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Mohamed N. Rahaman University of Missouri-Rolla, USA Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110720 International Standard Book Number-13: 978-1-4200-0705-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface...........................................................................................................................................xiii Author..............................................................................................................................................xv Chapter 1 Sintering of Ceramics: Fundamentals..........................................................................1 1.1 Introduction...............................................................................................................................1 1.2 The Sintering Process...............................................................................................................1 1.2.1 Characterization of Sintering.......................................................................................3 1.2.2 Approach to Sintering..................................................................................................3 1.3 Driving Force for Sintering......................................................................................................4 1.3.1 Surface Curvature.........................................................................................................5 1.3.2 Applied Pressure...........................................................................................................6 1.3.3 Chemical Reaction........................................................................................................6 1.4 Defects in Crystalline Solids....................................................................................................6 1.4.1 Point Defects and Defect Chemistry............................................................................7 1.4.2 Kroger–Vink Notation..................................................................................................8 1.4.3 Defect Reactions...........................................................................................................8 1.4.4 Defect Concentration....................................................................................................9 1.4.5 Intrinsic Defects..........................................................................................................11 1.4.5.1 Schottky Defect...........................................................................................11 1.4.5.2 Frenkel Defect.............................................................................................12 1.4.6 Extrinsic Defects........................................................................................................13 1.4.6.1 Nonstoichiometry........................................................................................14 1.4.6.2 Influence of Solutes....................................................................................15 1.4.6.3 Brouwer Diagram........................................................................................16 1.4.7 Defect Chemistry and Sintering.................................................................................18 1.5 Diffusion in Crystalline Solids...............................................................................................19 1.5.1 Diffusion Equations....................................................................................................19 1.5.1.1 Solutions of the Diffusion Equation...........................................................21 1.5.2 Atomistic Diffusion Processes...................................................................................22 1.5.3 Mechanisms of Diffusion...........................................................................................26 1.5.3.1 Lattice Diffusion.........................................................................................26 1.5.3.2 Grain Boundary Diffusion..........................................................................27 1.5.3.3 Surface Diffusion........................................................................................28 1.5.3.4 Trends in Diffusion Coefficients.................................................................28 1.5.4 Types of Diffusion Coefficients.................................................................................28 1.6 The Chemical Potential..........................................................................................................29 1.6.1 Chemical Potential of a Mixture of Gases................................................................30 1.6.2 Chemical Potential of Solids and Liquids.................................................................31 1.6.3 Chemical Potential of Atoms and Vacancies in a Crystal.........................................31 1.6.4 Chemical Potential of Atoms and Vacancies beneath a Curved Surface..................32 1.6.5 Equilibrium Vacancy Concentration beneath a Curved Surface...............................34 1.6.6 Vapor Pressure over a Curved Surface......................................................................35 1.7 Diffusional Flux Equations....................................................................................................36 1.7.1 Flux of Atoms.............................................................................................................36 1.7.2 Flux of Vacancies.......................................................................................................37 1.8 Diffusion in Ionic Crystals: Ambipolar Diffusion.................................................................37 1.9 Concluding Remarks..............................................................................................................41 Problems...........................................................................................................................................41 References........................................................................................................................................43 Chapter 2 Solid-State and Viscous Sintering..............................................................................45 2.1 Introduction.............................................................................................................................45 2.2 Mechanisms of Sintering........................................................................................................46 2.3 Effects of Grain Boundaries...................................................................................................48 2.4 Theoretical Analysis of Sintering...........................................................................................50 2.5 Herring’s Scaling Law............................................................................................................51 2.5.1 Derivation of the Scaling Law...................................................................................51 2.5.1.1 Scaling Law for Lattice Diffusion..............................................................52 2.5.2 Application and Limitation of the Scaling Law........................................................53 2.5.2.1 Relative Rates of Sintering Mechanisms....................................................53 2.5.2.2 Limitation of the Scaling Law....................................................................54 2.6 Analytical Models..................................................................................................................55 2.6.1 Stages of Sintering.....................................................................................................55 2.6.1.1 Initial Stage.................................................................................................56 2.6.1.2 Intermediate Stage.......................................................................................56 2.6.1.3 Final Stage...................................................................................................56 2.6.2 Modeling the Sintering Process.................................................................................57 2.6.3 Initial Stage Models...................................................................................................57 2.6.3.1 Geometrical Model.....................................................................................57 2.6.3.2 Kinetic Equations........................................................................................58 2.6.3.3 Summary of the Initial Stage Sintering Equations.....................................61 2.6.3.4 Limitations of the Initial Stage Sintering Equations..................................62 2.6.4 Intermediate Stage Models.........................................................................................63 2.6.4.1 Geometrical Model for Solid-State Sintering.............................................64 2.6.4.2 Mechanisms.................................................................................................64 2.6.4.3 Sintering Equations.....................................................................................64 2.6.4.4 Geometrical Model for Viscous Sintering..................................................68 2.6.5 Final Stage Models.....................................................................................................70 