase . . . . . . . . . . . . . . . . . . . . . . . . 309 9.1. Description of the zones of growth . . . . . . . . . . . . . . . . . . . . . 309

9.1.1. The initial solid is a single reactant . . . . . . . . . . . . . . . . . . . 310 9.1.2. The initial solid reacts with another phase . . . . . . . . . . . . . . . 311

9.2. Direction of the development of phase B during the growth . . . . . . 311 9.2.1. The initial solid is a single reactant . . . . . . . . . . . . . . . . . . . 312 9.2.2. The initial solid reacts with another phase . . . . . . . . . . . . . . . 312

9.3. Modes and models for growth . . . . . . . . . . . . . . . . . . . . . . . . 312 9.3.1. Modes of the growth of a crystal of B on support A . . . . . . . . . 312 9.3.2. Modeling the growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

9.4. Relationship between the motion velocities of the interfaces and the chemical growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

9.4.1. Inward development of formed solid . . . . . . . . . . . . . . . . . . 315 9.4.2. Outward development of the formed solid. . . . . . . . . . . . . . . 317

9.5. Methodology to model growth . . . . . . . . . . . . . . . . . . . . . . . . 318 9.5.1. Modeling the space function of growth . . . . . . . . . . . . . . . . 319 9.5.2. Modeling the reactivity of growth. . . . . . . . . . . . . . . . . . . . 319

9.6. Expressions of the space functions for the growth of a grain . . . . . . 320 9.6.1. Space functions in isotropic growth. . . . . . . . . . . . . . . . . . . 320 9.6.2. Space functions in radial anisotropic growth . . . . . . . . . . . . . 330 9.6.3. Introduction of a dimensionless time . . . . . . . . . . . . . . . . . . 335

Chapter 10. Transformation by Surface Nucleation and Growth . . . . . . 337 10.1. Nucleation, growth, and experimental rate . . . . . . . . . . . . . . . . 338 10.2. One-process model with instantaneous nucleation and slow growth . 339

10.2.1. Reaction of a single grain (or massive material) . . . . . . . . . . 340 10.2.2. Case of a monodispersed powder . . . . . . . . . . . . . . . . . . . 342 10.2.3. Shapes of kinetic and rate curves . . . . . . . . . . . . . . . . . . . 344 10.2.4. One-process model with slow nucleation and instantaneous growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10.2.5. Reaction of a single grain . . . . . . . . . . . . . . . . . . . . . . . . 345 10.2.6. Reaction of a powder . . . . . . . . . . . . . . . . . . . . . . . . . . 346

10.3. Two-process models: nucleation and growth . . . . . . . . . . . . . . . 347 10.3.1. General expression for the rate . . . . . . . . . . . . . . . . . . . . . 347 10.3.2. Influence of the past on the transformation rate . . . . . . . . . . . 350

10.4. Two-process model with surface nucleation-radial anisotropic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

10.4.1. Reaction of a single grain . . . . . . . . . . . . . . . . . . . . . . . . 351

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10.4.2. Construction of the model of evolution of a powder . . . . . . . . 352 10.4.3. Calculation of the free area (space function) for nucleation. . . . 353 10.4.4. Calculation of the rates and the fractional extents according to time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 10.4.5. Dimensionless rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 10.4.6. Conclusion on the surface nucleation and radial anisotropic growth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

10.5. Two-process model with surface nucleation and isotropic growth . . 361 10.5.1. Qualitative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 10.5.2. Quantitative approach . . . . . . . . . . . . . . . . . . . . . . . . . . 361 10.5.3. Modeling the evolution of a grain . . . . . . . . . . . . . . . . . . . 365 10.5.4. Modeling the evolution of a collection of grains . . . . . . . . . . 366 10.5.5. Application to the spherical grains: model of Johnson-Mehl and Mampel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

