Fundamentals of ceramic powder processing and synthesis

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Fundamentals

of Ceramic Powder Processing and Synthesis

This Page Intentionally Left Blank

Fundamentals of Ceramic Powder Processing and Synthesis Terry A. Ring Department of Chemical and Fuels Engineering and

Department of Materials Science and Engineering University of Utah Salt Lake City, Utah

?P A c a d e m i c Press San Diego New York Boston London Sydney Tokyo Toronto

Photo taken from Millot, G., La Science 20, 61-73 (1979). Please see Chapter 1 for more information.

This book is printed on acid-free paper. ( ~

Copyright 9 1996 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. A c a d e m i c Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495

United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Ring, Terry. Fundamentals of ceramic powder processing and synthesis / by Terry Ring. p. cm. Includes index. ISBN 0-12-588930-5 (alk. paper) 1. Ceramic powders. I. Title TP815.R56 1995 95-15418 666--dc20 CIP

PRINTED IN THEUNITED STATES OF AMERICA 96 97 98 99 00 01 MM 9 8 7 6 5

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To Susan's understanding

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Contents

Preface xxi

I

INTRODUCTION HISTORY, RAW MATERIALS, CERAMIC POWDER CHARACTERIZATION 1.1 General Concepts of Ceramic Powder Processing References 6

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4

Ceramic P o w d e r Processing History a n d Discussion of N a t u r a l R a w Materials 1.1 Objectives 7 1.2 Historical Perspective 8 1.3 Raw Materials 27 1.3.1 Natural Raw Materials 27 1.3.2 Synthetic Raw Materials 34 1.4 Selecting a Raw Material 40 1.5 Summary 41 References 41

2

Ceramic Powder Characterization 2.1 Objectives 43 2.2 Introduction 43

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2.3 Powder Sampling 44 2.3.1 Sampling Accuracy 44 2.3.2 Two-Component Sampling Accuracy 45 2.3.3 Sampling Methods 46 2.3.4 Golden Rules of Sampling 46 2.4 Particle Size 48 2.4.1 Statistical Diameters 48 2.4.2 Mean Particle Size 52 2.4.3 Size Distribution Accuracy 55 2.5 Particle Morphology 56 2.5.1 Shape Factors 57 2.5.2 Shape Analysis 59 2.5.3 Fractal Shapes 60 2.5.4 Internal Porosity 62 2.6 Powder Density 63 2.7 SurfaceArea 64 2.7.1 First Layer Adsorption~Langmuir Adsorption 64 2.7.2 Multilayer Adsorption~BET Adsorption 65 2.8 Particle Size Distributions 66 2.8.1 Normal Distribution 68 2.8.2 Log-Normal Distribution 69 2.8.3 Rosin-Rammler Distribution 72 2.9 Comparison of Two-Powder Size Distributions 73 Problem 2.5. Comparison of Two Size Distributions 74 2.10 Blending Powder Samples 75 Problem 2.6. Mixing Two Log-Normal Size Distributions 76 2.11 Summary 78 Problems 78 References 79

H 3

CERAMIC POWDER SYNTHESIS The Population Balance 3.1 Objectives 85 3.2 Microscopic Population Balance 86 3.3 Macroscopic Population Balance 87 Problem 3.1. Constant Stirred Tank Crystallizer 88

Contents

ix

3.4 Population Balances Where Length, Area, and Volume Are Conserved 89 3.4.1 Conservation of Length in the Batch Grinding of Fibers 89 3.5 Population Balances on a Mass Basis 91 3.5.1 Population Balances on a Discrete Mass Basis 91 3.5.2 Population Balances on a Cumulative Mass Basis 92 3.6 Summary 93 3.6.1 Microdistributed Population Balance 93 3.6.2 Macrodistributed Population Balance 93 3.6.3 List of Symbols 93 References 94

4

Comminution a n d Classification of Ceramic Powders 4.1 Objectives 95 4.2 Comminution 96 4.2.1 ComminutionEquipment 96 4.2.2 Energy Required for Size Reduction 101 4.2.3 Comminution Efficiency 102 4.2.4 Population Balance Models for Comminution Mills 103 4.2.5 Array Formulation of Comminution 110 4.3 Classification of Ceramic Powders 115 4.3.1 Dry Classification Equipment 115 4.3.2 Classifier Fundamentals 117 4.3.3 Size Selectivity, Recovery, and Yield 123 4.3.4 Classifier Efficiency 124 4.3.5 Wet Classification Equipment 127 4.4 Comminution and Classification Circuits 129 4.5 Summary 135 Problems 136 References 136

5

Ceramic P o w d e r Synthesis with Solid Phase R e a c t a n t 5.1 Objectives 139 5.2 Introduction 140 5.3 Thermodynamics of Fluid-Solid Reactions

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Contents

5.4 Oxidation Reactions 144 Problem 5.1. Free Energy of Oxidation 145 5.5 Reduction Reactions 147 5.6 Nitridation Reactions 148 5.7 Thermodynamics of Multiple Reaction Systems 148 Problem 5.2. What Is the Reaction Product When A1 Metal Is Exposed to Air at 800~ 149 5.8 Liquid-Solid Reactions 151 5.9 Fluid-Solid Reaction Kinetics 151 5.9.1 Shrinking Sphere Model 157 5.9.2 Comparison with Kinetic Models 158 5.9.3 Kinetic Models Where Nucleation and Growth Are Combined 161 5.10 Fluid-Solid Reactors 162 Problem 5.3. Conversion of a Size Mixture of Ceramic Powders 165 5.11 Solid-Solid Reactions 166 5.11.1 Vaporization of One Solid Reactant 167 5.11.2 Solid-Solid Interdiffusion 170 5.12 Summary 176 Problems 177 References 178

6

Liquid Phase Synthesis by Precipitation 6.1 Objectives 179 6.2 Introduction 180 6.3 Nucleation Kinetics 183 6.3.1 Homogeneous Nucleation 183 6.3.2 Heterogeneous Nucleation 189 6.3.3 Secondary Nucleation 192 6.4 Growth Kinetics 193 6.4.1 Stages of Crystal Growth 196 6.4.2 Diffusion Controlled Growth 196 6.4.3 Surface Nucleation of Steps 202 6.4.4 Two-Dimensional Growth of Surface Nuclei 203 6.4.5 Screw Disclocation Growth 204 6.4.6 Summary of Growth Rates 208 6.5 Crystal Shape 210 6.5.1 Equilibrium Shape 210

Contents

6.5.2 Kinetic Shape 212 6.5.3 Aggregate Shape 214 6.5.4 Crystal Habit Modification by Impurities 216 6.6 Size Distribution Effects--Population Balance and Precipitator Design 220 6.6.1 Continuous Stirred Tank Reactor 220 6.6.2 Batch Precipitation 226 6.6.3 Effect of Aggregation on the Particle Size Distribution 229 6.7 Coprecipitation of Ceramic Powders 244 6.7.1 True Coprecipitation 244 6.7.2 Simultaneous Precipitation and Coaggregation 246 6.8 Summary 249 Problems 249 References 250

7

P o w d e r Synthesis with Gas Phase R e a c t a n t s 7.1 Objectives 255 7.2 Introduction 256 7.3 Gas Phase Reactions 260 7.3.1 Flame 260 7.3.2 Furnace Decomposition 262 7.3.3 Plasma 262 7.3.4 Laser 262 7.4 Reaction Kinetics 263 7.4.1 Combination Reactions 265 7.4.2 Thermal Decomposition Reactions 267 7.4.3 Laser Reactions 268 7.4.4 Plasma Reactions 269 7.4.5 Complex Reaction Mechanisms 269 7.5 Homogeneous Nucleation 270 7.6 Collisional Growth Theory 275 7.7 Population Balance for Gas Phase Synthesis 278 7.8 Dispersion Model for Gas Synthesis Reactors 280 7.8.1 Single-Point Nucleation 284 7.8.2 Multipoint Nucleation 285 7.9 Population Balance with Aggregation 289 7.9.1 Rapid Flocculation Theory 290 7.9.2 A Physical Constraint on the Population Balance 292 7.9.3 Other Numerical Models 295

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7.10 Quenching the Aggregation 296 7.10.1 Heat Transfer Quench 298 7.10.2 Gas Mixing Quench 300 7.11 Particle Shape 301 7.12 Summary 303 Problems 303 References 304

8

Other Ceramic Powder Fabrication Processes 8.1 Objectives 307 8.2 Spray Drying 307 8.2.1 Atomization 309 8.2.2 Droplet Drying 315 8.2.3 Gas-Droplet Mixing 327 8.2.4 Spray Dryer Design 330 8.3 Spray Roasting 331 8.4 Metal Organic Decomposition for Ceramic Films 335 8.5 Freeze Drying 336 8.5.1 Problem: Freezing Time for a Drop 338 8.6 Sol-Gel Synthesis 340 8.6.1 Precursor Solution Chemistry 343 8.6.2 Film Formation 347 8.6.3 Gel Drying 349 8.6.4 Thermal Decomposition of Gels 350 8.6.5 Gel Sintering 350 8.7 Melt Solidification 351 8.8 Summary 353 Problems 353 References 354

III 9

CERAMIC P A S T E F O R M A T I O N - MISE-EN PATE Wetting, Deagglomeration, and Adsorption 9.1 Objectives 359 9.2 Wetting of a Powder by a Liquid 360 Problem 9.1. Spreading H20 on SiO2 364

Contents

9.3 9.4

9.5 9.6

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9.2.1 Smooth versus Rough Surfaces 366 Problem 9.2. Wetting of a Rough Solid Surface 368 9.2.2 Partial Wetting of a Solid 368 9.2.3 Internal Wetting 368 9.2.4 Heat of Wetting 370 Problem 9.3. Solvent Selection 373 Deagglomeration 374 9.3.1 Ultrasonification 375 Adsorption onto Powder Surfaces 379 9.4.1 Gibb's Adsorption Isotherm for the Liquid-Vapor Interface 380 9.4.2 Adsorption Isotherms for the Solid-Liquid Interface 382 9.4.3 Binary Solvent Adsorption 384 9.4.4 Adsorption of Ions 386 Problem 9.4. Surface Change 394 9.4.5 Adsorption of Ionic Surfactants 398 9.4.6 Adsorption of Polymers 403 9.4.7 Selection of a Surfactant 410 Chemical Stability of a Powder in a Solvent 414 9.5.1 Stability in Water 414 Summary 416 Problems 417 References 418

Colloid Stability of Ceramic Suspensions 10.1 Objectives 421 10.2 Introduction 421 10.3 Interaction Energy and Colloid Stability 422 10.3.1 van der Waals Attractive Interaction Energy 422 Problem 10.1. H a m a k e r Constant 427 10.3.2 Electrostatic Repulsion 428 10.3.3 Steric Repulsion 445 10.3.4 Total Interaction Energy 466 10.4 Kinetics of Coagulation and Flocculation 467 10.4.1 Doublet Formation 467 Problem 10.2. Determine the Half-Life for Doublet Formation for Various Initial Number Densities of Particles in Water 467

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Contents

10.4.2 Growth and Structure of Large Aggregate Clusters 475 10.4.3 Shear Aggregation 486 Problem 10.3. Critical Size for Shear Aggression 487 10.5 Colloid Stability in Ceramic Systems 488 10.6 Summary 489 Problems 489 References 491

11

Colloidal Properties of Ceramic Suspensions 11.1 Objectives 495 11.2 Introduction 496 11.3 Sedimentation 497 Problem 11.1. Terminal Settling Velocity 499 11.3.1 Nonspherical Particle Settling 500 11.3.2 Hindered Settling 500 Problem 11.2. Hindered Settling Velocity 502 11.3.3 Centrifugal Sedimentation 503 11.3.4 Sedimentation Potential 503 11.4 Brownian Diffusion 504 11.4.1 Nonspherical Particle Diffusion 504 11.4.2 Fick's Laws for Diffusion 505 11.4.3 Equilibrium between Sedimentation and Diffusion 505 Problem 11.3. Sedimentation Equilibrium 506 11.4.4 Rotational Diffusion 506 11.5 Solution and Suspension Colligative Properties 509 11.5.1 Osmotic Pressure of Electrolyte Solutions 511 11.5.2 Osmotic Pressure of Polymer Solutions 512 11.5.3 Osmotic Pressure of the Double Layer in a Colloidal Suspension 513 11.6 Ordered Suspensions 516 11.6.1 Osmotic Pressure (and Other Thermodynamic Properties) of a Ceramic Suspension 517

Contents

11.6.2 Measurement of Ordered Array Structure 526 11.6.3 Defects in Ordered Arrays 527 11.6.4 Processing Effects on Order Domain Size 529 11.6.5 Measurement of Ordered Domain Size by Light Diffraction 530 11.6.6 Effect of Ordering and Domain Size on Ceramic Processing 531 11.7 Summary 532 Problems 532 References 533

IV 12

G R E E N B OD Y F O R M A T I O N - MISE-EN FORME Mechanical Properties of Dry Ceramic Powders and Wet Ceramic Suspensions 12.1 Objectives 541 12.2 Introduction 542 12.3 Equations of Motion 543 12.3.1 Continuity Equation 543 12.3.2 Momentum Balance 544 12.3.3 Constitutive Equations for Dry Powders 545 12.3.4 Constitutive Equations for Fluids 545 12.4 Ceramic Suspension Rheology 550 12.4.1 Dilute Suspension Viscosity 551 12.4.2 Rheology of Concentrated Ceramic Systems 562 Problem 12.1. Hard Sphere Stress-Strain Curve 569 12.4.3 Ceramic Paste Rheology 585 12.5 Mechanical Properties of Dry Ceramic Powders 590 12.5.1 Coefficient of Pressure at Rest 592 12.5.2 Compact Body 594 12.5.3 Plastic Body 595

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12.5.4 Yield Criteria for Packings 596 12.5.5 The Coulomb Yield Criterion 597 12.5.6 Yield Behavior of Powders at Low Pressures 599 12.6 Summary 602 Problems 603 References 605

13

Ceramic Green Body Formation 13.1 Objectives 609 13.2 Introduction 610 13.3 Green Body Formation with Ceramic Suspensions 612 13.3.1 Slip Casting 613 13.3.2 Filter Pressing 618 13.3.3 Tape Casting 620 13.3.4 Sedimentation Casting and Centrifugal Casting 629 13.3.5 Electrodeposition 636 13.3.6 Dip Coating 638 13.4 Extrustion and Injection Molding of Ceramic Pastes 643 13.4.1 Flow in the Extruder 644 13.4.2 Flow in the Extrusion Die 646 13.4.3 Flow into the Injection Molding Die 651 13.5 Green Body Formation with Dry Powders--Dry Pressing 653 13.5.1 Tapped Density 654 13.5.2 DiePressing 656 13.5.3 Stress Distribution in the Ceramic Compact 661 13.5.4 Deformation of Visco-Elastic Solids and Fluids 667 13.5.5 Die Ejection and Breakage 667 13.5.6 Isostatic Pressing 671 13.5.7 Green Machining 673 13.6 Green Body Characterization 674 13.7 Summary 675 Problems 675 References 677

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V

PRESINTERING HEAT TREATMENTS OF DRYING A N D B I N D E R B URNO UT

14

Green Body Drying 14.1 Objectives 683 14.2 Introduction 683 14.2.1 Heat Transfer 686 14.2.2 Mass Transfer 687 14.2.3 Flow of Liquid in the Pores 689 14.2.4 Drying Shrinkage 690 14.2.5 Drying Induced Stresses 691 14.3 Sphere and Cylinder Drying 693 14.3.1 Boundary Layer Heat and Mass Transfer Giving the Drying Rate for the Constant Rate Period 693 14.3.2 Shrinkage during the Constant Rate Period 695 14.3.3 Diffusion and Heat Conduction in the Porous Network Giving the Drying Rate for the Falling Rate Period 698 Problem 14.1. Drying Time Calculation 700 14.3.4 Cylinder Drying 702 14.4 Drying of Flat Plates 703 14.5 Warping and Cracking during Drying 705 14.5.1 Thermal Stresses Induced during Drying 708 Problem 14.2. Temperature Difference Induced Tensile Stress 712 14.5.2 Flow Stresses during Drying 713 14.5.3 Capillary Stresses 716 14.6 Characterization of Ceramic Green Bodies 718 14.6.1 Green Density 719 14.6.2 Uniformity of Microstructure Mixedness 719 14.6.3 Green Body Strength 721 14.7 Summary 726 Problems 726 References 727

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Contents

Binder Burnout 15.1 Objectives 729 15.2 Introduction 730 15.2.1 Heat Transfer 731 15.2.2 Mass Transfer 732 15.3 Thermal Degradation of Polymers 733 15.3.1 Reaction Kinetics 737 15.3.2 Polymer Residues and Volatiles 738 15.4 Oxidative Polymer Degradation 738 15.4.1 Reaction Kinetics 749 15.4.2 Polymer Residues and Volatiles 750 15.5 Kinetics of Binder Burnout 752 15.5.1 Kinetics of Binder Oxidation 755 15.5.2 Kinetics of Volatiles Loss 758 Problem 15.1. 760 15.5.3 Kinetics of Binder Pyrolysis without Oxygen 761 15.5.4 Kinetics of Carbon Removal 762 Problem 15.2. 765 15.6 Stresses Induced during Binder Burnout 767 15.6.1 Thermal Stresses Induced during Binder Burnout 768 15.6.2 Stresses Due to Volatile Flow 770 15.7 Summary 771 Problems 772 References 775

VI 16

SINTERING AND FINISHING Sintering 16.1 Objectives 781 16.2 Introduction 782 16.3 Solid State Sintering Mechanisms 785 16.3.1 Driving Force for Sintering 786 16.3.2 Sintering Kinetics by Stage 788 16.3.3 Effect of Green Density of Sintering Kinetics 811

Contents

16.4

16.5

16.6 16.7 16.8

17

16.3.4 Effect of Particle Size Distribution on Sintering Kinetics 812 16.3.5 The Effect of Fractal Aggregates on Sintering Kinetics 817 Grain Growth 824 16.4.1 Normal Grain Growth 827 16.4.2 Abnormal Grain Growth 840 ReactiveSintering 844 16.5.1 Sintering wtih a Liquid Phase 844 16.5.2 Solid State Reactive Sintering 860 16.5.3 Gas-Solid Reactive Sintering 861 Pressure Sintering 864 Cool Down after Sintering 867 Summary 869 Problems 869 References 871

Finishing 17.1 Objectives 875 17.2 Introduction 875 17.3 Ceramic Machining 876 17.3.1 Effect of Machining on Ceramic Strength 877 17.3.2 Effect of Grinding Direction on Ceramic Strength 878 17.3.3 Effect of Ceramic Microstructure on Strength 879 17.3.4 Grinding and Machining Parameters 880 17.4 Coating and Glazing 882 17.5 Quality Assurance Testing 883 17.5.1 Proof Testing 884 17.6 Nondestructive Testing 886 17.7 Summary 888 References 889

Appendix Appendix Appendix Appendix Appendix Appendix

A B C D E F

Ceramic Properties 891 Gamma Function 893 Normal Probability Function t Test 901 Reduction Potentials 903 Thermodynamic Data 905

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Contents

Appendix G Summary of Differential Operations Involving the V-Operator in Rectangular Coordinates (x, y, z) 915 Appendix H Summary of Differential Operations Involving the V-Operator in Cylindrical Coordinates (r, 0, z) 917 Appendix I Summary of Differential Operations Involving the V-Operator in Spherical Coordinates (r, 0, d~) 919 Appendix J Liquid Surface Tensions 921 Appendix K Drago E and C Parameters 925 Appendix L Hildebrand Solubility Parameter and Hydrogen Bond Index 929 Appendix M Hydrated Cation Radii 935 Index 937

Preface

In the past 15 years ceramic powder processing and synthesis have undergone a transformation. Scientific and engineering methods have been applied in this field at a much higher level than ever before, allowing much greater control of properties than could be achieved previously. Ceramic systems are not simple and therefore these scientific and engineering methods had to achieve sufficient sophistication to be adaptable to this field. We now have many examples of the application of these scientific and engineering methods to ceramics. As a result these first examples can be explained to students of ceramics, who with this knowledge, can continue this evolution of sophistication in the fundamentals of ceramic powder processing and synthesis. This book was written in an attempt to do just that. The organization of this book is explained in the introduction. Basically, it is organized like a ceramic manufacturing facility starting with raw materials and ending with sintering and finishing. Various chapters contain problems within the text for illustration. At the end of each chapter, additional problems allow the reader to go into greater depth using the material presented in the chapter. These problems are not necessarily easy but the reader's efforts to resolve them will result in much greater knowledge of the material covered in the chapters. Finally, I acknowledge the help of others in writing this book. Many long nights over a period of more than six years were spent writing this book and my family has suffered as a result. This book is dedicated to my understanding wife, Susan. Many people have helped me with concepts and ideas. Professor Alain Mocellain critiqued the outline of this book and made many useful suggestions. Dr. Paul Bowen, Dr. xxi

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Preface

Dennis Gallagher, Dr. Jacques Lemaitre, and the LTP-EPFL students performed the very important task of proofreading the manuscript. Dr. Bowen provided gentle guidance in areas where rewriting was required. Academic Press provided a long list of anonymous reviewers, one for each chapter; I am indebted to them for many helpful suggestions. Elizabeth Burdet worked diligently to minimize the other work in my laboratory so that sufficient time was available to write this book. Silvia Yvette helped with typing the first draft of this manuscript, Wilma Bunners made many of the more complex drawings found in the text, and my wife, Susan, read each chapter for English corrections. Many thanks to all.

Terry A. Ring

PART

I I N T R OD UC TION: HISTORY, RAW MATERIALS, CERAMIC POWDER CHARACTERIZATION

Many options are to be considered in organizing a book on the fundamentals of ceramic powder processing. One could organize a book along phenomenological lines (e.g., similar thermodynamics and reaction or diffusion kinetics) or along material classifications lines (e.g., oxides, carbides, and nitrides) or along material properties lines (e.g., structural ceramics and electronic ceramics). After considering the many possibilities, this book has been organized as if the reader were following a ceramic process in a factory from powder to final finished piece. Ceramic powder processing can take two traditional routes: one is a wet powder processing route, where the powder is mixed in the liquid and cast into the green body before firing; the second is a dry ceramic powder processing route, which consists of pressing the dry powder (with binder) into the green body and then firing. Both of these processing routes are shown in the Figure 1.1. This is the flow sheet for the computer controlled tile making facility for the ' INAX Corporation in Japan. Here they use these two routes, a wet paste-extrusion route and a dry-pressing route, to make ceramic tiles. These processing routes are also used for modern ceramics such as tiles for the space shuttle's surface and electronic BaTiO3 capacitors. In this figure one sees all the steps that go into making ceramics, starting with grinding the ceramic powders to develop a very fine particle-size distribution (the grinding circuits contain classification and recycle loops). This is

2

Part I

Introduction

followed by putting the ceramic powder into liquid form, adding different additives to adsorb to the particle surface and prevent coagulation of the particles, as well as to adjust the rheology of the paste and provide a binder of the particles after consolidation. The paste is then dewatered to the best consistency for extrusion into the desired shape. The resulting green bodies are dried very slowly, then subject to binder burn-out treatment at higher temperatures followed by sintering. During sintering, pores are removed from the ceramic body, leaving behind a fully dense piece which must then be finished in some way (e.g., applying a glaze or grinding to size). This constitutes the wet route as shown in Figure 1.1. The outline of this book follows that sequence of events very closely. As a result we have the following parts of this book:

F I G U R E 1.1

Ceramictile manufacturingprocess. Photo courtesyof Inax Corp., Japan.

Part I Introduction

3

Part I. Introduction: History, Raw Materials, Ceramic Powder Characterization Part II. Ceramic Powder Synthesis Part III. Ceramic Paste Formation: Mise-en Pdte Part IV. Green Body Formation: Mise-en Forme Part V. Presintering Heat Treatments of Drying and Binder Burnout Part VI. Sintering and Finishing The parts are further broken down into chapters discussing the chemical, physical, and engineering fundamentals of each step of the process. The other route for ceramic manufacturing, starting with dry powders and pressing them with a polymer or binder, is discussed in this book in the different sections. This route will have in common with the wet processing methods the steps of powder synthesis and ceramic green body formation, binder burn-out, sintering, and finishing; thus, the reader interested in the dry powder processing route can follow this processing sequence by stepping over various materials which are not of interest. For the students particularly interested in ceramic part manufacture, Part II of this book, discussing ceramic powder synthesis, would be of less interest. As a result the student can start with the part three after reading the introductory chapters in Part I on raw materials and ceramic powder characterization. Each chapter is broken into sections with the first section always stating the objectives of the chapter, and the last section always providing a summary of the chapter. In the text, problems are worked to elucidate the points discussed. Finally at the end of each chapter there are unworked problems that the students can do for homework. The book attempts to provide a large list of references for specific concepts and ideas presented elsewhere, and we hope that the reader will refer to these references for the derivation of specific equations not presented. This book is highly mathematical in comparison with other texts in the field, because this field should be much more quantitative than heretofore presented. With these mathematics, the field of ceramic powder processing can become more quantitative in the future.

4

Part I Introduction

1.1 G E N E R A L C O N C E P T S POWDER PROCESSING

OF C E R A M I C

Several general ideas are associated with ceramic powder processing. These general ideas have been generated after many years of research and have resulted in a philosophy of ceramic powder processing. The first idea is that uniformity in the microstructure of a single phase ceramic is better for electrical and mechanical properties. This idea is based on the Griffith fracture theory for ceramics, where the strength of the ceramic is related to the largest flaw size. With a bigger flaw size, weaker single phase ceramics result. Uniformity is also important for electrical ceramics. For example, the final grain size distribution of BaTiO3 should be uniform to have the highest dielectric constant for ceramic capacitors or the highest piezoelectric coupling constants for actuators. In the case of the capacitor, the grains should be uniformly small; and in the case of the actuator, they should be uniformly large to achieve the orthorhombic crystal structure necessary for piezoelectricity, which is prevented for grains less than 1 micron in size for pure BaTiQ. The idea of uniformity remains for both large and small grains in the case of electrical properties. This is sometimes difficult when cannibalistic grain growth occurs during sintering, leading to a bimodal grain size distribution. For this reason, dopants are used to prevent cannibalistic grain growth. Another idea is that the microstructural inhomogeneities that occur in casting a green body remain (or even get larger) during drying, binder burn-out, and sintering. Therefore, to obtain the best uniformity the casting process must be performed very carefully with suspensions that contain no bubbles or large pieces of polymers. In addition, the uniformity produced in the green body should not be destroyed by rough handling. In the case of drying and binder burn-out, huge volumes of gas, many thousands of times that of the green body itself, must leave the green body. This process puts tensile stress on the green body which can cause cracks. To prevent these cracks, drying and binder burnout conditions which are very slow are desirable. Uniformity is also extended from the green body casting down to the ceramic suspension utilized for casting. In this case uniformity of the particles used is important because larger and smaller particles segregate into different parts of the mold during casting of monophase ceramics, leading to

1.1 General Concepts of Ceramic Powder Processing

5

nonuniformity. This is the same reason why stable colloidal suspensions are used for casting to prevent packing inhomogeneities caused by aggregates. With composite ceramics which consist of two or more different phases, uniform mixing in the suspension is also important. This may be impossible if the two powders utilized have either different densities or different particle-size distributions or both. For this reason, the suspension is often flocculated with polymers so that the well-mixed nature of a suspension is preserved in the flocs..These flocs, with their inhomogeneous packing of particles, are then broken into homogeneous green bodies by pressing at high pressure. The last general concept of ceramic powder processing is that smaller powders sinter to give smaller grains that give a stronger ceramic piece. This idea is again based on the Griffith fracture theory for ceramics, where the strength of the ceramic is related to the largest flaw size. Assuming homogeneity, a smaller grain size will result in a smaller flaw size, leading to a stronger ceramic. The sintering times tl and t2 for two powders with the same chemistry but different particles sizes rl and r2 is given by Herring's scaling law [1]:

t2 = [r2/rl]ntl where n is a constant depending on the sintering mechanism. In the case of volume diffusion, n = 3. From the Herring scaling law, we see that, as the mean particle size is decreased, the time needed to sinter a ceramic piece is decreased. These general concepts will play an important role in the selection of a process for the manufacture of a particular ceramic part and as a result these general concepts will be encountered again and again throughout this book.

Reference 1. Herring, C., J. Appl. Phys. 21, 301 (1950).

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1

Ceramic Powder Processing History and Discussion of ~~atural Raw ?daterials

1.1 O B J E C T I V E S This chapter will give the reader a historical perspective of the field of ceramic powder processing. This field has a long and rich history which in many ways is impossible to trace because it goes back to before writing. Nonetheless, there is a rich archaeological record of ceramic articles produced by different technologies from which we can learn a great deal. In addition, this chapter presents the raw materials used for ceramic manufacture both historically and in the present day. Finally an overview of the organization of this book is presented. This book is organized like a ceramic factory, with powder synthesis and preparation first followed by paste preparation, forming, drying, binder burnout, and sintering.

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Chapter I

Ceramic Powder Processing History

1.2 H I S T O R I C A L P E R S P E C T I V E The first ceramic objects in the archaeological record are fired clay figures appearing about 22,000 B.C. [1]. These figures were probably n a t u r a l clay pieces shaped by h a n d into a h u m a n o i d form, allowed to dry, and placed in a fire. This art form gradually became used for more practical objects such as bowls and storage vessels on a m u c h larger scale. This larger scale of production became an integral part of the Chinese villages about 6000 B.C., where the ceramic kiln played a central role [17]. As a result, the f u n d a m e n t a l s of ceramic powder processing, the title of this book, have been practiced for over eight millennia [1 ]. A highly developed ceramic technology was in place for e a r t h e n w a r e like t h a t shown in Figures 1.1 [2] and 1.2 [2] well before the Bronze Age (about 4000 B.C. [3]), at a time when silkworm cultivation was also invented. These red pottery vases have a complex shape and are painted with black o r n a m e n t a l patterns. Figure 1.2 was excavated at Pan-p'o,

FIGURE 1.1 Red pottery vase with a contracted waist, a black design on a red base

that covers the earthen ware (brown). It was excavated at Lan-chou, Kansu, in 1958, and is 18.3 cm in height, from the third millenium B.C. Taken from "The Genius of China" [2].

1.2 Historical Perspective

9

Yang-shao bowl excavated in 1954-57 at Pan-p'o, Shensi, made of red pottery painted with black (carbon) triangles over a slip of white clay, 12.7 cm in height, from the fifth or fourth B.C.Taken from "The Genius of China" [2].

FIGURE 1.2

Shensi, China* with other objects t h a t date it to 5000 B.C. or 4000 B.C. The gloss of the paint results from b u r n i s h i n g the clay before firing. The deep red color suggests the use of a clay (i.e., kaoline, an aluminosilicate mineral) containing hematite, a red iron oxide. This pottery was fired in a kiln of a relatively advanced design, capable of t e m p e r a t u r e s up to about 1000~ because only at t e m p e r a t u r e s above 900~ does kaolinite sinter to reasonable s t r e n g t h [4,5]. By 3000 or 2000 B.C., a b u r n i s h e d black pottery was produced as shown in Figure 1.3 [6]. This type of pottery, excavated at Wei-fang, S h a n t u n g , China, is either entirely black or has a black surface with a grey core. Analysis of the polished surface shows t h a t it has only a higher concentration of carbon t h a n at the core. Much debate has centered on how this polished surface was achieved. But, due to the presence of carbon in the clay body, the kiln design m u s t have been sufficiently advanced to give a reducing atmosphere. This pottery is similar to the present day S a n t a Clara pottery produced by the Indians in the s o u t h w e s t e r n USA. * Note that all Chinese names have been romanized by the Wade-Giles system.

10

Chapter I

Ceramic Powder Processing History

FIGURE 1.3

Tall beaker, tou, excavated in 1960 at Weifang, Shantung. It is of burnished black pottery, 16.1 cm in height, and from the third or second millenium B.C. Taken from "The Genius of China" [2].

The first archaeological record of bronze production in China comes from an Erlitou Culture (1700 B.C.) site in Henan, Shanxi, China [6]. Bronze is an alloy of copper and tin (although in ancient China, lead was also frequently used). The earliest known Chinese bronze object is shown in Figure 1.4, which has 92% copper and 7% tin. This wine cup displays the basic metalworking features of the Chinese Bronze Age, which are sharply different from Near Eastern and Western traditions. This alloy is not an accident but a deliberate choice and indicates that a complex metallurgical infrastructure was in place to mine the ores of both metals and then smelt each ore to its respective metals.

1.2 Historical Perspective

11

FIGURE 1.4 Wine cup with tripod feet, of bronze, from the seventeenth century B.C. (one of the earliest so far known). Taken from "Treasures from the Bronze Age of China" [6].

Most important, this bronze vessel has seams which show it to have been cast from a mold made in four separated sections. This wine cup required a complex ceramic mold, which sets the early Chinese bronze technology apart from the lost-wax process used in the West. The production of bronze is a major undertaking. Sources of copper and tin must be located and protected. The ore must be mined and the metal removed. In the case of copper, this is difficult because copper accounts for only a small fraction of the volume of the ore. In ancient China, the ores seems to have been crushed, liberated, separated, and smelted at the mines and then transported to communities for casting. The melting of large quantities of metal, primarily copper, Tm = 1085~ required elaborate kilns and huge fires of high intensity; skills that had developed out of the ceramic tradition. Casting required controlled cooling of the metal to avoid holes and cracks in the finished object; skills which also relied on the precise fitting ceramic molds. During the Shang Dynasty (1600 to 1027 B.C.), when writing was first developed, bronze metallurgy developed into a highly skilled technology. Shang bronze molds where made from loess, the wind-blown ochrecolored soil that covers much of the landscape of northern China. Loess is rich in micas, fine quartz, sodium feldspar, and alkaline minerals

12

Chapter I

Ceramic Powder Processing History

FIGURE 1.5 (a) Diagram showing how early Chinese bronzes were formed: (1) the model, (2) the sections of the mold, and (3) schematic of completed vessel.