2.6.5.1 Geometrical Model for Solid-State Sintering.............................................70 2.6.5.2 Sintering Equations.....................................................................................71 2.6.5.3 Geometrical Model for Viscous Sintering..................................................72 2.6.6 Limitations of the Analytical Models........................................................................73 2.7 Numerical Simulation of Sintering........................................................................................73 2.7.1 Numerical Simulation of Solid-State Sintering.........................................................74 2.7.2 Numerical Simulation of Viscous Sintering..............................................................76 2.8 Phenomenological Sintering Equations.................................................................................78 2.9 Sintering Diagrams.................................................................................................................79 2.9.1 Construction of the Diagrams....................................................................................80 2.9.2 Limitations of the Diagrams......................................................................................81 2.10 Sintering with an Externally Applied Pressure......................................................................81 2.10.1 Hot Pressing...............................................................................................................81 2.10.1.1 Hot Pressing Models...................................................................................81 2.10.1.2 Densification Rate in Hot Pressing.............................................................84 2.10.1.3 Hot Pressing Mechanisms...........................................................................85 2.10.2 Sinter Forging.............................................................................................................88 2.10.3 Hot Isostatic Pressing.................................................................................................88 2.11 Stress Intensification Factor and Sintering Stress.................................................................89 2.11.1 Stress Intensification Factor.......................................................................................89 2.11.2 Sintering Stress...........................................................................................................91 2.11.3 Measurement of the Sintering Stress and Stress Intensification Factor....................93 2.11.3.1 Zero-Creep Technique.................................................................................93 2.11.3.2 Loading Dilatometry...................................................................................93 2.11.3.3 Sinter Forging Technique............................................................................96 2.12 Alternative Derivation of the Sintering Equations................................................................97 2.12.1 General Isothermal Sintering Equation....................................................................100 2.12.2 General Isothermal Creep Equation for Porous Solids...........................................100 2.13 Concluding Remarks............................................................................................................101 Problems.........................................................................................................................................101 References......................................................................................................................................102 Chapter 3 Grain Growth and Microstructure Control..............................................................105 3.1 Introduction...........................................................................................................................105 3.2 General Features of Grain Growth.......................................................................................106 3.2.1 Grain Growth and Coarsening.................................................................................106 3.2.2 Occurrence of Grain Growth....................................................................................106 3.2.3 Driving Force for Grain Growth..............................................................................107 3.2.4 Normal and Abnormal Grain Growth......................................................................107 3.2.5 Importance of Controlling Grain Growth................................................................108 3.2.5.1 Effect of Grain Size on Properties...........................................................108 3.2.5.2 Attainment of High Density.....................................................................110 3.3 Ostwald Ripening.................................................................................................................110 3.3.1 The LSW Theory......................................................................................................112 3.3.1.1 Ostwald Ripening Controlled by the Interface Reaction.........................112 3.3.1.2 Ostwald Ripening Controlled by Diffusion..............................................114 3.3.2 Modifications to the LSW Theory...........................................................................115 3.3.3 Time-Dependent Ostwald Ripening.........................................................................117 3.4 Topological and Interfacial Tension Requirements.............................................................117 3.5 Normal Grain Growth in Dense Solids................................................................................119 3.5.1 The Model of Burke and Turnbull...........................................................................119 3.5.2 Mean Field Theories.................................................................................................121 3.5.3 Topological Analysis of Grain Growth....................................................................123 3.5.4 Computer Simulation of Normal Grain Growth......................................................125 3.6 Abnormal Grain Growth in Dense Solids............................................................................127 3.6.1 Causes of Abnormal Grain Growth.........................................................................128 3.6.2 Application of Controlled Abnormal Grain Growth................................................131 3.7 Grain Growth in Thin Films................................................................................................133 3.8 Mechanisms Controlling the Boundary Mobility................................................................135 3.8.1 Pinning by Fine Second-Phase Particles..................................................................135 3.8.1.1 The Zener Model......................................................................................137 3.8.1.2 Computer Simulations of Particle-Inhibited Grain Growth.....................138 3.8.1.3 Comparison with Experimental Results...................................................140 3.8.2 Solute Drag...............................................................................................................141 3.9 Grain Growth and Pore Evolution in Porous Solids...........................................................144 3.9.1 Pore Evolution during Sintering..............................................................................146 3.9.2 Thermodynamic Analysis of Pore Shrinkage..........................................................148 3.9.3 Kinetics and Mechanisms of Grain Growth in Porous Solids................................152 3.9.4 Grain Growth in Very Porous Solids.......................................................................152 3.9.5 Grain Growth in Less Porous Solids.......................................................................156 3.9.5.1 Pore Mobility............................................................................................156 3.9.5.2 Kinetics of Pore–Boundary Interactions...................................................159 3.9.5.3 Grain Growth Kinetics..............................................................................160 3.10 Simultaneous Densification and Grain Growth...................................................................160 3.10.1 Microstructural Maps...............................................................................................162 3.10.1.1 Brook Model.............................................................................................162 3.10.1.2 Yan, Cannon, and Chowdhry Model........................................................165 3.10.2 Computer Simulations of Microstructural Evolution..............................................168 3.11 Fabrication Principles for Ceramics with Controlled Microstructure.................................168 3.12 Concluding Remarks............................................................................................................171 Problems.........................................................................................................................................171 References......................................................................................................................................172 