10.6. Non-isobaric and/or non-isothermal kinetics . . . . . . . . . . . . . . . 370 10.6.1. One-process models . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 10.6.2. Two-process models . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

10.7. Powders with granular distributions . . . . . . . . . . . . . . . . . . . . 375 10.8. Return to the first and second kind of changes of laws . . . . . . . . . 376

10.8.1. First kind of changes of laws . . . . . . . . . . . . . . . . . . . . . . 376 10.8.2. Second kind of changes of laws . . . . . . . . . . . . . . . . . . . . 376

10.9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Chapter 11. Modeling and Experiments . . . . . . . . . . . . . . . . . . . . . . 379 11.1. The adequacy between the experimental conditions and modeling . . 379 11.2. Expressions of experimental speeds . . . . . . . . . . . . . . . . . . . . 381

11.2.1. Thermogravimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 11.2.2. Microcalorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 11.2.3. Manometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 11.2.4. Measurement of the amounts of solids A with X-ray diffraction. 384 11.2.5. Measurement of the amounts of the formed solid B . . . . . . . . 385 11.2.6. Thickness of the layer of a planar sample of B . . . . . . . . . . . 386 11.2.7. Relationships between experimental speeds . . . . . . . . . . . . . 387

11.3. Derivation of the kinetic curves. . . . . . . . . . . . . . . . . . . . . . . 388 11.4. The experimental verification of the assumptions . . . . . . . . . . . . 388

11.4.1. The pseudo-steady state mode test. . . . . . . . . . . . . . . . . . . 388 11.4.2. The test of the separable rate or the φΕ test . . . . . . . . . . . . . 391

11.5. Determination of the morphological model for growth . . . . . . . . . 395 11.5.1. Choice of the category of models: one-process or two-process model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 11.5.2. Determination of the model and its parameters . . . . . . . . . . . 396

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11.6. Calculations of the reactivity of growth and the specific frequency of nucleation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.7. Variations of the kinetic properties with the intensive variables . . . 399

11.7.1. Determination of the variation in the reactivity of growth starting from the morphological model . . . . . . . . . . . . . . . . . . . . 399 11.7.2. Direct determination of the variation in the reactivity of growth starting from the experiment . . . . . . . . . . . . . . . . . . . . 399 11.7.3. Comparison of the two obtained variations: new verification of the morphological model . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

11.8. Methodology of a study . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 11.8.1. Identification of the reaction . . . . . . . . . . . . . . . . . . . . . . 402 11.8.2. The separation of the models . . . . . . . . . . . . . . . . . . . . . . 402 11.8.3. Methodical approach of a study . . . . . . . . . . . . . . . . . . . . 404

Chapter 12. Granular Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . 407 12.1. Qualitative description of the model . . . . . . . . . . . . . . . . . . . . 408 12.2. Morphological modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

12.2.1. Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 12.2.2. Geometry of the neck . . . . . . . . . . . . . . . . . . . . . . . . . . 410 12.2.3. Relation between the fractional extent and the radius x of the bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

12.3. Structure of the coalescence mechanism . . . . . . . . . . . . . . . . . 413 12.3.1. Transport phenomenon and groups of elementary steps . . . . . . 413 12.3.2. Various kinetic modes with rate-determining steps. . . . . . . . . 414 12.3.3. Definition of the reactivity of coalescence . . . . . . . . . . . . . . 415

12.4. Determination of the space functions . . . . . . . . . . . . . . . . . . . 416 12.4.1. Mode with an interface reaction as the rate-determining step . . 417 12.4.2. Modes with diffusion as rate-determining step . . . . . . . . . . . 418 12.4.3. Recapitulation of the space functions . . . . . . . . . . . . . . . . . 420

12.5. Rate constants and radius of curvature. . . . . . . . . . . . . . . . . . . 420 12.6. Reactivity of coalescence of a solid with a single component . . . . . 423