[3]. The natural clay content (mostly illite) in loess ranges between 10 and 20%, which is enough to give it plasticity when mixed with water [3]. The unique property of loess as a ceramic molding material is that it does not shrink much as it is dried and fired [3] to 900~ Also, it is porous after firing, allowing the bronze to degas into the mold during solidification. By the Warring States Period (475 to 221 B.C.), there is evidence of the prolific use of ceramic multipart piece molds shown in Figure 1.5 in the direct casting of bronze vessels and weapons. In the Shang Dynasty, the first example of pottery covered with a high-fired feldspathic glaze [2] was observed. The body of this vase, shown in Figure 1.6, is of near stoneware hardness. The glaze, requiring kiln temperatures of 1200~ is spread uniformly over the whole body. This glaze technology then disappears from the archaeological record until the late fourth or early third century B.C. China's first emperor, Ch'in Shih Huang Ti, in about 221 B.C. united the various warring states of China by providing a uniform code of law, standards of currency, written language, and weights and measures

1.2 Historical Perspective

F I G U R E 1.5

13

(b) Actual completed vessel. Taken from "Treasures from the Bronze Age

of China" [6].

and completed the separate ramparts of the Great Wall of China, some 1000 km long, as protection from northern invaders. During his reign, ceramic and bronze arts were also practiced to perfection. In his mausoleum 7000 life size terracotta soldiers (one shown in Figure 1.7) and horses made of fired loess were discovered. On the eve of the Western Han Dynasty (206 B.C. to 8 A.D.), low-fired lead-fluxed glaze made its first appearance. This is the predecessor to the "polychrome" lead glasses of the Sui (581 to 618 A.D.) and T'ang Dynasties (618 to 906 A.D.). The colors of the lead glazes (i.e., brown, yellow, green, and blue) were produced by adding refined metal ores to the glaze mixture. A three-colored T'ang Dynasty vase, shown in Figure 1.8, is an example of this technology. These glazes were generally

14

Chapter I

Ceramic Powder Processing History

FIGURE 1.6 Glazed pottery vase of high fired stoneware hardness, excavated in 1965 at Ming-Kung-lu, Cheng-chou, Honan. It is covered with a high-fried feldspatic glaze, 28.2 cm in height, and from the sixteenth or fifteenth century B.C. Taken from "The Genius of China" [2].

applied over a slip of white clay. During the T'ang Dynasty, the feldspathic glazes evolved into what is to become a long tradition of white porcelaneous ware like that shown in Figure 1.9. This glaze required firing at 1300~ This body has a glassy phase, filling the pores, giving a nonporous fired body. For this development, purified raw materials had to be used with a specific narrow range of chemical composition. This glaze was further refined into the subtle celedon green prolific in the Sung Dynasty (960 to 1279 A.D.). At this point the Chinese ceramics had reached one of its technological objectives, which was a porcelain

1.2 Historical Perspective

15

Life-sizefigure of a terracotta military commander in the mausoleum of the first emperor of China. Taken from "Treasures from the Bronze Age of China" [6].

FIGURE 1.7

body fused with a glossy, translucent green glaze that r u n g when struck and looked like jade. With the porcelaneous ware of the Yuan Dynasty (1271 to 1368 A.D.) underglaze painting of cobalt blue (cobalt oxide fired to give cobalt silicate, which is blue [7]) and copper red (copper oxide fired to give metallic copper, which is red [7]), present in the delicate

Tang Dynasty covered bowl, with lead glaze in green, brown, and yellow, excavated in 1958 at Loyang, Honan, 21 cm in height, from the first half of the 800s A.D. Taken from "The Genius of China" [2].

F I G U R E 1.8

White porcelaneous bowl, clear glazed with applied medallions, excavated in 1956 at Han-sen-chai, Near Sian, Shensi, 23 cm height, from the T'ang dynasty 667 A.D. Taken from "The Genius of China" [2].

F I G U R E 1.9

1.2 Historical Perspective

17

leaf and floral motifs of the vase with cover shown in Figure 1.10, indicate a new technological level of excellence. This was to be exploited in the Ming D y n a s t y (1368 to 1644 A.D.) and transferred to Europe to become the Meisen and Delft pottery of the early 18th century. Control of the a t m o s p h e r e during firing of these glazes (i.e., reducing conditions) was necessary to give the desired blue or red color and not simply black. The glazing technology culminated in the polychrome overglaze paints of the Ch'ing D y n a s t y (1644 to 1911 A.D.), Figure 1.11 [8].

White porcelain vase and cover with underglaze decoration of incised dragons and blue glaze waves, excavated in 1964 in Pao-ting, Hopei, 51.5 cm height, from the Yuan dynasty, late 14th century A.D. Taken from "The Genius of China" [2].

F I G U R E 1.10

18

Chapter 1 Ceramic Powder Processing History

FIGURE 1.11 Polychrome glazed vase decorated with flowers and insects in a peach branch, 51.4 cm height, from the Ch'ing dynasty, with a Ch'ien Lung mark, 1736-95. Taken from "A Handbook of Chinese Ceramics" [8].

Ceramic powder processing technology is discussed in the T'ao Shuo [9]. This text describes how kaolin raw materials had to be found and ground to the desirable size distribution. After grinding, the earth was washed and purified. This was done by mixing it with water in a large

1.2 Historical Perspective

1

earthen ware jar and stirring the mixture until all the organic impurities had floated to the top and were poured off. The resulting paste was next passed through a fine horsehair sieve and then into a bag made of two thicknesses of finely woven silk. Afterward, the paste was poured into several earthenware vessels, so that the excess water could run off. The paste was then allowed to sediment. The settled paste was further dewatered by wrapping it in a fine cotton cloth, and placing it in a bottomless wooden box resting on dry earthenware bricks. More bricks were piled on top of the cotton bag of paste to press and absorb more of the water, using both hydrostatic and osmotic pressure. When free of excess water, the paste was thrown on large stone slabs and turned over and over until it was ductile. The paste was worked into the green body shape by various techniques: coiled and layered by hand, thrown on a potter's wheel, slip cast or pressed into molds, or stamped. The green body was then dried slowly, so as not to crack it, and fired in a wood-burning kiln under oxidizing conditions at more than 900~ A stunning example of this type of technology are the 7000 life-size terracotta statues, each with a different face, of the army of the First Emperor of China, Emperor Ch'in (221 to 207 B.C.), at his grave site in Xian, China (see Figure 1.7). Historians believe there is an official document describing this ceramic powder processing technology that was among the official documents of the Ch'Hi state incorporated in 140 B.C. Indeed, updated copies of the Chou Li, an early encyclopedia of art and technology, shows wood block prints of the various processing steps. The wood block prints shown here are from "T'ao Shuo," Description of Pottery [9], in six books by Chu Yen. This work describes 20 woodblock prints dating from 1743 with T'ang Ying, director of the Imperial Factory at Ching-t~, narrating a description of each print. Several of these woodblock prints are reproduced in Figures 1.12 and 1.13 showing green bodyt shaping methods, decorating, and firing in a wood-burning kiln. Closely interacting with this earthenware technology were developments in metallurgy. Some of the metallurgical operations are described in the T'ien Kung K'ai Wu published in 1637 A.D. from which Figures 1.14 and 1.15 come [10]. Much of the glaze technology that sealed the outside of the porous earthenware structure and gave the body color and texture was a result of ceramic alloying of metal oxides, which were made available by metallurgical operations. The lead oxide flux glazes of the Han Dynasty (202 B.C. to 220 A.D.) and the T'ang Dynasty (618 to 906 A.D.) funeral ware were refined to give colors that t Unfired ceramic body. Green due to color of purified kaolin when wet. Also from the Chinese word Qing with the definitions (1) fresh and (2) green.

20

FIGURE 1.12

Chapter I

Ceramic Powder Processing History

Woodblock prints of the stamping of a pattern on (a) and the firing of a "dragon kiln" with several chambers (b). named because they snaked their way at a constant gradient up the Taken from T'ien Kun K'ai Wu, 1637 print from "Description of Porcelain" [9].

the surface of a bowl These kilns were so contours of a hillside. Chinese Pottery and

included white, amber, yellow, green, and violet blue with a minutely cracked texture, analogous to the lead oxide glazes used today. The colors were obtained by adding refined metal oxides to the basic glaze formula and controlling the oxidizing or reducing conditions in the kiln. This history of ceramic powder processing technology is only a brief description of the events in China. In fact, other parts of the world also contributed to the technological developments of ceramic powder processing. Table 1.1 lists the roots of the ceramic technologies throughout the world. Egypt played an important part in the development of faience about 4000 B.C. and glass making about 1500 B.C. In the 9th

1.2 Historical Perspective

FIGURE 1.12

21

(Continued)

century Baghdad played a role in developing tin glaze ware, to cite some examples. One of the most interesting developments is that of porcelain manufacture [11]. Crude porcelain was first made during the T'ang dynasty in China (618 to 908 A.D.). This technology was carefully guarded by the Chinese but finally spread to Korea by the ll00s and to Japan by the 1600s. Marco Polo and other Western travelers described the Chinese porcelain to the Italian ruling class upon their return from the Far East, and they started importing pieces. Under royal patronage, alchemists tried to discover how the material could be manufactured, but without chemical analytical methods success came only from trial and error. In 1575, under the sponsorship of de Medicis in Florence, soft paste porcelain was developed, a mixture of clay and ground glass, fired at 1200~ The French also produced soft paste porcelain at Rouen and St. Cloud in the 1600s. Later in the 1600s, this technology spread

22

Chapter I

Ceramic .Powder Processing History

FIGURE 1.13 Woodblock prints of painting the ceramicware with cobalt underglaze patterns (a) and using a wheel for painting a circle on the rim of a bowl (a). In (b), two men are dipping the painted ware into a great bowl of glaze prior to firing. Taken from T'ien Kun K'ai Wu, 1637. Print from "Description of Chinese Pottery and Porcelain" [9].

to other parts of France (Chantilly, Mennecy, Vincennes, and Sevres) and to England (Chelsea, Bow, and Derby) in the mid-1700s. The secret of true porcelain was not rediscovered in Europe until 1707 by von Tschirnhaus (a mathematician) and B~ttger (a kidnapped alchemist), who were "employed" by Augustus the Strong of Saxony. Augustus the Strong's fascination with collecting Oriental porcelain nearly bankrupted his kingdom. Using the crude scientific analysis of BOttger, Tschirnhaus recognized that true porcelain must be a mixture of natural materials and not ground glass as in soft paste porcelain. They ordered samples of clay from various parts of the kingdom and finally substituted ground feldspar for ground glass of the soft paste with a natural kaolin clay. Tschirnhaus and B~ttger established a true porcelain factory at Meissen near Dresden. The first major sales from this

1.2 Historical Perspective

F I G U R E 1.13

23

(Continued)

factory took place at the Leipzig Fair in 1713. This technology spread quickly across Europe, fueled by the demand of the new fad of drinking tea, coffee, and chocolate. Throughout this history the purity of the raw materials has been of the utmost importance. In ancient China the purification of raw materials was practiced for 8000 years with the use of purified white clay as a wash coat under designs. Ground feldspar of a closely controlled chemical composition was used for the very first glaze of the Shang Dynasty. During the Bronze Age, ceramic raw materials were first synthesized (e.g., lead oxide used for the basic glaze formula). In modern times, ceramic powder raw materials are still purified and synthesized by separate processes (e.g., Bayer process alumina, flame synthesis of titania, dead-burned magnesia) so that their purity is better controlled. This attention to raw material purity coupled with attention to the

24

Chapter I

Ceramic Powder Processing History

FIGURE 1.14 Distillation of mercury in a retort, from T'ien Kun K'ai Wu, 1637. Re-

printed by permission of the publishers from "Science in Traditional China" [10], copyright 9 1981 by the Chinese University of Hong Kong. details of each process step are important attributes of the development of ceramic powder processing. In this brief overview of the first recorded ceramic powder processing, we find all of the attributes of this technology still with us today; for example, raw materials selection, grinding, size classification, raw material purification and blending, paste preparation, dewatering, green body formation, drying, and firing. In this regard ceramic powder processing is a very old art. Yet, in the last 10 years, enormous development has taken place in the scientific understanding of this very old art. Currently now that the speed with which these new developments emerge appears to be slowing down, it is time to reflect on what has been accomplished and where we stand scientifically. This book will give the state of the art of ceramic powder processing in the early 1990s. We should keep in mind that this field is still progressing and this book, like the official documents of the Ch'hi state incorporated

Liquidation process for the separation of silver from copper by lead, which is later cupelled, from T'ien Kun K'ai Wu, 1637. Reprinted by permission of the publishers from "Science in Traditional China" [10], copyright 9 1981 by the Chinese University of Hong Kong.

F I G U R E 1.15

TABLE 1.1

Outline of Ceramic History Showing the Main Lines of Technological Development

Prehistory of Ceramics -22,000 B.C. earliest known fired clay figures -8,000 B.C. fired earthenware vessels in Near East -6,000 B.C. slip coatings and clays prepared by decanting suspensions, ochre red and black decoration, manganese and spinel black pigments, control of oxidation, reduction during firing, impressed designs, rouletting, incised decoration, coil and slab construction, burnishing, joining paddle and anvil shaping, carving, and trimming -4,000 B.C. Egyptian faience 4,000-3,500 B.C. wheel throwing, earthenware molds, craft shops - 1 6 0 0 B.C. vapor glazing, prefritted glazes, lead glazes - 1 5 0 0 B.C. glass making, alkaline glazes - 1 0 0 0 B.C. glazed stoneware in China - 7 0 0 B.C. Greek black and red wares

Developments toward particular ceramic products Soft-paste porcelain

900s clay quartz ware in Egypt 1200s enameled minai ware 1400s white tile 1500s Isnik tile blue on white wares 1575-1587 Medici porcelain 1600s Gombroon ware 1695 soft paste porcelain at St. Cloud 1742 soft paste porcelain at Chelsea 1796 Spode's English bone china 1857 Beleek frit porcelain

Hard-paste porcelain 206 B.C.-221 A.D. (Han Dynasty in China) White porcelain 618-906 (Tang Dynasty in China) extensive porcelain exports to Europe 960-1279 (Sung Dynasty in China) celadon and J u n ware, cobalt blue and white porcelain 1368-1644 (Ming Dynasty in China) blue and white, reduced copper red and white porcelain extensively exported to Europe 1600s Arita ware 1600s Bottinger porcelain 1700s fine white semivitrous ware in England 1800s Parian porcelain

Tin-glazed ware

Jasperware

Stoneware 6000 B.C. earthenware 600 B.C. terracotta in Greece

900s tin-glazed ware in Baghd a d - l u s t e r painting 1300s majolica ware in Spain and Italy 1500s polychrome painting 1600s paintings of history and stories 1700s faience in Europe 1700s blue and white Delftware 1900s hand-crafted tin glaze ware

1400s German stoneware, salt glazing 1400s English sipware 1600s fine terracotta 1700s turning by steam engine 1700s basalite cane ware 1764 Wedgewood jasperware

1400s German stoneware, salt glaze, English slipware 1600s fine terracotta 1800s engine turning 1900s hand-crafted stoneware

1.3 Raw Materials

27

140 B.C., is only a description of the state of the art. Many problems have yet to be solved before ceramic powder processing can be developed into a mature field for all ceramics. Part of the reason for constant technological evolution is that this field will never be c o m p l e t e n n e w ceramic compositions are always being developed (i.e., high temperature superconductors, piezoelectrics, varistors). In this regard ceramic powder processing will never be without challenging frontiers.

1.3 R A W M A T E R I A L S Since the Bronze Age both natural ceramic raw materials and synthetic raw materials have been used. Today synthetic raw materials are referred to as industrial minerals or specialty chemicals. Natural raw materials are those to which only physical separations are performed (e.g., clay soils from which organic raw materials are floated, feldspar rock ground to a particular size distribution). With this classification, a description of common ceramic raw materials will be given in the next part of this chapter.

1.3.1 N a t u r a l R a w M a t e r i a l s

1.3.1.1 Clays Clays were probably the first ceramic raw materials. Clay minerals are fine-particle hydrous aluminum silicates, like those shown in Figure 1.16 [12], which develop plasticity when mixed with water. They have a wide range of chemical and physical characteristics but the common attribute of a crystalline layer structure consisting of electrically neutral aluminosilicate layers as shown in Figure 1.17. The platelike morphology gives easy cleavage, which leads to a fine particle size and a narrow particle-size distribution and allows the particles to easily move over one another. Clays perform two important functions in ceramic bodies. First, the plasticity of clay suspensions is basic to many of the forming processes commonly used to fabricate ceramic bodies; the ability of clay-water suspensions to be dewatered to give a shape with strength during drying and firing is unique. Second, clays fuse over a temperature range, depending on composition, that can be economically attained, to become dense and strong without losing their shape. The most common clay minerals of interest to ceramists are based on the kaolin structure, A12(Si2Os)(OH)4. (The term kaolin comes for the name, Kao Ling, of a mountainous district, 20 miles northeast of Chingtechen, China, famous during the Tang and Sung Dynasties as a strong hold for outlaws [7]). The reason why kaolin is such a useful

28

FIGURE 1.16

Chapter 1 Ceramic Powder Processing History

Photomicrograph of kaolinite platelets, A12Si20~(OH)4. From Millot [12].

FIGURE 1.17 A platelet of kaolinite consists of a tetrahedral layers and octahedral layers superimposed. The summits of the tetrahedral layer and octahedral layers form a plane of oxygen atoms. The distance between the two units is 7/k. From Millot [12].

1.3 Raw Materials

~

~0

Chapter I

Ceramic Powder Processing History

FIGURE 1.18 Commoncompositions in the ternary system MgO-A1203-SiO2.Taken from Kingeryet al. [13], reprinted with permission from John Wiley & Sons, Inc. 9 1976,

New York. raw material is that above 500~ the crystallization of water evolves. Then it decomposes at 980~ to form fine-grained mullite, A16Si2013, in a silica matrix. F u r t h e r heating of kaolin gives rise to a growth of mullite crystals, crystallization of the silica matrix as cristobalite, and formation of a eutectic liquid at 1595~ as shown in the phase diagram in Figure 1.18 [13]. Reasonable strengths of the sintered kaolin ceramics can be obtained on firing between 900 and 1000~ where one can observe the viscous sintering [4,5] of the kaolin polymorphs produced above 900~ Common natural impurities (i.e., MgO, CaO, Na20, K20) in kaolin act as mineralizers, which promote the crystallization of different mineral phases and enhance strength in this temperature range [5]. Other clay minerals important in ceramics (and their chemical formulas) are Kaolinite Halloysite

Ale(Si2Os)(OH)4 A12(Si2Os)(OH)42H20

1.3 Raw Materials Pyrophyllite Montmorillonite Mica Illite

31

A12(Si2Os)2(OH) 2 All.67Nao.33Mgo.33(Si2Os)2(OH)2 A12K(Sil.sAlo.5Os)2(OH)2

A12_xMgxKl_x_y(Si1.5_yAlo.5§

These minerals have different stacking of the silica and alumina layers, as well as, incorporating metal hydrates of Na, K, Mg, A1, or Fe between the silica and alumina layers. Clay minerals can also be characterized according to their morphological features including crystal habit (i.e., plates, rods, or rolled-up platelets) stacked in either a house of cards or blocklike aggregates giving a particle-size distribution. 1.3.1.2 Talc A related natural raw material is talc, a hydrous magnesium silicate with a layer structure similar to clay minerals [14]. Talc has the chemical formula Mg3(Si2Os)2(OH)2 and is used as a raw material for making tile, dinnerware, and electronic components. Talc decomposes to give a mixture of fine-grained protoenstatite crystals, MgSiO3, in a silica matrix at 1000~ Further heating leads to crystal growth of enstatite (MgO 9SiO2), which has a high thermal expansion coefficient. A liquid is formed at 1547~ At this temperature almost all of the talc melts because its composition is not far from the eutectic composition in the MgO-SiQ system (Figure 1.18). Reasonable sintered strengths can be obtained when talc is sintered at 1000~ The high thermal expansion coefficient of enstatite is used in glaze formulations to put the glaze into compression after firing, which prevents crazing. In some cases, saponite a hydrous magnesium aluminum silicate is used in place of talc because saponite is cheaper than talc. 1.3.1.3 F e l d s p a r Feldspar is an anhydrous aluminosilicate containing K, Na, or Ca. The value of feldspar in ceramics is due to it being an inexpensive and water insoluble source of alkali. The major minerals of interest in this area are orthoclase, K(A1Si3)Os albite Na(A1Si3)Os, and anorthite Ca(A12Si2)Os. These minerals are widely abundant in nature. Feldspars are a major constituent of igneous rocks (e.g., granite contains about 60% feldspar). These minerals are used as a flux which forms a glass phase in either the ceramic body or the glaze. Figure 1.19 [15] shows the phase diagram for the ternary system K20-A1203-SiO2. In this phase diagram orthoclase (potassium feldspar) is shown to give a near eutectic composition which melts between 800 and 1000~ Feldspar provides alumina and alkali for the glass batch that is used for bottles, fiber glass, and television picture tubes. Feldspar is also the most widely used fluxing agent for ceramics and can be found in formulation for both bodies and glazes, as well as enamels.

32

Chapter 1 Ceramic Powder Processing History

FIGURE 1.19 Common compositions in the ternary system K20-A1203-SiO2 from Schairer and Bowen [15].

1.3.1.4 Silica

Silica is both abundant and widespread in the earth's crust. In addition, it is one of the purest of the abundant minerals. The most commonly used forms of natural silica are quartz, cristobalite, diatomite. Quartz is the most common form of silica, which in natural form can have very large crystals > 10 cm with very high purity. Common sand is high in quartz. Silica sand shows sharply angular fragments of quartz [14]. It is an important constituent of igneous rocks such as granite and diorite. It is also found in most metamorphic rocks, constituting a major portion of sandstone. Quartz as a pure form is often found in veins in other rocks. Diatomite consists of the skeletons of diatoms, an ancient microbe about 10 ftm in diameter, see Figure 1.20. This material is widely found in bogs throughout the world, however, large deposits of diatomite are rather rare. Diatomite is used in insulating bodies to give pores smaller than the mean free path of air and for catalysts to provide a controlled microporous diffusion pathway. Silica is a very important raw material for ceramics. Its extensive use is due to its hardness, high melting point, low cost, and ability to form glass. Silica

1.3 R a w Materials

F I G U R E 1.20

33

Silica, Si02, diatom skeleton, from a Cellite Corporation | advertisement.

is used in a ceramic body like aggregate is used in concrete: to provide a solid mass around which the glass phase can be used to bind the body together.

1.3.1.5 Wollastonite Another source of water insoluble calcium is wollastonite (CaO 9 SiO2). Wollastonite is found in either a pure form or in association with garnet or calcite and dolomite. The impure wollastonite deposits must be beneficiated by optical sorting, high intensity magnetic separation, or froth flotation. In glazes, wollastonite may be used as a substitute for calcite, which reduces the volatiles and increases the gloss and texture of the glaze. Wollastonite deposits are known for their low iron content, which gives a glaze with an excellent fired color. In enamels, wollastonite acts as a n a t u r a l frit to reduce gas evolution. It is also

34

Chapter 1 Ceramic Powder Processing History

used in ultralow ceramic insulating bodies and as an auxiliary flux in electrical insulators. 1.3.1.6 A l u m i n u m M i n e r a l s

Corundum (A1203) in its impure form is also known as emery, a common abrasive. Natural corundum has sharp highly angular particles [14]. Sillimanite minerals with the theoretical composition A12SiO5 are also a source of alumina for refractories. They include alusite, sillimanite, and kayanite, which are common metamorphic minerals found in slates and schists. Alusite is the aluminosilicate mineral which is stable at low pressure and low temperature. Sillimanite is stable at high temperature, and kayanite is stable at high pressure. Kayanite (A1203 9SiO2) is commonly used for mullite (3A1203 92SIO2) refractories and porcelain sparkplug insulators and has prismatic crystals with steplike fracture surfaces. 1.3.1.7 L i t h i u m M i n e r a l s

The important lithium minerals are spodumene (Li2A12Si4012), lepidolite (LiKA12F2Si3Og), amblygonite (Li2F2A12P2Os), and petalite (LiA1 Si4010). Spodumene has prismatic and lath-shaped crystals. In a few cases, it is used in glaze formulations in a ground form; in other cases, a lithium salt is extracted and used in a pure form in glass, glazes, and acid resistant enamels. Lithium minerals are most often used as network modifiers in glass to increase melting efficiency and lower the thermal expansion coefficient of the glass, which increases durability. 1.3.1.8 F l u o r i n e M i n e r a l s

For ceramic use, the most important mineral containing fluorine is fluorite (CaF2) which occurs in fluorspar. Natural deposits have a purity of 90-98% with silica as the principal impurity. Fluorite mineral powders have angular surfaces which result from cleavage and conchoidal fracture of the mineral [14]. Fluorspar is used in many forms of optical glass of low index of refraction and in enamels.

1.3.2 Synthetic R a w Materials Synthetic raw materials are those produced by the chemical treatment of natural raw materials or by the chemical transformation of synthetic materials. 1.3.2.1 T r a n s f o r m e d N a t u r a l R a w Materials M a g n e s i t e and Calcite Magnesite is the mineral form of magnesium carbonate which has particles composed of aggregates of wellcrystallized 1 t~m rhombohedra, many of which are in parallel align-

1.3 Raw Materials

35

ment [14]. It is often associated with calcite (CaCO3), which is a mineral with well-defined rhomboidal or prismatic crystals [14]. The mixture of magnesium and calcium carbonate is the mineral dolomite [14]. Dolomite particles are rounded agglomerates composed of rhombohedral subunits. Magnesite, calcite, and dolomite can be calcined to drive off the CO2, leaving the respective metal oxides. Magnesite is used for refractories because it has one of the highest fusion points known (2800~ and is resistive to many metal slags. Calcite is used in ceramics, as well as ground limestone in glazes, enamels, and glass. Another source of calcium in ceramics is calcined gypsum a natural hydrous calcium sulfate mineral. Calcium and magnesium are network modifiers in glass, which improve the glass's resistance to chemical attack.

Barium Minerals Barite (BaSO4)and witherite (BaCO3) are commonly used to supply barium in ceramic formulations. Purified barium carbonate, made by dissolution and reprecipitation, is used most frequently in ceramic processes and as fluxing compounds in the glazes, glass, and enamels of electronic ceramics and in heavy clay products to prevent scumming. The use of these minerals have the drawback that upon heating they give off gas, which can cause cracks. L e a d M i n e r a l s The most common lead-containing mineral is galena (PbS) followed by anglesite (PbSO4) and cerussite (PbCO3). The two latter minerals result from the weathering of galena. The occurrence of galena deposits is unexpectedly high and spread throughout the world. Galena is roasted in air to give lead oxide. Red lead (Pb304) and white lead (PbCO3 9Pb(OH)2) are commonly used as a basic flux. From the point of view of health, the lead should be reacted with silica to give insoluble PbSi205 . Lead is used in "crystal" stemwear, electrical glass for lighting and television picture tubes, and radiation-absorbing glass and in sanitary ware for enamels and glazes.

1.3.2.2 Synthetic Raw Materials--Specialty Chemicals Alumina Alumina used in ceramics today is commonly obtained via the Bayer process. The Bayer process starts with gibbsite (A1203 9 3H20), which is a common soil mineral often found in association with hematite (Fe203). This raw material is leached with sodium hydroxide at high temperature and pressure and separated from the hematite which is insoluble. The resulting sodium aluminate solution is then allowed to precipitate gibbsite. This purified gibbsite is calcined to give alumina, A1203, which contains both well-formed hexagonal crystals and rounded agglomerated masses attached to the surfaces of the hexagonal surfaces [14]. Another synthesis method for alumina is to mix the sodium aluminate solution with an acid to lower its pH and thereby

36

Chapter 1 Ceramic Powder Processing History

FIGURE 1.21 Pseudo-boehmite gel produced by precipitation of alumina by acid-base neutralization. Versal | a Kaiser Aluminum Corp. product. Photo courtesy Ron Rigge.

precipitate a microcrystalline boehmite, which forms gel agglomerates similar to those shown in Figure 1.21. Aluminas are commonly used as catalytic substrates and as silicon chip substrates, as well as additives to glass. High-alumina ceramics are used as refractories for ladle metallurgy. C h r o m i a Chromite C r 2 F e O 4 is the most commonly used chromium-containing mineral for ceramic formulations. This mineral has a spinel crystal structure, where the iron may be replaced by magnesium and aluminum. Chromite is used in ceramics largely as a refractory in the form of burned and chemically bonded bricks. For this purpose, a low-silica material is desired. When low silica is desired, chromic oxide is extracted from chromite by dissolution in acid, removal of the iron impurity by liquid-liquid extraction, and precipitation of the hydroxide, which is subsequently calcined to the oxide. Chromic oxide is used as a color additive to glazes and enamels and in ferrite production to give magnetic materials.

M a g n e s i a Magnesia (MgO) is produced from seawater or brine. In one process, the chloride brine is sprayed into a reactor where hot gasses convert the MgC12 solution to MgO and HC1. The MgO is slurried

1.3 Raw Materials

37

with water, which reacts to form Mg(OH)2. The Mg(OH)2 is washed, thickened, filtered, and then calcined to produce magnesia. Magnesia produced in this way is composed of agglomerates of well crystallized 1 ~m platelets [14]. In another process, the magnesium chloride brine is reacted with strong base to precipitate Mg(OH)2, which is washed, thickened, filtered, and then calcined to produce magnesia. With increasing calcination time and temperature, the MgO crystallites increase in size. Magnesia calcined at <900~ has a relatively high chemical reactivity and is a useful industrial chemical. Dead-burned magnesia calcined at >1400~ has a low chemical reactivity and is used exclusively in refractories because it has a high resistance to basic metallurgical slags.

Soda Ash, Caustic Soda In the Solvay process, soda ash is produced by reaction of salt (NaC1) with limestone (CaCO3) to produce soda ash and a calcium chloride salt solution. Ammonia enters the reaction process at various steps but is not consumed. Caustic soda is produced by electrolysis of NaC1 brine solutions, giving C12 gas and Na metal, which forms an amalgam with the Hg of the cathode. The amalgam is decomposed using water to form a sodium hydroxide solution, which is concentrated and precipitated to give anhydrous caustic soda. The glass container and flat glass industries use an extensive quantity of both soda ash (NaCO3) and caustic soda (NaOH) as a network modifier to decrease the working temperature of the glass. T i t a n i a Pigment grade titania is produced by the oxidation of titanium tetrachloride. Titanium tetrachloride is produced by the chlorination and selective distillation of ilmenite (FeTiO2) ore. The powder produced by the oxidation process consists of spherical particles (0.2-0.3 ~m in diameter [14]. Titania's high refractive index of 2.5 and its narrow submicron size distribution makes it a very good white pigment in glass and glaze. Zinc Oxide Zinc oxide has a specific gravity of 5.6, sublimes at 1800~ is photoconductive, insoluble in water, soluble in strong alkali solutions, and in acid solutions. It is produced by one of two processes. One process vaporizes zinc metal and burns the vapor in air to give a fine spherical zinc oxide particles [14]. In the other process, the mineral form of zinc sulphide is roasted with carbon to reduce the ore to zinc metal, which in turn vaporizes to give a gas which is burned in air. Zinc oxide powder is used in the manufacture of glass, glazes, porcelain enamels, varistors, and magnetic ferrites. In glass, glazes, and enamels, zinc oxide offers great fluxing power, reduction of expansion, prevention of cracking and crazing, and enhanced gloss and whiteness.

38

Chapter I

Ceramic Powder Processing History

Zirconia Common zirconium-containing minerals include baddeleyite (ZrO2) and zircon (ZrSiO4). Most of the zirconium-containing materials used in ceramics are extracted from zircon sands. This extraction is performed by chlorination of the silicious raw material and distillation of the mixed metal chloride gases. The separated zirconium chloride is then mixed with water and precipitated as the hydroxide or the hydroxychloride. Upon calcination, both the hydroxide and the hydroxychloride decompose to zirconium oxide. Calcined zirconia particles are composed of 0.1 t~m granules agglomerated into rounded ~20 t~m particles [14]. Zirconium produced in this way has 4% halfnia in it. To remove the halfnia, a liquid-liquid extraction must be performed on the zirconium chloride solution before precipitation. Zirconium oxide is used as an opacifier in glazes and enamels or as a refractory after fusion with lime, which acts as a stabilizer of the crystal phases present. Mixed with yittria the tetragonal phase of zirconia is stabilized, which transforms to monoclonic undergoing a 15% volume change, allowing ceramics to be transformation toughened by the presence of this phase. S i l i c o n Carbide The Acheson process is used to produce large quantities of SiC. This process carbothermically reduces SiO2 to give SiC and CO(g) in a resistance furnace. In 36 hr at 2400~ the chemical reaction is complete. The SiC produced is 1 to 5 mm crystals of a-SiC and must be ground to the desirable particle size distribution. Lowpurity silicon carbide is used in abrasive and refractory applications. High-purity silicon carbide is used for reaction bonded ceramics that require strength at high temperatures, high thermal conductivity, high thermal shock resistance, and a low thermal expansion coefficient. For the manufacture of high-performance ceramics by sintering or hot pressing, other methods of powder synthesis are used. Such processes include plasma-arc synthesis, batch reaction of silica and carbon in CO or inert gas, decomposition of polycarbosilanes, and chemical vapor decomposition. In addition, SiC whiskers are manufactured by the carburization of molten silicon. These single crystal whiskers are used in ceramic matrix composites. Other Metal C a r b i d e s A host of other metal carbides are used in ceramic formulations. These include TaC, TiC, Cr3C2, VC, Mo2C, B4C, WC, and ZrC. These metal carbide powders are produced by carbothermal reduction of the relevant metal oxide or reaction of the relevant metal with carbon in CO or an inert atmosphere. These metal carbides are used as abrasives and in high-temperature wear applications. S i l i c o n N i t r i d e Silicon nitride is a synthetic raw material which is synthesized by various high-temperature reactions between 1000

1.3 Raw Materials

39

and 1600~ The three most important methods of silicon nitride powder synthesis are 9 reacting silicon metal powder with nitrogen 9 reacting silica, nitrogen, and carbon 9 reacting chlorosilanes with a gas containing nitrogen (e.g., ammonia). Silicon nitride whiskers are also produced by variations of processing conditions in these synthesis methods. Silicon nitride is used for toolbits for cutting cast iron and other high-temperature wear parts including burner nozzels. O t h e r Metal N i t r i d e s Many other metal nitrides are used in ceramic formulations. These include A1N, TiN, VN, and BN. These metal nitride powders are produced by carbothermal reduction of the relevant metal oxide in a nitrogen-containing atmosphere or reaction of the relevant metal with a nitrogen-containing reducing atmosphere. These metal nitrides are used as abrasives and in high-temperature wear applications. B o r i d e s Metal borides form another important class of ceramic powders, which include TiB2, BC, W2B, and MoB. Borides have metallic characteristics, with high electrical conductivity and positive coefficient of electrical resistivity. They are produced either by reaction of the relevant metal with boron at a suitable temperature, usually in the range of 1100-2000~ or by reaction of a mixture of the relevant metal oxide and boron oxide with aluminium, magnesium, carbon, boron, or boron carbide followed by purification. Borides are used for electrically heated boats for aluminum evaporation and sliding electrical contacts, as well as abrasives and wear parts, including sandblast nozzels, seals, and ceramic armor plates. TiB 2 has been investigated for use as a nonconsumable replacement for the consumable graphite anode in the electrolytic reduction of alumina to aluminum metal. O t h e r R a w M a t e r i a l s This list of ceramic raw materials is by no means complete. A myriad of other raw materials are presently being used in ceramic formulations. New raw materials are being developed all the time to fulfill the need for better material properties and tailor ceramic powder properties to meet different ceramic processes. The tailoring of ceramic powders usually involves altering particle morphology or particles size distribution for use in a new ceramic powder process. Such new ceramic materials include metal silicides (e.g., NbSi2, V3Si, WSi2, and MoSi2) and metal sulphides (e.g., CdS2). These materials are synthesized in various ways by small-scale batch methods and are used for highly specific applications.