Chapter 4 Liquid-Phase Sintering.............................................................................................177 4.1 Introduction...........................................................................................................................177 4.2 Elementary Features of Liquid-Phase Sintering..................................................................178 4.2.1 Enhancement of Densification.................................................................................178 4.2.2 Driving Force for Densification...............................................................................179 4.2.3 Formation of the Liquid Phase................................................................................179 4.2.4 Microstructures.........................................................................................................179 4.3 Stages of Liquid-Phase Sintering.........................................................................................180 4.4 Thermodynamic and Kinetic Factors...................................................................................182 4.4.1 Wetting and Spreading of the Liquid.......................................................................182 4.4.2 Dihedral Angle..........................................................................................................184 4.4.2.1 Liquid Penetration of the Grain Boundary...............................................185 4.4.2.2 Shape of the Liquid and the Grains.........................................................186 4.4.3 Effect of Solubility...................................................................................................187 4.4.4 Capillary Forces........................................................................................................187 4.4.5 Effect of Gravity.......................................................................................................191 4.5 Grain Boundary Films..........................................................................................................192 4.6 The Basic Mechanisms of Liquid-Phase Sintering.............................................................196 4.6.1 Stage 1: Rearrangement and Liquid Redistribution................................................197 4.6.1.1 Liquid Redistribution................................................................................197 4.6.1.2 Particle Rearrangement.............................................................................200 4.6.2 Stage 2: Solution-Precipitation................................................................................202 4.6.2.1 Densification by Contact Flattening.........................................................203 4.6.2.2 Densification Accompanied by Ostwald Ripening...................................205 4.6.2.3 Assessment of the Densification Models..................................................206 4.6.2.4 Grain Shape Accommodation...................................................................206 4.6.2.5 Coalescence...............................................................................................208 4.6.3 Stage 3: Ostwald Ripening.......................................................................................209 4.6.3.1 Densification by Pore Filling....................................................................209 4.6.3.2 Microstructural Coarsening.......................................................................211 4.7 Numerical Modeling of Liquid-Phase Sintering..................................................................214 4.8 Hot Pressing with a Liquid Phase........................................................................................215 4.9 Use of Phase Diagrams in Liquid-Phase Sintering.............................................................215 4.9.1 Zinc Oxide................................................................................................................216 4.9.2 Silicon Nitride..........................................................................................................217 4.10 Activated Sintering...............................................................................................................220 4.11 Vitrification...........................................................................................................................221 4.11.1 The Controlling Parameters.....................................................................................222 4.11.2 Vitrification of Silicate Systems...............................................................................223 4.12 Concluding Remarks............................................................................................................224 Problems.........................................................................................................................................226 References......................................................................................................................................227 Chapter 5 Special Topics in Sintering.......................................................................................231 5.1 Introduction...........................................................................................................................231 5.2 Inhomogeneities and Their Effects on Sintering.................................................................232 5.2.1 Differential Densification.........................................................................................232 5.2.2 Control of Inhomogeneities......................................................................................234 5.2.3 Correction of Inhomogeneities.................................................................................234 5.2.4 Shrinkage of Large Pores.........................................................................................236 5.3 Constrained Sintering I: Rigid Inclusions............................................................................240 5.3.1 Volume Fraction of Inclusions.................................................................................241 5.3.2 Densification Rate of the Composite and the Matrix..............................................242 5.3.3 The Rule of Mixtures...............................................................................................242 5.3.4 Transient Stresses during Sintering..........................................................................244 5.3.4.1 Composite Sphere Model..........................................................................244 5.3.4.2 Stress Components....................................................................................245 5.3.4.3 Effect of Transient Stresses on Sintering.................................................246 5.3.4.4 Calculation of Transient Stresses and Strain Rates..................................247 5.3.5 Percolation and Network Formation........................................................................251 5.3.5.1 The Concept of Percolation......................................................................253 5.3.5.2 Effect of Percolation on Sintering............................................................254 5.3.5.3 Numerical Simulations..............................................................................254 5.3.6 Factors Influencing the Sintering of Ceramic Composites......................................255 5.4 Constrained Sintering II: Adherent Thin Films...................................................................258 5.4.1 Models for Constrained Sintering of Thin Films....................................................258 5.4.2 Experimental Observations of Sintering of Thin Films..........................................260 5.4.3 Crack Growth and Damage in Constrained Sintering of Films..............................262 5.5 Constrained Sintering III: Multilayers.................................................................................264 5.6 Constitutive Models for Porous Sintering Materials...........................................................265 5.7 Morphological Stability of Continuous Phases...................................................................266 5.7.1 Rayleigh Instability and Microstructural Evolution................................................267 5.7.2 Morphological Stability of Thin Films....................................................................269 5.8 Solid Solution Additives and the Sintering of Ceramics.....................................................272 5.8.1 Effect of Additives on Kinetic Factors....................................................................273 5.8.2 Effect of Additives on Thermodynamic Factors......................................................274 5.8.3 Segregation of Additives..........................................................................................275 5.8.3.1 Elastic Strain Energy.................................................................................275 5.8.3.2 Electrostatic Interaction and the Space Charge Concept.........................276 5.8.4 The Role of MgO in Al O ......................................................................................280 2 3 5.9 Sintering with Chemical Reaction: Reaction Sintering.......................................................282 5.9.1 Influence of Process