12.6.1. Case of vacancies diffusion in the solid . . . . . . . . . . . . . . . 423 12.6.2. Case of gas diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . 430 12.6.3. Summary of the reactivities. . . . . . . . . . . . . . . . . . . . . . . 435

12.7. Extensions to the coalescence of solids with several components . . 436 12.7.1. Coalescence of anatase in the presence of water vapor . . . . . . 437 12.7.2. Coalescence of anatase in the presence of oxygen and hydrogen chloride with or without water vapor . . . . . . . . . . . . . 441 12.7.3. Coalescence of ceria in presence of oxygen and water vapor. . . 442

12.8. Relations between experiments and modeling . . . . . . . . . . . . . . 443 12.8.1. Experimental measurement of coalescence . . . . . . . . . . . . . 443 12.8.2. Determination of the variations of the reactivity with intensive quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

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12.8.3. Relation between experiment and space function in the model of tangential spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

12.9. Oswald ripening and reduction in porosity . . . . . . . . . . . . . . . . 448

Chapter 13. Decomposition Reactions of Solids . . . . . . . . . . . . . . . . . 449 13.1. Classifications of decomposition reactions . . . . . . . . . . . . . . . . 450

13.1.1. Classification according to the sign of the enthalpy . . . . . . . . 450 13.1.2. Classification according to the origin of the gas molecule . . . . 450

13.2. Extent measurement with the change of the mass . . . . . . . . . . . . 451 13.2.1. Stoichiometric solids. . . . . . . . . . . . . . . . . . . . . . . . . . . 452 13.2.2. The produced solid is not stoichiometric . . . . . . . . . . . . . . . 453 13.2.3. The initial solid is not stoichiometric . . . . . . . . . . . . . . . . . 454

13.3. Observed experimental results . . . . . . . . . . . . . . . . . . . . . . . 456 13.3.1. Rate-time and extent-time curves . . . . . . . . . . . . . . . . . . . 456 13.3.2. Influences of the gas pressures . . . . . . . . . . . . . . . . . . . . . 457 13.3.3. Influence of temperature . . . . . . . . . . . . . . . . . . . . . . . . 459 13.3.4. Non-isothermal decomposition reactions. . . . . . . . . . . . . . . 460

13.4. Kinetics of growth in decomposition reactions of solids . . . . . . . . 462 13.4.1. Qualitative analysis of the growth . . . . . . . . . . . . . . . . . . . 463 13.4.2. Basic growth mechanism with gaseous diffusion . . . . . . . . . . 464 13.4.3. Basic mechanism of growth with diffusions of defects . . . . . . 473 13.4.4. Smith-Topley’s Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 475

13.5. Nucleation in decomposition reactions of solids . . . . . . . . . . . . . 478 13.5.1. Experimental approach of nucleation . . . . . . . . . . . . . . . . . 479 13.5.2. Example of the dehydration of kaolinite . . . . . . . . . . . . . . . 481 13.5.3. Nucleation and Smith-Topley’s effect . . . . . . . . . . . . . . . . 483

13.6. Total kinetic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 13.7. Influence of the granular distribution . . . . . . . . . . . . . . . . . . . 484 13.8. Normal and abnormal growth . . . . . . . . . . . . . . . . . . . . . . . . 486

Chapter 14. Reactions Between Solids . . . . . . . . . . . . . . . . . . . . . . . 489 14.1. Classification of the reactions between solids . . . . . . . . . . . . . . 490

14.1.1. Simple addition reactions . . . . . . . . . . . . . . . . . . . . . . . . 490 14.1.2. Addition reactions involving decomposition . . . . . . . . . . . . 490 14.1.3. Addition reactions involving a redox reaction. . . . . . . . . . . . 491 14.1.4. Exchange reactions or double decompositions . . . . . . . . . . . 491

14.2. The modeling assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . 492 14.3. The experimental measure of the extent of the reactions . . . . . . . . 493 14.4. Reactivities of reactions between solids . . . . . . . . . . . . . . . . . . 494