40

Chapter I

Ceramic Powder Processing History

1.4 S E L E C T I N G A R A W M A T E R I A L To select a ceramic raw material, it is necessary to know the final material properties demanded of the ceramic product and the ceramic process by which it will be fabricated. With the physical property information, it is possible to develop a list of raw materials that, after high temperature fabrication, will give the desired chemical formulation. This list of raw materials will next have to be considered in light of a particular ceramic process to be used, which may include, for example, powder mixing, slurry formation, slip casting, drying, binder burn-out, and reactive sintering. To prevent segregation in the ceramic green body, raw materials with similar particle morphology and size distributions should be used. Thus, the different raw materials necessary for the process must be compared to one another for particle size and shape compatibility. Sometimes surface chemistry compatibility is also important. The particle morphology and particle size distribution of a particular raw material depends on the method of powder synthesis. The fundamental principles of many of these powder synthesis methods will be discussed in the balance of this book to explain the reasons for the various particle morphologies and particle size distributions observed in natural and synthetic raw materials. These powder characteristics influence in what ceramic processes these powders can be used. For the simple case of the processing a single ceramic powder, what

FIGURE 1.22 Micrograph of silicon nitride powder, SN-E10, from UBE Industries, Ltd. [16] with sedigraph size distributions for various grades of this powder with 3 m2/ gm (E03), 5 m2/gm (E05), and 10 m2/gm (El0).

References

41

type of powder should be chosen? Reflecting on the "General Concepts of Ceramic Powder Processing" discussed in the introduction to this chapter, a ceramic powder with high chemical purity and a uniform size distribution and particle morphology is the best choice. Only synthetic ceramic powders provide these characteristics. A good example of such a powder is shown in Figure 1.22. Here a silicon nitride powder with a >97% s-phase purity and a narrow size distribution of spherical particles is shown. Various grades of this powder corresponding to different particle size distributions are commercially available from UBE Industries, Ltd. [16].

1.5 SUMMARY This chapter has reviewed the field of ceramic powder processing from a historical perspective. In addition, it has catalogued the various ceramic powder raw materials used to produce ceramics.

References 1. Smith, B., and Weng, W.-Go, "China--A History in Art," Gemini Smith Inc. Book. Doubleday, New York, 1972. 2. "The Genius of China," an exhibition of the archaeological finds of the People's Republic of China held by the Royal Academy, London 29 September 1973 to 23 January 1974. 3. Wood, N., New Sci. February, pp. 50-53 (1989). 4. Lemaitre, J., and Delmon, B., Am. Ceram. Soc. Bull. 59(2), 235 (1980). 5. Lemaitre, J., and Delmon, B., J. Mater. Sci. 12, 2056-2065 (1977). 6. "Treasures from the Bronze Age of China," an exhibit from the People's Republic of China, The Metropolitan Museum of Art, Ballantine Books, New York. 7. Brankston, A. D., "Early Ming Wares of Chingtechen," p. 64. Vetch and Lee Ltd., Hong Kong, 1938. 8. Valenstein, S. G., "Handbook of Chinese Ceramics." Weidenfeld & Nicholson, London, 1989. 9. Bushell, S. W., "Description of Chinese Pottery and Porcelain" (translation of "T'ao Shuo"). Oxford Univ. Press, Oxford, 1977. 10. Needham, J., "Science in Traditional China." Harvard Univ. Press, Cambridge, MA, 1981. 11. Anderson, K. J., M R S Bull., July, pp. 71-72 (1990). 12. Millot, G., La Science 20, 61-73 (1979). 13. Kingery, W. D., Bowen, H. K., and Ulhmann, D. R., "Introduction to Ceramics," 2nd ed. Wiley (Interscience), New York, 1976. 14. McCrone, W. C., and Delly, J. G., "The Particle Atlas." Ann Arbor Sci. Publ., Ann Arbor, MI. 15. Schairer, J. F., and Bowen, N. L., Am. J. Sci. 245, 199 (1947). 16. UBE Industries, Ltd., Ceramic Div., Tokyo Head Office, ARK MORI Building, 1232, Akasaka 1-chome, Minato-ku, Tokyo, 107 Japan. 17. Hobson, R. L., "Chinese Pottery and Porcelain." Dover, New York, 1976.

This Page Intentionally Left Blank

2

Ceramic Powder Characterization

2.1 O B J E C T I V E S To characterize a ceramic powder, a representative sample must be taken. Methods of sampling and their errors therefore are discussed. Powder characteristics, including shape, size, size distribution, pore size distribution, density, and specific surface area, are discussed. Emphasis is placed on particle size distribution, using log-normal distributions, because of its importance in ceramic powder processing. A quantitative method for the comparison of two particle size distributions is presented, in addition to equations describing the blending of several powders to reach a particular size distribution.

2.2 I2VTROD UCTION In all ceramic raw materials, both natural and synthetic, a powder with a particular chemical formula is the primary objective. Chemical

43

44

Chapter 2

Ceramic Powder Characterization

analysis of ceramic powders is performed by many techniques from X-ray fluorescence spectrometry and atomic absorption spectrometry to the wet chemical methods of titration. All of these techniques are subjects unto themselves, covered by other books and not be discussed further here. After satisfying this primary objective of chemical purity, other powder characteristics are important to optimize the powder to the requirements of the ceramic process in which it will be used. The diversity of these production methods calls for the choice of ceramic powders to be based on different characteristics. Beyond chemical purity, the most important characteristics for subsequent ceramic processing are particle morphology, particle size distribution, and surface chemistry. The surface chemistry of ceramic powders is extremely important for wet and dry processing methods and will be discussed in detail in a separate chapter. The characteristics of ceramic powders corresponding to their size and shape are discussed in this chapter. An excellent book that treats particle size measurement is one by Allen [1], from which many concepts used in this chapter are taken.

2.3 POWDER SAMPLING Before any characteristics of a powder can be measured it is imperative to have a representative sample of the powder. This problem can be viewed in its true magnitude by considering that several tons of material will be analyzed on the basis of less than 1 gm of material. The ultimate that may be obtained in a representative sample is called the perfect sample; the difference between this perfect sample and the bulk can be established by a statistical method, described in the following problem.

2.3.1 Sampling Accuracy P r o b l e m 2.1. D e t e r m i n e t h e S a m p l i n g E r r o r

A glaze formulation has poor color when a finely ground silica powder has a fraction of iron impurities larger than 50 ppm by weight. Let us assume that a 10 gm sample is taken from a 10,000 kg batch. In this 10 gm sample, we find 40 ppm iron particles greater than 44 ftm by sieving. The maximum sampling error, E, can be expressed as [1]

E-- +20i _ ~

(2.1)

2.3 Powder Sampling

45

where (r~ is the standard deviation intrinsic in the sample due to the sampling of 10 gm from a 10,000 kg batch and P is the weight fraction of material greater than 44/xm measured in the sample. The standard deviation due to sampling is determined by [1]

~ =[

W-~

( s)112

. (Pw~ + ( 1 - P)w2) . 1-- Wbb

(2.2)

where Ws and Wb are the weight of the sample and the bulk, respectively; w~ and w2 are the weights of individual grains, the metal impurity particles 1 assumed to be the density of iron with a diameter of 44 txm and 2 assumed to be silica 0.5 txm in diameter. In this example, P = 40 x 10 -6, Ws = 10 -2 kg, Wb = 10,000 kg, Wl = 3.5 x 10 -1~ kg, and w2 = 1.7 x 10 - ~ kg, giving a value of cri = 2.37 x 10 -7 and an error of 1.19%. In addition to the error caused by using a small sample, we have the error in our analytical technique. For our example, when multiple iron impurity measurements were made on the same sample, the standard deviation of the impurities was ___4 ppm with a m e a n value of 40 ppm as before. This 10% analysis error will have an effect on the total error since the total standard deviation o't is O.t

=

(0.2 + ~'nJ __2,1/2

(2.3)

where (rn is the standard deviation of the analysis technique (i.e., 4 • 10 -s for this example). Accounting for the error in our analysis in this problem, we find a total error (using equation (2.1) with (rt replacing (r~) is 20.0%. The analysis error (rn is, therefore, the most significant error in this example.

2.3.2 Two-Component Sampling Accuracy Any powder can be considered to be made up of two components, the fraction above and below a certain size and assumptions made as to the weights of the individual grains in each of the two components. Equation (2.2) may then be used to determine the sampling accuracy of a single powder. Furthermore, if the particles are counted instead of weighed, a more general equation is applicable [1]:

(2.4) where p is the fraction of particles greater t h a n a certain size, Ns is the number of particles counted, and Nb is the number of paraticles in the bulk. (This equation is also used to determine the accuracy of public opinion poles.) It is obvious from the preceding equations, that the larger is the sample, the smaller is the sampling standard deviation.

46

Chapter 2

Ceramic Powder Characterization

2.3.3 Sampling Methods Unfortunately, the size of a practical analytical sample is often minuscule compared to the bulk material being sampled and even the analytical sample is subject to a large degree of sampling variation. There are two ways to reduce this variation. One way is to make up a large laboratory sample from many increments of the bulk and divide the laboratory sample to produce an analytical sample. This laboratory sample is often retained for replicate analyses to determine the standard deviation of the analytical method. The second way to reduce sampling variation is to take a number of replicate samples and mix them together to make an analytical sample. A representative sample is difficult to obtain when one considers that 1. Particles encounter many types of segregation that will bias the sample. 2. Many different conditions are to be sampled. Frequently, one must sample a continuous stream, batches, bags, heaps, hoppers, or trucks. The most important segregation-causing property is particle size, and this problem is exacerbated with flowing material. In a heap, the fine particles tend to concentrate at the center of the heap as shown in Figure 2.1. In a vibrating container, coarse material tends to concentrate at the surface, even if the coarse material is denser than the fine material. This problem was observed to the chagrin of a farmer who ordered a railroad car of wheat seed. When the car arrived it looked from the top surface as if he had received a carload of beans. The 1% impurity, beans, had segregated to the top of the load with the gentle railroad vibrations in shipping. An understanding of these tendencies of particles to segregate prevents careless sampling practices.

2.3.4 Golden Rules of Sampling [1] For the many possible situations in which sampling has to be performed, two principles can be given that will decrease segregation: Rule 1. A powder should be sampled when in motion. Rule 2. The whole stream of powder should be taken for many short increments of time in preference to part of the stream being taken for the whole time. In many possible situations a sample has to be obtained under conditions that often necessitate the use of inferior sampling techniques, however, observance of these Golden Rules will lead to the best sampling procedure.

2.3 Powder Sampling

47

FIGURE 2.1 Cross-section of a pile of binary powder showing demixing of larger black particles ( - 1 ram) to the periphery and finer white particles (-0.2 ram) to the center of the pile. Taken from Figure 1.11 in Allen [1].

For bag sampling it is best to select the bags at random and repeatedly use a splitter to homogenize the sample, taking a portion of one of the splits as the sample is in motion. For heap or hopper sampling it is important to note that the cross-section is likely to contain large degrees of segregation with fine particles concentrated near the axis and coarse particles concentrated at the periphery of the heap. Since this segregation is a common occurrence, static sampling is not suggested unless all the particles are <44 ~m, where segregation is greatly redued. Once a representative sample has been obtained, the particle size distribution and other powder characteristics can be determined.

48 2.4

Chapter 2

PARTICLE

Ceramic Powder Characterization

SIZE

2.4.1 S t a t i s t i c a l D i a m e t e r s Imagine an irregularly shaped particle like that shown in Figure 2.2. For this particle, an infinite number of statistical diameters radiating from the center of gravity of the particle exist. The average unrolled diameter of the projected contour is the integrated average defined by [2]

-dR

=

f~~ r -dO -.

(2.5)

This calculation is tedious to perform for each particle of a distribution even with the advent of computer aided image analysis. Faster techniques simply measure a linear dimension parallel to some fixed direction and assume that the particles are oriented randomly so that these measurements average out when a sufficiently large population has been sized. Several linear dimensions typically are used. Feret's diameter is the mean value of the distance between pairs of parallel tangents to the projected outline. Martin's diameter is the mean cord length of the projected outline of the particle. In addition, the maximum

200

.150

n,.

E(R)

(b)

,

0

,..........

I

~"

50

, ......

~R

0

2x

FIGURE 2.2 Multitude of particle diameters: (a) definition of unrolled radius R; (b) unrolled curve. Taken from Figure 4.1 in Allen [1].

2.4 Particle Size Feret's

49

Diameter

P a r t i c l e Outline Martin's Diameter Projected Area Diameter Maximum H o r i z o n t a l I n t e r c e p t

F I G U R E 2.3

Different particle diameters. Taken from Stockham and Fortman [3].

and minimum horizontal intercepts are also used as linear dimensions. All of these linear dimensions are shown in Figure 2.3 [3]. Another commonly used measure of a particle's diameter is the projected area diameter, which is the diameter of a circle having the same area as that projected by the particle. Other particle size definitions are given in Table 2.1. A commonly used size dependent property is the equivalent spherical diameter. The equivalent spherical diameter is the diameter of a sphere with the same volume as the particle. For a cube this sphere would have a diameter 1.24 times the edge length of the cube. Another common equivalent spherical diameter is the Stokes diameter. The Stokes diameter is the diameter of a sphere that has the same terminal settling velocity as an irregular particle. (Note: Settling has to be under laminar flow [i.e., Reynolds number less than 0.2] in both cases and the density of both the particle and the sphere are assumed to be the same). The sieve diameter is the length of a side of the minimum square aperture through which a particle will pass. An irregularly shaped particle will pass through the smallest possible mesh only if it is presented in the optimum orientation. Sieving times for elongated particles approach infinity because only two orientations will allow the particle to pass the smallest sieve. Long sieving times can cause problems since some amount of particle breakage will inevitably result during this time. A common solution to this dilemma is to sieve for a specific period of time with all samples of the same type and live with the error and breakage that results. For distributions of irregular-shaped particles, the size distribution measured is dependent on the method of measurement. For this reason, the sizing method employed should duplicate the property of interest (e.g., for pigments, the projected area diameter is of interest; for catalyst substrates, the surface to volume diameter is of interest, etc.). There are many different methods of measuring size distribution. Table 2.2 gives the most common methods utilized and the type(s) of size dimension(s) measured. These methods of size measurement have to be coupled with either some counting or weighing method to determine the

T A B L E 2.1

Symbol

Definitions of Particle Size a

Name

dv

Volume d i a m e t e r

ds

Surface d i a m e t e r

dV$

Surface volume d i a m e t e r

dd

Drag diameter

df

Free-falling d i a m e t e r

dstk

Stokes's d i a m e t e r

Formula

Definition 77

Diameter of a sphere having the same volume as the particle Diameter of a sphere having the same surface as the particle

V=gd3v

Diameter of a sphere having the same external surface to volume ratio as a sphere Diameter of a sphere having the same resistance to motion as the particle in a fluid of the same viscosity and the same velocity (d d approximates to ds when R is small) Diameter of a sphere having the same density and the same free-falling speed as the particle in a fluid of the same density and viscosity

dsv =

The free-falling diameter of a particle in the l a m i n a r flow region (Re < 0.2)

S = 7rd2s de~

FD = CDApwV2/2, where COA = f(dd), FD = 37r dd~Y Re < 0.2

de~ d2 = ~d

da

Projected area diameter

d~

Projected area diameter

de

Perimeter diameter

dA

Sieve diameter

dR

Feret's diameter

dm

Martin's diameter

dur

Unrolled diameter

a

Diameter of a circle having the same area as the projected area of the particle resting in a stable position Diameter of a circle having the same area as the projected area of the particle in random orientation Diameter of a circle having the same perimeter as the projected outline of the particle The width of the minimum square aperture through which the particle will pass The mean value of the distance between pairs of parallel tangents to the projected outline of the particle The mean chord length of the projected outline of the particle The mean chord length through the center of gravity of the particle

77 A=~d2a

Mean value for all possible orientations d~ = ds for convex particles

dur = E(dR ) = _1 f ~ dRdO R

Taken from Table 4.1 in T. Allen, Particle Size Measurement (London: Chapman and Hall, 1981).

7]"

52

Chapter 2

Ceramic Powder Characterization

TABLE 2.2 Methods of Size Distribution Analysis Method

Dimension measured

Gas absorption Coulter counter (Electro-zone) Inertial separation Light scattering Photon correlation spectroscopy Permeability

Projected Area diameter Feret's diameter Martin's diameter Perimeter diameter Unrolled diameter Sieve diameter Stokes's diameter Drag diameter Free-fall diameter Surface to volume diameter Volume diameter Stokes's diameter Scattering diameter Sixth moment diameter Surface to volume diameter

Microscopy

Sieve Settling

amount of particles in each size range. In the case of microscopy, the n u m b e r of particles is counted. With sieving, the weight of particles in a size range is measured. With settling, either the optical density of the suspension of particles in the size range is measured or the percent mass in the size range is measured by X-ray absorption. With Coulter electro-zone counting, the n u m b e r of particles in a particular volume range is counted. All other techniques of sizing do not have separate m e a s u r e m e n t s of either the weight and size or number and size of particles in a particular size range. With light scattering, the number density and the size distribution are best-fit parameters for a series of scattered light intensity m e a s u r e m e n t s at different angles. With gas adsorption and gas permeability, the average surface to volume diameter of the particles is determined.

2.4.2 Mean P a r t i c l e Size The mean particle size measured by these various methods is different depending on the method used. The mathematical definition of the various mean sizes, with their mathematical definitions, follow [1]. Number, length mean diameter or arithimatic diameter: di -~lQi 9

_ ZNi"

dgL = dav -

(2.6)

Geometric m e a n diameter by number: deN = aloglo

~;-~//

.

(2.7)

2.4 Particle Size

53

Number, surface mean diameter: ( X N i ' d ~ ) 112 d n s = \ ~]QI

(2.8)

Number, volume mean diameter: dNy= \

ZNi

] "

(2.9)

Length, surface mean diameter: ZNi " d~ dsn = ZNi " d i "

(2.10)

Length, volume mean diameter: ( X N i ' d ~ ~/2 dvL = \ E n i ~ / "

(2.11)

Surface, volume mean diameter: dvs = ENi " d 2. ENi d i

(2.12)

Volume, moment mean diameter and weight (or mass), moment mean diameter: ENi " d4 dv = dw = dM = ENi " d~"

(2.13)

P r o b l e m 2.2. M e a n P a r t i c l e S i z e

An example of microscopic size analysis is given in Table 2.3 (taken from Stockham and Fortman [3]). Here the number of particles found in various particle size ranges is measured. From this data, determine various mean sizes, including the arithmetic mean size, dav ; the geometric mean by number, dgN; the surface mean, dNs ; the volume mean, dNy; the volume to surface mean, dys; and the weight or mass mean, dw. For the various mean sizes, the mathematical definitions are given in equations (2.6) to (2.12). The relevant columns of sums are given also in Table 2.1, allowing the calculation of various mean sizes as follows:

day -

12248 1000

= alo

= 12.2/zm

10(

)

= 100 m

T A B L E 2.3

Calculation of Various Average Diameters a ,,,,,,

Particle size interval (Ixm)

1.0-1.4 1.4-2.0 2.0-2.8 2.8-4.0 4.0-5.6 5.6-8.0 8.0-12.0 12.0-16.0 16.0-22.0 22.0-30.0 30.0-42.0 42.0-60.0 60.0-84.0 Totals

Middle size (txm), d

1.2 1.7 2.4 3.2 4.8 6.8 10.0 14.0 19.0 26.0 36.0 51.0 72.0

Frequency of occurrence n

Count frequency of n (%)

2 5 14 60 100 190 250 160 110 70 28 10 1 1000

0.2 0.7 2.1 8.1 18.1 37.1 62.1 78.1 89.1 96.1 98.9 99.9 100.0

a Taken from Stockham, J. D. and Fortman, E. G. [3].

nd

2 8 34 192 480 1292 2500 2240 2090 1820 1008 510 72 12248

n log d

0.16 1.15 5.32 30.31 68.12 158.18 250.00 183.38 140.66 99.04 43.58 17.08 1.86 998.84

nd 2

nd a

3 14 81 614 2304 8785 25000 31360 39710 47320 36288 26010 5184 222673

3 25 194 1966 11059 59742 250000 429040 754490 1230320 1306368 1326510 373248 5752965

Mass frequency of nd 3 (%)

n

0.1 0.3 1.3 5.6 13.2 26.3 47.7 70.4 93.5 100.0

nd t

4 42 464 6291 53084 406246 2500000 6146560 14335310 31988320 47029248 67652010 26872856 196991436

55

2.4 Particle Size

(222673~ 1/2

dNS = \ ~ - ~ ]

= 14.9/~m

d y y = (5 750-~065)1/3

= 17.9/~m

5752965

dvs = 222673

dw =

196991436 5752965

= 25.4/xm

= 34.2/~m.

As you can see from this set of data, m a n y different m e a n sizes can be calculated. Therefore, reporting a mean size without reference to the statistical method used is utterly useless.

2.4.3 Size Distribution Accuracy With sizing techniques t h a t count the particles, it is desirable to know the accuracy of the size distribution after counting a given n u m b e r of particles. Figure 2.4 gives the number of particles to be counted to give a specific accuracy [3]. For a less t h a n 1% error in a given size

% w g t in Size Interval

z~

2.0-

2%

5%

10% 15% 20%

!.._

0

I,,, I,,

!11 "0 0 .,., 1.0 0 (I) e,J X III

0

1

10

100

,

10 ,000

Number of Particles C o u n t e d in S i z e Range

FIGURE 2.4 Expectederror in the size distribution based on the number of particles counted in the size range and the weight percent in the size interval. The curves in this figure have been calculated with the equation Ei, j = Wj/V~ii.

56

Chapter 2 CeramicPowder Characterization

range where 20% of the mass appears, more than 400 particles must be sized. For a less than 0.1% error in a given size range where 20% of the mass appears, more than 40,000 particles must be sized. For size intervals where fewer numbers of particles were counted, the expected error increases. This sizing error should be kept below the sampling error (see Section 2.1) so that the total error is not dominated by the analysis error. P r o b l e m 2.3. D e t e r m i n e t h e E r r o r of t h e P a r t i c l e Size D i s t r i b u t i o n G i v e n in T a b l e 2.3

For each size interval, we must first determine the weight percent. Then using the number of particles counted in that size range, we can determine the error for that size range. For the size interval 16-22 t~m, 110 particles were counted out of 1000. The weight fraction, Wj, of particles in this size interval is 13%. Using Figure 2.4 [3] with values of 13% for the weight percentage in the size interval and 110 for the number of particles counted, Nj, we find an error, Ej, of -1.2% for the 16-22 ftm size interval. The curves in this figure have been calculated with the equation Ej

= ~Nj"

(2.14a)

This process is repeated for each size interval to complete the analysis. The largest error in this size distribution occurs in the 60-84 t~m size interval, where one particle accounted for 6.5% of the mass. This is the largest error (~5%) for any size category. The total error for the size distribution, ET, c a n be calculated by a sum of the products of the error for each size interval Ei with the weight percent in each size interval, Wi" N

ET = ~ Ei Wi. i=l

(2.14b)

For this problem the total error was calculated to be 2%.

2.5 PARTICLE MORPHOLOGY The size of a cubic particle is uniquely defined by its edge length. The size of a spherical particle is uniquely defined by its diameter. Other regular shapes have equally appropriate dimensions. With some regular particles more than one dimension is necessary to specify the geometry of the particle as, for example, a cylinder, which has a diameter and a length. With irregularly shaped particles, many dimensions

2.5 Particle Morphology

57

are required to completely specify the shape of the particle. As we have seen in Figure 2.2, a single particle can have an infinite number of statistical diameters, hence these diameters are meaningful only when a sufficient number of particles have been measured to give average statistical diameters in each size range. The numerical relation between the various diameters of a single particle depends on the particle's shape.

2.5.1 Shape Factors A dimensionless combination of different average diameters of a distribution of particles is called a shape factor. Shape factors have three functions: 1. Proportionality factors between different particle size determination methods (e.g., the average size determined by surface area is proportional to the average size determined by volume and the proportionality constant is a shape factor). 2. Conversion factors for expressing results in terms of an "equivalent sphere." 3. Transformations of the square and cube of the measured particle diameter into the particle surface area and particle volume, respectively. Shape factors can be applied to an individual particle or to a distribution of particles. They are not sensitive to the size distribution discussed later in this chapter. For a distribution of particles, the shape factor measured is an average value. Shape factors are always understood relative to two methods of particle size determination. The volume shape factor ~v is defined as [4]

Yt Yt

_

(2.15a)

Nt

%

ZNid~ ZNid~ Nt where Vt is the total volume of the sample, Nt is the total number of particles in the sample, Ni is the number of particles with a diameter di. For a population of spheres av is 7r/6 or 0.52, whereas for a population of cubes av is 1.0. For all other shapes, values greater than 0.52 are observed when the major axis of the particle is used for di. The surface shape factor, a~, is defined as [4] St St as

~ , N i d i2

_

N t ....... 2 ~N id i ...........

Nt

(2.15b)

58

Chapter 2

Ceramic Powder Characterization

where St is the total surface area of the sample. For a population of spheres as is 7r and for a population of cubes as is 6.0. Another useful shape parameter is the ratio of as to av [4]: St

% _ ENid~ av Vt

(2.15c)

EN~d~ This ratio is obtained easily from experimental gas adsorption data for the surface area per unit volume, Sv = St/Vt (which equals 6/dv~ for a sphere, where dvs is the volume to surface mean diameter defined by equation (2.11)). Because spheres have a minimum surface area per unit volume, all values of %/% will be larger than 6 when the major axis is used for di. When the minor axis is used for di, values of %/% can be both smaller and larger than 6. Table 2.4 [1] gives calculated shape factors for various geometric shapes. Their values cover a broad range. A particular value of the shape factor does not specify a particular geometry because many geometries can give the same shape factor. The dynamic shape factor, K, is defined as the resistance of a particle to motion divided by the resistance of a sphere to motion when the particle and the "equivalent sphere" have the same volume. When the particle is settling under laminar flow, K is given by [4] (dNv~2 K= \-~t ]

(2.16)

where dNy is the mean volume diameter (equation (2.7)) and dst is the Stokes diameter. The dynamic shape factor, K, is equal to the reciprocal TABLE 2.4

Calculated Values of Shape Coefficients a

,,,,,

Form Sphere Spheroid

Ellipsoid Cylinder

Proportions

1 : 1:2 1:2:2 1:1:4 1:4:4 1:2 : 4 height height height height height

= = = = =

diameter 2 diameters 4 diameters 89 diameter 88 diameter

Linear dimension used as D

Ols

OLv

OLs/Olv

Diameter Minor axis Minor axis Minor axis Minor axis Shortest axis Diameter Diameter Diameter Diameter Diameter

3.14 5.37 8.67 10.13 28.50 15.86 4.71 7.85 14.14 3.14 2.36

0.52 1.05 2.09 2.09. 8.38. 4.19 0.79 1.57 3.14 0.39 0.20

6.00 5.13 4.14 4.83 3.40 3.79 6.00 5.00 4.50 8.00 12.00

a Taken from Table 4.6 in T. Allen, Particle Size Measurement (London: Chapman and Hall, 1981).

2.5 Particle Morphology

59

of the square root of the sphericity, Cw, one of the earliest shape factors used: K = Cwu2

(2.17)

Sphericity [5], Cw, is defined by Cw =

Surface area of a sphere having the same volume as the particle Surface area of the particle [dyv~2 (2.18)

The volume to surface mean diameter is given by (2.19)

dvs - d~vs - CwdNv

which is directly related to sphericity. At a low Reynolds number, the drag diameter equals the surface diameter of convex particles. For this case, the Stokes diameter, defined as the free-fall diameter in the laminar flow region, is related to sphericity as follows: {d~rv~ u2 "--

u4n W

LbNV

o

(2.20)

Shape factors do not provide specific information on the geometry of the particles but simply give a number for comparison purposes. To provide particle geometry information, a model geometry (i.e., cube, tetrahedron, sphere, etc.) must be selected and the particles of the population compared to that geometry to see how closely the particles correspond to it. This approach, pioneered by Heywood [6], can be used only when three mutually perpendicular dimensions of the particle can be determined. This amount of information on each particle is typically not available, which prevents the common use of this technique to give detailed particle shape information.

2.5.2 Shape Analysis Shape regeneration can be performed by Fourier transformation techniques. First the particle's center of gravity is found from its outline. Then a polar coordinate system is set up. Schwarcz and Shane [7] traced the outline of the particle and generated a plot of R n v e r s u s On, where n is the number of equal angles utilized. Meloy [8-11] used a Fourier transform to generate the Fourier coefficients, An, associated

60

Chapter 2

Ceramic P o w d e r Characterization

with the angles 0 n and found that particles have signatures that follow the empirical equation A n -

Aon-s

(2.21)

A plot of log A n v e r s u s log n gives a straight line with a slope o f - s , which is the shape signature of the particle. Generalizations of this method to three-dimensional particles are also possible. Particle shapes can be analyzed and reproduced using this Fourier transform method.

2.5.3 Fractal Shapes When a particle's surface is not uniformly smooth like that of a sphere but has a texture like that of broccoli or cauliflower, the particle is said to be fractal [12]. With the two-dimensional projection of a fractal particle, we find that the perimeter of the particle increases without limit as the size of the ruler used as a measure decreases in length. The circumference, C, estimated with a ruler of size x is proportional to C x OL X 1-D

(2.22)

where D is the fractal dimension of the particle. An example of a computer generated fractal particle [13] is shown in Figure 2.5. To determine the fractal dimension of a particle, a series of circles are drawn with different centers and different radii. A plot of the average number of the particle subunits within the circle is plotted versus the radius of the circle on log-log paper and the slope is the fractal dimension, D. Fractal shapes are traditionally produced by agglomeration

Two-dimensional computer generated fractal aggregate. Taken from Hurd [13], copyright 9 1986 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

F I G U R E 2.5

2.5 Particle Morphology

61

FIGURE 2.6 Electron micrograph of a silica cluster. Bar = 5000 /~. Taken from Schaefer, D. W., Martin, J. E., Wiltzius, P., and Cannell, D. S., Phys. Rev. Lett. 52(26), 2371-2374 (1984).

processes occurring in precipitation of sol-gel particles and in flame, plasma, and laser synthesis of ceramic powders. A classic example of a silica sol aggregate is shown in Figure 2.6 with a fractal dimension of 2.1. Flame synthesized carbon black also has a fractal geometry with fractal dimensions between 1.7 and 2.5. The properties of fractal ceramic powders depend on their fractal dimensions. The density, p, of a fractal particle depends on its radius, R, and the fractal dimension: po~R D-3.

(2.23)

The surface area per unit mass, A, of a fractal particle depends on the radius of the individual particles, ro, the radius of the aggregate, R, and the fractal dimension: Aa(ro) -1 9R D.

(2.24)

Fractal particles modify external phenomena in the surrounding space (e.g., a conducting fractal modifies the electric field around it, an absorbing fractal modifies the concentration of the diffusing species around it, and a fractal immersed in a fluid modifies fluid flow around

62

Chapter 2 CeramicPowder Characterization

it). The basic laws governing these external processes can be expressed in terms of the fractal dimension, D [14]. Another interesting property of fractals is the intersection probability between two fractals. The number of intersections, M~2, between two fractal particles 1 and 2, both of size R, are placed independently in the same region of space is M120lR D1+D2-3

(2.25)

where D~ and D2 are the fractal dimension of each particle, respectively. If the power of R is negative, the probability of intersection decreases as the size of the fractal, R, increases. For this reason, two fractals with the same fractal dimension will have a low intersection probability if D1 = D2 > 1.5. As a result, fractals with fractal dimensions greater than 1.5 act as individual spheres in dilute solution. In concentrated solution, fractals with a fractal dimension less than 1.5 can be forced to interpenetrate strongly, unlike hard spheres, during settling and dewatering of fractal sol-gel ceramic powders.