Variables.................................................................................284 5.9.1.1 Particle Size of Reacting Powders............................................................284 5.9.1.2 Sintering Temperature...............................................................................284 5.9.1.3 Applied Pressure.......................................................................................285 5.9.1.4 Processing Trajectories in Reaction Sintering..........................................285 5.9.2 Experimental Observations of Reaction Sintering...................................................286 5.10 Viscous Sintering with Crystallization.................................................................................289 5.10.1 Effect of Process Variables.......................................................................................290 5.10.1.1 Viscosity....................................................................................................290 5.10.1.2 Pore Size...................................................................................................290 5.10.1.3 Applied Pressure.......................................................................................291 5.10.1.4 Heating Rate..............................................................................................291 5.10.2 Analysis of Viscous Sintering with Crystallization.................................................291 5.10.2.1 TTT Diagrams...........................................................................................291 5.10.2.2 Analysis in Terms of Sintering with Rigid Inclusions.............................294 5.10.3 Experimental Observations of Sintering with Crystallization.................................296 5.11 Concluding Remarks............................................................................................................297 Problems.........................................................................................................................................297 References......................................................................................................................................298 Chapter 6 Sintering Process Variables and Sintering Practice.................................................305 6.1 Introduction...........................................................................................................................305 6.2 Sintering Measurement Techniques.....................................................................................305 6.2.1 Furnaces....................................................................................................................306 6.2.2 Shrinkage and Density.............................................................................................307 6.2.3 Grain Size.................................................................................................................309 6.3 Conventional Sintering.........................................................................................................309 6.3.1 Particle and Green Compact Effects........................................................................310 6.3.1.1 Particle Size...............................................................................................310 6.3.1.2 Particle Size Distribution..........................................................................310 6.3.1.3 Particle Shape and Particle Structure.......................................................311 6.3.1.4 Particle Packing.........................................................................................312 6.3.1.5 Effect of Green Density............................................................................314 6.3.2 Anisotropic Sintering Shrinkage..............................................................................315 6.3.2.1 Pore Shape Anisotropy..............................................................................315 6.3.2.2 Particle Alignment.....................................................................................318 6.3.3 Heating Schedule......................................................................................................320 6.3.3.1 Stage 1: Binder Burnout...........................................................................321 6.3.3.2 Stage 2: Low-Temperature Soak..............................................................321 6.3.3.3 Stage 3: Heat-Up to the Sintering Temperature.......................................321 6.3.3.4 Stage 4: Isothermal Sintering...................................................................321 6.3.3.5 Stage 5: Annealing....................................................................................322 6.3.3.6 Stage 6: Cool-Down to Room Temperature.............................................322 6.3.3.7 Isothermal Sintering..................................................................................322 6.3.3.8 Constant Heating Rate Sintering..............................................................323 6.3.3.9 Effect of Heating Rate on Sintering.........................................................323 6.3.3.10 Master Sintering Curve.............................................................................329 6.3.3.11 Multistage Sintering..................................................................................330 6.3.3.12 Fast Firing.................................................................................................332 6.3.3.13 Rate-Controlled Sintering.........................................................................333 6.3.4 Sintering Atmosphere...............................................................................................334 6.3.4.1 Gases in Pores...........................................................................................334 6.3.4.2 Vapor Transport.........................................................................................338 6.3.4.3 Water Vapor in the Sintering Atmosphere................................................338 6.3.4.4 Volatilization and Decomposition.............................................................340 6.3.4.5 Oxidation Number.....................................................................................342 6.3.4.6 Defect Chemistry and Stoichiometry.......................................................346 6.3.5 Production of Controlled Sintering Atmospheres....................................................346 6.4 Microwave Sintering............................................................................................................348 6.4.1 Interaction of Microwaves with Matter...................................................................349 6.4.2 Microwave Sintering Techniques.............................................................................352 6.4.3 Microwave Sintering of Ceramics............................................................................353 6.4.4 Plasma Sintering.......................................................................................................355 6.4.5 Plasma-Assisted Sintering........................................................................................356 6.5 Pressure-Assisted Sintering..................................................................................................357 6.5.1 Hot Pressing.............................................................................................................357 6.5.1.1 Hot Pressing Process Variables.................................................................359 6.5.1.2 Analysis of Hot Pressing Data..................................................................359 6.5.1.3 Reactive Hot Pressing...............................................................................360 6.5.2 Sinter Forging...........................................................................................................361 6.5.3 Hot Isostatic Pressing...............................................................................................361 6.5.3.1 Hot Isostatic Pressing: Process Variables.................................................363 6.6 Concluding Remarks............................................................................................................364 Problems.........................................................................................................................................365 References......................................................................................................................................366 Appendix A Physical Constants.........................................................................................................................371 Appendix B SI Units—Names and Symbols.....................................................................................................373 Appendix C Conversion of Units.......................................................................................................................375 Appendix D Ionic Crystal Radii (in units of 10–10 m).......................................................................................377 Appendix E Density and Melting Point of Some Elements and Ceramics......................................................379 Index..............................................................................................................................................383