14.4.1. Position of the problem and experimental approach . . . . . . . . 494 14.4.2. Structures of the reaction mechanism of growth . . . . . . . . . . 495

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14.4.3. Expression of the reactivities, reaction of titanium dioxide with barium carbonate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

14.5. Rates of the reactions between powders . . . . . . . . . . . . . . . . . . 508 14.5.1. Problems of designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 14.5.2. Rates of a two-grain level . . . . . . . . . . . . . . . . . . . . . . . . 515 14.5.3. Rate of a granular cell . . . . . . . . . . . . . . . . . . . . . . . . . . 516 14.5.4. Rates on the scale of the powder. . . . . . . . . . . . . . . . . . . . 532

14.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

Chapter 15. Gas-Solid Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 543 15.1. Classification of gas-solid reactions . . . . . . . . . . . . . . . . . . . . 544

15.1.1. Class 1: synthesis reactions . . . . . . . . . . . . . . . . . . . . . . . 544 15.1.2. Class 2: double-decomposition reactions. . . . . . . . . . . . . . . 544

15.2. Pure metal gas reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 15.2.1. Experimental data of oxidation of metals . . . . . . . . . . . . . . 546 15.2.2. Reaction zones and elementary reactions . . . . . . . . . . . . . . 554 15.2.3. Pure modes with interface rate determining step . . . . . . . . . . 566 15.2.4. Pure diffusion modes . . . . . . . . . . . . . . . . . . . . . . . . . . 569 15.2.5. Mixed modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

15.3. Growth process in the reduction of metallic oxides by hydrogen . . . 585 15.3.1. Mechanism with diffusion of gases through the pores . . . . . . . 586 15.3.2. Mechanisms with diffusion of defect in the formed solid phase . 591 15.3.3. Conclusion about the reduction of oxides by hydrogen . . . . . . 593 15.3.4. Example of the reduction of a uranium oxide . . . . . . . . . . . . 594

15.4. Growth process of oxidation of metals by water vapor . . . . . . . . . 596 15.4.1. General approach of mechanism. . . . . . . . . . . . . . . . . . . . 596 15.4.2. n-type formed oxide with interstitial cations. . . . . . . . . . . . . 597 15.4.3. n-type formed oxide with anion vacancies . . . . . . . . . . . . . . 598 15.4.4. p-type formed oxide with cation vacancies . . . . . . . . . . . . . 599 15.4.5. p-type formed oxide with interstitial anions . . . . . . . . . . . . . 600

Chapter 16. Transformations of Solid Solutions. . . . . . . . . . . . . . . . . 603 16.1. General information on transformations of solid solutions. . . . . . . 603

16.1.1. Various types of transformations of solid solutions . . . . . . . . 603 16.1.2. Variations of concentrations in solid solution . . . . . . . . . . . 604

16.2. Oxidation of metal alloys . . . . . . . . . . . . . . . . . . . . . . . . . . 606 16.2.1. Selective oxidation of single-phase binary metal alloys . . . . . . 607 16.2.2. Internal oxidation of single-phase binary alloys. . . . . . . . . . . 620 16.2.3. Oxidation of single-phase binary alloys with miscibility of formed oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 16.2.4. Oxidation of single-phase binary alloys with formation of two superimposed oxide layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

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16.3. Variations of the composition of a solid solution with gas formation 640 16.3.1. Fractional extent and rate . . . . . . . . . . . . . . . . . . . . . . . . 640 16.3.2. Spatial structure of the model . . . . . . . . . . . . . . . . . . . . . 642 16.3.3. Pure diffusion mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 16.3.4. Example: variation of stoichiometry of an oxide by reaction with hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

16.4. Superposition of a variation of stoichiometry and decomposition . . 648

Chapter 17. Modeling of Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 651 17.1. Non-stoichiometry of iron oxide . . . . . . . . . . . . . . . . . . . . . . 651