2.5.4 Internal Porosity Porosity within a particle is a manifestation of the shape of a particle. Fractal particles will have internal porosity as a result of their shapes. Fractal particles with low fractal dimensions (i.e., <2.0) will have a broad pore size distribution, where the largest pore approaches the size of the aggregate. Fractal particles with large fractal dimensions (i.e., >2.0) will have narrower pore size distributions with most of the porosity occurring at a size much smaller than that of the aggregate. Calcination of metal salt particles or metal hydroxides to produce oxides is another common method to produce internal porosity. In the gas evolution that takes place in transformation to the oxide, pores are opened up in the particle structure. The opening of pores in a hydrous alumina powder can increase its surface area from 0.5 m2/gm (its external area) to 450 m2/gm (its internal pore area). Two methods are used to measure the pore size distribution in a powder: mercury porosimetry and adsorption-desorption hysteresis. Both methods utilize the same principle: capillary rise. A nonwetting liquid requires an excess pressure to rise in a narrow capillary. The pressure difference across the interface is given by the Young and Laplace equation [15]. AP = - 2T cos 0

(2.26)

2.6 Powder Density

63

where T is the surface tension of the liquid, r is the capillary radius and 0 is the contact angle between the liquid and the capillary walls and is always measured within the liquid. For contact angles greater than 90 ~, the pressure difference is negative and the level of the meniscus in the capillary will be lower than the level in a surrounding liquid reservoir. The pressure difference is the pressure required to bring the level of the liquid in the capillary up to the level in the surrounding reservoir. Therefore, the absolute pressure required to force a nonwetting liquid into a pore of radius r is AP when we start with the powder in an evacuated state. The volume of liquid entering the pores is measured separately at each applied pressure giving a pore volume versus radius plot. One of the disadvantages of applying equation (2.26) is that pores are not cylindrical in cross-section. Therefore, pore size distribution results can only be comparative. In addition, "ink bottle" pores will give different sizes depending on whether the size is measured upon filling or emptying. Porosity can be an advantage or a disadvantage in ceramic powders, depending upon the processing and final application. Tailoring of the pore size distribution is very important for catalytic substrates, because access to the catalytic sites depends on these diffusional pathways.

2.6 POWDER D E N S I T Y The weight of a powder divided by the volume it occupies is its bulk density. The bulk density of a powder is often much less than the density of the individual grains that make up the powder. The true density of the individual grain is determined by pyncnometry. In pyncnometry a given mass of powder is placed in a vessel with a calibrated volume. Then a fluid is used to fill the vessel. The volume of the fluid filling the vessel and the actual volume of the powder are measured by the difference between the volume of the empty calibrated vessel and the actual volume of the fluid used to fill the vessel containing the powder. This volume combined with the powder weight allows the true density to be calculated. When the fluid is a gas, the actual volume of the gas used to fill the void space of the vessel to a given pressure is measured directly. Inert gases are used for this measurement to minimize adsorption. When the fluid is a liquid, the mass of the added liquid is measured and the liquid density at a given temperature is used to calculate the volume of the liquid filling the void space in the vessel. Care must be exercised in the selection of the liquid because nonwetting liquids will not enter the pores of the powder particles.

64

Chapter 2 Ceramic Powder Characterization

2.7 SURFACE AREA The specific surface area of a ceramic powder can be measured by gas adsorption. Gas adsorption processes may be classified as physical or chemical, depending on the nature of atomic forces involved. Chemical adsorption (e.g., H20 and A120a) is caused by chemical reaction at the surface. Physical adsorption (e.g., N2 on A12Oa) is caused by molecular interaction forces and is important only at a temperature below the critical temperature of the gas. With physical adsorption the heat of adsorption is on the same order of magnitude as that for liquefaction of the gas. Because the adsorption forces are weak and similar to liquefaction, the capillarity of the pore structure effects the adsorbed amount. The quantity of gas adsorbed in the monolayer allows the calculation of the specific surface area. The monolayer capacity (Vm) must be determined when a second layer is forming before the first layer is complete. Theories to describe the adsorption process are based on simplified models of gas adsorption and of the solid surface and pore structure.

2.7.1 First Layer Adsorption--Langmuir Adsorption The first theoretical equation relating the quantity of gas adsorbed to the equilibrium pressure of the gas was proposed by Langmuir [16]. Using the kinetic theory of gases, Langmuir balanced the rates of condensation and evaporation of the gas molecules at the solid surface, giving the volume adsorbed, V, as a function of the gas pressure, P:

V bP = 0=~ Vm 1 +bP"

(2.27)

In this equation, Vm is the monolayer capacity, 0 is the fractional surface coverage, and b is defined as

b = X/27rmkT where ro is the molecular vibration time, m is the molecular mass, k is Boltzmann's constant, T is temperature, and Q is the energy evolved when a gas is adsorbed. The Langmuir equation is usually written in the form P

1 -

t

P

VbVmVm"

(2.29)

A plot of P/V versus P is linear and gives the monolayer capacity Vm as the reciprocal of the slope. To relate this to the specific surface

2.7 Surface Area

65

area, it is necessary to know the area occupied by a molecule, (r, and use the following equation: S - NAv~

(2.30)

where NAV is Advogadro's number and 17 is the molar volume = 22,410 cm3/mole. The Langmuir analysis is limited to monolayer adsorption for both physical and chemical adsorption. This analysis can sometimes be applied to the adsorption of solutes from solvents.

2.7.2 Multilayer Adsorption--BET Adsorption An extension of the Langmuir approach to multilayers adsorption was made by Brunauer, Emmett, and Teller, BET [17]. They assumed that the Langmuir equation applies to each layer. The heat of adsorption of the first layer was assumed to have a special value, but for the subsequent layers, the heat of adsorption was assumed to be equal to the heat of condensation of the gas. The volume adsorbed is then a summation of the adsorbed volumes of each layer. Upon evaluation of the summation, the BET equation results: V V~

-

cP (Po - P)[1 + (c - 1)(P/Po)]

(2.31)

where P0 is the saturation pressure at the temperature of adsorption and c is defined as c - ~albi exp[(Q1 - Qv)/RT]

(2.32)

where ai and bi are the condensation and evaporation rate constants, respectively, for the ith layer, Q~ is the heat of adsorption of the first layer, and Qv is the heat of vaporization. The BET equation is often written as P _ 1 c-1 V(Po - P) Vmc + Vmc (P/Po).

(2.33)

A plot of P/[V(Po - P)] versus P/Po gives a straight line from which the monolayer capacity, V~, and the constant c can be determined. The specific surface area of a powder can be determined using equation (2.30) and the adsorbed area of a gas molecule. Nitrogen with an area of 16.2/~2 at its boiling point, 77 K, is the gas of choice for powder surface area analysis because this gas has a high c value. For low surface area powders, krypton gas is used at its boiling point of 120 K,

66

Chapter 2

Ceramic Powder Characterization

because it has a higher heat of vaporization t h a n nitrogen. Theoretical problems are associated with the interpretation of krypton adsorption isotherms due to a lack of knowledge of (r for krypton on different surfaces and ai and bi. Once the average shape of the particles in a powder has been established by one of several means and its ramifications on shape factors, fractal dimensions, surface area, and porosity are determined, the distribution of particle sizes is the next piece of information necessary to characterize the powder.

2.8 P A R T I C L E

SIZE DISTRIBUTIONS

The microscopic size data for a population of particles shown in Table 2.3 can be plotted as a histogram like that shown in Figure 2.7, where the number of particles counted in a size range is proportional to the height of the rectangle. Another plotting method is to construct the rectangles so that the area of the rectangle is proportional to the number of particles. For this histogram, the total area is proportional to the total number of particles. If a sufficient number of particles is counted and sized, a smooth curve can be plotted when a large number of intervals are used for size classification. Often, a cumulative distribu0.50

0.40 c 0 (D 0 . 3 0 13..

-> 9 0.20

a-a

O.lO

o.oo

1.4

. ~ r--!

2.8

F;. s'.6

~-] r--1

,

4'2 6b 84

Size ( m i c r o n s )

FIGURE 2.7 Size distribution histogram: % mass versus size.

2.8 Particle Size Distributions

6~

%

100 90

8O c

|

o

k--

70

D.

60

>

50

==m

ID

lz

30

20 10 0

1.4

2

2.8

4

5.6

8

Size

FIGURE 2.8

12

16

22

30

42

60

84

(microns)

Cumulative size distribution: Relative % less than versus size.

tion gives a more convenient display of the data. Two types of cumulative distribution are possible: a percent greater than distribution and a percent less than distribution. The percent greater than distribution is a summation of all the particles larger than a given size plotted as a function of size. The percent less than distribution is a summation of all the particles smaller than a given size plotted as a function of size. The cumulative distribution of the microscopic size data given in Table 2.3 is shown in Figure 2.8. The cumulative distribution has the advantage that the mean size and the percentage between any two sizes can be easily read off the plot. If the range of particle size is very large for either the histogram or the cumulative plot, a logarithmic size scale can be used. Cumulative distributions can be fitted by a linear function if the data fit a suitable mathematical function. This curve fitting gives no insight into the fundamental physics by which the particle size distribution was produced. Three common functions are used to linearize the cumulative distribution: the normal distribution function, the log-normal distribution function, and the Rosin-Rammler distribution function. By far athe most commonly used is the log-normal distribution function.

68

Chapter 2 Ceramic Powder Characterization

50,

402; 30-

.-%

/

20-

100

20

0

l'U

40

60

80

(microns)

Diameter

FIGURE 2.9 Normal distribution histogram: % mass versus size.

2.8.1 N o r m a l D i s t r i b u t i o n This type of distribution occurs when the measured value of the size is determined by a large number of small additive effects, each of which may or may not operate. This distribution gives the well-known bellshaped curve shown in Figure 2.9. It might be expected that many distributions follow this function but only narrow size ranges of classified material follow this distribution. Real distributions are skewed to larger sizes. The equation representing a normal distribution is [18] f(x)

1 (r~exp_

[-(x_

2~r2

j

(2.34)

where r is the standard deviation with units of length and E is the arithmetic mean size giving f ( x ) units of length -1. The distribution is normalized so that

f~

f(x)dx=

1.

(2.35)

oc

The cumulative distribution is the integral of the distribution function. F(x) =

f

X

f(x) dx.

(2.36)

-- o~

Tables of values of the normal distribution function, f(Z), and the cumulative distribution function, F(Z), are given in the appendix of

2.8 Particle Size Distributions 90

,

,,,

69

,,,,,,

80 .~60 9 ~o

~5 20

0

o.ol

FIGURE 2.10

~b io 3'o ~ ~;o6"07b 8'0 9'o 9's 9'8 s'9 9~8 Cumulative Percent M A S S (%)

. . . . . . . . . . . . . . . . . . . . . . . .

,,

Linearized normal distribution: Cumulative

99.99

% l e s s t h a n v e r s u s size.

this book. In these tables Z = (x - ~)/(r. A plot of the cumulative normal distribution is linear on normal probability paper, like that shown in Figure 2.10. A size distribution that fits the normal distribution equation can be represented by two parameters, the arithmetic mean size, ~, and the standard deviation, or. The mean size, ~, is the size at 50% of the distribution, also written as xs0. The standard deviation is easily obtained from the cumulative distribution as O" -- X84.13 -- X50 :

X50 -- X15.87.

(2.37)

The normal distribution has the disadvantage t h a t finite fractions of the distribution occur at sizes less than zero, which is physically unrealistic.

2.8.2 Log.Normal Distribution The log-normal distribution is frequently observed in ceramic powder processing. The log-normal distribution is skewed to larger sizes compared to the normal distribution and has no finite probability for sizes less t h a n zero as seen with the normal distribution. It is obtained by replacing x with z = In d in the normal distribution, which gives the following distribution function [18]:

f(z) =

1 [ - (z - ~)2 (rz-----~ exp ~-~ j

(2.38)

70

Chapter 2 CeramicPowder Characterization

where (rz is the standard deviation of z and 5 is the mean value of z. The cumulative distribution is again simply the integral of the distribution function: F(z) =

fz f ( z ) d z

(2.39)

--or

This distribution can be rewritten in terms of size, d, as follows:

1

(ln(

In o-eX/~ exp

l) 2

2(ln %)2

'

(2.40)

where d e is the geometric mean size and % is the geometric standard deviation. The normal probability function table given in the appendix of this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/de)/(ln %)] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, d e , and the geometric standard deviation, %. The geometric mean size is the size at 50% of the distribution, dso. The geometric standard deviation is easily obtained from the following ratios: ds4.1a _ dao _> 1.0. (rg- dso d15.87

(2.4:1)

If the number distribution follows a log-normal distribution then the surface area and the weight distributions also follow log-normal distributions with the same geometric standard deviation. Conversion from one log-normal distribution to another is easily done using the following equations for the various mean sizes [18]: In day = In dNs= In dNv = In dNM = In dvs = In dw =

In dgN + In dgN + In dgN + In dgN + In dgN + In deN +

0.5 In 20-g 1.0 In 20"g 1.5 In 20"g 2.0 In 2 o'g 2.5 In 20"g 3.5 In 2 o-e

(2.42) (2.43) (2.44) (2.45) (2.46) (2.47)

where dgN is the geometric mean size of the number distribution and the other means sizes are defined by equations (2.6) through (2.12).

2.8 Particle Size Distributions 10.

71

,,,,,

E 0 N

1

o.m

CO

0.1 0.01

....

FIGURE 2.11 versus size.

lb

2'0 3'0 Lo,~ 6'o 7be'o %

9b s's 9'8 9'9 9~.8'

99.99

Linearized log-normal distribution plot: Cumulative % less than

To transform from the geometric mass mean, dgM, to other mean sizes the following equations are used [18]: In dgy = In d a v In dy8 = In dNy= In d w = In d y s =

In dgM In dgM In dgM In d~M -In dgM + In dgM -

3.0 In 2 (rg 2.5 In 2 (rg 2.0 In 2 (rg 1.5 In 2 (rg 0.5 In 2 (rg 0.5 In 2 (rg.

(2.48) (2.49) (2.50) (2.51) (2.52) (2.53)

P r o b l e m 2.4. D e t e r m i n e the M e a n S i z e s [18] from a Log-Normal Distribution

Using the best fit of a log-normal distribution in Figure 2.11, calculate the mean sizes, d g y , dys, dNy, dgM, dsv with equations (2.41) through (2.46) and compare these results to those obtained in the problem in Section 2.3.1 for the same data. Using Figure 2.11, we find t h a t the dgy = d s o = 10.0/~m and the value of (rg = ds4.~3/dso = 18.5/10.0 = 1.85. With these values, we can substitute into equations (2.41) through (2.46) and get the following

72

Chapter 2 CeramicPowder Characterization

values that are compared to those obtained in the problem in Section 2.3.1"

dgN day

dNs dNy

dgM dvs

Equations (2.41)-(2.46) ftm

Problem in Section 2.3.1 ftm

10.0 12.1 14.6 17.6 37.4 25.8

10.0 12.2 14.9 17.9 34.2 25.4

The differences between the two methods of calculation seen here are a result of the linearization of the data. The results from the problem in Section 2.3.1 are the more accurate because the data is not linearized for these calculations.

2.8.3 R o s i n - R a m m l e r D i s t r i b u t i o n For materials that have undergone comminution, the Rosin-Ramruler distribution [4] is frequently applicable. The Rosin-Rammler weight distribution is given by f ( x ) = nbx n-1 e x p ( - b x n)

(2.54)

where n is a characteristic of the material and b is a measure of the range of particle size, x, being analyzed. Integration of this distribution gives the cumulative weight distribution F ( x ) = e x p ( - b x n)

(2.55)

which gives a straight line when the double log ofF(x) is plotted versus log x. (Note: Equation (2.5) is also the Weibull distribution.) The ratio of tan-~(n) to x36.s is a form of variance of the size distribution. The mass mean size, X M -- ~ x i W i / ~ , W i , is given by F(n -1 + 1) XM=

(2.56)

(nN/-b)

and the volume to surface mean diameter by mass, Xys = ~x~Wi/~,x iWi, 2 is given by X,s = [nV~F(1 - n-~)] -~

(2.57)

These values can be evaluated from tabulated values of the gamma function, F, given in the appendix of this book and experimentally determined values of n.

2.9 Comparison of Two-Powder Size Distributions

73

2.9 C O M P A R I S O N OF TWO-POWDER SIZE D I S T R I B U T I O N S For quality control purposes, ceramists are often required to determine if the particle size distribution of one batch of powder is the same or different from another. This determination is difficult when the two batches of powder have similar mean sizes. A statistical method [19] must be used to make this distinction. To determine if two particle size distributions are the same or different, Student's t-test is used by applying the null hypothesis to the two sample means. For normal distributions the t-statistic is defined as the ratio of the difference between the two sample arithmetic means (A1 and A 2) to the standard deviation of the difference in the means [20]" t =

A1 - A 2 . o~1-~2)

(2.58)

Using the definitions of the normal size distributions, the t-statistic can be formulated as follows [20]" v

~1 + ~2/

(~1 - 1)(r~ + (~2 - 1)(r~

(2.59)

where V~ and V2 are the n u m b e r of size classifications used to determine the size distributions of samples 1 and 2, respectively; (r~ and (r2 are the standard deviations of samples 1 and 2, respectively; and v is the number of degrees of freedom defined by [20] v = ~ + ~72- 2.

(2.60)

When two samples are very similar, t approaches zero; when they are different, t approaches infinity. The value of t is used to calculate the P value using Student's t-test tables, given in the appendix of this book. The P value is the probability that the two distribution means are the same; that is, A~ = A 2. When the P value is greater t h a n a critical accepted value (typically 5% [21] or the experimental error due to both sampling and size determination if it is larger) then the null hypothesis (Ho" A~ = A2) is accepted (i.e., the two populations are considered to be the same). Ceramic powder size distributions are often represented by log-normal distributions and not by normal distributions. For this reason the t statistic must be augmented for use with lognormal distributions. Equation (2.59) can be modified for this purpose to

74

Chapter 2 CeramicPowder Characterization

give [19]

T~2]'t

t = (ln ~gl _ in ~g2) [ (T~1~1.__~2_[_

(2.61)

(~1 - 1)(ln o-e~)2 + (r12 - 1)(ln %2)2 In the preceding equation the subscripts refer to samples 1 and 2, Dg is the geometric mean size (size that 50% is less than), % is the geometric standard deviation of the particle size distribution ((rg -Ds4.13%/Dso% -= Dso%/D15.s7~ >- 1.0), ~?is the number of the size classifications used to determine the particle size distribution, and v is the number of degrees of freedom defined by equation (2.60), V~ and V2 have the same meaning as before. Again, the lower is the value of t, the more likely the two particle size distributions are the same. With a value of the t statistic, Student's t-test tables can be used to determine the P value and test the null hypothesis (Ho" In Dgl = In Dg2) (i.e., when the P value is greater than a critically accepted value, then the two populations are considered to be the same). Please note that the two geometric mean sizes must be on the same basis (i.e., count or mass) for this analysis to be valid.

Problem 2.5. Comparison of Two Size Distributions [19] The particle size distributions of each of these two samples was determined by sieve analysis using five sieves. The cumulative size distribution is plotted on log probability paper in Figure 2.12. Compare the two ceramic powders, 1 and 2. A best fit of the data gave the following parameters: Sample

Dg(ftm)

%

1 2

7.0 7.8

1.2 1.5

5 5

Using the previous equations, we find that the number of degrees of freedom is 8 and the t value is 0.49. Using the standard t-test, a P value of 64% is obtained. This means that the probability that the two populations are the same is 64%. Because the P value is gTeater than 5%, we can say that these two powders are essentially the same. For most applications, powder 1 may be substituted for powder 2. For more exacting applications, a higher P value may be necessary for successful powder substitution. As a result, for each ceramic powder application a critical P value for powder substitution should be determined.

75

2.10 Blending Powder Samples [1] 300

200

100" 90" 80-

~

60 50

a

4O

10

0

1

tO 2'0 3'0''4'0 5'06"0 7'0 8'0 9"0

9'8

99:9

99.99

% Less Than FIGURE 2.12 Comparison of two log-normal size distributions.

2.10 BLENDING POWDER SAMPLES [1] The blending of two batches of powder is often performed to yield a new batch with a different mean size. The blending of two nonintersecting log-normal distributions is shown in Figure 2.13(a) where 40% of one powder with D ~ = 7.2 ~m and r = 1.6 is mixed with 60% of another powder with Dg2 = 30.6 ~m and ~g2 = 1.3. This gives a mixture with a size distribution that asymptotically approaches the 60% parent size distribution at large sizes and asymptotically approaches the 40% parent size distribution at small sizes. The location of the S-shaped intermediate size distribution depends on the relative fractions of each of the parent size distributions used. A relative percentage plot of the mixture size distribution and of the two parent size distributions is shown in Figure 2.13(b). Here we see two intersecting size distributions that when mixed show a bimodal distribution. The blending of two intersecting log-normal distributions is shown in Figure 2.14(a), where 40% of one powder with D_~1 = 14.4 and %1 = 1.3 is mixed with 60% of another powder with D~2 = 12.6 ~m and crg2 = 2.8. This gives a mixture with a size distribution that asymptotically approaches the 60% parent size distribution at both large and

76

Chapter 2 a

Ceramic Powder Characterization

lO0 -

E .N (/)

60% 10-

40% 0

..........

0.01 1() 50 9'0 99.99 Bimodal n o n - i n t e r s e c t i n g distributions.

b F(d)

0.5

0 0

10

20

30

40

50

60

70

80

d, s i z e ( ~ m )

FIGURE 2.13 Mixing two size distributions, bimodal nonintersecting distributions: (a) linearized log-normal plot, (b) relative % versus size. Taken from Figures 4.14 and 4.15 in Allen [1].

small sizes and approaches the 40% parent at the point of intersection of the two parent distributions. A relative percentage plot of the mixture size distribution and of the two parent size distributions is shown in Figure 2.14(b). Here we see two overlapping size distributions that when mixed show a monomodal distribution.

Problem 2.6. Mixing Two Log-Normal Size Distributions [22] Determine the relative mass fraction of a mixture of two powders, one with Dgl = 3.0 ftm and %1 = 2.0 and the other with Dg2 = 5.0 t~m and (rg2 = 1.5, which has a composite of D5o%mix = 4 ftm. The mixture size distribution, Fro(Z), is simply the sum of the cumulative distributions of each of the parents (i.e., FI(Z) and F2(Z), see equa-

2.10 Blending Powder Samples [1] a

7~

lOO size

(pro)

10-

0% ~~60%

distributions

cure.% u n d e r s i z e

0 0.01

lb

s'0

9'o

9~.99

Bimodal intersecting distributions.

b 1.5 -

f(d) 1=

0.5

0 0

10

20

30

40

50

60

70

80

d, size(~m)

FIGURE 2.14 Mixing of two size d i s t r i b u t i o n - m o n o modal intersecting distributions: (a) linearized log-normal plot, (b) relative % versus size. T a k e n from Figures 4.16 and 4.17 in Allen [1].

tions (2.38) and (2.39)) multiplied by the mass fraction of the parent distributions: Fro(Z) =

0.5

= Y~ . F ~ ( Z ) +

(1

-

Y~) . F 2 ( Z )

(2.62)

where Z = l n ( D s o ~ m i J D g ) / l n ( ( r g ) for their respective distributions. In this example we want a Ds0~m~ of 4/~m, so F r o ( Z ) = 0.5. Using the normal distribution table in the appendix of this book, we find F I ( Z 1 = 0.42) = 0.6628 and F 2 ( Z 2 = -0.55) = 0.2912. Using the preceding equation, we find that the mass fraction of powder 1 in the mixture, Y~, is 0.56. Blending more than two powder size distributions follows the same general rules outlined here.

78

Chapter 2 Ceramic Powder Characterization

2.11 SUMMARY This chapter has described the various techniques of ceramic powder characterization. These characteristics include particle shape, surface area, pore size distribution, powder density and size distribution. Statistical methods to evaluate sampling and analysis error were presented as well as statistical methods to compare particle size distributions. Chemical analytical characterization although very important was not discussed. Surface chemical characterization is discussed separately in a later chapter. With these powder characterization techniques discussed, we can now move to methods of powder preparation, each of which yields different powder characteristics.

Problems 1. Calculate the sphericity form factor for a cylinder having a diameter D and a length K • D when the cylinder changes from a needle to a flake (i.e., K = ~, 89 1, 2, 4, and 8). 2. Two TiO2 powders have been analyzed for their size distributions, which follow. Can powder B be substituted for powder A? Justify your answer. D (~m)

Powder A % GT

Powder B % GT

10.0< 10.0-9.0 9.0-8.0 8.0-7.0 7.0-6.0 6.0-5.0 5.0-4.0 4.0-3.0 3.0-2.0 2.0-1.0 <1.0

100 99.9 99.2 98.7 97.5 96.6 92.2 88.5 71.7 43.2 0

100 99.9 99.5 99.2 98.4 97.8 94.6 91.6 76.6 47.6 0

3. The following table represents a distribution of size made by screening a powder of density 2500 kg/m 3. Calculate the surface area per unit mass assuming that the particles are spherical. Using microscopic counting of the same particles, a surface area per unit mass was directly determined to be 2.95 cm2/gm. Calculate the ratio between the mean diameter found by screening and that determined by microscopy. d(~m) Weight %

295 0

211 5

152 25

104 40

75 25

33 5

References

4. For and that 5. You

79

a powder with a log-normal size distribution Dg N = 1.05 t~m (rg = 1.84 measured by number, calculate the mean diameter would be measured by screening. have two powders: A B

5.2 tLm DgM = 25 t~m

D g N --

(rg = 1.9 % = 1.9

If you mix 20% in mass of A with 80% in mass of B what is the mean diameter of the mixture. 6. For a 10,000 kg batch of powder (size distribution given in problem 3) we have analyzed 20 gm. If we are concerned with the fraction of particles larger than 211 t~m in size, what is the sampling error? 7. For the powder with the size distribution determined by microscopy, which follows, determine the error in the size distribution. Assume that the particles are cubic with a density of 2500 kg/m 3. Size interval (t~m) 300-200 Number counted 3

200-150 5

150-100 25

100-75 40

75-30 25

30-0 5

References 1. Allen, T., "Particle Size Measurement." Chapman & Hall, London, 1981. 2. Tsubaki, J., and Jimbo, G., Powder Technol. 22, 161-178 (1979). 3. Stockham, J. D., and Fortman, E. G., "Particle Size Analysis." Ann Arbor Sci. Publ., Ann Arbor, MI, 1977. 4. Herdan, G., "Small Particle Statistics," p. 173. Butterworth, Stoneham, MA, 1960. 5. Wadell, H., J. Geol. 43, 250-280 (1935). 6. Heywood, H., J. Pharm. Pharmacol., Suppl. 15, 56T (1963). 7. Schwarcz, H. P., and Shane, K. C., Sedimentology 13, 213-231 (1969). 8. Meloy, T. P., "Screening." AIME, Washington, DC, 1969. 9. Meloy, T. P., in "Advanced Particulate Morphology" (J. K. Beddow and T. P. Meloy, eds.) pp. 85-99. CRC Press Inc., Boca Raton, FL, 1977. 10. Meloy, T. P., Powder Technol. 16(2), 233-254 (1977). 11. Meloy, T. P., Powder Technol. 17(1), 27-36 (1977). 12. Mandelbrot, B. P., "Fractal Form Chance and Dimension." Freeman, San Francisco, 1977. 13. Hurd, A. J., in "Physics of Complex and Supermolecular Fluids" (S. Safran and N. Clark, eds.), p. 501. Wiley, New York, 1986. 14. Witten, T. A., and Cates, M. E., Science 232, 1607-1611 (1986). 15. Adamson, A. W., "Physical Chemistry of Surfaces." Wiley (Interscience), New York, 1967. 16. Langmuir, I., J. Am. Chem. Soc. 40, 1361 (1918). 17. Brunauer, S., Emmett, P. H., and Teller, E., J. Am. Chem. Soc. 60, 309 (1938). 18. Irani, R. R., and Callis, C. F., "Particle Size: Measurement, Interpretation and Application." Wiley, New York, 1963. 19. Shen, A. T., and Ring, T. A., Aerosol Sci. Technol. 5, 477-482 (1986). 20. Meyer, S. L., "Data analysis for Scientists and Engineers," pp. 280-281. Wiley, New York, 1975. 21. Schefler, W. C., "Statistics-Concepts and Applications," p. 246. Benjamin/Cummings, Menlo Park, CA, 1988. 22. Orr, C., "Particulate Technology." Macmillan, New York, 1966.

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PART

!! CERAMIC POWDER SYNTHESIS In this part of the book we will discuss the various methods used to synthesize ceramic powders. Ancient ceramic powder synthesis techniques are shown in Fig. II.1. Up to now, we have discussed raw materials that are either natural in origin or a natural mineral after thermal decomposition. These materials require crushing and grinding to obtain a desirable particle size distribution for most ceramic processing. The fundamentals of comminution and classification will be discussed in Chapter 4. To make the best use of the potential properties of ceramics, new ceramic powders with high chemical purity and fine particle size are necessary. For this reason, methods have been developed to synthesize ceramic powder raw materials. Table II.1 gives an overview of the various methods for powder synthesis. These methods are generally broken into three categories: 1. Solid phase reactant 2. Liquid phase reactant 3. Gas phase reactant. Chapter 5 will be devoted to solid phase synthesis of ceramic powders; Chapter 6, to liquid phase synthesis; and Chapter 7, to gas phase synthesis. Other miscellaneous methods of ceramic powder synthesis are discussed in Chapter 8. All of these ceramic powder synthesis methods have one thing in common, the generation of particles with a particular particle sized distribution. To predict the particle size distribution a population balance is used. The concept of population balances on both the micro and

82

Part H

Ceramic Powder Synthesis

F I G U R E II.1

Ancient ceramic powder synthesis.

macro scale are presented in Chapter 3 for repeated use in the other chapters in this part of the book. In solid phase reaction synthesis, there are three types of chemical reactions: oxidation or reduction of a solid, thermal decomposition of a solid, and solid state reaction between two types of solid. With liquid phase synthesis of ceramic powders, there are five different methods: drying of a liquid, precipitation, sol-gel synthesis, hydrothermal synthesis, and reactions of a liquid metal melt with a gas to give a solid ceramic. There are basically three operational principles for precipitation; temperature change, evaporation, and chemical reaction. Sol-gel

TABLE ILl

Ceramic Powder Synthesis Methods.

Solid phase reactant

Liquid phase reactant

Gas phase reactant

Solid state reactant Thermal decompositions of a solid Oxidation or reduction of a solid

Precipitation Solution heating or cooling Evaporative salting-out Chemical reaction with insoluble product Hydrothermal synthesis Forced insolubility Dissolution reprecipitation

Evaporative Condensation Gas phase reaction with solid product Thermal decompositions Oxidation or reduction reactions Combination reactions with a solid product

Miscellaneous synthesis methods Solvent removal Spray drying Freeze drying Spray roasting Sol-gel synthesis Melt solidification

Part II Ceramic Powder Synthesis

83

synthesis is essentially a precipitation method; however, the solid precipitated is of nanometer size and can be organized into a gel network or sol particle depending on conditions. Hydrothermal synthesis methods use high pressure to make a specific solid phase insoluble. Gas phase ceramic powder synthesis methods include evaporation-condensation and chemical reactions in the gas phase. These gas phase reactions include thermal decomposition, oxidation or reduction, as well as chemical combination reactions. With all of these ceramic powder synthesis techniques, chemical and crystal phase purity are desired in addition to controlled particle size distributions. As with all techniques some processes are better suited than others for a specific type of ceramic powder. In addition to technical advantages of various processes, advantageous production costs also influence process selection. Some examples of commercial ceramic powder processes include solid phase synthesis of Si3N4 and SiC, liquid phase synthesis of A1203 and Zr02, and gas synthesis of Ti02 and Si02.

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3

The Population Balance

3.1 O B J E C T I V E S The population balance is used as a method of accounting for particles as they go through a process, such as grinding, classification, crystallization, aggregation, or grain growth. This chapter is devoted to the development of population balances, because it is of fundamental importance to several other of the chapters in this book. The chapter draws heavily on the excellent text Theory of Particulate Processes by Randolph and Larson [1]. The number density of particles N(L) (with units of number of particles per unit volume) is equal to the integral of the population ~?0(L) from size L to L + hL and is defined as f L+hL

N(L) = ~

~?o(L)dL.

(3.1)

JL

The units on the population ~o(L) are number of particles per unit volume per particle length, L. The population can be decomposed into

85

86

Chapter 3 The Population Balance

the product of the particle size distribution, f0(L), and the number density, n, with units of number of particles per unit volume

Vo(L) = nfo(L).

(3.2)

The number density, n, is, in general, a function of location (x, y, z) within the process equipment and time, t, because particles can be produced or destroyed so n ~ n(x, y, z, t). The particle size distribution, fo(L), with subscript zero is based on number and has the units (size) -1 as discussed in Chapter 2. This particle size distribution must be normalized such that 1 = f o fo(L) dL, to be used in population balances. Before completing this introduction, it is necessary to discuss moments of the distribution. The number density of the population, N T = f : ~o(L) dL, is essentially the zero moment, mo, of the distribution function ~?o(L). The j t h moment is described as or

mj -- fo Lj v~

dL.

(3.3)

The average size of the population is simply equal to the ratio of the first moment, m~, to the zeroth moment, mo. These moment definitions are slightly different for the population, ~0(L), when compared to the size distribution, f(L), discussed in Chapter 2.

3.2 M I C R O S C O P I C

POPULATION

BALANCE

The microscopic population balance is obtained by accounting for all the particles in a differential volume AV = hx hy hz as shown in Figure 3.1. The population for particles in a differential volume can be stated as Accumulation = I n p u t - Output + Net generation.

(3.4)

The accumulation term is the change in the population with time, d~?o/dt. The input and the output terms are considered together. For the input-output terms, we have one term due to flow and another (X+AX,y+Ay,z+Az)

X

(x,y,z) "~! FIGURE 3.1 Control volume for population balance.

3.3 Macroscopic Population Balance

87

due to internal variables like particle growth. For input-output due to flow we have d(vx ~o) d(vy ~o_____~) d(vz ~o) V . v ~?o = ~ + dy +T

(3.5)

where Vx, Vy, v z are the velocity components in the x, y, and z directions. For output-input due to growth or other internal variables x~ we must add V" Pi "1~0-"

d(vi ~?o)

(3.6)

dx i

where vi is the velocity of an internal variable for example the grow rate, G, and xi is an internal variable for example the size of the particle. For the net generation terms we have birth, B, minus death, D. Combining all these terms we have the microscopic population balance: 070 ~ 0 (VxTO) + 0 O----'t-- O--X

0

~ y ( p y T~O) -~" ~Z ( Pz T~O)

m 0 + /~'=~X~X[V~ / ~?0] -- B + D = 0.