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Mohamed N. Rahaman University of Missouri-Rolla, USA Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110720 International Standard Book Number-13: 978-1-4200-0705-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface...........................................................................................................................................xiii Author..............................................................................................................................................xv Chapter 1 Sintering of Ceramics: Fundamentals..........................................................................1 1.1 Introduction...............................................................................................................................1 1.2 The Sintering Process...............................................................................................................1 1.2.1 Characterization of Sintering.......................................................................................3 1.2.2 Approach to Sintering..................................................................................................3 1.3 Driving Force for Sintering......................................................................................................4 1.3.1 Surface Curvature.........................................................................................................5 1.3.2 Applied Pressure...........................................................................................................6 1.3.3 Chemical Reaction........................................................................................................6 1.4 Defects in Crystalline Solids....................................................................................................6 1.4.1 Point Defects and Defect Chemistry............................................................................7 1.4.2 Kroger–Vink Notation..................................................................................................8 1.4.3 Defect Reactions...........................................................................................................8 1.4.4 Defect Concentration....................................................................................................9 1.4.5 Intrinsic Defects..........................................................................................................11 1.4.5.1 Schottky Defect...........................................................................................11 1.4.5.2 Frenkel Defect.............................................................................................12 1.4.6 Extrinsic Defects........................................................................................................13 1.4.6.1 Nonstoichiometry........................................................................................14 1.4.6.2 Influence of Solutes....................................................................................15 1.4.6.3 Brouwer Diagram........................................................................................16 1.4.7 Defect Chemistry and Sintering.................................................................................18 1.5 Diffusion in Crystalline Solids...............................................................................................19 1.5.1 Diffusion Equations....................................................................................................19 1.5.1.1 Solutions of the Diffusion Equation...........................................................21 1.5.2 Atomistic Diffusion Processes...................................................................................22 1.5.3 Mechanisms of Diffusion...........................................................................................26 1.5.3.1 Lattice Diffusion.........................................................................................26 1.5.3.2 Grain Boundary Diffusion..........................................................................27 1.5.3.3 Surface Diffusion........................................................................................28 1.5.3.4 Trends in Diffusion Coefficients.................................................................28 1.5.4 Types of Diffusion Coefficients.................................................................................28 1.6 The Chemical Potential..........................................................................................................29 1.6.1 Chemical Potential of a Mixture of Gases................................................................30 1.6.2 Chemical Potential of Solids and Liquids.................................................................31 1.6.3 Chemical Potential of Atoms and Vacancies in a Crystal.........................................31 1.6.4 Chemical Potential of Atoms and Vacancies beneath a Curved Surface..................32 1.6.5 Equilibrium Vacancy Concentration beneath a Curved Surface...............................34 1.6.6 Vapor Pressure over a Curved Surface......................................................................35 1.7 Diffusional Flux Equations....................................................................................................36 1.7.1 Flux of Atoms.............................................................................................................36 1.7.2 Flux of Vacancies.......................................................................................................37 1.8 Diffusion in Ionic Crystals: Ambipolar Diffusion.................................................................37 1.9 Concluding Remarks..............................................................................................................41 Problems...........................................................................................................................................41 References........................................................................................................................................43 Chapter 2 Solid-State and Viscous Sintering..............................................................................45 2.1 Introduction.............................................................................................................................45 2.2 Mechanisms of Sintering........................................................................................................46 2.3 Effects of Grain Boundaries...................................................................................................48 2.4 Theoretical Analysis of Sintering...........................................................................................50 2.5 Herring’s Scaling Law............................................................................................................51 2.5.1 Derivation of the Scaling Law...................................................................................51 2.5.1.1 Scaling Law for Lattice Diffusion..............................................................52 2.5.2 Application and Limitation of the Scaling Law........................................................53 2.5.2.1 Relative Rates of Sintering Mechanisms....................................................53 2.5.2.2 Limitation of the Scaling Law....................................................................54 2.6 Analytical Models..................................................................................................................55 2.6.1 Stages of Sintering.....................................................................................................55 2.6.1.1 Initial Stage.................................................................................................56 2.6.1.2 Intermediate Stage.......................................................................................56 2.6.1.3 Final Stage...................................................................................................56 2.6.2 Modeling the Sintering Process.................................................................................57 2.6.3 Initial Stage Models...................................................................................................57 2.6.3.1 Geometrical Model.....................................................................................57 2.6.3.2 Kinetic Equations........................................................................................58 2.6.3.3 Summary of the Initial Stage Sintering Equations.....................................61 2.6.3.4 Limitations of the Initial Stage Sintering Equations..................................62 2.6.4 Intermediate Stage Models.........................................................................................63 2.6.4.1 Geometrical Model for Solid-State Sintering.............................................64 2.6.4.2 Mechanisms.................................................................................................64 2.6.4.3 Sintering Equations.....................................................................................64 2.6.4.4 Geometrical Model for Viscous Sintering..................................................68 2.6.5 Final Stage Models.....................................................................................................70 