17.1.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 17.1.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 17.1.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 17.1.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

17.2. Stability of calcium carbonate. . . . . . . . . . . . . . . . . . . . . . . . 658 17.2.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 17.2.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 17.2.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 17.2.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

17.3. Thermodynamics of a solid-solid reactions . . . . . . . . . . . . . . . . 665 17.3.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 17.3.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 17.3.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 17.3.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

17.4. Hydrates of alumina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 17.4.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 17.4.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 17.4.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671 17.4.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671

17.5. Point defects in a metal sulfide . . . . . . . . . . . . . . . . . . . . . . . 679 17.5.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 17.5.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 17.5.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680 17.5.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681

17.6. Point defects of an alkaline bromide . . . . . . . . . . . . . . . . . . . . 689 17.6.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 17.6.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 17.6.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 17.6.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689

17.7. Diffusion of a metal into another metal . . . . . . . . . . . . . . . . . . 694 17.7.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 17.7.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

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17.7.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 17.7.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

17.8. Generation of atmospheres with very low pressures . . . . . . . . . . 701 17.8.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 17.8.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 17.8.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 17.8.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

Chapter 18. Mechanisms and Kinetic Laws . . . . . . . . . . . . . . . . . . . 709 18.1 Coalescence of anatase grains . . . . . . . . . . . . . . . . . . . . . . . . 709


18.2. Reaction of a cubic sample . . . . . . . . . . . . . . . . . . . . . . . . . 713 18.2.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 18.2.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 18.2.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 18.2.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

18.3. Anisotropic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 18.3.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 18.3.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 18.3.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 18.3.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

18.4. Gas-solid reaction with one-process model . . . . . . . . . . . . . . . . 732 18.4.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 18.4.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 18.4.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 18.4.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734

18.5. The direction of the development of a layer . . . . . . . . . . . . . . . 738 18.5.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 18.5.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 18.5.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 18.5.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 740

18.6. Mampel modeling by way of the point of inflection . . . . . . . . . . 747 18.6.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 18.6.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 18.6.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 18.6.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748

18.7. Nucleation in a reaction of dehydration . . . . . . . . . . . . . . . . . . 753 18.7.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 18.7.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

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18.7.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754 18.7.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754

18.8. Influence of particle size in nucleation-growth approach. . . . . . . . 759 18.8.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 18.8.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 18.8.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 18.8.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760

18.9. Decomposition with slow nucleation and slow anisotropic growth determined by diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 767


19. Mechanisms and Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 19.1. Competition oxidation – volatilization by TGA . . . . . . . . . . . . . 779


19.2. Controlled rate thermal analysis (CRTA) . . . . . . . . . . . . . . . . . 783 19.2.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 19.2.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 19.2.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785 19.2.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785

19.3. Sulfurization of a metal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 19.3.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 19.3.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 19.3.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 19.3.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

19.4. Oxidation of a metal and some of its alloys. . . . . . . . . . . . . . . . 794 19.4.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 19.4.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 19.4.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796 19.4.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

19.5. Reduction of octo-oxide of triuranium by dihydrogen . . . . . . . . . 804 19.5.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 19.5.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 19.5.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 19.5.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806

19.6. Dehydration of kaolinite . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 19.6.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813

Table of Contents xix

19.6.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 19.6.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 19.6.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815

19.7. Decomposition of a carbonate of a metal . . . . . . . . . . . . . . . . . 823 19.7.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 19.7.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 19.7.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 19.7.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824

19.8. Reaction between two solids . . . . . . . . . . . . . . . . . . . . . . . . 837 19.8.1. Key words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 19.8.2. Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837 19.8.3. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838 19.8.4. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839

Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

Appendix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

Appendix 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

Appendix 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 861

Appendix 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867

Appendix 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873

Appendix 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

Appendix 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881

Appendix 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899

Appendix 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919

Preface

The content of this book is a result of the courses given to the third-year students in École Nationale Supérieure des Mines de Saint-Étienne and the master’s students in chemical engineering in Saint-Etienne and Grenoble.