(3.7)

This differential equation is the fundamental population balance. This equation together with mass and energy balances for a system form a dynamic multidimensional accounting of a process where there is a change in the particle size distribution. This equation is completely general and is used when the particles are distributed along both external and internal coordinate space. External coordinate space is simply the position x, y, and z in Cartesian coordinates. Internal coordinates xi are, for example, the shape, chemical composition, and the size of the particles. More convenient and more restrictive forms of the population balance will be subsequently developed.

3.3 M A C R O S C O P I C

POPULATION

BALANCE

A population balance over a macroscopic region has many engineering applications. For this type of balance the general population balance developed in equation (3.7) can be simplified. Into a macroscopic volume, we can have an arbitrary number of inputs and outputs at flow rates: Qk. In addition, if we assume that the suspension is well mixed within the volume, multiplying (3.7) by d V and integrating over the volume of the system gives j~ (~~-~ + V " ve ~o + V " Vi ~o + D - B ) d V = O.

(3.8)

88

Chapter 3 The Population Balance

The terms involving the velocity of the moving surfaces are essentially equal to the sum of the input-output streams multiplied by the population contained in them: f~ V " Ve ~?0d Y = ~k Qk ~ok +

7o d_____V dt "

(3.9)

In addition the moving surfaces of the system account for the change in the volume of the system with time. This special population balance may be written as 0~?o ~ V.v~ ~o + Ot

~?od(log V ) ( Q ~ o k ) = B -D- ~ dt

k

(3.10)

after dividing by the volume and rearranging. This is the macroscopic population balance, which is a more useful form of population balance for describing transient and steady state particle size distributions in well-mixed vessels. This population balance in conjunction with mass and energy balances gives a complete description of particulate processes in well-mixed vessels.

P r o b l e m 3.1. Constant S t i r r e d Tank Crystallizer Simplify the macroscopic population balance to describe the particle size distribution in a continuous constant volume isothermal wellmixed crystallizer with mixed product removal operating at steady state. Assume the crystallizer feed streams are free of suspended particles, that the crystallizer operates with negligible breakage, and that agglomeration and crystallization cause no change in the volume of the system. The macroscopic population balance gives a(G~o) 0~o+~ +~o ot

OL

d(log V) = B - D - ~ dt

k

~okVQk

(3 "11)

where G is the growth rate. Due to the conditions imposed, the birth, B, and death, D, due to breakage or agglomeration, the volume charge term, d(log V ) / d t , and volume accumulation, dVo/dt, terms are set equal to 0. The right-hand side of equation (3.11) represents only the mixed suspension discharge, Vo(t, L) Q/V. Assuming McCabe's law for the growth rate (i.e., G # f(L)), we may extract it from the differential, giving 0"00 70 G~-E + - - = 0 T

(3.12)

where r is the mean residence time of the vessel and is equal to the volume of the reactor, V, divided by the flow rate, Q (i.e., r = V/Q).

3.4 Population Balances and Property Conservation

89

The boundary condition for this equation results for the nucleation of particles of size Lo and is defined as v0(L) = B(Lo)/G

a t L = Lo

(3.13)

where B(Lo) is the nucleation rate. Therefore, the solution after integration of equations (3.12) is Vo(L) = Vo(Lo) exp[-(L - Lo)/(Gr)] = B(Lo)/G exp[-(L - Lo)/(G~)]

(3.14)

Here we see an exponential size distribution is predicted by the population balance. (B(Lo) is also the birth term due to nucleation of particles of size L o. It could also be used in the population balance substituting for B directly, but this approach requires a LaPlace transform solution, which also results in equation (3.14).) Many inorganic precipitations operate in this way with small supersaturation; that is, nearly all the mass is precipitated in one pass through the precipitation. This equation for a well-mixed constant volume crystallizer will be discussed in further detail in Chapter 6.

3.4 POPULATION BALANCES WHERE LENGTH, AREA,

AND

VOLUME

ARE

CONSERVED

Both the macro and micro population balances just derived conserve the number of particles. In some cases, it is appropriate to perform balances where the particles' length, area, or volume (or mass) is conserved. For example, length conservation is critical in grinding fibers and volume conservation is critical in grinding other particle shapes. Such conservation equations can also be developed under the umbrella of a population balance, but this population balance must be different than those previously derived, where particle number is conserved. The way to make them different is to couple to the population balance an appropriate conservation equation. The population based on length, area, and volume (or mass) can be derived from the population based on number as shown in Table 3.1. Let us illustrate this idea of property conservation with an example showing conservation of length.

3.4.1 Conservation of Length in the Batch Grinding of Fibers In this case, the conservation of length is maintained by assuring that

OLT(t) O f~ T~I(L, t) dL = f : Ov~(L't) ....... Ot. . . ._ O~ O ~ dL = O

(3.15)

~

Chapter 3 The Population Balance

TABLE 3.1 Population Balance Substitutions for Conservation of Length, Area, and Volume

Quantity conserved Number

Length

Area

Volume

Mass

Population, units

Normalization, units

Conversion from number based population

Size distribution, f i(L)

1

fo(L) = ~o(L)/NT

~?o(L), no./vol./m,

NT = f : ~o0(L)dL,

vl(L), length/vol./m,

LT = f : ~I(L) dL,

~2(L), area/vol./m,

AT = f : v2(L) dL,

v3(L), solid vol./vol./m,

VT = f: ~3(L)dL,

~3(L) = fig L 3

solid vol./vol.

~o(L)/NT

~m(L), solid mass/ vol./m,

no./vol. ~I(L) = flL L

Vo(L)/NT

length/vol.

"02(L)= flA L2

~o(L)/NT

area/vol.

MT = f : ~m(L) dL = ~m(L) = fig L 3 ~o(L )/NT f: Ps~3(L) dL,

fl(L) = 71(L )/L T f2(L) = v/2(L)/AT f3(L) =

~3(L)/VT f3(L) =

~Om(L )/M T

mass/vol.

in addition to the population balance equation. To cope with this situation, first of all the macroscopic population balance appropriate for a batch grinding

OVo(L, t) =B -D Ot

(3.16)

is solved for ~?o(L, t). The population, Vo(L, t), is then used to construct the population based on length, ~ (L, t), which from Table 3.1 is given by ~ (L, t) = flLL ~?o(L, t)/NT(t)

(3.17)

where flL is the length conversion factor (which is 1.0 for fibers where L is their length). Using the conservation of length, equation (3.15), we note that ~?~(L, t) must not vary with time. Hence, temporal changes in the population, ~o(L, t), must be just compensated for by increases in NT(t) with the grinding time. As a result the population, Vo(L, t), must be separable into a function of time equivalent to NT(t) and a function of size. But this is not surprising because ~o(L, t) was defined in equation (3.2) as

77o(L, t) = fo(L, t) n(t)

(3.18)

3.5 Population Balances on a Mass Basis

91

where fo(L, t) is the size distribution at a particular time and n(t) is the number of particles per unit volume or NT(t). Combining equations (3.15), (3.17), and (3.18), we find that

f : O~l(L , t) dL = (~ O[flLL ~?o(L, t)/NT(t)] dL at Ot ~o

_- f: O[flLLotfo(L,t)] dL

= 0

(3.19)

which is the governing equation for grinding fibers. As the length of the fibers decreases with time according to the population balance, the size distribution accommodates this increase, assuring that equation (3.19) is obeyed. Therefore, the conservation equation plays an important role in the population balance by placing limits on the population. In general, these conservation equations are sometimes coupled into the definitions of the birth and death functions, B and D, in the population balance, thus assuring both the conservation of a property and the balance of the population simultaneously, without the necessity of two separate differential equations--population balance and property conservat i o n - t o be solved simultaneously. Such birth and death formulations will be discussed in Chapter 4 for grinding.

3.5 P O P U L A T I O N

BALANCES

ON A MASS

BASIS

3.5.1 P o p u l a t i o n B a l a n c e s on a Discrete Mass Basis In some cases population balances are performed on a cumulative basis. One reason for such a population balance is so that theory can be easily compared to experimental data obtained from sieve analysis. To predict the mass on a sieve with size L, which came from the sieve above of size L + AL, we must perform the integral L+AL

m(L) = (

"L

L+AL

psfiV L3 '~o(L) dL = Ps fi

'~3(L)dL.

(3.20)

The mass is related to the cumulative (percent less than) size distribution by volume (or mass), F3(L), as follows:

m(L) = MT[Fa(L + AT.) - Fa(L)]

= M T

fa(x) dx

(3.21)

where MT is the total mass per unit volume and f3(L) is the size distribution by volume (or mass). Therefore if either the microscopic or macroscopic population balance is multiplied by psflyL3 and integrated over

92

Chapter 3 The Population Balance

the domain L to L + AL, we obtain population balances based on mass, which for the microscopic population balance is given by

Om(L,x,y,z,t) + V. vemm(L,x,y,z, t) at + V. v~m(L, x, y, z, t) =Bm - D m

(3.22)

and for the macroscopic population balance is given by

Om(L, t) + V . vim(L, t) + m(L, t)d(log V) Ot dt (3.23) where B~ and Dm are the birth and death functions on a discrete mass basis using sieves of a particular size.

3.5.2 P o p u l a t i o n B a l a n c e s on a C u m u l a t i v e Mass Basis In addition, population balances can be performed using the cumulative population on a mass basis. To obtain the cumulative population on a mass basis, M(L), from the population based on number, ~o(L), we must perform the following integration: L

M(L)= fo

L

L o(L)dL : fo .s (L)dL

r

The cumulative population on a mass basis is related to the cumulative (percent less than) size distribution by mass, FM(L), as follows:

M(L) = MT[FM(L)] = VTPs [ f : f3(x) dx ]

(3.25)

where MT is the total mass per unit volume and f3(L) is the size distribution by volume. Therefore, if either the microscopic or macroscopic population balance is multiplied by psflyL3 and integrated over the domain L = 0 to L, we obtain population balances based on the cumulative population on a mass basis, which for the microscopic population balance is given by

OM(L, x, y, z, t) + V. veM(L , x, y, z, t) Ot + V" viM(L, x, y, z, t)

= B M -

D M

(3.26)

and for the macroscopic population balance is given by

OM(L, t) + V . viM(L, t) + M(L, t)d(log V) ot dt (3.27)

3.6 S u m m a r y

93

where B M and DM are the birth and death functions on a cumulative mass basis. Population balances on a discrete mass basis and on a cumulative mass basis are frequently used in grinding circuits, which are discussed in Chapter 4.

3.6 S U M M A R Y In this chapter we have developed both predictive and descriptive methods for particle size distributions in various types of processes using the population balance. The population balance can be used to conserve one of the following properties: number, length, area, or volume (or mass), when appropriate conservation equations are used in conjunction with the population balance. The population balance together with (1) a property conservation equation and (2) associated mass, momentum, and energy transport equations and their associated boundary conditions provide a unified predictive theory for particulate systems. The population balance was represented in two general forms: microscopic and macroscopic. Each has advantages in the description of specific systems. These forms of the population balance together with their restrictions and advantages follow for easy reference. These two forms of the population balance can also be rewritten on either a discrete mass or cumulative mass basis.

3.6.1 Microdistributed Population Balance ~7~0 ~- V " VeT~O "~- V " ViTrO : B - D

(3.28)

ot

Uses Assumption

Spatially distributed systems Particles are numerous enough to approximate a continuum, and each particle has an identical trajectory in particle phase space.

3.6.2 Macrodistributed Population Balance 0~____~o+ V" Vi~o + Ot Uses Assumptions

dt

k

+ B -D

(3.29)

Well-mixed systems Same as microdistributed balance. Only internal coordinate particle velocities are considered.

3.6.3 List of Symbols AT f/(L) Fi(L)

= -~

Total area per unit volume (area/m 3) Size distribution (mass/m 3) Cumulative size distribution

94

Chapter 3 The Population Balance

L

Size (m)

LT Total length per unit volume (length/m 3) m(L) Population density on a mass basis for a discrete size interval L to L + AL (mass/m 3) Total mass per unit volume (mass/m 3) Cumulative population density on a mass basis (mass/m 3) Number of particles of size L to L + AL per unit volume (no./m 3) Total number of particles per unit volume (no./m 3) V Volume t Time v Velocity vector (subscript i is internal variable, subscript e is external variable) xi Internal variable vector components G Growth rate (m/time) B Birth rate D Death rate vi(L) Population density [i = 0, no./(m 3 9m)] my j t h moment of the population

MT M(L) N(L) NT

3.6.3.1 Greek Symbols flL Conversion factor for length flA Conversion factor for area fig Conversion factor for volume r

Mean residence time (time)

3.6.3.2 Subscripts e i i = i = i = i = flh m M

External variable Internal variable or basis for population 0 Number basis 1 Length basis 2 Area basis 3 Volume basis moment Discrete mass basis Cumulative mass basis

Reference 1. Randolph, A. D., and Larson, M. A., "Theory of Particulate Processes." Academic Press, New York, 1971.

4

Comminution a n d Classification of Ceramic Powders

4.1 O B J E C T I V E S To use natural minerals it is necessary to grind them down to a desired particle size distribution. Grinding can be performed with the minerals dry or slurried in liquid. In most laboratories, this process is performed in a batch jar mill while on an industrial scale, continuous comminution equipment is used in conjunction with size classification equipment to recycle the coarse material. Figure 4.1 shows a typical comminution circuit with classification and recycle steps, as well as separation of the mineral from the conveying fluid. Classification can be performed by centrifuges, cyclones, or air classifiers. In this chapter, the fundamentals of comminution and classification will be discussed in addition to a review of the different types of equipment used for these two steps. Comminution and classification are the most important methods of transformation of minerals to ceramic powders. They are also used for synthetic ceramic powders, because 95

96

Chapter 4

Comminution and Classification of Ceramic Powders

FEEDMATERIAL

~ FLUID

CLASSIFIED R:)WDER RECYCLE

I su,soo .u,o ~ PRO~JCT FIGURE 4.1 Grindingcircuit. powders produced by chemical reaction often have undesirable particle size distributions. For this reason, comminution and classification are often used for thermally decomposed minerals and synthetically produced ceramic powders.

4.2 C O M M I N U T I O N

4.2.1 C o m m i n u t i o n E q u i p m e n t This description of communition equipment draws heavily from the same materials in the Chemical Engineering Handbook [1]. Different types of particle size reduction equipment are used, depending upon the type of material and its initial and desired particle size distribution. Table 4.1 lists these different types. When considering ceramic powders, we generally find relatively narrow size distributions are desired with the mean size less than 5 fLm. To meet these needs various types of mills are used, including (1) mills with balls, pebbles, and rods as the medium; (2) high-speed peripheral mills; and (3) fluid energy mills. Fluid energy mills are able to produce the very finest powders. Figure 4.2 shows the various types of grinding equipment. The jaw

4.2 Comminution T A B L E 4.1

97

Types of Size-Reduction

Equipment Jaw crushers (continuous) Gyratory crushers (continuous) Heavy-duty impact mills (continuous) Rotor breakers Hammer mills Cage impactors Roll crushers and shredders (continuous) Tumbling media mills (batch and continuous) Ball mills Rod mills Autogneous Stirred media mills (batch and continuous) Stirred ball mills Stirred sand mills Vibratory media mills (batch and continuous) Fluid shear mills (batch and continuous) Colloid mills Microatomizer Fluid impact mills Opposed jet Jet with anvil Centrifugal jet

crusher has an articulating arm that moves the jaws back and forth, crushing material to a size small enough to fall from the bottom of the jaws. This type of crusher can be as large as 3 m across, producing powders of 5 cm at the outlet, or as small as 10 cm across, producing powders of 1 mm at the outlet. It is commonly used in large-scale mining operations. A more efficient grinding mill for the same duty is that of a gyratory crusher. The gyratory crusher can also be as large as 3 m in diameter, producing material of 5 cm in diameter at the outlet. It consists of a cone-shaped pedestal, oscillating within a larger cone-shaped bowl. The angles of the cone are set such that the width of the passage decreases toward the bottom of the working faces. The pedestal consists of a mandrel that is free to turn on its spindle. The spindle is driven from an eccentric bearing below. Differential motion causing attrition can occur only when pieces are caught simultaneously at the top and bottom of the passage due to the different radii at these points. Gyratory crushers generally require about one-fourth of the energy input per ton of material for grinding similar to that of the jaw crusher [4]. For smaller particle sizes, the hammer mill is often used. Here the material is impacted with a heavy duty hammer

Crushing and grinding equipment. (a) Jaw crusher, (b) gyratory crusher, (c) vibratory mill, (d) fluid shear mill, (e) pin mill, (f) fluid jet mill, (g) hammer mill, and (h) ball mill. Redrawn from various figures in Perry and Chilton [1], Lowrison [2], and Van Cleef [3].

F I G U R E 4.2

98

F I G U R E 4.2

(Continued)

99

100

Chapter 4 Comminution and Classification of Ceramic Powders

FIGURE 4.2

(Continued)

mounted on a horizontal shaft. Material is forced at high speed into the breaker plate and then falls through a grating at the bottom of the device. H a m m e r mills often clog and as a result are designed so as the motor can be reversed to unclog them. By far the most common mill for fine particle sizes is a ball mill. Here material is continuously fed to grinding media that are in constant motion, tumbling one over another. The impact of balls (media) against each other grinds the material trapped between them. After the material has been in the mill for a certain period of time, it is flushed out of the system by a flowing over a central part of the cone at the exit of the mill. The balls, which are generally larger, are retained inside the mill by a grate. Other media mills use the same comminution concept as the ball mill (i.e., collision of media with material trapped between) but apply the energy in the form of vibrational energy (e.g., vibro-energy mill) or mechanical stirring energy (e.g., a colloid--or attrition--mill). The fluid energy mill consists of a high-speed jet of particle laden gas that impinges on either another jet directed in the opposite direction or a hard wall. Particles are broken by the impact of high-speed collision. Fluid energy mills are able to produce the very finest powders, because with this method all the energy is absorbed by the particles to be communited. None is lost to other grinding media.

101

4.2 Comminution

4.2.2 E n e r g y R e q u i r e d f o r S i z e R e d u c t i o n The energy required to grind a metric ton of material with the different types of mills is given in Figure 4.3. For the very smallest grind, between 10 and 1 ftm typically used for the production of ceramic powders, ball mills, pin mills, vibratory ball mills, fluid energy mills and attrition mills are the most frequently used. The energy required for these types of mills can vary drastically from 10 kWhr/ton to 1000 kWhr/ton. The largest values are associated with the fluid energy and attrition mills. This energy is a sum of the energy required (1) to move the machine, its kinetic energy, and friction; (2) to move the material, its kinetic energy, plastic and elastic deformation, and internal friction; and (3) to break the material into smaller particles. In almost all cases, the energy required by the mills shown in Figure 4.3 is very poorly utilized. The energy required to break the material is often less t h a n 1% of the total energy needed to run the mill [4,5]. The energy, E, required to grind material into a smaller particle size, L, from size, L 0, can be described by [6]:

E =-

-~dL

(4.1)

0

where C is a constant. This equation has an exponent, n, which is given by the following "laws" [7]:

,

1000

o

: e.

LIJ >,, O)

0 Q.

r

"laws"

1 2 1.5

Kick's Ritenger's Bond's

~Attrition mill : Fluid-energy mill

100

10.

C 0

9

n

r

-

~ ~

~.Vibratory : ="-~-Ball" mill Rod

mill=

1

0.1

,

o:,

mill

--Autogenous-mill -Roller mill

% R o l l crusher = "Jaw crusher ~Hammer mill ~. -: Gyratory crusher

c o

........ 0 .... o obo, o~o~o:o~

ball

,

1.0

..........'

v

10:0 100'.0 1000.0

Feed-product size, cm

F I G U R E 4.3 Average energy required for size-reduction equipment, 9 is typical product size, @ is typical feed size.

102

Chapter 4 Comminution and Classification of Ceramic Powders

Ritenger's law suggests that the energy required to fracture material is simply the energy required to generate new surface area. Kick's law on the contrary suggests that the energy required to fracture a particle is related to the energy stored within the volume of the particle. Bond developed a theory where n = 1.5, corresponding to a best fit of the data from many ball mill runs. Peterson [7] has suggested that there is a critical diameter above which the energy required to comminute a particle is related to the Kick's law and below which the energy required to fracture a particle is related to the Ritenger's law. The energy required to fracture material given in equation (4.1) corresponds to less than 1% of the energy required to run the mill, as shown in Figure 4.3. In comparing dry grinding with wet grinding, the value of constant C in equation (4.1) for dry grinding is 30% higher than that for wet grinding the same material [1]. For this reason, the energy utilization is 30% better for wet grinding than for dry grinding.

4.2.3 Comminution Efficiency Grinding material to smaller and smaller sizes is a process that requires more and more energy. Part of the reason is that smaller particles have more strength than larger particles, because the smaller material has fewer defects than the larger particles. An example of this phenomenon is shown in Figure 4.4 [8], where the strength of quartz glass spheres is measured at different diameters. At 80 t~m diameter, the strength is on the order of 103 N / m m 2 but at 10 tLm the strength has increased threefold simply as a result of their being fewer volume defects in the smaller particles. The Griffith law [9] of failure applies to both large and small particles. Chemical treatments can also affect particle strength. Soda glass treated with molten LiNO3 is weaker in tension by an order of magnitude [7] (the Weibul modulus increases also). Lithium ions diffuse into the glass replacing the sodium ions with a classic ~v/time dependence. This causes microcracking of the glass surface. These weakened glass particles are ground to smaller particle size faster than the untreated beads in identical ball mill tests. The other reason for the high energy required to grind to smaller sizes is the inefficiencies of grinding [6]. These inefficiencies can be separated into two major categories: mechanical, having to do with the transfer of useful energy from the grinding media to the particles being comminuted, and fluid, having to do with the transport of particles through the grinding zone. Other mechanical losses are also present, such as mechanical drive losses, but these are either minor or easily remedied. The mechanical losses in a conventional ball mill stem largely

4.2 Comminution

4.10 3

t

103

strength of quartz glass spheres, tempered for 2 hours at 1010~

3.1o3

04

E E z

v

2.1

e-

l-

,.

10~

1-

N ='10 N=22

I

N=20

........

.

0

20

40

N=20

N=20

60

80

i

l

1

100

sphere diameter x/pm F I G U R E 4.4

Strength of glass spheres. Taken from Leschonski [8].

from the fact that the energy is put into the grinding media rather than into the material to be ground. When a ball tumbles one diameter, trapping material between the impacting balls, more than 10 times the energy necessary to fracture these particles is available [7]. As a result, most of the energy is absorbed into the grinding media through losses of frictional heating or stored elastic energy. Fluid losses are harder to quantify but the movement of fluid in a ball mill drags the particles to be ground out of the way of the impacting balls, as shown in Figure 4.5.

4.2.4 P o p u l a t i o n B a l a n c e Models for Comminution Mills Generally, two types of grinding mills are used in ceramic processing. One is a batch mill or jar mill, used frequently in ceramic powder processing laboratories, and the other is a continuous mill where material is continuously fed and removed, used in industry. These continuous mills are often used in conjunction with classification circuits to recycle large material for regrinding. In this section, we consider the application of the population balance to both the batch and the continuous open-circuit mill without classified recycling. With the population balance we can describe the change in the particle sized distribution that results upon passage through the mill. For these mills we start with the macroscopic population balance from Chapter 3 which is given as follows.

104

Chapter 4

F I G U R E 4.5

Comminution and Classification of Ceramic Powders

Fluid-particle interaction at grinding media impact.

4.2.4.1 M a c r o s c o p i c P o p u l a t i o n B a l a n c e o n a D i s c r e t e Mass Basis

Om(L, t) = _ ~ (Qkm~L, t!) + B m - Dm Ot

(4.2)

k

where Qk is the flow rate of various s t r e a m s into ( - ) and out of (+) the mill, V is the volume of the mill, Bm and Dm are the birth and death functions on a discrete mass basis using sieves of a particular size, and m(L) is the m a s s of material t h a t will fall onto a sieve of size L from the sieve above at L + AL, defined as L+AL

m(L) = ~

JL

psfiy L3 To(L) dL.

(4.3)

4.2.4.2 P o p u l a t i o n B a l a n c e s o n a C u m u l a t i v e Mass Basis

OM(L, t) = _ ~ (Qk Mk(L, t)) + BM_ DM c~t

k

(4.4)

Y

where L

M(L) = fo PsflyL3 v~

dL = MT[FM(L)]

(4.5)

and FM(L) is the cumulative (percent less than) size distribution by mass and B M and DM are the birth and death rates on a cumulative mass basis.

4.2 Comminution

105

In these population balance equations, the volume change term is zero and the internal velocity term is also zero and therefore are not included. When the population balance is performed on a number basis, it is necessary to have a conservation equation, which for grinding is usually the total particle volume. 4.2.4.3 C o n s e r v a t i o n of V o l u m e in C o m m i n u t i o n

For conservation of volume, the constraining equation is

dVTdt - fly_f: d[L3~~

t)] dL

= lpsfo d[m(L,dt t)] dL =0

(4.6)

where fly is the volume conversion factor, which is 7r/6 for a sphere where L is the sphere diameter. For conservation of volume to be applied, the population balance must be written in terms of either number or mass and either discrete or cumulative. This conservation of volume is then used to develop the birth and death functions appropriate for these population balances. 4.2.4.4 B i r t h a n d D e a t h F u n c t i o n s

The death rate in grinding is also referred to as the breakage rate and, for the population balance on a discrete mass basis, is given by [11]

DIn(L) = S(L) m(L).

(4.7)

The function S(L) is the specific breakage rate, which is dependent on particle size because smaller particles are usually stronger than larger ones, see Figure 4.4. This death rate has been shown by Sedlatschek and Bass [12] to be in accord with probability theory. The specific breakage rate, S(L), has several definitions in the literature. The simplest is [13]

S(L) = k (L/Lo) ~

(4.8)

where k and fl are constants for a given material in a particular mill under stated operating conditions and L o is a reference size. The more complex form of the specific breakage rate is [14]

S(L) = k (L/Lo)~ (1 + L/L1) A

A> 0

(4.9)

where L~ is another reference size, corresponding to the size where the grinding media is too small to nip the particle or the size of particle where fines cushion it, preventing breakage. The birth function for conservation of volume must be consistent with the death function because each particle death results in the birth of smaller particles, resulting from communition. When particles of size L break, they produce a suite of daughter particles, called the primary progeny, with a size distribution p(x, L). This function applies

106

Chapter 4 Comminution and Classification of Ceramic Powders

strictly to the primary progeny before rebreakage. The term p(x, L) may be described as the mass fraction broken from size L that appears in size x. The birth function can now be constricted as follows: l"

BIn(L) = ly=L S(y)p(L, y) m(y, t) dy

(4.10)

where S(y) is the specific breakage rate. Combining the birth and death rates into the population balance on a discrete mass basis, we have

(

) s;

Ore(L, t) = _ ~, .Qk ink(L, t) + S(y)p(L y) re(y, t) dy Ot k V -L ' -

S(L)

(4.11)

m(L, t).

The resulting particle size distribution therefore depends to a large degree on the primary progeny function p(x, y) and to a lesser extent on the specific breakage rate. Much effort has been extended in describing the primary progeny function in a cumulative form:

P(x, y) =

f:p(z, y) dz

(4.12)

where P(x, y) is described as the cumulative mass fraction of particles ofsizey broken into sizes belowx. The t e r m P ( x , L ) m u s t be a normalized function; thus [15]

fo'P(x, y) dx = 1.

(4.13)

For mass conservation, the following must also apply [15]:

2

f2xP(x,3,) dx = 3'

(4.14)

assuming one particle breaks into two pieces. Several mathematical representations of the cumulative primary progeny function are given in the literature [11, p. 282]:

P(x,y) = ~j Ix a- y1] + [1 P(x, y) =

-- (I~j]

[ x -~l y]

1 - exp( -x/y) 1 - exp(- 1)

P(x, y) = P~(y/x)-*

with O > 0

note, nonnormalized!

The progeny function has been shown to be independent of mill diameter, ball density, ball loading in a limited range, powder charge, and hold-up but dependent on media diameter and shape as well as mill

4.2 C o m m i n u t i o n

107

linear geometry. The cumulative primary progeny function can be used with the population balance on a cumulative mass basis as follows [11, p. 329]" aM(L , t ) = _ Ot

k

/

~ l

\

v

t) } + BM -- DM /

(4.15)

where the birth and death terms are given by f

BM - DM = ly =L S ( y ) P(L, y) M ( y , t) dy - S(L) M(L).

(4.16)

Using integration by parts, the population balance can be rewritten as OM(L, t) = _ ~ Ot k

Qk _

, t)

+

s;

:L

S(y)P(L, y) OM(y, _t_____dy _~) Oy

(4.17)

which is a more condensed way to write the population balance for comminution. 4.2.4.5 B a t c h C o m m i n u t i o n

In batch comminution, the summation term in equation (4.11) is zero because there are no flows into or out of the mill. Therefore the population balance on a discrete mass basis is simplified to Om(L,ot t) = fy=L S ( y ) p ( L , y ) m(y, t) dy - S(L) ra(L, t).

(4.18)

This integro-differential equation has a similarity solution [16] for certain cases corresponding to specific forms to the functions S(y) and p(x, y). The similarity solution is of the form m(L' t) =

l--~-Z

(-4.19)

where the similarity function, Z, depends only upon the dimensionless particle size, L/X(t). The size scaling factor or the characteristic particle size, X(t), is the first moment of the normalized particle size distribution X(t) =

fl

L re(L, t) d L

(4.20)

and it varies in time according to dX(t) - - k Ko[X(t)] ~-~ dt

(4.21)

where Ko is a dimensionless constant. The preceding equation holds when the specific breakage rate, S(x), and the primary progeny function,

108

Chapter 4 Comminution and Classification of Ceramic Powders

p(x, y), have the following definitions:

(4.22)

S(x) = k x ~ p(x,y) =

(4.23)

thus providing the definitions of k and ft. The resulting solution to the population balance is given by [17] m(L,t)=~t)~-~)

~-~ exp

[ - ~ o1

(4.24)

where K~ is a normalizing factor. Inspection of this equation shows it is self preserving. This means that when X(t) m(L, t) is plotted as a function of (L/X(t)) the function generated at different grinding times should collapse on a single curve, which is characterized by Z(L/X(t)), see equation (4.19). Employing the cumulative distribution, M(L, t) = MTFM(L, t), the similarity can be written as m

(4.25)

M(L, t) = Z(L/Lso ~)

or any other percent mass for that case where Z is the appropriate similarity function on a cumulative mass basis. This similarity analysis has been performed on data for the grinding of agglomerated A1203 platelet particles as is shown in Figure 4.6. In Figure 4.6(a), the cumulative particle size distributions are provided as a function of ball mill grinding time. Figure 4.6(b) shows the same data replotted as a function ofL/Lso~. In this figure, the data collapses to one curve for all grinding times beyond 0.5 hr and the Lso~ shows the time dependence shown in Figure 4.6(c), which suggests that a is 2.0, see equation (4.22). For the case when a = fl and a > 0, the similarity solution is given by m(L, t) = e x p [ - k L ~ t] M(L, O) + k t ~ L ~-1

M(x, t = O) dx .

(4.26)

This can be rewritten on a cumulative greater t h a n mass basis as m(x, t) dx =

m(x, t = 0) dx e x p [ - k L" t]

(4.27)

which is a R o s i n - R a m m l e r size distribution [19]. Over the years this distribution has been found to be reasonably satisfactory for many grinding systems [11, p. 330], including minerals and ceramics. In general, the similarity solution for comminution is of considerable

~ 2 Comminution

109

F I G U R E 4.6 Grinding data for Atochem A1203 Platelets Grade 1. Cumulative percent mass versus diameter for various grinding times. (a) Batch grinding, average of three measurements using Horiba Capa-700. Data from Mulone and Bowen [18]. (b) Ball mill grinding. (c) L50%versus time for grinding the platelets.

110

Chapter 4 Comminution and Classification of Ceramic Powders C

10

E ::::L

o~ o i.o

_..!

1

0

.....

'

1'0

..............

9

....

20

Time (hrs)

FIGURE 4.6 (Continued)

importance because it gives valuable insight into batch comminution. It shows t h a t the initial size distribution is smoothed out after a sufficiently long time and the size distribution given by the similarity solution is independent of the initial conditions. For this reason, the similarity solution is not valid for the initial comminution periods.

4.2.5 Array Formulation of Comminution Generally, these simplified progeny functions have been shown to be inadequate. As a result, the more recent approach is to use the population balance on a discrete mass basis for a series of sieves of a given range of sizes. These sieve sizes correspond to the sieves used in the analysis of the particles on a routine basis to characterize and control the product of the mill. In this case, the progeny function is averaged over L to L + AL, corresponding to the size range between sieves. This tends to smooth the progeny function, giving

Omj(t) [Qk vJk(t) ] i-1 ~ +~ -: -Sjmj(t) + [Sjp~;j my(t)] (4.28) Ot j=li>l where my(t) is the mass of material on the j t h sieve as a function of time, t (asj increases the size of the particles in the size range increases), Sj is the specific breakage for material in the size range j, and P i;j is the progeny for the j t h size range broken into the ith size range. The

4.2 Comminution

111

summation on the right-hand side is simply the birth function and replaces the integral in equation (4.28). The right-hand side of equation (4.28) can be rewritten in a r r a y nomenclature as follows"

o~

(4.29)

where re(t) is a time dependent vector of masses corresponding to each size range with elements my(t), I is the identity matrix (Iij = 0 when i ~ j, and Iij = 1, when i = j), P is the cumulative progeny matrix with elements corresponding to Pi, j, and S is the vector of specific breakage rages with elements Sj.

4.2.5.1 Batch Comminution For a batch mill, the summation of the left-hand size of the above equation is zero. In this case, an analytical solution can be obtain to the population balance (equation 4.29)" re(t) = e x p [ - (I - P) S t] m(t = 0)

(4.30)

where m(t = 0) is the vector of masses corresponding to the feed size distribution. For the case where no two specific breakage rates are the same, the matrix exponential is easily simplified by a similarity transformation into [20] mbatch(t)

--

T J(t) T -1 m(t = 0)

(4.31a)

where

{exp(-Sit); Jij(t) = {0;

T~j =

i =j} i ~j}

(4.31b)

{0;

i <j}

{1;

i =j}

i~ [ eikSk Tkj]" i>j}.