2.6.5.1 Geometrical Model for Solid-State Sintering.............................................70 2.6.5.2 Sintering Equations.....................................................................................71 2.6.5.3 Geometrical Model for Viscous Sintering..................................................72 2.6.6 Limitations of the Analytical Models........................................................................73 2.7 Numerical Simulation of Sintering........................................................................................73 2.7.1 Numerical Simulation of Solid-State Sintering.........................................................74 2.7.2 Numerical Simulation of Viscous Sintering..............................................................76 2.8 Phenomenological Sintering Equations.................................................................................78 2.9 Sintering Diagrams.................................................................................................................79 2.9.1 Construction of the Diagrams....................................................................................80 2.9.2 Limitations of the Diagrams......................................................................................81 2.10 Sintering with an Externally Applied Pressure......................................................................81 2.10.1 Hot Pressing...............................................................................................................81 2.10.1.1 Hot Pressing Models...................................................................................81 2.10.1.2 Densification Rate in Hot Pressing.............................................................84 2.10.1.3 Hot Pressing Mechanisms...........................................................................85 2.10.2 Sinter Forging.............................................................................................................88 2.10.3 Hot Isostatic Pressing.................................................................................................88 2.11 Stress Intensification Factor and Sintering Stress.................................................................89 2.11.1 Stress Intensification Factor.......................................................................................89 2.11.2 Sintering Stress...........................................................................................................91 2.11.3 Measurement of the Sintering Stress and Stress Intensification Factor....................93 2.11.3.1 Zero-Creep Technique.................................................................................93 2.11.3.2 Loading Dilatometry...................................................................................93 2.11.3.3 Sinter Forging Technique............................................................................96 2.12 Alternative Derivation of the Sintering Equations................................................................97 2.12.1 General Isothermal Sintering Equation....................................................................100 2.12.2 General Isothermal Creep Equation for Porous Solids...........................................100 2.13 Concluding Remarks............................................................................................................101 Problems.........................................................................................................................................101 References......................................................................................................................................102 Chapter 3 Grain Growth and Microstructure Control..............................................................105 3.1 Introduction...........................................................................................................................105 3.2 General Features of Grain Growth.......................................................................................106 3.2.1 Grain Growth and Coarsening.................................................................................106 3.2.2 Occurrence of Grain Growth....................................................................................106 3.2.3 Driving Force for Grain Growth..............................................................................107 3.2.4 Normal and Abnormal Grain Growth......................................................................107 3.2.5 Importance of Controlling Grain Growth................................................................108 3.2.5.1 Effect of Grain Size on Properties...........................................................108 3.2.5.2 Attainment of High Density.....................................................................110 3.3 Ostwald Ripening.................................................................................................................110 3.3.1 The LSW Theory......................................................................................................112 3.3.1.1 Ostwald Ripening Controlled by the Interface Reaction.........................112 3.3.1.2 Ostwald Ripening Controlled by Diffusion..............................................114 3.3.2 Modifications to the LSW Theory...........................................................................115 3.3.3 Time-Dependent Ostwald Ripening.........................................................................117 3.4 Topological and Interfacial Tension Requirements.............................................................117 3.5 Normal Grain Growth in Dense Solids................................................................................119 3.5.1 The Model of Burke and Turnbull...........................................................................119 3.5.2 Mean Field Theories.................................................................................................121 3.5.3 Topological Analysis of Grain Growth....................................................................123 3.5.4 Computer Simulation of Normal Grain Growth......................................................125 3.6 Abnormal Grain Growth in Dense Solids............................................................................127 3.6.1 Causes of Abnormal Grain Growth.........................................................................128 3.6.2 Application of Controlled Abnormal Grain Growth................................................131 3.7 Grain Growth in Thin Films................................................................................................133 3.8 Mechanisms Controlling the Boundary Mobility................................................................135 3.8.1 Pinning by Fine Second-Phase Particles..................................................................135 3.8.1.1 The Zener Model......................................................................................137 3.8.1.2 Computer Simulations of Particle-Inhibited Grain Growth.....................138 3.8.1.3 Comparison with Experimental Results...................................................140 3.8.2 Solute Drag...............................................................................................................141 3.9 Grain Growth and Pore Evolution in Porous Solids...........................................................144 3.9.1 Pore Evolution during Sintering..............................................................................146 3.9.2 Thermodynamic Analysis of Pore Shrinkage..........................................................148 3.9.3 Kinetics and Mechanisms of Grain Growth in Porous Solids................................152 3.9.4 Grain Growth in Very Porous Solids.......................................................................152 3.9.5 Grain Growth in Less Porous Solids.......................................................................156 3.9.5.1 Pore Mobility............................................................................................156 3.9.5.2 Kinetics of Pore–Boundary Interactions...................................................159 3.9.5.3 Grain Growth Kinetics..............................................................................160 3.10 Simultaneous Densification and Grain Growth...................................................................160 3.10.1 Microstructural Maps...............................................................................................162 3.10.1.1 Brook Model.............................................................................................162 3.10.1.2 Yan, Cannon, and Chowdhry Model........................................................165 3.10.2 Computer Simulations of Microstructural Evolution..............................................168 3.11 Fabrication Principles for Ceramics with Controlled Microstructure.................................168 3.12 Concluding Remarks............................................................................................................171 Problems.........................................................................................................................................171 References......................................................................................................................................172 Chapter 4 