We wish through this work to make the synthesis of two extremely different approaches of heterogenous kinetics and reactivity of solids.

The examination of literature shows that heterogenous kinetics has developed, thanks to the works of two groups of researchers, a priori not very dependant on one another: on the one hand, metallurgists, specialists in corrosion of metals and alloys by gases at high temperature, and on the other hand, chemists, specialists in thermal analysis and who are more focused on reactions of salt decompositions.

Those in the first group usually work on massive metals and were not often confronted with the variations of reaction rates with time, the laws encountered being rather simple; they thus were not much concerned with the effects of morphology on the kinetics; on the other hand, they helped deepen the understanding of reaction mechanisms quite a lot, their study being based on point defects of the solids, which helps with the understanding of influences of temperature and gas pressures.

Those in the second group most often work on powders and the essence of their concerns relates to the influence of morphologies on evolutions, integrating the superposition of the two processes of nucleation and growth; on the other hand this group is not so concerned with the reaction mechanisms in the strict sense of the term.

xxii Handbook of Heterogenous Kinetics

Each one of these two groups produced works more specialized on a particular type of reaction without connections between the approaches always being explicit. Among them we can mention the following:

For the first point of view, we can mention the books edited by P. Kofstad [KOF 66], P. Sarrazin, A. Galerie, and J. Fouletier [SAR 00], A.M. Huntz-Aubriot and B. Pieraggi [HUN 03], and David Young [YOU 08].

For the second point view, the most recent book is that by A.K. Galwey and M.E. Brouwn [GAL 99].

Perhaps, it is the merit of a true French School of heterogenous kinetics to have mixed these two populations, a school that was born and developed around the 40 annual “Heterogenous Days Kinetics” that have taken place since we initiated them in 1968 and that has seen the work of the teams from Cluny, Compiegne, Dijon, Grenoble, Limoges, Marseilles, Orsay, Rennes, Saclay, Saint-Etienne, and Toulouse

This work was generally carried out under the instigation of industrial companies (such as ECA, CEZUS, COGEMA, COMURHEX, FRAMATOME, IFP, IRSID, LAFARGE, PECHINEY the ALCAN, RHÔNE-POULENC then RHODIA, and USINOR) which, giving concrete problems, obliged the researchers to progress in the formulation of the concepts and in deepening the fundamental aspects. These works have shown that many industrial problems require very fundamental research when they touch fields of knowledge that are not developed enough compared with the needs and this is the case with heterogenous kinetics.

Heterogenous kinetics is not a completed science but the aim of this book is to put in perspective the concepts and methods common to a great number of types of transformations. We hope we have succeeded, thanks mainly to the introduction of two new properties: (1) reactivity – primarily a function of intensive variables (temperature, partial pressures, concentrations) and related to the chemical mechanism; and (2) space function, related to the morphology of the system at a given time. This introduction now makes it possible to realize that metallurgists were especially interested in the reactivity and chemists concentrated their efforts primarily on the space function.

This book is concerned with the modeling of transformation of solid gas systems under the action of temperature. It is divided into 19 chapters, which we have gathered, after an introduction describing the main experimental data (Chapter 1), into four parts.

The first part (Chapters 2 to 6) presents the basics that seem necessary to the comprehension of heterogenous kinetics and talks about point defects in solids

Preface xxiii

(Chapter 2), recalls thermodynamics, which is always very related to kinetics (Chapter 3), an introduction to the concept of elementary step reactions in solid state (Chapter 4), a study of diffusion (Chapter 5), and an approach to chemisorption, always present insofar as the solids are constantly placed in an external gas medium (Chapter 6).