(4.31c)

In this array form, these equations can easily be solved on a computer. But, for m a n y cases, the n u m b e r of sieves used in analysis is relatively small, often less t h a n five, making m a n u a l manipulation also possible. In addition, the coefficients of the T J(t) T -~ matrix can be determined for a mill if the input and output size distribution on a mass basis is known [21]. In addition to its usefulness in batch mill simulation, this equation plays a role in the description of continuous grinding as we will see in the next section.

112

Chapter 4 Comminution and Classification of Ceramic Powders FEED QF = FLOW RATE "rio(L) = FEED POPULATION

MILL

V3.LlVlF_,V

PRODUCT Q =FLOW RATE q(L) = PRODUCT POPULATION

FIGURE 4.7

O p e n - c i r c u i t g r i n d i n g mill.

4.2.5.2 Continuous Grinding Figure 4.7 is a schematic diagram of a well-mixed open-circuit grinding mill. Here a powder is fed to a grinding mill with a volume V. The outlet flow rate Q is equal to the inlet flow rate QFat steady state. The mean retention time, z, is the ratio of the mill volume to the flow rate (i.e., 9 = V/Q). We have two options to attack this problem. The first is to directly use the population balance:

and a t t e m p t a solution. To do this we must know the effective volume of the mill, the flow rate in and out, as well as the masses in each size range entering and leaving the mill. If we assume that the mill is well mixed, the masses in each size range leaving the mill are the same as those in the mill, re(t). Furthermore, if the flow rate in is equal to t h a t leaving the mill, we can rewrite the population balance as

where minput(t) is the inlet masses in each size range and r is the m e a n residence time. For steady state operation, neither the inlet, outlet, nor the mill mass vector varies with time, giving upon r e a r r a n g e m e n t the simplification I n = [ I -- ( I -- P ) S T ] - 1 m i n p u t .

(4.34)

This solution, however, is only good for well mixed mills. In Figure 4.8 [22,23], the change in the particle size distribution is shown for

4.2 Comminution

1.0 ! fl=1.0 (D

113

~k~=4018

"

~

j

k~:=

1.0

k't':

0

o,

~E ~r

-5"~ EE

0.4

~/a

0.2

3 ~ 0.0 0.00

0.02

0.04

0.06

0.08

Particle size, in Cumulative size distribution of ground material after various dimensionless times in a continuous open-circuit ball mill, size selectivity S(x) = kx#, ]3 = 0. Taken from AIME [22] and Randolph and Larson [23].

F I G U R E 4.8

different values of kr with a size selectivity, fl, of 1.0 (see equations (4.22) and (4.23) with a = fl). Here we see the size distribution becomes finer as the time in the mill (or the grinding rate constant) increases. At long times, the curvature of the size distribution also changes. Figure 4.9 shows the effect of the size selectivity parameter, fl, for k~ = 1.0 1 o0 .

k~'=1.0 .

.

.

.

.

!

.

08

o

~N

=4.0

~~

"~9 0.6

~a

,>~ 04

;11

~s 0.0,

0.00

,

0.02

,

i

0.04

,

!

0.{)6

0.68

Particle size, in Cumulative size distribution of ground material at a dimensionless time of 1.0 in a continuous open-circuit ball mill with various size selectivities of fl, S(x) = kx#. Taken from AIME [22] and Randolph and Larson [23].

F I G U R E 4.9

114

Chapter 4 Comminution and Classification of Ceramic Powders

starting with the same initial size distribution as in Figure 4.8. Values of fl less t h a n 1, corresponding to the idea that small particles are stronger t h a n large particles (see Figure 4.4), gives selective grinding of the smaller particles whereas values offi larger t h a n 1 gives selective grinding of the larger particles. The second approach to a solution is appropriate for other types of mixing. The mixing is characterized by a residence time distribution, E(t), which is the distribution of time that the material is in the mill. When linear breakage kinetics occur and all particle sizes are characterized by a single residence time distribution, a general relationship between input and output of the mill can be established, using the following equation: l

moutput -- ~J

mbatch(t) 0

(4.35)

E(t) dt

where the output is simply a sum of the responses of the mill to the feed size distribution treated as a sequence of impulses based on batch grinding weighted by the distribution of residence times in the mill. Making the substitution for the batch residence time and rearranging we get [21] moutput =

T

[So

]

J(t) E(t) dt T -1 m i n p u t

--

T Jc(~) T -1 minput (4.36a)

where r is the mean residence time and

J~(t) =

fo

E(O) exp{-(Si~')O} dO; i =j

{0;

(4.36b)

i C-j}

where 0 is the dimensionless time variable 0 = t/r. In this ease, any residence time distribution can be used for the mill, and the mill output can be determined from the mill input. In array form, these equations can easily be solved on a computer. But for many cases the number of sieves used in analysis is relatively small, often less than five, making manual manipulation also possible. In addition, the coefficients of the T J(r) T -1 matrix can be determined for a mill if the input and output size distribution on a mass basis is known [21]. For the well-mixed ease the residence time distribution is given by

E(O) = exp(-0)

giving

Jcij(r)

= (1 + S i r ) -1

for i = j

(4.37)

which is exactly the same as the preceding approach for a well-mixed mill; and for a plug flow case, the residence time distribution is given by

E(O) = 8(0 - 1)

giving Jcij(r) = exp(Sir) for i = j

(4.38)

4.3 Classification of Ceramic Powders

115

In addition, several other two-parameter residence time distributions have been formulated [24]. With these or any other residence time distribution, the specific breakage rate, and the progeny array, the mill output can be determined from the mill input thus the comminution step is mathematically described.

4.3 CLASSIFICATION OF CERAMIC POWDERS Classification is the separation of particulates into a coarse and fine fractions. Classification should be distinguished from solid-fluid separation (a step also shown in Figure 4.1), although the two unit operations overlap. Classification is usually by size, but may also depend on other properties of the particles: density, particle shape, electric, magnetic, and surface properties. Classification of particulates usually takes place in a conveying fluid either liquid or gas. Classification equipment generally operates in the 1000-0.1 ftm range by the selective application of any of the following forces: gravity, drag, centrifugal, and collision. Table 4.2 gives a listing of various classification equipment.

4.3.1 Dry Classification Equipment Dry classification equipment uses a gas stream to convey the solids. The gas used most often is air, and for that reason the term air classifiers is often used to describe this type of equipment (see Fig. 4.10). Air classifiers evolved from two sources, the original simple expansion chamber and the Mumford and Moodie separator, patented in 1885.

TABLE 4.2

Classification Equipment

Classification

Size range

Wet Screens Sedimentation Classifiers Hydrocyclones Elbow Classifier Centrifuge

1 1 500 100 50

m - 4 4 ftm m m - 1 0 t~m tLm--0.1 ftm ftm-0.1 t~m t~m-0.1 t~m

1 100 1000 500

m - 4 4 ftm ftm-10 t~m ftm-0.1 ftm ftm-0.1 ftm

Dry Screens Expansion chamber Air Classifier Gas Cyclone

Air classification equipment: (a) cyclone, (b) expansion chamber, (c) modern complex air classifier, and (d) classifier based on particle inertia.

FIGURE 4.10

116

4.3 Classification of Ceramic Powders

117

In the former, coarser particles drop out of a gas stream as its velocity is decreased upon expanding to a larger space. Baffles, vanes, or other directional and impact devices were later incorporated into the expansion chamber to change the flow direction and provide collision surfaces to knock out coarser particles, as in the Mumford and Moodie separator. In the Mumford and Moodie separator, shown in Figure 4.11, solids are fed into a rising gas tream with a rotating distributor plate that imparts a centrifugal force to them. While the coarser particles drop into an inner cone, the fines are swept upward by the action of an internal fan, move with the gas between the vanes in the expansion section of the outer cone, and are collected at its bottom. The gas is then recirculated up toward the distributor. Many types of classification equipment are commercially available. Klumpar et al. [25] discussed the major designs in a review article. Basically, there are classifiers with and without rotors that collide with the particles, those with an updraft, those with a side draft, and other miscellaneous equipment. Equipment designed for classification takes advantage of a number of different phenomena: small particles settle more slowly in a fluid than large particles; small particles have less inertia and can change their direction with gas flow more easily than large particles; larger particles require a higher conveying velocity; larger particles have a greater centrifugal force in cyclonic flow than small particles; and large particles have a larger probability of collision with a rotating blade. A classifier is designed to minimize particle-particle interactions in the classification zone allowing the fluid-particle interactions to facilitate classification.

4.3.2 Classifier F u n d a m e n t a l s 4.3.2.1 F o r c e s

Forces acting on the individual particles are responsible for directing large and small particles into their respective collection chambers. The forces that act on the particles in the classification zone are gravity, aerodynamic drag, centrifugal force, and collision. Each type of equipment listed in Table 4.2 uses one or more of these forces. The interplay among these forces is complex and not very well understood. Hence, comprehensive mathematical models for classification equipment are few and far between. Gas classifiers with internal rotors, as shown in Figure 4.11, will be used in the following discussion. The individual expressions described for the forces here are valid for all other classifiers, both dry and wet. However, the directions of

118

Chapter 4

Comminution and Classification of Ceramic Powders

F I G U R E 4.11 Forces acting on a particle in a gas classifier. T a k e n from K l u m p a r et al. [25, pp. 17-19].

the force vectors may be different and some forces may not be applicable. In this classifier design, the feed is introduced at the center of a horizontal rotating feed plate. Friction with the plate accelerates the particles radially. Once the particles have an angular velocity, their centrifugal force accelerates them, and they bounce and roll to the outside edge of the feed plate, where they fall off. The final velocity attained by the falling particles will approach that of the angular velocity of the outside edge of the feed distributor plate. The direction of this velocity vector is tangential to the circumference of the distributor plate. Once in the gas, the particles encounter gravity, aerodynamic drag,

119

4.3 Classification of Ceramic Powders

and centrifugal force vectors, ]/, that change the velocity vector, 2, both in direction and in magnitude according to Newton's Second Law [26]: ~ ~i - m d Ct i dt

(4.39)

where m is the mass of the particle (pTrd3/6 for a sphere of diameter d and density p). The gravity force, fv, will be directed downward in the classifier, as shown in Figure 4.11, and will be given by [13] [cv = m ( p -

(4.40)

Pfluid)g

where g is the gravitation constant. Fluid enters the classifier tangentially and then gradually turns radially into the rotor. The net slip velocity vector, ~ (= Uga~- /~particle), between this flow and the velocity of the particle will lead to an aerodynamic drag force vector, fD, that will act in a direction opposite to the slip velocity vector, as shown in Figure 4.11. leD =

88

1 " ~Pfluid

~2 " CD.

(4.41)

In this expression 88 2 is a characteristic area of a spherical particle, 1 -2 is a characteristic kinetic energy for the flow, and CD is the ~PnuidU drag coefficient [27]. For spheres, the drag coefficient is given as a function of the Reynolds number (Nae Ctpd/t~), as shown in Figure 4.12 [28]. At low Reynolds -

-

102

O

10 - "

r

~9

+-

.

'

.

.

.

.

.

.

.

.

~

Ellipsoid 1:0.75

Stokes,s l a : \ ~

u--~d [~ Disk

,-

~

0

...........

u---~d (~

~ ........

~.J---

u

o m 10-1_

l Sphere

m L_

!

'

u__ ~ ,~_.~,.~.

'

/

Ellipsoid 1:1.8 10-2 2 x 10-1

I

1

I

10

. I. . . . . . I . . . . . .I . 102 103 104

I

10 s

106

Reynolds number (Re) F I G U R E 4.12 Drag coefficient for particles moving relative to a fluid. Adapted from Eisner [28] and K l u m p a r et al. [25, pp. 17-19].

120

Chapter 4 Comminution and Classification of Ceramic Powders

numbers, (i.e., NRe < 1.0), Stokes's law applies, reducing the drag force to the simple formula [14] ]~D= 3~rt~t2.

(4.42)

Owing to the forces acting on the particles, there may be a component of angular velocity, Va, in the particles' motion. This component will give rise to the centrifugal force vector [13],/~c"

[c = mv2a r

(4.43)

directed radially as shown in Figure 4.11 from the radial position, r, of the particle. The magnitude of all these forces is highly dependent upon the diameter of the particle, either through its dependence on the mass of the particle, which is equivalent to pTrd3/6, or through the drag coefficient and characteristic area. As a result, the large particles will be affected more by gravity and centrifugal forces and less by aerodynamic drag, and they will end up in the coarse particle chamber. Intermediatesized and fine particles will be affected more by aerodynamic drag and less by gravity and centrifugal forces and will be directed towards the rotor. Sufficiently small particles will pass through the pins without contact and out through the fine particle chamber with the bulk of the gas flow. (Fine particles are separated from the gas flow by an external cyclone.) The intermediate-sized particles are deflected by the pins into the coarse particle chamber. Numerical techniques have been used to predict the gas flow fields in air classifiers. These flow fields are used to compute particle trajectories using the preceding equations, giving rise to the prediction of performance of air classifiers [29]. 4.3.2.2 C o l l i s i o n

For a collision to occur, the particle must be aerodynamically captured by the rotating pin shown in Figure 4.11. There are primarily three ways [30] for this to occur (Figure 4.13): 1. direct interception, 2. inertial deposition, 3. electrostatic deposition. Electrostatic deposition and other mechanisms of capture such as diffusional deposition and thermal deposition will not be discussed in this section, as they affect only the smallest particles, which might remain stuck to the pins until being knocked off by large colliding particles. The efficiency of collision, Ek, is given by the ratio of the crosssectional area of the fluid stream from which all the particles are removed to the cross-sectional area (projected in the flow direction) of

4.3 Classification of" Ceramic Powders

121

F I G U R E 4.13 Streamlines and particle trajectories approaching a pin. Taken from Klumpar et al. [25, pp. 17-19].

the pin. Each collision mechanism has its specific collision efficiency relationship. For direct interception the collision efficiency Ek is given by -1

=

-

.

(4.44)

For other geometries, other collision efficiency expressions will apply [31]. A discussion of flat plate geometry is given by Rajhans [32], and one for spheres is given by Ottavio and Goren [33]. For inertial collision, the collision efficiency is a function of the Stokes number, as shown in Figure 4.14 [34], where the Stokes number is given [1] by N8 t

=

(Ypin - /~particle)(P -- Pfluid) d2 18/-tdpin

(4.45)

where Vapin is the angular velocity of the pin with diameter dpi n (-- Db in Figure 4.14). Other pin geometries are also presented in Figure 4.14. The total collision efficiency for both mechanisms is simply the sum of the efficiency from all the active mechanisms. For classification using cylindrical pins, the collision efficiency is dependent on the angular velocity of the pins, the number of pins in the circle, the diameter of the pins, and the diameter and density of the particles. As a result, both the feed characteristics and the operating conditions will determine what intermediate sizes will be deflected by the pins back toward the coarse particle chamber. The force imparted to a particle during contact with a pin will determine its rebound velocity. Once a particle is in free flight again, gravity, drag, and centrifugal forces must be

122

Chapter 4 Comminution and Classification of Ceramic Powders 1.0 O O.89" -

.!

0.7-

..-...O=o

Oz

0.1-

(9

01 0.3(~ 0.2I"= 0.10

0.01

Intercepts: Ribbon or

cylinder:

//0.~.1

--

1

Sphere:~ i

",

''I

9

01

,

,

,

i'"'~ ............... ,'

I0

1.

I

,

i

i

w

!

I

10

'

'

'

'

'

100

Nst FIGURE 4.14 Capture efficiency by inertial impaction for spheres, cylinders and ribbons. Taken from Langmuir and Blodgett [34].

accounted for to establish the dynamics of the particle. With a sufficiently large collision force, particles may fragment, resulting in the comminution of the feed.

4.3.2.3 Nonspherical Particles In the preceding sections, it has been convenient to define a spherical shape equivalent in volume to irregularly shaped particles. This is a typical simplification used throughout the classification industry, but it is fundamentally in error. For irregularly shaped particles, the specific drag force will not be parallel to the motion of the particle unless the particle has a certain symmetry or a specific orientation. For most real particles, the drag force will cause the particle to rotate as well as change translational velocity. Therefore, both force and moment analyses must be performed for precise accuracy. Moment analysis adds a level of complication that is often neglected. The drag on irregularly shaped particles is discussed in detail by Clift et al. [35]. Basically, the irregularly shaped particle gives rise to a different drag coefficient versus Reynolds number expression, as shown in Figure 4.12 for simple geometries. For aspect ratios less than 1, the drag coefficient is less than that of an equivalent sphere, but for aspect ratios greater than 1, the drag coefficient is greater than that of an equivalent sphere, as shown in Figure 4.12. Equations describing the gravity and centrifugal forces for irregularly shaped particles will not differ from an equivalent sphere if the movement of the center of mass of the particles is considered.

4.3 Classification of Ceramic Powders

123

4.3.3 Size Selectivity, Recovery, a n d Yield Size selectivity is the best measure of classifier performance under a given set of operating conditions. Size selectivity, SS(d), is defined as the ratio of the mass of particles of size d entering the coarse stream to the quantity of size d in the feed. The equivalent mathematical expression is [36] SS(d) =

WcFc(d) WcFc(d ) + WwFw(d)

(4.46)

where We is the mass flow rate of coarse fraction, WWis the mass flow rate of the fine fraction, Fc(d) is the cumulative percent mass less t h a n size d of the coarse stream, and Ff(d) is the cumulative percent mass less t h a n size d of the fine stream. Selectivity of a typical classifier is plotted as a function of size in Figure 4.15. Selectivity monotonically increases from 0 to 1 as size increases (Curve b-b'). Even though size selectivity is a complete measure of classifier performance, the user is often required to take a shortcut method of expressing performance on a specific feed material. A practical m e a s u r e m e n t of overall classification performance for a given application can be obtained by calculating recovery and yield. Recovery is the relative amount of material in the feed that is finer t h a n size d that is recovered in the product. Recovery, R(d), expressed

100

>,, >

,,,,,,,

(,,)

a) 50

m,,,

q) w

ffl

L

Dso%

_--

O0

Particle diameter, D (pm) F I G U R E 4.15 17-19].

Types of size selectivity curves. Taken from Klumpar et al. [25, pp.

124

Chapter 4 Comminution and Classification of Ceramic Powders

as a fraction of the feed, can be calculated from the cumulative particle size distribution data as follows. When the fine stream is the product [11, p. 329] WfFf(d) Rf(d) = WwFw(d) + WcFc(d)

(4.47)

and when the coarse stream is the desired product Re(d) =

WcFc(d) WwFw(d) + WcFc(d)"

(4.48)

The difference between the fine and coarse stream recovery is sometimes called classifier efficiency (i.e., E ( d ) = Rf(d) - Re(d)). Yield, on the other hand, is a measure of the product obtained regardless of quality and calculated as a fraction of the feed. When the fine stream is the product [11, p. 329], WW Yf-- w f + w c

(4.49)

and when the coarse stream is the product Yc =

We

wr+ Wc"

(4.50)

4.3.4 Classifier Efficiency The perfect classifier would send all particles in the feed larger than a designated "cut size" to the coarse stream and all the particles smaller than the designated cut size to the fine stream (curve a - a ' Figure 4.15). This assumes that size is the only characteristic influencing particle trajectories. Other characteristics such as specific gravity and shape will also affect the forces acting on the particles and influence their trajectories and therefore will significantly affect a classifier's performance. Real classifiers suffer from two types of inefficiency. The first type occurs because the trajectory taken by a specific particle size varies from particle to particle. The probability that a particle smaller than the "cut size" will end up in the coarse stream is not 0. Instead, the probability increases monotonically from 0 for particles much smaller than the cut size to 1.0 for particles much larger, resulting in misplaced material (curve b - b ' , Figure 4.15). The identity of the cut size is, therefore, lost in real classifiers. A substitute cut size is defined at SS(d) = 0.5 and is the size, d, of the particles whose probability of entering either stream is 50%. A measure of the slope of the probability function at the cut size is the Sharpness index, s, which is the ratio of size of the particles whose

4.3 Classification of Ceramic Powders

125

probability of entering the coarse stream is 25% to the size of the particles whose probability of entering that stream is 75%. s = dss~d)=O.25

(4.51)

dss(d)=O.75"

An ideal classifier would have a sharpness index of 1.0; real classifiers have values less t h a n this. Industrial classifiers operating properly will have sharpness index values between 0.5 and 0.8. Actual sharpness index values will change as a function of the properties of the feed and operating conditions. The other type of classifier inefficiency is apparent bypass, a. If, because of mutual interference or other reasons, some of the feed material bypasses the separation and reports to either the fine or the coarse streams, then a certain percentage of one of the product streams will have the same particle size distribution as the feed material. Both the apparent bypass and the sharpness index dictate the performance of classifiers.

4.3.4.1 Effects of Apparent Bypass and Sharpness Index A comparison of the effects of apparent bypass and the sharpness index on the particle size distributions of the coarse and fine streams is given in Figure 4.16(a)-(d). In each figure, the product is defined by a single control point (i.e., 95% less t h a n 150 t~m) and produced from the same feed ( i.e., 50% less than 150 ftm). Figure 4.16(a) shows the results from an ideal classifier, where the apparent bypass is 0 and the sharpness index is 1.0 (a = 0, s = 1.0). Figure 4.16(b) shows the results from a real classifier with an apparent bypass into the coarse stream of 30% (a = 0.3, s = 1.0). Bypass into the coarse stream is essentially the only type of bypass observed in classification equipment. The apparent bypass does not affect the size distribution of the fine stream. However, it does affect that of the coarse stream: the ratio of the coarse stream mass to the fine stream mass increases and the recovery of 150 t~m particles decreases. Figure 4.16(c) shows the results from a real classifier with a sharpness index of 0.6 (s = 0.6, a = 0.0). The cut size, ds0, must be lowered to achieve the desired control value of 95% less t h a n 150 ftm. This trend is typical of classifier behavior. The lower the sharpness index, the smaller is the cut size required to produce the desired single-point control value. Figure 4.16(d) represents a typical industrial air classifier having both types of inefficiency (a = 0.3 and s = 0.6). The ratio of the coarse stream mass to the fine stream mass is further increased and the recovery value of the 150 ftm particles is further decreased by the combination of both types of inefficiency.

126

Chapter 4

35

a=O

....... s = 1.0

""I

Comminution and Classification of Ceramic Powders '

'

'

' ' " ' 1

/'

a=O

3 0 - _ d5o= 150 ~m

s=0.6

d5o=122.51~m

25 ~o

20 Fine

r

arse

-~ 9 15

0 a

~,~Feed~

10

|

35 30

i

' ' a=0.3

25

~

,

,

,

, , , I

...... s=l.0

C

i

,

i

i

Jli,

l

1 , , , ..... 1 dso =15011m

Feed~

~

I

J

:

J

,

JiJl

i

I

i

i

i

|ii

a = 0.3

~

_

, I

s = 0.6

i

,,

~

I

I

,t

!

I , , i

i

,

i

,

i,,

I

d5o = 122.5 l~m

Fines A

2o

Fine

arse

~ 15

Feed

10

b

, , l. , , , i i 100

D (~m)

l

, I t i ,,,,I 1000 10

1O0

D (llm)

10()0

F I G U R E 4.16 Size distributions for various types of classifier performance. Taken from Klumpar et al. [25, pp. 17-19].

The effects of bypass on size selectivity are shown in Figure 4.15. Bypass reporting to the fine stream changes curve b to curve c. Bypass reporting to the coarse stream moves from curve b' to curve c'. Comminution of the particles in a classifier will give rise to a size selectivity curve such as d. Analysis of various types of industrial classifiers has led to the observation that the sharpness index is essentially constant for a classifier (with a fixed geometrical configuration) over its normal operating range. Assuming that bypass is minimal, only two things affect the size distribution of the fine stream: the size distribution of the feed and the cut size. Hence, if the size distribution of the feed is constant, only the cut size (dso) will affect the size distribution of the fines. Bypass can be minimized by proper design and operation of the classifier.

4.3 Classification of Ceramic Powders

127

4.3.5 Wet Classification Equipment Wet classification is performed by filtration, settling, centrifugation, and hydrocyclones. When operated in conjunction with grinding equipment, the wet classification equipment must operate continuously and give a pumpable fluid. This is often accomplished in practice with hydrocyclones because the other methods are unsuitable (e.g., settlers and centrifuges are used for dilute suspensions, filters and screens produce a nonpumpable cake.) 4.3.5.1 H y d r o c y c l o n e s (and Cyclones) Hydrocyclone operation has been reviewed in two books, one by Svarovsky [37] and the other a conference proceedings edited by Svarovsky and Thew [38]. Hydrocyclones are generally geometrically similar to one of two families described by Rietema [39] and Bradley [40]. The geometric characteristics of these two families are given in Figure 4.17. In hydrocyclone design, the particle laden flow enters radially and rotates within the body of the hydrocyclone. Forces of gravity, centrifugal and drag, act on the particles to force a separation. The particles larger than the cut size are sent to the underflow, and the particles smaller than the cut size are sent to the overflow along with most of

Overflow Inlet

r

im

I

il

L',

c

\

\

Ou

Hydrocyclone Family

DI/Dc

Do/D c

Rietema

0.28

.34

Bradley

1/7

1/5

F I G U R E 4.17

I/D e

/

/I'

t

Underflow

L1/D c

I./D c

0

.4

-

5.0

10"-20" ....

1/3

1/2

Cyclone dimensions.

9" .........

128

Chapter 4 Comminution and Classification of Ceramiv Powders

the liquid. The cut size of a hydrocyclone has been shown by Svarovsky [41] and Medronho [42] to be determined from the Stokes number, Nsts0 = [(Ps - Pf)Vcdso~]/18ttDc, where Vc is the superficial velocity in the cyclone body and D e is the diameter of the cyclone. The product of the Stokes number and the Euler number, NEu = 2AP/pfV2c, where AP is the pressure drop across the cyclone, is a constant for the geometrically similar hydrocyclones. For Rietema's hydrocyclones the product is a function of the volumetric flow rate underflow to throughput ratio, Rw, and feed solids volume fraction, ~bs: NstsoNEu = 0.474 [ln(1/Rw)] ~

exp(8.96 ~bs).

(4.52)

Two other dimensionless equations 42 describe the operation of Rietema's hydrocyclones. 371.5 x,~70"ll6Reexp(-2.124~s)

NEu =

(Du~ 4"75 ~r-o.3o Rw = 1218 \De/ X'Eu

where the hydrocyclone Reynold's number is given by N R e -- D cVcPW

tt and the superficial velocity in the cyclone body, Vc = 4Q/~rD2c, where Q is the volumetric feed flow rate. The second design equation for the flow underflow to throughput ratio, Rw, is given in terms of the underflow diameter, Du, and the Euler number, which is a useful aspect of this engineering correlation. Similar equations for the Bradley family of hydrocyclones with different geometries are [43] NstsoNEu

NEu

=

0.055[ln(1/Rw)] ~ exp(12(bs)

=

258 N~e87

(4.53) ~ D u ~ 2"63 ~7-1.12 Z'Eu 9

Rw = 1.21 • 10 s \Dcc]

The size selectivity, SS(d), of these types of hydroeyelones is given either by Lynch and Rao [45]: 1

SS(d) . . . . . . . . . . . . . . . . . . . . exp 3'

(4.54) + exp(7) - 2

where dso~ is the "cut size" defined earlier, d is the particle diameter, and 7 is an experimentally determined parameter typically 4.9 [45] to

4.4 Comminution and Classification Circuits

129

5.1 [26], or by Plitt [46],

[

SS(d) = 1 - exp - 0.693

d

n

(4.55)

where n is an experimentally determined constant typically n = 3.12. Other size selectivity curves have also been published [44, 47]. Apparent bypass is minimized by the section of the overflow tube that extends inside the cyclone to a level below the feed port as shown in Figure 4.17. These equations with the operating parameters give the performance characteristics of a specific hydrocyclone design. A similar series of equations can also be used to describe the performance of gas cyclones of different, geometrically similar design. Gas and liquid cyclones are also often used for size classification in grinding circuits. Much more sophisticated models for hydrocyclones have been developed by Rajamani and Miln [48] and Heiskanen [49]. They have used numerical solutions to the Navier-Stokes momentum balance equation for the swirling flow of the hydrocyclone [50]. They also used laser doppler velocimetry to validate this velocity profile, as shown in Figure 4.18(a). They then used a force balance on the different types of particles in the feed to determine the particle trajectories in this swirling flow, shown in Figure 4.18(b). These particle trajectories determine the spacial particle concentration profile, one of which is shown in Figure 4.18(c). This spacial particle concentration profile shows the high density of particles at the wall. This plot also shows some apparent bypass is occurring for this relatively high inlet volume fraction as there is a return of particles that have once migrated to the wall at the bottom of the cyclone. With the spacial particle concentration established, the viscosity can be readjusted because suspension viscosity depends upon the volume fraction of particles. With several interactions of this numerical scheme the size selectivity curve can be predicted for the hydrocyclone under these operating conditions as is shown in Figure 4.18(d).

4.4 C O M M I N U T I O N A N D C L A S S I F I C A T I O N

CIRCUITS Grinding with classified recycled material is commonly practiced in industry. Some industrial grinding circuits contain many mills of different types that operate in conjunction with classifiers of different types. By far the most common type of grinding mills and classifiers is schematically shown in Figure 4.19(a), where material is fed to a grinding mill that is followed by a classifier. The classifier is used to

(a) Left: Measured and predicted tangential velocities in a 75-mm hydrocyclone; right: measured and predicted axial velocities in a 75-ram hydrocyclone. (b) Predicted fluid streamlines and particle trajectories in a 75-ram hydrocyclone.

F I G U R E 4.18

130

FIGURE 4.18 (Continued) (c) Predicted particle volume fraction for 35% weight CaCO3 powder in a 75-mm hydrocyclone. (d) Experimental and predicted size selectivity curves for 35% weight CaCO3 powder in a 75-mm hydrocyclone (the interactions correspond to viscosity corrections made for the particle volume fraction distribution within the hydrocyclone). From Rajamani and Miln [48].

131

132

Chapter 4 Comminution and Classification of Ceramic Powders a FEE) QF =FLOW RATE 1TIF(L) = POPULATION FLOW RATE = Q (L)

CLASSIFIER I:EC'Ys

QR=FLOW RATE

II

mR(L) =POPULATION

PRODUCT

QP=FLOW RATE raP(L) =POPULATION

FEED

QR=FLOW RATE

QF =FLOW RATE

mR(L)= POPULATION

mF(L) = POPULATION CLASSIFIER

PRODUCT v

QP=FLOW RATE raP(L) =POPULATION

FLOW RATE

-Q

GRINDING MILL VOLUME ,V

POPULATION=rlI(L)

FIGURE 4.19 Schematic of (a) postclassification closed circuit grinding and (b) preclassification closed circuit grinding.

4.4 Comminution and Classification Circuits

133

separate the desired particle size distribution and return the larger particles to the grinding mill. This return stream is called the recycle. The recycle ratio, R, is given by

R - QR_ QR QP

(4.56)

QF"

Performing a population balance on the classifier, we find (4.57)

Qm(L) = Qpmp(L) + QRmR(L)

which can be simplified if we define a classification function, C(L), as

C(L)=QRmR(L)_ R mR(L)_ R (SS(L)) Qm(L) R + 1 m(L) R +1

(4.58)

where SS(L) is the size selectivity function shown in Figure 4.15, which is characteristic of the operation of the classifier. Several classification functions, C(L), are given in Figure 4.20. Here, the fraction of particles by mass reporting to the recycle stream is given as a function of particle size, L, for a screen and a cyclone. Several authors have used empirical classification functions instead of classifier performance curves with reasonable results for the overall comminution-classification circuit control. The steady state (i.e., dm/dt = O) macroscopic population balance on a discrete mass basis over the grind-

PJ(R+I) ,i,

1.0 SIEVE

OR CLASSIFIER C(L)

0

L(cut) FIGURE 4.20

SIZE, L

Size selectivity function for sieves and classifiers.

134

Chapter 4

Comminution and Classification of Ceramic Powders

ing mill is given by 0 = QFmF(L) + QRmR(L) _ Qm(L_____~)+ BIn(L) - DIn(L) V V V

(4.59)

where B ~ ( L ) and D ~ ( L ) are the birth and death functions for the total system. This equation can be simplified using the classification function C(L), the definition of the recycle ratio, R, and the mean residence time r = V/(QR + QF),

0

= mR(L)

~(R + 1 ) -

1 -- C ( L ) m ( L ) + BIn(L) - Dm(L)

r

(4.60)

or upon rearrangement, mR(L) + r m ( L ) = [1 - C(L)](R + 1) [1 - C(L)] [BIn(L) - DIn(L)].

(4.61)

At this point we must use the birth and death functions described in Section 4.2.4.4 and solve this equation for the population in the mill. Then using the population balance over the classifier equation (4.57), the product population, mR(L) , c a n be determined from the population inside the grinding mill, m(L), as follows: mR(L) = (1 - C(L))(1 + R ) m ( L ) .

(4.62)

This analysis can be used in segments to describe the behavior of multistage comminution classification systems. The cumulative distribution of particles for the feed to the mill, the material in the mill, and the classified product are given in the accompanying Figure 4.21. This figure was constructed by using a R o s i n - R a m m l e r size distribution (L o = 1.0 ~m) both entering and exiting the mill (fl = 1 and kr = 0.5 ~ m - i ) and entering the 5 ~ m sieve classifier with a recycle ratio of 10. In this figure you can see the effect of grinding and the effect of classified product removal. This two step population balance for communition and classification can be rewritten in array form [21]: m

R

[I - C][TJ(t)T-i]{I - C[TJ(t)T-i]}-imF

=

(4.63)

where C is the classification array and [TJ(t)T -i] is the operational a r r a y for a continuous mill operating on the mill feed equal to the sum of the process feed, mR, and the recycle, mR, which is described by equations (4.31c) for T and (4.36b) for J(t). The classification array is defined by a population balance around the classifier: m

R -

Qm [I - C ] ~ = [I - C](1 + R)m. ~F

(4.64)

4.5 Summary

FIGURE 4.21

135

Size distribution for a ball mill grinding circuit.