Liquid-Phase Sintering.............................................................................................177 4.1 Introduction...........................................................................................................................177 4.2 Elementary Features of Liquid-Phase Sintering..................................................................178 4.2.1 Enhancement of Densification.................................................................................178 4.2.2 Driving Force for Densification...............................................................................179 4.2.3 Formation of the Liquid Phase................................................................................179 4.2.4 Microstructures.........................................................................................................179 4.3 Stages of Liquid-Phase Sintering.........................................................................................180 4.4 Thermodynamic and Kinetic Factors...................................................................................182 4.4.1 Wetting and Spreading of the Liquid.......................................................................182 4.4.2 Dihedral Angle..........................................................................................................184 4.4.2.1 Liquid Penetration of the Grain Boundary...............................................185 4.4.2.2 Shape of the Liquid and the Grains.........................................................186 4.4.3 Effect of Solubility...................................................................................................187 4.4.4 Capillary Forces........................................................................................................187 4.4.5 Effect of Gravity.......................................................................................................191 4.5 Grain Boundary Films..........................................................................................................192 4.6 The Basic Mechanisms of Liquid-Phase Sintering.............................................................196 4.6.1 Stage 1: Rearrangement and Liquid Redistribution................................................197 4.6.1.1 Liquid Redistribution................................................................................197 4.6.1.2 Particle Rearrangement.............................................................................200 4.6.2 Stage 2: Solution-Precipitation................................................................................202 4.6.2.1 Densification by Contact Flattening.........................................................203 4.6.2.2 Densification Accompanied by Ostwald Ripening...................................205 4.6.2.3 Assessment of the Densification Models..................................................206 4.6.2.4 Grain Shape Accommodation...................................................................206 4.6.2.5 Coalescence...............................................................................................208 4.6.3 Stage 3: Ostwald Ripening.......................................................................................209 4.6.3.1 Densification by Pore Filling....................................................................209 4.6.3.2 Microstructural Coarsening.......................................................................211 4.7 Numerical Modeling of Liquid-Phase Sintering..................................................................214 4.8 Hot Pressing with a Liquid Phase........................................................................................215 4.9 Use of Phase Diagrams in Liquid-Phase Sintering.............................................................215 4.9.1 Zinc Oxide................................................................................................................216 4.9.2 Silicon Nitride..........................................................................................................217 4.10 Activated Sintering...............................................................................................................220 4.11 Vitrification...........................................................................................................................221 4.11.1 The Controlling Parameters.....................................................................................222 4.11.2 Vitrification of Silicate Systems...............................................................................223 4.12 Concluding Remarks............................................................................................................224 Problems.........................................................................................................................................226 References......................................................................................................................................227 Chapter 5 Special Topics in Sintering.......................................................................................231 5.1 Introduction...........................................................................................................................231 5.2 Inhomogeneities and Their Effects on Sintering.................................................................232 5.2.1 Differential Densification.........................................................................................232 5.2.2 Control of Inhomogeneities......................................................................................234 5.2.3 Correction of Inhomogeneities.................................................................................234 5.2.4 Shrinkage of Large Pores.........................................................................................236 5.3 Constrained Sintering I: Rigid Inclusions............................................................................240 5.3.1 Volume Fraction of Inclusions.................................................................................241 5.3.2 Densification Rate of the Composite and the Matrix..............................................242 5.3.3 The Rule of Mixtures...............................................................................................242 5.3.4 Transient Stresses during Sintering..........................................................................244 5.3.4.1 Composite Sphere Model..........................................................................244 5.3.4.2 Stress Components....................................................................................245 5.3.4.3 Effect of Transient Stresses on Sintering.................................................246 5.3.4.4 Calculation of Transient Stresses and Strain Rates..................................247 5.3.5 Percolation and Network Formation........................................................................251 5.3.5.1 The Concept of Percolation......................................................................253 5.3.5.2 Effect of Percolation on Sintering............................................................254 5.3.5.3 Numerical Simulations..............................................................................254 5.3.6 Factors Influencing the Sintering of Ceramic Composites......................................255 5.4 Constrained Sintering II: Adherent Thin Films...................................................................258 5.4.1 Models for Constrained Sintering of Thin Films....................................................258 5.4.2 Experimental Observations of Sintering of Thin Films..........................................260 5.4.3 Crack Growth and Damage in Constrained Sintering of Films..............................262 5.5 Constrained Sintering III: Multilayers.................................................................................264 5.6 Constitutive Models for Porous Sintering Materials...........................................................265 5.7 Morphological Stability of Continuous Phases...................................................................266 5.7.1 Rayleigh Instability and Microstructural Evolution................................................267 5.7.2 Morphological Stability of Thin Films....................................................................269 5.8 Solid Solution Additives and the Sintering of Ceramics.....................................................272 5.8.1 Effect of Additives on Kinetic Factors....................................................................273 5.8.2 Effect of Additives on Thermodynamic Factors......................................................274 5.8.3 Segregation of Additives..........................................................................................275 5.8.3.1 Elastic Strain Energy.................................................................................275 5.8.3.2 Electrostatic Interaction and the Space Charge Concept.........................276 5.8.4 The Role of MgO in Al O ......................................................................................280 2 3 5.9 Sintering with Chemical Reaction: Reaction Sintering.......................................................282 5.9.1 Influence of Process Variables.................................................................................284 