The second part (Chapters 7 to 11) presents the modeling of the reactions of solids by the introduction of the general concepts with the installation of the mechanisms and their resolutions in a single process (Chapter 7), the study of the nucleation process of a new solid phase (Chapter 8), the growth of the nucleus (Chapter 9), and the superposition of the two processes of nucleation and growth (Chapter 10). This part finishes with Chapter 11 which makes it possible to connect the concepts introduced by modeling to the experimental data. This part is largely devoted to space function.

The third part (Chapters 12 to 16) is devoted to the application of the general concepts of modeling to a certain number of families of transformations such as the transformations of coalescence of grains (Chapter 12), decompositions of solids (Chapter 13), reactions between solids (Chapter 14), and reactions between gases and solids (Chapter 15). Finally, we approach the treatment of transformations involving solid solutions, a field still largely in the waste land (Chapter 16). Essentially, this part is concerned with the function reactivity.

Finally, the fourth part is made up of three chapters of exercises and problems with their solutions. Every chapter refers to the one of the preceding parts:

– Chapter 17: Modeling of Mechanisms (Chapters 1 to 7);

– Chapter 18: Kinetic Mechanisms and Laws (Chapters 8 to 11);

– Chapter 19: Mechanisms and Reactivity (Chapters 12 to 16).

The solution of an exercise requires the knowledge presented in the corresponding chapters and possibly discussed in the preceding ones.

Each exercise present four parts: the aim gives the list of concepts involved in the problem, the problem with the questions, the numerical data, and the solution.

We strongly advise the readers to try and think about the answers of questions with only the statement, without taking note of the data. This will enable them to define by themselves the data that will be necessary, a situation that researchers or engineers face in their daily practice.

xxiv Handbook of Heterogenous Kinetics

Numerical calculations and the layouts of the curves were all carried out using a traditional spreadsheet provided with its usual mathematical and statistical functions. The corresponding Excel sheets for each problem can be downloaded from the following website: www.emse.fr/~soustelle.

In the appendices, we gather the main equations that can be developed from the modeling. To use these formulas, the reader can obtain the software “CIN3”, which is available on request by sending an e-mail to the following address: [email protected]

The bibliographical references at the end of the book do not claim to be exhaustive; they are there to illustrate the chronology of the appearance of the main concepts and can be used as a starting point for a targeted bibliography search.

As for any science based on experimentation, modeling in heterogenous kinetics requires assumptions, approximations and simplifications but these should not in any case be synonymous with a lack of rigor in the reasoning, calculations, and the control of the experiments. We hope to have been faithful to this policy.

Since this book obviously owes much to the whole French school, it could not have been possible without the fundamental contribution of all the people of my lab in Saint-Étienne and especially the team directed by Michele Pijolat. I want to acknowledge all the young researchers who developed original experimental methodologies and brought with them integrating concepts. I am greatly indebted to them and their boss for their help during numerous and fruitful discussions.

In addition, readers will notice active collaborations with Patrice Nortier for the writing of Chapter 8 about nucleation and Gerard Thomas for Chapter 14 about reactions between solids. The particular competence of each of them in the field concerned allied to their general knowledge of heterogenous kinetics enabled us to profit from contributions completely essential for the coherence of the whole of the work. We address to them our most sincere thanks.

Finally, I must not forget the hard work of Marie Prin-Lamaze who had the great task of translating the text from the French edition. I am greatly indebted to her for her help.