By analogy between equations (4.62) and (4.64), we can see that L+AL [I

-

C] ~ f JL

[1 - C(L)] dL

(4.65)

for each element of the C array. In some cases, the feed powder has a sufficient fraction of material within the desired size range to w a r r a n t its separation before grinding, as shown in Figure 4.19(b). In this case, the classifier is placed before m R --

{ ( I - C ) [ I - (TJ(t)T-1)C]-I(TJ(t)T-1)C

+

I}m F.

(4.66)

This case can also be easily calculated with knowledge of the various arrays that have the same definitions as here except that the classification array has a different definition, which can be determined from the following equation: mR = (I -- C) (Q m + QFmF)

(4.67)

4.5 S U M M A R Y In this chapter, the fundamentals of classification and comminution of ceramic powders have been described. Comminution is described by birth and death functions in a population balance. These birth and

136

Chapter 4 Comminution and Classification of Ceramic Powders

death functions depend on two parameters characteristic of the material being ground (i.e., the specific breakage rate and the primary progeny function). Classification of ceramic powders according to size can be performed by m a n y different types of equipment depending on whether the powder is being ground wet or dry. Each type of equipment has its own characteristics for operation. These operating characteristics can be reduced to a selectivity curve. Combining this selectivity curve and the population balance for the grinding mill a model of a comminution circuit can be developed which is useful for their design and control.

Problems 1. Experiments have shown that 750 kWhr/ton is required to grind an A1203 powder form 1 mm to 10 t~m. Determine the energy required to grind the same powder from 10 t~m to 1 t~m using the laws of Kick, Ritenger, and Bond. Note that C is different for each grinding law. 2. For a hydrocyclone with De, of 10 cm operating on an aqueous suspension of ZrO2 fed at a velocity of 5 m/sec, determine the size selectivity function. 3. For an open circuit ball mill 5 liters in volume operating at a flow rate of 1 liter per hr the product is a ZrO2 suspension with a Rosl i n - R a m m l e r size distribution with a weight mean size of 0.5 tLm and a volume to surface mean diameter of 0.7 t~m. Determine the feed distribution to the mill assuming the value of fl the grinding selectivity factor is 1.0 and k is 0.5 (hr/tLm) -1. 4. If a 0.8 t~m screen with recycle ratio fixed at 5 was used as a classifier in combination with the preceding ball mill, what would be the size distribution leaving the mill? What would be the size distribution leaving the classifier?

References 1. Perry, R. J., and Chilton, C. H., "Chemical Engineers' Handbook," 5th ed. McGrawHill, New York, 1973. 2. Lowrison, G. C., "Crushing and Grinding." Butterworth, London, 1974. 3. Van Cleef, J., Am. Sci. 79, 309 (1991). 4. Committee on Comminution and Energy Consumption, "Comminution and Energy Consumption," Rep. No. NMAB-364. National Materials Advisory Board, National Academy of Science. Washington, DC, 1981. 5. MacPherson, A. R., Society of Mining Metallurgical Engineers-American Institute of Mining Engineers Fall Meet. Exhib., Denver, CO, Prepr. No. 6184 (1963). 6. Bond, F. C., Min. Eng. (Littleton, Colo.) March, p. 315 (1953). 7. Peterson, C. R., Weiss, M. A., Klumpar, I. V., and Ring. T. A., "Shale Oil Recovery Systems Incorporating Ore Beneficiation," Final Rep. DOE/ER/30013. Dept. of Energy, Washington, D.C. October 1982.

References

137

8. Leschonski, K., "Possibilities and Problems encountered in the Mechanical Production of Submicron Particles," in Ceram. Powder Process. Sci., Proc. Second Ann. Mtg. on Ceram. Powder Process. Sci., Bergtesgaden, FDR, Oct. 1988, H. Hausner, G. L. Messing, S. Hirano, Eds., Deutsche Keramische Gesellschaft, Koln, Germany, p. 521-534. 9. Griffith, Philos. Trans. R. Soc. London, Ser. A 221, 163 (1920). 10. Ikazaki, F., Kamiya, K., Uchida, K., Gotoh, A., and Kawamura, M., Proc. World Congr. Partic. Technol. 2nd, Kyoto Japan, 1990. (1990), p. II-345-351. 11. Prasher, C. L., "Crushing and Grinding Process Handbook," pp. 267-285. Wiley, Chickchester, 1987. 12. Sedlatschek, K., and Bass, L., Powder MetaU. Bull. 6, 148-153 (1953). 13. Austin, L. G., Shoji, K., Bhatia, V. K., Jindal, V., Savage, K., and Klimpel, R. R., Ind. Eng. Chem. Process Des. Dev. 15(1), 187-196 (1976). 14. Austin, L. G., Klimpel, R. R., and Beattie, A. N., in "Design and Installation of Communinution Circuits (A. L. Mular and G. V. Jergensen, eds.), pp. 301-324. AIME, New York, 1982. 15. Ramkrishna, D., Rev. Chem. Eng., July, p. 58 (1986). 16. Kapur, P. C., Chem. Eng. Sci. 25, 899-901 (1970). 17. Kapur, P. C., Chem. Eng. Sci. 27, 435-431 (1972). 18. Mulone, R., and Bowen, P., Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992), private communication. 19. Rosin, P., and Rammler, E., J. Inst. Fuel 7, 29-36 (1933). 20. Gandy, G. A., Gumtz, G. D., Herbst, J. A., Mika, T. S., and Fuerstenau, D. W., Trans. Aime 241, 538-549 (1969). 21. Herbst, J. A., Rajamani, K., and Kinneberg, D. J., "ESTIMILL, A Program for Grinding Simulation and Parameter Estimation with Linear Models--Program Description and User Manual." Utah Comminution Center, University of Utah, Salt Lake City, 1988. 22. AIME, AIME Meet., New York, Prepr. 71-B-78 (1971). 23. Randolph, A. D. and Larson, M. A. "Theory of Particulate Processes." Academic Press, New York, 1971. 24. Levenspiel, O., "Chemical Reaction Engineering," pp. 270-300. Wiley, New York, 1972. 25. Klumpar, I. V., Currier, F. N., and Ring, T. A., Chem. Eng., March 3, (1986) p. 77. 26. Housner, G. W., and Hudson, D. E., "Applied Mechanics and Dynamics," 2nd ed., p. 2. Van Nostrand-Rheinhold, New York, 1959. 27. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena," p. 192. Wiley, New York, 1960. 28. Eisner, F., Proc. Int. Congr. Appl. Mech., 3rd, 1930, p. 32 (1930). 29. de Silva, S. R., Walsh, D. C., Johansen, S. T., BergstrOm, T., and Bernotat, S., Kona (Hirakata, Jpn.) 9, 131-138 (1991). 30. Wong, J. B., Ranz, W. E., and Johnstone, H. F., J. Appl. Phys. 27, 161-169 (1956). 31. Licht, W., "Removal of Particulate Matter from Gaseous Wastes--Filtration." Am. Pet. Inst., Washington, DC, 1961. 32. Rajhans, G. S., in "Air Sampling Instruments" (P. J. Lioy, and M. J. Y. Lioy), 6th ed. p. Q-1. 1983. 33. Ottavio, T. D., and Goren, S. L., Aerosol Sci. Technol. 2, 91-108 (1983). 34. Langmuir, I., and Blodgett, U.S. Army Air Forces Tech. Report 5418, (U.S. Dept. Comm., Off. Tech. Serv., Rep. PB27565) (1946). 35. Clift, R., Grace, J. R., and Weber, M. E., "Bubbles, Drops and Particles," p. 69. Academic Press, New York, 1978. 36. AIChE Equipment Testing Procedures Committee, "Particle Size Classifiers--A

138

Chapter 4 Comminution and Classification of Ceramic Powders

Guide to Performance Evaluation," AIChE Equipment Testing Procedure, AIChE, New York, 1980. 37. Svarovsky, L., "Hydrodyclones. " Technomic Publ. Co., Inc., Lancaster, PA, 1984. 38. Svarovsky, L., and Thew, M. T. eds., "Hydrocyclones: Analysis and Applications." Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. 39. Rietema, K., Chem. Eng. Sci. 15, 298-325 (1961). 40. Bradley, D., "The Hydrocyclone," p. 129. Pergamon, Oxford, 1965. 41. Svarovsky, L., "Hydrocyclones," pp. 1-11 and 44-57. Holt, Rinehart & Winston, Eastbourne, PA, 1984. 42. Medronho, R. A., Ph.D. Thesis, University of Bedford (1984). 43. Antunes, M., and Medronho, R. A., in "Hydrocyclones: Analysis and Aplications" (L. Svarovsky and M. T. Thew, eds.), p. 8. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. 44. Rosin, P., Rammler, E., and Intelmann, D., Z. VDI 76, 433-437 (1932). 45. Lynch, A. J., and Rao, T. C., Proc.--Int. Miner. Process. Congr. 11th, Cagliari, Italy, 1975, pp. 1-25 (1975). 46. Plitt, R. A., CIM Bull., 69(776) 114-123 (1976). 47. Yoshioka, N., and Hotta, Y., Chem. Eng. Jpn. 19(12), 632-640 (1955). 48. Rajamani, R. K., and Miln, L., Proc. Int. Conf. Hydrocyclones, 4th, South Hampton, UK, 1992 (published by BHRA-Fluid Engineering Center, Cranefield, Bedford, England). 49. Heiskanen, K., Kona, (Hirakata, Jpn.) 9, 139-148 (1991). 50. Hsieh, K. T., and Rajamani, R. K., AIChE J. 37, 735-746 (1991).

5

Ceramic Powder Synthesis with S o l i d Phase Reactant

5.1 O B J E C T I V E S This chapter discusses the fluid-solid and solid-solid reactions used to produce ceramic powders. The first aspect of this discussion is the spontaneity of a particular reaction for a given temperature and atmosphere. Thermodynamics is used to determine whether a reaction is spontaneous. The thermodynamics of the thermal decomposition of minerals and metal salts, oxidation reactions, reduction reactions, and nitridation reactions is discussed because these are often used for ceramic powder synthesis. After a discussion of thermodynamics, the kinetics of reaction is given to determine the time necessary to complete the reaction. Reaction kinetics are discussed in terms of the various rate determining steps of mass and heat transfer, as well as surface reaction. After this discussion of reaction kinetics, a brief discussion of the types of equipment used for the synthesis of ceramic powders is presented. Finally, the kinetics of solid-solid interdiffusion is discussed.

139

140

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

FIGURE 5.1

Shrinkingcore model.

5.2 I N T R O D U C T I O N A solid is a reactant in two general types of powder synthesis reactions. One type is a fluid-solid reaction, where the fluid is either a liquid or a gas. The other type is a solid-solid reaction. Fluid-solid reactions can be represented by A(fluid) + b B(solid)--~ d D(solid)

(5.1)

A(fluid) + b B(solid)--* d D(solid) + e E(fluid)

(5.2)

b B(solid)--* d D(solid) + e E(fluid).

(5.3)

With each of these reactions, a solid of one type (B) is the reactant and a solid of another type is the product (D). A fluid is also a reactant or a product of the reaction. In some cases, the solid product (D) forms a shell on the outside of particle B, giving a diffusion barrier for further reaction. This type of reaction is modeled as a shrinking core, as seen in Figure 5.1. In other cases, the product D flakes off the surface of particle B, because there is a large difference in the molar volume of reactant B and product D. This type of reaction is modeled as a shrinking sphere as seen in Figure 5.2 [1].

Shrinkingsphere model. From Levenspiel [1], copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

FIGURE 52,

5.3 Thermodynamicsof Fluid-Solid Reactions

141

Fluid-solid reactions include thermal decomposition of minerals, roasting (oxidation) of sulfide ores, reduction of metal oxides with hydrogen, nitridation of metals, and carburization of metals. Each type of reaction will be discussed from the thermodynamic point of view. Then reaction kinetics for all of the various rate determining steps in fluid-solid reactions will be discussed for two general models: shrinking core and shrinking particle. Solid-solid synthesis reactions operate by different mechanisms, which include solid state diffusion and chemical reaction. Diffusion in ceramic solids is always ionic in nature and depends on defect or hole diffusivity, as well as, electron conductivity. Once the ionic reactants are in close association, chemical reactions can take place. Before the reaction kinetics can be discussed, the thermodynamics must be discussed to see if the reactions are either spontaneous or at equilibrium. To determine if a reaction is either spontaneous or nonspontaneous as written, the Gibbs free energy for the reaction must be determined. The Gibbs free energy of reaction may be calculated from the free energy of formation for each of the species in the balanced reaction equation after correction for the reaction conditions (i.e., temperature and pressure). When the free energy of reaction is negative, the reaction is spontaneous. When the free energy is positive, the reaction is nonspontaneous; and when the free energy is zero, the reaction is equilibrium. A discussion of the thermodynamics of fluid-solid, thermal decomposition and solid-solid reactions important to ceramic powder synthesis is discussed next.

5.3 T H E R M O D Y N A M I C S REACTIONS

OF FLUID-SOLID

The thermodynamics discussed in this and the following section draws heavily from the book Physical Chemistry by Castellan [2]. A classic example of a solid-fluid ceramic powder synthesis reaction is that of calcination and dehydration of natural or synthetic raw materials. Calcination reactions are common for the production of many oxides from carbonates, hydrates, sulfates, nitrates, acetates, oxalates, citrates, and so forth. In general, the reactions produce an oxide and a volatile gaseous reaction product, such as CO2, SO2, or H20. The most extensively studied reactions of this type are the decompositions of magnesium hydroxide, magnesium carbonate, and calcium carbonate. Depending on the particular conditions of time, temperature, ambient pressure of CO2, relative humidity, particle size, and so on, the process may be controlled by a surface reaction, gas diffusion to the reacting

142

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

surface, or by heat transfer to or from the reacting surface. The kinetics of each of these rate limiting steps is considered later. Let us first consider the thermodynamics of decomposition of calcite (CaCO3). CaCO3(s)--~ CaO(s) + CO2(g)

kcal A~o-29S = 44.3 ~ ~"RXN mole

(5.4)

where ~AR~Z0-29S X N is the enthalpy of reaction at standard conditions (i.e., 1 atm pressure and 298 K). This reaction is strongly endothermic (i.e., AH RXN ~ is +), which is typical of most salt decompositions. This means that heat must be supplied to the reaction for it to continue. At different temperatures, the heat of reaction, AHRxN (T), is given by [2]" FT AHRxN(T) = AHRxN(298)+ ~ ACpd T (5.5) J298 where hCp is the sum of the molar heat capacities (at constant pressure) of all the products of the reaction times their stoichiometric coefficients minus the sum of the molar heat capacities of all the reactants times their stoichiometric coefficients (i.e., CpcaO -~ Cpc02 - Cpcac03 ). The standard free energy, AG O, of reaction is determined by noting [2] AGRxN = AHRxN -- T ASRxN, where ASRxN is the entropy change of the reaction. Tables of AGRxN, AHRxN, and ASRxN are available in the appendix of this book. For the thermal decomposition of calcium carbonate (barium carbonate, magnesium carbonate, and magnesium hydroxide), the standard free energy, AGO, of reaction is plotted as of function of temperature in Figure 5.3 [3]. The standard free energy, AGO, is only part of the total free energy of reaction, AGRxN: AGRxN = AGO + RgT In K

(5.6)

where Rg is the gas constant. A second term contains the distribution coefficient [2], K, which is defined as follows for the decomposition of calcium carbonate: K = fCO2 aca~ aCaC03

Pc~

(5.7)

PTOT

where fco2 is the fugacity of CO2 (= Pco2/Pwowfor a n ideal gas) and a is the activity of either CaO or CaCO3. The activity of the solids is always assumed to be 1.0, giving the preceding simplification. The second term (with the sign reversed) is plotted in Figure 5.3 as the dashed lines for various values of Pco2When the free energy of reaction, AGRxN, is positive, the reaction is nonspontaneous. When it is negative, the reaction is spontaneous.

143

5.3 Thermodynamics of Fluid-Solid Reactions

30-

PCOz or PH20(atm)

/,

~~.

22-

o

\\~. .

~'~c;~,,.~

,.>~.Q

O

e"

I

/

/

1

i

-

o

.c: rr

-6

.

.

.

/

i"

.

i

.i

11

'"

~

,,"10 T / / " ///10-'

1 /

.,~

..G-' .-" .--"~%,"

x. /

I

,/""

.

.

,I-"

.

~."

O

~

I U

,

~' ..... -----~"'-"~"""""--

,

. ....

i .......- "~

-..-"" ,

9

.

i

_ ...........

_" _

I 10

-14

.

---"

.......

.

/-"

///"

......

,.O' o 1O -, .._ ---...,. 9 102

~ "

..

"~ ~'-..

"~ " "

~"

" - - .. 1 0 3 "-...

-18 -22

-24

, , , , , 2 0 0 300 4()0 500 " ....... 600 7()0 800 " go0 1000 11'001200 1300

Temperature

(K)

F I G U R E 5.3

S t a n d a r d free energy of reaction as a function of temperature: --equilibrium gas pressure above oxide/carbonate or oxide/hydroxide. D a t a from Kingery et al. [3], with additions by T. A. Ring.

When it is zero, the reaction is in equilibrium and the standard free energy, AGO, is related to the equilibrium constant [2] AGO = - R g T In

K e.

(5.8)

For the calcite decomposition reaction, the equilibrium constant, Ke, has the same definition as the distribution coefficient, K, given

in equation (5.7) giving the equilibrium partial pressure of CO2. The dependence of the equilibrium constant on temperature is given by the Clausius-Claperon equation [2]" d l n K e AHRx N (T) = dT RgT 2 "

(5.9)

When AGObecomes zero, the equilibrium partial pressure of CO2, Pco2, above CaCO3 becomes 1 atmosphere. The temperature at which

144

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

this occurs is 1156 K (for MgCO3, T = 672 K, and Pn2o = 1 atm for Mg(OH)2, T = 550~ The free energy associated with the partial pressure of CO2 in air in contact with calcium carbonate at different t e m p e r a t u r e s is shown in Figure 5.3. From the CO2 partial pressure, we can determine the temperature at which the mineral becomes unstable when heated in air. For example, calcium carbonate becomes unstable above 810 K (MgCO3 T > 480 K, Mg(OH)2 T > 445 K, depending on relative humidity). At other partial pressures of CO2 different destabilization temperatures are applicable. The thermal decomposition of metal sulphates, acetates, oxalates, nitrates, and so forth can also be considered with similar thermodynamic considerations. Because the partial pressure of these gaseous decomposition products are minuscule in air, these salts are unstable at all t e m p e r a t u r e s where AG Ois negative. The kinetics of decomposition of these salts, however, is slow at low temperatures. These reaction thermodynamic fundamentals are applicable to all other reactions discussed in this chapter. 5.4

OXIDATION

REACTIONS

Two types of oxidation reactions are of interest in ceramics: oxidation of metals and oxidation of sulfides. The oxidation of sulphides is a common extractive metallurgical process, generating an oxide ceramic powder. The oxide product is usually an intermediate product on the way to metal production but if sufficiently pure it can be used directly as a ceramic powder. A common example is the roasting of zinc sulphide to form zinc oxide, O2(g) + 2 ZnS(s)-* 2 ZnO(s) + 2 SO2(g) kcal AH ~ - - 166.9 ~ RXN -mole

(5.10 )

or the roasting of iron pyrite, FeS2, by the reaction O2(s) + 2FeS2(s)--* Fe203(s) + 4SO2(g) AH ~ RXN

- - 5 9 2 kcal mole --

(5.11)

These reactions are strongly exothermic, which is typical of these types of oxidation reactions. This means that the heat produced by the reaction will heat up the particle and further increase the reaction rate. The equilibrium constant for the oxidation of zinc sulphide is given by 2

Ke =

1

P~o2P~ow Po2

(5.12)

5.4 Oxidation Reactions

145

assuming all gasses are ideal and the fugacities are equal to the partial pressures. The equilibrium constant is related to the standard free energy, AGO, as shown in equation (5.8). When the ratio of partial pressures, P~so2/Po2,is less t h a n that at equilibrium, the reactant, ZnS, is unstable at that temperature. The oxidation of metal powders is a method to produce relatively pure oxides. A common metal oxidation is O2(g) + ~4 A1 ~ ~2 A1203

A~Z0-29S_ _ 268.4 kcal ~'~ RXN -mole"

(5.13)

This reaction, like all metal oxidation reactions, is strongly exothermic. The standard free energy of this and m a n y other oxidation reactions are given in Figure 5.4. Written in this form, the distribution coefficient for all metal oxidation reactions is given by -1

K = Po2 Pwow

(5.14)

assuming an ideal gas. The distribution coefficient and the standard free energy make up the total free energy of reaction according to equation (5.6). The term -RgT In Po2 is also given in Figure 5.4 [4,5] on the outside scale of the graph (pt O and scale). When the standard free energy, AG~ is less t h a n -RgT In Po2/Pwowthe oxide is stable. In gas fired metal oxidations, the fuel gives a combustion gas of a particular CO2 and H20 composition. At these high temperatures, the decomposition of CO2 CO2 ~ CO + 89 02

A~0-2~S _ _ 94.2 kca___ll ~'- RXN -mole

(5.15)

creates an equilibrium partial pressure of oxygen that influences the metal oxide stability at temperature, as does the decomposition of water H20(g) ~ H2 + 89 02

kcal A~z0-2~S_ --57.8 ~ ~'- RXN -mole"

(5.16)

Using the axes exterior to the graph in Figure 5.4, different Po2 values can be easily accounted for by noting either the ratio of H2/H20 using the water decomposition equilibrium (pt. H and H2/H20 scale) or the ratio CO/CO2 using the CO2 decomposition equilibrium (pt. C and CO/CO2 scale).

Problem 5.1. Free Energy of Oxidation Using Figure 5.4 determine the Gibbs free energy of reaction for the oxidation of Mn at 600~ in air (Note" air has aPo2 = 0.21) and combustion gases with the ratio CO/CO2 = 0.1.

146

Chapter 5

Ceramic Powder Synthesis with Solid Phase Reactant Hz/Hz0 ratio 1/1G'1/1/0 ? 1/1"06

Temperature 200

(~

400

CO/CO'rati;1/(O?1/1061/iOS

600

800

1000

1200

1400

1/~0 s

1 / ; 0 z'

1/~0+ 1600

1/1"0:3

11~03

1800

2000

1: lO'

l/;OZ 2200

20

1-

2400

1

,

r'o

-20-~

161-~

'

I0 z. M

"

-4o-i -60

1

lO -~ ~6L 3"

i

+'1

-80

-

11~ 6 -

9

..~-100-

103. I0!

1~~.

,q

~-120-

R

r -140-

%

=

:s ,,,0

+

I~1 - 1 6 0 " lO'

.~-18o. // -~,oo ~

-

-

./

.~>~"

/

,:~,1

"T- T-1

/

. "

10 ? I0611~#"

T " "Tt

" ]-

1~ lO?110'I

--o

10' 10is. -T - 10 s TElement lOxide'

o,

-

2

"

8

~

0

-

- 3 0 0 I....... 473 .l_A._b__Ab solt ute zero

I Melting point

~

I Boiling point

I Sublimation point I Transition point

673

8"t3

10'73

1273

Temperature

14'73

(~

16'73 18'73 20'73 C0/COzratio 1 ~ + HzlHzO ratio

.....1~oo"'i~1s .. o l

,

1_ ~o_'~176 109 110, -2o 1":

.

16100169010 e~ 1(~?~ ~

~

~

~

1060 ~

10 sO

10 42

~

~

ld3e' *,

I

,

2 2 7 3 2473 lOj~

ld;

,

2673-T"

11012

+

1 ~f 1p- ~

+da+1-22 ++ 0 . I

16ze

....

102 ~

1~ +

FIGURE 5.4

S t a n d a r d free e n e r g y of f o r m a t i o n of oxides as a f u n c t i o n of t e m p e r a t u r e . D a t a t a k e n f r o m R i c h a r d s o n a n d Jeffes [4], modified by T. A. Ring, as well as D a r k e n a n d G u r r y [5].

At 600~ the oxidation of Mn line gives AG o = - 1 5 3 kcal/mole. For the partition coefficient term we have + R g T In K = - R g T In PoJPToT = 2.7 kcal/mole

giving a total free energy of reaction of AGRxN = h G ~ + R g T l n K

= - 150 kcal/mole

5.5 Reduction Reactions

147

which is nicely spontaneous. For the combustion gases case, we have a partition coefficient term given by

+RgT In K = - R g T In Po2/PToT. The P02 can be determined for the equilibrium

2CO 2 ~ 2CO

+

0 2

which has the equilibrium equation

Po2P~o

AG O=

-RgT In Ke = -RgT In P~o----~"

Drawing a line between point C and the ratio of 0.1 on the scale outside the graph gives the value of the term -RgT in Po2 for the oxidation of Mn at to 600~ of ~ - 9 0 kcal/mole. This gives the total free energy of Mn oxidation of AGRxN = AGO +

RgT In K = - 6 3 kcal/mole

which is still spontaneous. Such calculations predict that most metals want to oxidize in air, and they do at the surface. Fortunately, this oxide layer grows very slowly in some cases like A1. This slow growth is due to the low electrical conductivity of the oxide layer as electrons, necessary for the oxidation of the metal, must also be transfered across the oxide layer.

5.5 REDUCTION REACTIONS The reduction of oxides in reducing atmospheres is also an important industrial fluid-solid reaction that produces a powder. Because these types of reactions can affect ceramic powder synthesis, they are included in this chapter. However, these reduction reactions are frequently used to produce metal powders and are not often used to produce ceramic powders. These reduction reaction can, however, be the first step in a sequence of steps to produce carbide and nitride powders. Several examples of fluid-solid reduction reactions are Fe304(s) + 4H2(g)-o 3Fe(s) + 4H20(g)

L~/0-298 +36.6 ~kcal RXN =

CuO(s) + H2(g)--> Cu(s) + H20(g)

./0-298_-20.2

RXN --

mole

kcal mol-----e"

(5.16)

(5.17)

These reduction reactions are treated thermodynamically and kinetically in the same way as other fluid-solid reactions in this chapter.

148

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

5.6 N I T R I D A T I O N

REACTIONS

The direct nitridation of metal powders is commonly used to produce Si3N4, BN, A1N, and other nitrides: N2(g) + 3Si(s)--* 89

ATzo-29s 89.5 kca___ll ~'~ RXN = mole

(5.18)

N2(g) + 2Al(s) --* 2A1N(s)

~'~ RXNA ~Z0-298 = 152.8 molekCa---ll

(5.19)

kcal A~zo-298= 1 2 0 . 4 ~ ~'~ RXN mole"

(5.20)

N2(g) + 2B(s)--* BN(s)

These and other nitrogen reactions are strongly endothermic, requiring energy to continue. The standard free energy of several nitridation reactions are given in Figure 5.5. The distribution coefficient is defined by

-1PTOT9 K=PN2

(5.21)

This and the s t a n d a r d free energy make up the total free energy of reaction as shown in equation (5.6). In a gas mixture w h e r e PN2 = 0.79 atm and the rest is an inert gas, all the metal nitrides are stable with respect to their metals, except Fe above 250 K and Cr above 1325 K. In air where PN2 = 0.79 atm, this result is not true because the metals may also oxidize. Due to the presence of oxygen in air, we must also consider the oxidation reactions at the same time as the nitridation reactions. This is done in the next section. In addition to metal nitridation, metal carbides may be reacted as follows: N2(g) + TiC(s)-~ TiN(s) + C(s) N2(g) + MgC2(s)--* MgCN2(s) + C(s)

(5.22) (5.23)

to produce nitrides or carbonitride powders.

5.7 T H E R M O D Y N A M I C S REACTION SYSTEMS

OF M U L T I P L E

In a given system that consists of a solid and an atmosphere, several reactions are possible. For example, a metal powder could react with the nitrogen or the oxygen in air to form the metal nitride or metal oxide. This reaction can be predicted by determining the Gibbs free energy of the various reactions possible and selecting the reaction with most negative Gibbs free energy.

5.7 Thermodynamics of Multiple Reaction Systems

149

60-

40-

0

"~ E

20

0

600

800

1000

1200

1400

ir

7 r

i-.n,"

-20

I|

-40

-60

-8o

I

f "t ~ / ~

~ ~

~'~"

-140 160

FIGURE 5.5 S t a n d a r d free e n e r g y of f o r m a t i o n o f n i t r i d e s as a f u n c t i o n of t e m p e r a t u r e .

P r o b l e m 5.2. W h a t Is the R e a c t i o n P r o d u c t When A1 M e t a l Is E x p o s e d to A i r a t 800~ Two possible reactions are possible: 4 2 N2(g) + 2Al(s)--* 2A1N(s) and 02(g) + ~ A1--, ~ A1203(s)

150

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

The Gibbs free energy is calculated from the equation

AGRxN = AGO+ RgT In K for both reactions. For the oxidation reaction K = Po~ PTOT and for the nitridation reaction K = PN~ PTOT.Using Figure 5.4, the standard Gibbs free energy, AG ~ for the oxidation reaction can be determined at 800~ giving - 2 1 2 kcal per mole. The [RgT In K] term for the nitridation is given by

RgTlnK=l.98cal/mole.lO98Kln[

~J=3.4kcal/mole

yielding

AGRxN = AG O + RgT In K

= - 2 1 2 kcal/mole + 3.4 kcal/mole = - 2 0 8 . 6 kcal/mole of 02

which corresponds to the oxidation of ] moles of A1. Thus the Gibbs free energy per mole of A1 is -156.4 kcal/mole for oxidation. Using Figure 5.5, the standard Gibbs free energy, AG~ for the nitridation reaction can be determined at 800~ giving - 112 kcal per mole. The [RgT In K] term for the nitridation is given by

RgTlnK

= 1.98 eal/mole 91098~

[lO] ~

= 0.5 keal/mole

yielding AGm,e~ = AGo + RgT In K = - 1 1 2 keal/mole + 0.5 keal/mole = -:-111.5 kcal/mole of N2 which corresponds to the nitridation of 2 moles of A1. Therefore, the Gibbs free energy per mole of A1 is -55.75 kcal/mole for nitridation. Considering these two reactions, A120 a will be the equilibrium product because it has the most negative Gibbs free energy of reaction. The competition of metals for oxygen to form their oxides and for carbon to form their carbides is also a common problem for complex equilibrium calculations. In principle, multiple oxide and formation reactions can be considered simultaneously. This is the case for the reactions [6] W + 02 ~ W02

W02 + 102 ~-- W03 C+

89

~CO

2W + 2 C 0 ~ W2C + C02 W2C + 2CO ~ 2WC + CO2

5.9 Fluid-Solid Reaction Kinetics

151

2CO ~ C + CO2 W2C + C ~ 2WC. Which will be considered later in Figure 5.17, which is a plot of log

[Pco2/Pco] versus 1/T for Pco = 1 atm. Here we will see regions of the diagram that show that a particular solid product (e.g., WO2, WO3, W2C, or WC) is stable compared to all the other solid products. The most stable one is determined from the most negative Gibbs free energy of formation. The lines between two regions are constructed when the reaction that transforms one solid into the other is in equilibrium, i.e. hG = 0.

5.8 L I Q U I D - S O L I D

REACTIONS

Several reactions between solids and liquids produce ceramic powders: Ba(OH)2 (1, solution) + TiO2(s)--~ BaTiO3(s) + H20(1)

(5.24)

3H20(1) + 2 A1N(s)--* A1203(s) + 2 NH3(1, solution) (5.25) These reactions take place at ambient temperature and follow shrinking core kinetics [7,8] similar to the solid-gas reactions discussed earlier. These reactions have reasonably fast reaction kinetics at low temperatures because the liquid has a very high concentration of reactant compared to the gas phase.

5.9 F L U I D - S O L I D

REACTION KINETICS

This section draws heavily from the excellent book Chemical Reaction Engineering by Levenspiel [1]. Extensions of this basic theory to heat transfer have been made by the author. For more detail on the effects of heat transfer on the reaction kinetics, please see Wen et al. [9-13]. To consider fluid-solid reaction kinetics a generalized reaction will be considered: A(g) + bB(s) ~- rR(g) + sS(s)

(5.26)

A fluid for our consideration here is either a liquid or a gas. F l u i d solid reactions can be kinetically limited by several steps [8]: 1. Surface reaction, 2. Mass transfer in the boundary layer surrounding the particle, 3. Diffusion in the product layer,

152

Chapter 5

Ceramic Powder Synthesis with Solid Phase Reactant

4. Heat transfer in the boundary layer surrounding the particle, 5. Heat conduction in the product layer. These rate determining steps are shown in Figure 5.6. As the reaction is written in equation (5.26), mass transfer in the boundary layer and mass transfer by diffusion in the product layer can be limiting for the reactant gas, A, making its way in from the bulk gas to the unreacted core, or for the product gas, R, making its way out. In the case of thermal decomposition of a mineral, there is only the solid B on the left-hand side of equation (5.26). These thermal decompositions can also be treated by the same rate limiting steps as given previously. Although the product layer is often porous, it can produce a slower rate of either heat conduction or diffusion t h a n the boundary layer. As a result fluid-solid reactions occur at a sharply defined reaction interface, at a position r within the particle of size R. The mass flux associated with boundary layer mass transfer is given by

J1 = 47r R2K~(CAB - CAR) ~ 47r R2Kg CAB

(5.27)

where Kg is the mass transfer coefficient (given by the Colburn analogy for a sphere),* CAB is the concentration of A in the bulk gas CAB = PAB/ RgT CAR is the concentration of A at the surface of the particle of radius

R(CAR = PAR/RgT). If the concentration of A at the surface of the sphere, R, is near zero, the boundary layer mass transfer is the rate determining step. Simultaneously, there is a diffusive flux of A through the product layer, J2, given by

dCA

J2 = 47r r 2DAE - ~ r

= constant

(5.28)

r

where DAE is the effective diffusion coefficient of A in the product layer. The effective diffusion coefficient for a porous layer is given by (5.29) where OK (= a ~/i8RgT/TrMw) is the Knudsen [14] diffusion coefficient (a is the pore radius), D A is the molecular diffusion coefficient through the gas in the pores, s is the void fraction of product layer, ~ is the tortuosity of the pores (typical value is 2.0). Without a gaseous product, rR(g), the equilibrium concentration of A at the reaction surface, r, is given by CAE = (RgT) -1 exp(-AG~xN/RTr) (5.30) * Colburn analogy: 2 KgDRAA= 2.0 + 0.6 Re1/28cl/3, where Re is the Reynold's number for flow around the sphere of radius R, Sc is the Schmidt number, and DAis the diffusion coefficient for A in the bulk gas.