5.9.1.1 Particle Size of Reacting Powders............................................................284 5.9.1.2 Sintering Temperature...............................................................................284 5.9.1.3 Applied Pressure.......................................................................................285 5.9.1.4 Processing Trajectories in Reaction Sintering..........................................285 5.9.2 Experimental Observations of Reaction Sintering...................................................286 5.10 Viscous Sintering with Crystallization.................................................................................289 5.10.1 Effect of Process Variables.......................................................................................290 5.10.1.1 Viscosity....................................................................................................290 5.10.1.2 Pore Size...................................................................................................290 5.10.1.3 Applied Pressure.......................................................................................291 5.10.1.4 Heating Rate..............................................................................................291 5.10.2 Analysis of Viscous Sintering with Crystallization.................................................291 5.10.2.1 TTT Diagrams...........................................................................................291 5.10.2.2 Analysis in Terms of Sintering with Rigid Inclusions.............................294 5.10.3 Experimental Observations of Sintering with Crystallization.................................296 5.11 Concluding Remarks............................................................................................................297 Problems.........................................................................................................................................297 References......................................................................................................................................298 Chapter 6 Sintering Process Variables and Sintering Practice.................................................305 6.1 Introduction...........................................................................................................................305 6.2 Sintering Measurement Techniques.....................................................................................305 6.2.1 Furnaces....................................................................................................................306 6.2.2 Shrinkage and Density.............................................................................................307 6.2.3 Grain Size.................................................................................................................309 6.3 Conventional Sintering.........................................................................................................309 6.3.1 Particle and Green Compact Effects........................................................................310 6.3.1.1 Particle Size...............................................................................................310 6.3.1.2 Particle Size Distribution..........................................................................310 6.3.1.3 Particle Shape and Particle Structure.......................................................311 6.3.1.4 Particle Packing.........................................................................................312 6.3.1.5 Effect of Green Density............................................................................314 6.3.2 Anisotropic Sintering Shrinkage..............................................................................315 6.3.2.1 Pore Shape Anisotropy..............................................................................315 6.3.2.2 Particle Alignment.....................................................................................318 6.3.3 Heating Schedule......................................................................................................320 6.3.3.1 Stage 1: Binder Burnout...........................................................................321 6.3.3.2 Stage 2: Low-Temperature Soak..............................................................321 6.3.3.3 Stage 3: Heat-Up to the Sintering Temperature.......................................321 6.3.3.4 Stage 4: Isothermal Sintering...................................................................321 6.3.3.5 Stage 5: Annealing....................................................................................322 6.3.3.6 Stage 6: Cool-Down to Room Temperature.............................................322 6.3.3.7 Isothermal Sintering..................................................................................322 6.3.3.8 Constant Heating Rate Sintering..............................................................323 6.3.3.9 Effect of Heating Rate on Sintering.........................................................323 6.3.3.10 Master Sintering Curve.............................................................................329 6.3.3.11 Multistage Sintering..................................................................................330 6.3.3.12 Fast Firing.................................................................................................332 6.3.3.13 Rate-Controlled Sintering.........................................................................333 6.3.4 Sintering Atmosphere...............................................................................................334 6.3.4.1 Gases in Pores...........................................................................................334 6.3.4.2 Vapor Transport.........................................................................................338 6.3.4.3 Water Vapor in the Sintering Atmosphere................................................338 6.3.4.4 Volatilization and Decomposition.............................................................340 6.3.4.5 Oxidation Number.....................................................................................342 6.3.4.6 Defect Chemistry and Stoichiometry.......................................................346 6.3.5 Production of Controlled Sintering Atmospheres....................................................346 6.4 Microwave Sintering............................................................................................................348 6.4.1 Interaction of Microwaves with Matter...................................................................349 6.4.2 Microwave Sintering Techniques.............................................................................352 6.4.3 Microwave Sintering of Ceramics............................................................................353 6.4.4 Plasma Sintering.......................................................................................................355 6.4.5 Plasma-Assisted Sintering........................................................................................356 6.5 Pressure-Assisted Sintering..................................................................................................357 6.5.1 Hot Pressing.............................................................................................................357 6.5.1.1 Hot Pressing Process Variables.................................................................359 6.5.1.2 Analysis of Hot Pressing Data..................................................................359 6.5.1.3 Reactive Hot Pressing...............................................................................360 6.5.2 Sinter Forging...........................................................................................................361 6.5.3 Hot Isostatic Pressing...............................................................................................361 6.5.3.1 Hot Isostatic Pressing: Process Variables.................................................363 6.6 Concluding Remarks............................................................................................................364 Problems.........................................................................................................................................365 References......................................................................................................................................366 Appendix A Physical Constants.........................................................................................................................371 Appendix B SI Units—Names and Symbols.....................................................................................................373 Appendix C Conversion of Units.......................................................................................................................375 Appendix D Ionic Crystal Radii (in units of 10–10 m).......................................................................................377 Appendix E Density and Melting Point of Some Elements and Ceramics......................................................379 Index..............................................................................................................................................383

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