Michel Soustelle

Saint Vallier, January 2010

List of Symbols

[A]: (voluminal or areal) concentration of component A {A}: amount of component A a: lattice parameter Ag: model nucleation-growth parameter ai: activity of component i C: concentration c≠: activated species concentration D: diffusion coefficient d: dielectric constant e: growth spatial function of a nucleus e0: thickness of a plate ED: activation energy of diffusion E′a and E″a: apparent energies of activation F: faraday F: magnetic force being exerted on the solid A fg (r0): granular frequency of distribution fi: fugacity of gas i ft: function of partition of translation G: Gibbs energy

GD: space number of the flux of diffusion gi: Gibbs energy of pure component i h: Planck’s constant H: enthalpy of reaction h: height of a cylindrical grain I: intensity of a peak of X-ray diffraction I∞: intensity at the end of the reaction of a peak of X-ray diffraction I0: initial intensity of a peak of X-ray diffraction J: flux of diffusion K: equilibrium constant k: Boltzmann’s constant

ock : constant of corrosion

k', k'': rate constants ka: Erofeev’s nucleation constant kc: cubic kinetic constant kg: Avrami’s nucleation constant kl: linear kinetic constant kp: parabolic kinetic constant

xxvi Handbook of Heterogenous Kinetics

Ks: product of solubility L: number of degree of freedom m: total mass of solid at the time t m0: initial mass of solid MA: mass molar of component A mf: final mass of solid mi: initial mass of solid, cold N: number of grains not nucleated at the time t N0: initial number of grains N0: number of potential sites for nucleation n0: initial amount of the reactant solid nC(0): initial amount of matter of C Nc: index of coordination of a grain C ni: amount of i (number of moles) p: number of intensive variables P: pressure P0: equilibrium pressure Q: concerned heat

Aℜ : rate with respect to the component A R: constant of perfect gases R: ratio between the constants of corrosions of an alloy and a metal r: absolute speed of a reaction r0: initial radius of a cylindrical or spherical grain rA(0): initial radius of the grains of A R1, R 2: principal radii of curvatures ℜ : rate of reaction rA: reaction speed with respect to the component A ri, re: radii of the internal and external interfaces S: entropy, supersaturation

si, se: areas of the internal and external interfaces sp: area of the active interface T: temperature t: time ti: time of incubation tl: latency time tm: experimental time of detection of a signal u: electric mobility v: variance v: voluminal speed of reaction V: volume V0: initial volume of a reactant VA: volume of the phase A at the moment t Ve: volume of oxide equivalent to a mole of metal VmA: molar volume of A

iX : partial molar property xi: mole fraction of i xi: variable of composition z: coefficient of expansion

0z ,ijijz : co-ordination numbers of

spheres of i and j z : average co-ordination number Z: electrovalence of an ion α ∞: fractional extent at equilibrium αA: fractional extent with respect to component A αf: apparent fractional extent αμ: fractional extent just detected βj,ρ : arithmetic stoichiometric number of the component Xj in the ρth reaction βi: arithmetic stoichiometric number of Ai

List of Symbols xxvii

β′ and β″ : order of the direct and reverse reaction

χ: ratio of grains having nucleated Δ: distance from stoichiometry Δi(H): enthalpy associated with reaction i Δm: increment of solid mass between times 0 and t ΔR (X): operator δB: distance from stoichiometry with respect to B ε: switch (+1), (–1) ε: distance from equilibrium conditions φf: number of phases φ: reactivity γ: surface tension; specific frequency of nucleation γi: coefficient of activity κ' and κ": voluminal speed constants η: dimensionless date for nucleation χA: magnetic susceptibility of solid A κ: real speed constant of an elementary step μ: electric mobility μi: chemical potential of i

νjρ: algebraic stoichiometric number of the component Xj in the ρth reaction νA: algebraic stoichiometric number of component A νi: algebraic stoichiometric number of Ai

Π: tension of dissociation of an oxide θ: degree of covering of a surface θ: dimensionless time for growth θ∞ : degree of covering at equilibrium ρ: ratio between the reactivities of an alloy and a metal σ: area of a surface σ: electric conductivity σA: ratio of initial amounts of two powders

Aσ : average ratio of compositions Ω: solid angle ωexp: experimental reduced rate ωtheor: theoretical reduced rate

:ℵ affinity of a reaction Ξ: energy of a potential barrier ξA: extent of reaction with respect to the component A

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