153

5.9 F l u i d - S o l i d Reaction Kinetics

Moving

Gas

r eJa c t i o_ n

s_u r f a_c e ~ - .- . ~ ( '.f i l m

ffime, i--~-~

,

/

yProduct

I-..

I', IN

I ~

I

\1

i'

B

"

.~

~

'

I

R

f t-:

.,i

I" V I

,/1\ "\

I

- T - - -T - - ---l " ~= E ~;A~--, CRc__~_-] o -a o=,= - - er,~ , ~, - - 7 , 9, ~0~C o , ----4/-- ~..- - 4 ..1 R. c.~-

layer..__

,~.;-.,~i~.i:i:-Yi/!~ A

'

i~ !'\! ",LI .:~"~4 I-'-''~'1IA 14 i/i\\I i i',,

-~..~,.o 8~<

_

i ~ c t _ d d _ ' ~ ~, IXi~'e._ ! t ~ c e'...-'.'] | !-"~ - ~ i

,\ i\,

I

,'

/ /

..---

particle

of

~/Surface

I

I

/I

I

\

~"-~"--

\

~ ~"'

t----"-+~-l---,c i t i , .. I---- ~T-T--i--"'r T s ~x . ' s . . .l / I -

.

.

"~i~

~

I I

,

i

',

'

o

Tg

I

~

'= "~ '--

E

(:~

r 0 rcr R tRadial p o s i t i o n F I G U R E 5.6 Representation of temperatures and concentrations of reactants and products for the reaction A(g) + bB(s) --. rR(g) + sS(s) for a particle of unchanging size.

where A G ~ x N is the standard Gibbs free energy of reaction at the temperature of the reaction plane. The flux due to surface reaction is given by J 3 = 4~r r 2 k r ( e A r -

CAE) ~ 47r r 2 k r

CAr

(5.31)

w h e r e k r is t h e f i r s t - o r d e r r a t e c o n s t a n t for t h e s u r f a c e r e a c t i o n ( T a b l e 5.1 g i v e s v a l u e s c o r r e s p o n d i n g to t h e r m a l d e c o m p o s i t i o n of v a r i o u s

TABLE 5.1

Surface Reaction Rates, a k r - ko exp

R e ac t a n t

Ag2CO3 CuSO4- 5H20 CuSO4- 3H20 KCr(SO4)2- 12H20 KAl(SO4)2- 12H20 NiSO4- 7H20

k o (cm/sec)

1.3 3.3 2.8 2.8 1.2 8.0

x x x • x x

105 107 102 1011 106 108

EA

EA (kcal/mole)

23.4 18.3 15.6 22.6 16.6 19.0

a Data from Shultz, R. D. S., and Dekker, A. O., J. Phys. Chem., 60, 1095 (1956).

154

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

solids) and CAr is the concentration of A at the reaction plane r. If the reaction is far from equilibrium, the concentration of A in equilibrium with the solid B is essentially zero allowing the previous simplification. The total mass transfer is related to the total heat transfer, making sure that the amount of mass reacted is equal to the heat available for reaction. This balance gives J =

Q AH~

(5.32)

The heat flux can be composed of two parts: the heat flux in the boundary layer, Q~ = 47r R 2 h ( T s - TR)

(5.33)

where h is the boundary layer heat transfer coefficient (given by the Colburn analogy* for a sphere), TB is the bulk gas temperature, and TR is the temperature at the surface of the sphere; and the heat flux through the product layer, dT Q2 = 47r r 2 K e-~r

= constant

(5.34)

r

where Ke is the effective thermal conductivity of the product layer. If the product layer is porous the effective thermal conductivity is given by Ke-[1-~+ ks~ f ]

-1

(5.35)

where e is the void fraction in the porous product layer, ks is the thermal conductivity of the solid, and k f is the thermal conductivity of the fluid (liquid or gas) in the pores. To complete the picture, the flux of gas, A, must be related to the change in B and the size shrinking core radius, r. This can be accomplished by considering that the change in moles of B is equal to b times the change in the moles of A, from the reaction stoichiometry, which is also equal to the flux J described by equations (5.27), (5.28), (5.31), and (5.32): -dNB = -bdNA = -bJ.

(5.36)

The change in the number of moles of B is related to the change in the volume of B in the core: - d N B = - - P B d V = - - P B 47r r 2 d r

(5.37)

where PB is the molar volume of the solid (i.e., moles per cc). Using the relation between the flux J and the change in core radius given in * Colburn analogy: 2hR/kf = 2.0 + 0.6 Re 1/2Pr 1/3,where kf is the thermal conductivity of the bulk gas, Re is the Reynolds number, and Pr is the Prandlt number.

5.9 F l u i d - S o l i d Reaction Kinetics

155

equations (5.36) and (5.37) with the definitions of the fluxes for mass and heat transfer equations (5.27), (5.28), (5.31), (5.32) (with eq. (5.33) and (5.34)), it is possible to determine the time dependence of the fractional conversion, X B : ,,:1

'

for the shrinking core model (see Fig. 5.1) for a sphere (cylinder and plate) as given in Table 5.2. For all of these rate limiting steps the fractional conversion is a function of the time, t, divided by the time for complete conversion of the particle, r.

T A B L E 5.2 C o n v e r s i o n , XB, v e r s u s Time for S h r i n k i n g C o r e Model: A ( g ) + b B ( s ) ~rR(g) + sS(s) ,

,,,,

,

B o u n d a r y layer

Pore diffusion

Surface reaction

S p h e r e XB = 1 -- (r/R )3 Rate Controlling Step--Mass Transfer t/r = XB t/r = 1 -- 3(1 -- XB)2/3 + 2(1 -- XB)

pBR r = 3b K g C ~

pB R2 r = 6b De-----~

Rate Controlling Step--Heat Transfer t/r = X s t/r = 1 - 3(1

AH~xN PBR r = 3bh (TB - Ts)

-

X B)2/3

~_

t/r = 1 - (1 - XB) 1/3 T--

2(1

-

PBR bkrCAB

X B)

A H ~ ps R2 r = 6bKe (T B - T S)

P l a t e XB = 1 -- (x/L), L is t h e p l a t e t h i c k n e s s . Rate Controlling Step--Mass t/T = XB

Transfer t/r =

PBL = 3bgg-----~

pB L2 T = 2bDe'----~

Rate Controlling Step--Heat

T

~

t/r = X s AH~xN ps L b h ( T s - Ts)

t i t = XB PBL T-bkrCAs

Transfer

T--

t/r = X 2 AH~xN ps L 2 2 b k e ( T s - Ts)

C y l i n d e r XB = 1 -- (r/R)2 Rate Controlling Step--Mass

Transfer

t/r = XB PBR

t/r = XB + (1 -- XB) In (1 - XB) psR 2

~" = 2bgg-----~

~" = 4bDe-----~

Rate Controlling Step--Heat

t/r = XB AH~zcN pBR r = 2bh(TB - T s) ,,,,

Transfer

t/r = XB + (1 -- XB) In (1 -- X B) AH~xN ps R2 r = 4bke(T s _ Ts) ,

,

,,

,,

t i t = 1 - (1 - XB) 1/2 7"---

PBR bkrCAs

156

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

1.0

Particles of constant size Gas film diffusion

0.8

f~ C h e mcontrols ical

0.6

.x/

reaction

~fcontrols /~ Ash layer diffusion 'controls

,\\

fl0

x i

~

\

X \\ ~

0.4 ~k

<~

_ \

\,

Shrinking particles ~ fStokes regime 1x ~ L a r g e , turbulent regime

~

'" ~V_Reaction "~ ~ ' > ( ~ x cOntrOls

0.2

0

0

0.2

0.4

0.6

0.8

1.0

t/'~

FIGURE 5.7 Conversion versus time of a single spherical particle reacting with a surrounding fluid, r = time for complete conversion. From Levenspiel [1], copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

The results of these various models for a sphere are plotted Figure 5.7. All these models have similar trends with respect to conversion, XB versus dimensionless time, t/r. When the time scale has dimensions, then it is much easier to determine the rate controlling mechanism, because the time for maximum conversion, r, is so different for the different rate determining steps. The time for all fractional conversions, t, that are the same, has a very different dependence on the particle size, depending on the rate controlling step, as follows: t ~ R 15-2~ for boundary layer mass transfer or boundary layer heat transfer (the exponent drops as the Reynolds number rises, i.e. turbulent flow), t a R 2 for product layer diffusion control or product layer heat conduction control, t a R for chemical reaction control. Hence, kinetic runs with different particles of the same size can be used to distinguish between different rate determining steps. In the equations given in Table 5.2 the surface temperature is not known. The surface temperature is obtained by making the heat and mass transfer fluxes equal as given in equation (5.32). This is equivalent to equating the expressions for the values of~ for heat and mass transfer

5.9 Fluid-Solid Reaction Kinetics

157

given in this table. The largest r value is that of the rate determining step. There is always one rate determining step for heat transfer and another for mass transfer. Neglecting heat transfer as others [1] have done requires that the surface temperature be assumed. Only by equating heat and mass transfer steps can the surface temperature be calculated. These conversion time expressions assume that a single rate resistance controls the reaction of the particle. For a sphere these conversions are plotted in Figure 5.7 as a function of time for different regimes of control. The relative rate of each of the resistances changes as reaction time progresses. As shown schematically in Figure 5.8, initially the product layer provides no rate resistance because it is very thin. After some degree of reaction, however, the product layer grows thicker and can become rate controlling. For this reason, it is not reasonable that just one resistance controls the reaction rate throughout the whole reaction. Accounting for the simultaneous action of these resistances is straightforward because all are linear in reactant concentration. Thus a sum of the time values for each step is a means to obtain the total reaction time" Ttota1 :

TMT -Jr- TpD -~- TSR -F- THT ~- THC

(5.39)

where the r values are those given in Tables 5.2 and 5.3, where the subscripts correspond to mass transfer, MT; pore diffusion, PD; surface reaction, SR; heat transfer, HT; and heat conduction, HC.

Chemical reaction step controls

jFilm

diffusion controls fusion controls

o

Temperature T

FIGURE 5.8 Rate of reaction versus temperature. Because of the series relationship among resistances to reaction, the net or observable rate is never higher than for any individual steps acting alone. From Levenspiel [1], copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

158

Chapter 5

Ceramic Powder Synthesis with Solid Phase Reactant

5.3 S p h e r e C o n v e r s i o n , XB = 1 (r/R)3, v e r s u s T i m e for S h r i n k i n g S p h e r e :

TABLE

A(g) + bB(s) ~

rR(g) + sS(s)

Boundary layer

Surface reaction

Rate Controlling Step--Mass

XB) 2/3 T psR2 r = 2bKg----~ t = 1 - (1 -

Rate Controlling Step--Heat

Transfer t = 1 -- (1 -- XB) 1/3

r

PBR r = bkrCAB Transfer

XB) 2/3 AHRxN PsR2 2 b h ( T s - Ts)

t = 1 - (1 -

r T"-

5.9.1 Shrinking Sphere Model When the product layer flakes off as fast as it is formed, the reaction may be considered to be occurring at the surface of a shrinking particle (see Figure 5.2). This type of reaction is described by the following steps: 1. 2. 3. 4.

Mass transfer of A to the surface of the particle, Surface reaction between A and B, Mass transfer of any product gas away from the particle, Heat transfer to or from the particle surface, depending on whether the reaction is endothermic or exothermic.

As with the shrinking core model, boundary layer mass and heat transfer fluxes are applicable as well as the surface reaction flux. The fluxes are combined in a way similar to that of the shrinking core model to give the results in Table 5.3 for a shrinking sphere model. When this model is applicable, the particle morphology changes drastically during reaction from particles to flakes of particles. In the equations given in Table 5.3 the surface temperature is not known. The surface temperature is obtained by making the heat and mass transfer fluxes equal, as given in equation (5.32). This is equivalent to equating the expressions for the values of 9 for heat and mass transfer given in this table. The largest ~ value is that for the rate determining steps. There is always one rate determining step for heat transfer and another for mass transfer.

5.9.2 Comparison with Kinetic Models Thermal decomposition of CaCO3 has been studied by Satterfield and Frales [15]. They found that at low temperatures the crystallite size had a strong effect on the decomposition rate, indicating that pore

159

5.9 F l u i d - S o l i d Reaction Kinetics 1800 A

'

,

982oc

1760

o

1720 933

E

1680

Center

~

temperature

1640

1600

0

,,

2'0 4'0

6'0

8'0 1;0 120

883~

(rain)

Time

F I G U R E 5.9 Comparison of furnace temperature to center-line temperature of a cylindrical CaCO ~sample thrust into a preheated oven. Taken from Satterfield and Feales [15].

diffusion was the rate determining step. At high temperatures, the rate controlling step was heat transfer. Figure 5.9 shows the centerline temperature of a pressed cylinder of CaCO3 powder with time after being inserted into a furnace at 1780~ (= 971~ The center-line temperature increases to a maximum at 1680~ (= 915~ where CaO nucleation takes place. The subsequent decrease in temperature is a result of the endothermic heat of reaction. For the balance of the reaction, the center-line temperature is ~150~ (= 65~ less than the furnace temperature. At the end of thermal decomposition, the temperature increases to the furnace temperature. The effect of ambient CO2 partial pressure on reaction kinetics was studied by Hyatt et al. [16] and is shown in Figure 5.10.

6.0~

~

I\ I

-=

.~,~ E~

~._~ 9 3.o oo~ 2.0 o

I i

O,

---

\~

Theoretical

\.

O0~ \ss0oc ~ ~ 9

0

o o Experimental

~'.

0.1

0.2

,

0.3 C02

o

,

0.4

h

0.5

"

,

0.6

, .......

0.7

,

0.8

'

-

0.9

pressure (atm)

F I G U R E 5.10 Rate of CaCO 3 decomposition in C O 2 atmosphere. From Hyatt et al. [16]. Reprinted by permission of the American Ceramic Society.

160

Chapter 5 CeramicPowder Synthesis with Solid Phase Reactant

40 .............. x

1-2p

-!

3o

4-6p i 2o

lO

=5 0 0

'

2{~0 '

400

' 6{]0

800

10'00

12'00

Time (minutes)

FIGURE 5.11 Kaolinite decomposition at 400~ in a vacuum. From Holt et al. [17]. Reprinted by permission of the American Ceramic Society.

These results show that the rate of CaCO3 decomposition follows: 1 Surface reaction rate . . . . .

Pc02 P$02 1

(5.40)

SP~o 2 + R--~o where P~02 is the equilibrium CO2 partial pressure, Ro is the decomposition rate in an atmosphere void of CO2, and B is a constant. This surface reaction rate has the asymptotic behavior of rate = kr(CRE -- e R r ) given in equation 5.31 when Ro is negligible and BP~o 2 is a constant for a particular temperature. Here the subscript R corresponds to the product gas rR(g) in reaction (5.26) with its concentration e R r - - PRr/RgT at the reaction surface, r, and at equilibrium, e R E -PRE/RgT.

Some of the clay minerals, kaolin in particular, decompose in a twostep manner. Above 500~ the water of crystallization evolves by a shrinking core model, assuming a flat plate particle structure. Figure 5.11 [17] shows the typical X~ versus t plot for the decomposition of three size fractions of kaolinite at 400~ in vacuum. The product layer is a pseudomorph of the original crystal structure with vacancies at anion sites. This pseudomorph structure remains until 980~ when it collapses into crystalline mullite and silica releasing energy. The kinetics of the second solid state reaction is controlled by solid state diffusion. A similar situation is observed for the decomposition of A1203 93H20 gibbsite. The thermal decomposition of magnesium hydroxide was studied by Gordon and Kingery [18]. Their results, given in Figure 5.12, show

161

5.9 F l u i d - S o l i d Reaction Kinetics 1.0 m

x

~D r 0

o.

E

0.5

0.2

0 r

lo C

O

0.1

C

o

t~ L It.

0.5

"1

I 0

O

o 40

80

120

160

200

240

280

Time (min)

Decomposition of Mg(OH)2 at various temperatures, showing first-order reaction kinetics. From Gordon and Kingery [18]. Reprinted by permission of the American Ceramic Society.

FIGURE 5.12

that either boundary layer mass transfer or heat transfer controls the rate of thermal decomposition.

5.9.3 K i n e t i c M o d e l s W h e r e N u c l e a t i o n a n d G r o w t h Are Combined When nucleation takes place throughout a reacting solid particle and it is of the same speed as the growth of these grains, a different kinetic mechanism must be used. The mechanism called nucleation kinetics has been developed to combine these two steps into a single step. An example of this mechanism is the carboreduction of boron oxide to boron carbide. This overall reaction is written to imply that it follows a liquid-solid reaction pathway: 7C(s) + 2B203(1)-* B4C(s) + 6CO(g) However, it proceeds by a two step process of (1) a solid-liquid reaction to form a gaseous suboxide, C(s) + B203(1)--~ B202(g) + CO(g) and (2) a gas-solid reaction, 5C(s) + 2B202(g)--~ B4C(s) + 4CO(g). This reaction mechanism is limited by both the nucleation and growth of B4C(s). As a result it follows "nucleation kinetics." An extensive explanation of this mechanism is treated by Avrami [19-21], Tompkins

162

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

[22], and Erofeyev [23] and contains complicated mathematical treatment that is beyond the scope of this book. However, the simplified results of these two rate determining steps is that the kinetics follow: m

ln(1 - X B) = -(kt) m.

(5.41)

This nucleation kinetic mechanism is based on the activation of reaction sites, followed by growth of the product nucleii (B4C, in this case) through chemical reaction. The global rate constant, k, describes either of these two rate determining steps for the reaction mechanism. The values of m corresponds to m

Product crystal geometry

Rate determining step

4 3 3 2 2 1

Polyhedra Polyhedra Platelet Platelet Needle Needle

Nucleus activation Isotropic growth Nucleus activation Crystal growth Nucleus activation Crystal growth

In the case the carbo-reduction of B203 to B 4 C , w e find that m = 3 [24] and k is 3.86 • 106 s -1 exp(-301 kJ/mol/RgT) for the temperature range 1803 K to 1976 K and 2 • 1020 s -1 exp(-820 kJ/mol/RgT) for the temperature range 1976 K to 2123 K. The B4C particles produced are platelets. These experimental results are typical of other experimental data [22].

5.10 F L U I D - S O L I D

REACTORS

This section draws heavily from the book Chemical Reactor Engineer-

ing by Levenspiel [1]. Various methods of contacting fluids with particulate solids are shown in Figure 5.13. These contacting methods include countercurrent, crosscurrent, and cocurrent plug flow as well as mixed solids flow-intermediate gas flow, and semi-batch operations. Consideration of the residence time distribution for each type of fluid-solid contact is necessary to understand its effect on the conversion. As a result of a given residence time distribution, E(t), the average conversion of B, XB, is given by oc

1 -XB =

-

f0

[1 -XB(t)]E(t)dt

(5.41)

whereXs(t) is the conversion function given in Tables 5.2 or 5.3, depending on the model applicable. The residence time distribution can take

5.10 F l u i d - S o l i d Reactors

163

Various contacting patterns in fluid-solid reactors: (a-c) countercurrent, crosscurrent, and cocurrent plug flow; (d) intermediate gas flow, mixed solid flow; (e) semi-batch operations. From Levenspiel [1], copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

F I G U R E 5.13

any form. However, two simplified residence time distributions are frequently used: back mixed flow, E(t)

-

e -t/o 0

where 0 is the m e a n residence time; and plug flow,

where 8 is a delta function centered at t = O.

(5.42)

164

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

Equation (5.41) assumes that the gas is of a uniform composition throughout the reactor at all times. If the gas composition changes with the time or position within the reactor, a different equation must be used. To account for the effect of particle size distribution in addition to the residence time distribution is difficult because different size particles can remain in the reactor for different periods of time. To account for these effects completely a population balance must be performed, where the conversion is an internal variable (see Chapter 3). This type of treatment is beyond the scope of this chapter. A simplified method of accounting for the effects of a particle size distribution, m(R), on the mean conversion, XB, is by 1 -Xs =

-

f0

[1

-Xs(R)]m(R)dr

(5.44)

where m(R) is the normalized population weight distribution. This equation assumes that all the particles of different size have the same residence time within the reactor. This is not always a good assumption because fine particles follow the gas stream lines much

100 ....

8 7

6 5 4

~

Gas dilffusion controls

3

fReaction controls _Product layer ,~diffusion controls

108

CD

24 1 7 64 4" 3 2 0.1 0.01

2

3

4 56780.1

2

3

4 5678

1.0

1-X B

FIGURE 5.14 M e a n c o n v e r s i o n v e r s u s m e a n r e s i d e n c e t i m e in m i x e d flow r e a c t o r s , single size solid.

165

5.10 Fluid-Solid Reactors

50

,,

,, .............

, . . . . . . .

,,

Sin~lle Stage ~~. ~ x = 10 ii

N ~Gas

film diffusion controls

~Reaction controls ~~>~-Ash layer diffusion controls

"---2_'.-......

i ,,l,,,, I "~'-'

\ "

controls-"

-,

~'~

Reaction "" ~.~ controls -~~ Two Stages (t'=2t i) " ~ ~ ~ m , ~ 1

0.01

.........

I

I

I ....

I

I

I

I

"~"

0.1

I

I "'

I

'

I

l

I

I

!

9

1.0

1--X B u

Comparison of holding times needed to obtain a given conversion for mixed flow and plug flow of single size solid. From Levenspiel [1], copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

F I G U R E 5.15

better than the large particles, which tend to settle. For single sized particles, the mean conversion for a mixed flow reactor is given in Figure 5.14. The mean residence time, 0, must be much larger than the maximum time for reaction, r, for the average conversion, XB, to be complete (i.e., XB = 1.0) for all of the rate determining steps. In a plug flow reactor, however, the mean residence time, 0, needs to be only slightly larger than the maximum time only for reaction, r, for complete reaction. A comparison of the time for a certain conversion in a mixed and a plug flow reactor is given in Figure 5.15. For a specific conversion XB, the mixed flow reactor time is always larger than the plug flow reaction time.

P r o b l e m 5.3. Conversion o f a Size M i x t u r e o f Ceramic Powders A batch of spherical ZnS particles is to be oxidized in air in a crucible placed in a tube furnace at 800~ The mixture of 30% 0.5/~m particles, 40% 1.0 ftm particles, and 30% 2.0/~m particles is spread thinly over the bottom of the crucible so that the particles will have good contact

166

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

with the air. Taking each of these size fractions separately, previous experiments have determined the time for complete conversion as 5, 10, and 20 min for the particles size fractions in increasing size order. Find the average conversion for the mixture if a residence time of 8 min is used. Equation (5.44) is applicable for finding a solution: or

1 --XB =

-

fo

[1 - X B ( R ) ] m ( R ) d R

= [1 - x B ( o . a ~ m ) ] * 3 0 %

+ [1 - X B ( 1 . 0

txm)]*40%

+ [1 - XB(2.0/zml]*30% Because for the three sizes of particles, RI'R2"R 3 --TI'T2"T 3

we see from Table 5.2 that chemical reaction controls the conversion time characteristic. Thus, t/~ = 1 - (1 - XB) 1/3

which can be rearranged to give 1 - X s = (1 - t / r ) 3

allowing the average conversion to be calculated as 1 -XB

= [1.0-

1.o]*3o% +

= o + o.oo32 +

1 -i-0

*4:0%+

1 -2-0

*30%

0.0648 = 0.068

yielding X B = 93.2%.

5.11

SOLID-SOLID

REACTIONS

This section draws heavily from the book I n t r o d u c t i o n to C e r a m i c s by Kingery et al. [3]. The reaction between two (or more) types of solid is frequently practiced to produce multicomponent ceramic powders. Several examples include NiO(s) + AleO3(s)--) NiO(s) + SreO3(s)--* MgO(s) + FeeO3(s)~ ZnO(s) + A1203(s)--) BaCO3(s) + SiO2(s)-*

NiAleO4(s) NiCreO4(s) MgFeeO4(s) ZnA1204(s) BaSiO3(s) + CO2(g)

5.11 Solid-Solid Reactions 1

• 2

167

+ 2BaCO3(s) + 3CuO(s)--~ YBa2Cu3065(s) + CO2(g) 4B(s) + C(s)--* B4C(s) 7C(s) + 2 B203(1, g)--~ B4C(s) + 6CO(g) SiO2(s) + C(s)--* SiC(s) + CO2(g) WO2(s) + C(s) = WC(s) + CO2.

The first six reactions form mixed oxide ceramic powders. The last three reactions are carbothermal reductions to produce different metal carbides. The most famous is the Atcheson process for synthesis of SiC from SiO2 and carbon, where the carbon in the mixture of reactant powders is used as a resistive electrical conductor to heat the mixture to the reaction temperature. This reaction is performed industrially in a 10-20 m long bunker fixed with two end caps that contain the source and sink for the dc current. The reactant mixture is piled to a height of 2 m in the bunker and a current is applied. The temperature rises to the reaction temperatures, and some of the excess C reacts to CO, providing further heat. The 10-20 m bunker is covered with a blue flame for most of the reaction period. The resulting SiC is loaded into grinding mills to produce the ceramic powders and abrasives of desired size distributions. Carbothermic reduction can also be used in combination with other reactants as follows: 3SiO2(s) + 6C(s) + 2N2(g)--* Si3N4 + 6CO(g) Solid-solid reactions proceed by two different mechanisms. One mechanism is solid interdiffusion, where the two solid state reactants interdiffuse at the points of powder particle contact. This mechanism is applicable for the first six reactions given earlier and many others that form mixed oxide ceramic powders. The second mechanism is not truly a solid-solid reaction. It entails the vaporization of one of the reactants (by one of several mechanisms) and then reaction of this vapor with the other solid.

5.11.1 Vaporization of One Solid Reactant The carbothermic reduction of silica is believed to follow [25] a reaction mechanism given by SiO2(s) + C(g)--* SiO(g) + CO(g) SiO(g) + 2C(s)---> SiC + CO(g)

(5.45) (5.46)

Figure 5.16 [26] shows the free energy of these two reactions and that of the carbothermic reduction as a solid-solid reaction mechanism. At a temperature greater than 1900 K the solid-solid reaction becomes

168

Chapter 5

500 '-I

Ceramic Powder Synthesis with Solid Phase Reactant

Si02*C -- SiO(g)+CO (g)

300

10~

7

-,OOl

.......

!

-30( SiO(g)+2C-- SiC+CO (g) -500

300

1500

1700

~go0

T(K) FIGURE 5.16 Free energy change for carbothermic reduction of SiO2 as a function of temperature and pressure. From Kvorkijan et al. [26].

spontaneous (i.e., AG < 0). By comparison the vaporization mechanism to produce silicon monoxide is spontaneous at -1550 K, and the reaction that produces SiC from the monoxide is spontaneous at all temperatures. Almost all carbothermic reactions that involve silica proceed via the monoxide because it is produced at low temperature and is highly reactive. As another example, we look at the carbothermic reduction of tungsten oxide, which follows a reaction mechanism C(s) + 1/2 O2(g)--~ CO(g) WO2(s) + 3 C O ( g ) ~ WC(s) + 2CO2(g).

(5.47) (5.48)

Figure 5.17 gives the phase diagram for the system tungsten oxidetungsten carbide. WC is formed above 630~ than 1.0

i.e., log

\Pco/<0

when the \ P c o ] is less

. This is equivalent to the case where

there is free carbon in the system (see the dashed line). Yet another example of the vaporization of a solid is shown in the following reaction mechanism [24]:

I ) ~ B202(g) + 2CO(g) 5C(s) + B202(g)-->B4C(g) + 4CO(g).

C(s) + B203(s ,

5.11 Solid-SolidReactions

....

850 I

750

....

169

650 Temp./oC

I

I

+10 0

I:1..o 0 O1 O

-10.9

1

750

650 Temp./~

I

|

............

WO 3

".

~

w2,,

o ~

o

_o -1

lw2c

vvc+c

wo2

2-L+ c ....

+c

=

!

0.9

1

!

1.1 10 "Y 3 / K

1

FIGURE 5.17 P h a s e diagrams of the system WO3-WO2-W2C-WC-Clog Pco2/Pco versus 1/T a s s u m i n g Pco = 1 atm. F r o m Lemaitre et al. [6].

These vaporization and solid-fluid reactions have kinetics which are limited by either the vaporization reactions or the subsequent solid-fluid reaction. The vaporization reaction was discussed in Section 5.3 in the paragraph on the thermal decomposition of minerals and follows the traditional solid-fluid reaction kinetics discussed in this chapter, where the bulk gas concentration in Tables 5.2 and 5.3 is replaced by the equilibrium gas concentration at the temperature of decomposition. The vaporization reaction is, however, often at equilibrium, giving a vapor concentration that may be small but is sufficient to keep pace with the solid-fluid reaction. Therefore solid-fluid reaction kinetics can be used to analyze solid-solid reaction when a vapor phase is formed from one of the reactants.

170

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

FIGURE 5.18 Extent of solid state particle reaction in a particle assembly.

5.11.2 S o l i d - S o l i d

lnterdiffusion

A mixture of ceramic powders that interdiffuse at the points of contact is illustrated in Figure 5.18. In the shaded regions, interdiffusion and reaction are taking place. As time progresses, the region ofinterdiffusion will increase. As an example, let us consider the diffusion couple between one oxide, AO, and another, B203, giving a mixed oxide product, AB204. Interdiffusion can take place with different ions limiting the speed of interdiffusion. Figure 5.19 gives several examples of the types of diffusion species: cations, anions (i.e., oxygen), and electrons. Depending on the relative rates of diffusion of these species, the reaction can take place at either the AO/AB204 or the B203/AB204 interface. When diffusion is slower than the rate of reaction, the thickness of the product layer follows a parabolic growth law like that observed in Figure 5.20 for NiA1204. The flux of a species i, Ji, is a result of the electrochemical potential gradient, d~i/dx.

d~i Ji = CiBi dx

(5.49)

where C~ is the concentration of species i; Bi is the ion mobility, = Di/ kBT; and ~i = ~i + ZiF&, where ft i is the chemical potential of species i of valence Z i given by kBTdCi

dt~ = ~

Ci

(5.50)

5.11 Solid-Solid Reactions

AO

AB2041

1 71

B203

A2+ _1

3

B2o,) B2o3

AO

1_ 2B3+1

0e-I

AO

AB204

B203

L.. 2B3+ ! A2+..._I 02 -] (2) Yl

2B3+ I i-~-~

302-

(3)

FIGURE 5.19 Representation of several mechanisms that may control the rate of AB2Ot (e.g., Spinel) formation. From Kingery et al. [3], copyright 9 1970 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

where F is Faraday's constant, k B is Boltzmann's constant, T is the absolute temperature, Z~ is the valence of species i, and (b is the electrical potential at the location of the flux to obtain electrical neutrality given by

Jo + Je'

= JMe"

(5.51)

For the case of oxidation of a metal shown in Figure 5.21 the net flux is given by

Jox = IJ01 + IJMel which is also the oxidation rate,

Jox.

(5.52)

1 72

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant .........

1500oC

20-

-140

15-

-122

P~

%O

r

tO L O

-.

O

"~: 10 -

-100

.9 r. l-v

.9 tI-

9 1400~

5-

0

~ r-

71

r

0

100

'" ' "'''"

200

Time (Hr) FIGURE 5.20 Thickness of NiA1204 formed in NiO-AI203 couples as a function of time heated in argon at 1400 and 1500~ From Pettit et al. [27]. Reprinted by permission of the American Ceramic Society.

Ambient atmosphere

Metal

Oxide

3M~

Pg,

0e

[ P(~z=e(+AG~

~~]o

J F I G U R E 5.21 Chemical potential gradients across an oxide layer on a metal. From ~ngery et al. [3], copyright 91970 by John Wiley & Sons, Inc. Repented by permission

of John Wiley & Sons, Inc.

5.11 Solid-Solid Reactions

173

The general results can be expressed as Jox =

Crte'

]ZMe1F2 (to +

d ~{~1Tie tMe)

dx

(5.53) o't e, d/z o -iZol F e (to + tMe) - - ~ where (r is the electrical conductivity of the oxide and t i is the transference number (= (ri/(r) equal to the fraction of the total conductivity carried by species i. Assuming that t i and (r are average values for the layer and do not vary with composition, gives a result that is the parabolic rate law: dx K = -dt x

-

-

(5.54)

where K = ~-[e/IZMe I F 2 (to + tme)[A~(Lmel- Recalling that

ti(r =

CiZ~e2F 2 kBT Di

(5.55)

we can see t h a t the oxidation rate is controlled by atomic diffusivities as well as electrical conductivity of the solid AB204. Calculation of K requires knowledge of the diffusion coefficient for all ionic species, together with the chemical potential for each species as related to their position in the reaction product layer. The most rapidly moving ions (or ions plus electrons) arriving at the interface control the reaction rate. For the example of NiO-A1203, Pettit et al. [27] found the parabolic law explains the experiment data well as shown in Figure 5.20. Reconsidering a mixed assembly of spheres shown in Figure 5.22 reacting by a parabolic diffusion couple, the volume of unreacted material at time, t, is 4 V = ~ 7r(R - y)3. (5.56) The volume of unreacted material is also given by 4 V = ~ 7rR3(1 - XB)

(5.57)

where XB is the conversion (or volume fraction reacted). Combining the two preceding equations and solving for y we obtain y = r[1 - (1

-ZB)l/3].

Combining with equation (5.54), rewritten as follows,

dy2_2K dt

(5.58)

1 74

Chapter 5 Ceramic Powder Synthesis with Solid Phase Reactant

FIGURE 5.22 Schematic representation of reaction product layers forming on the surface of particles in a powder assembly. From Kingery et al. [3], copyright 9 1970 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

and integrating, we find [ 1 - [1 -XB)I/3] 2 = ~-~ 2K t

(5.59)

where 2 K / R 2 is essentially a reaction rate constant that is characteristic of reaction conditions with K, given by equation (5.54). This relationship has been found to hold for many solid-state powder reactions, including silicates, titanates and ferrites. Figure 5.23 illustrates the applicability of this equation to the solid state powder reaction between SiO2 and BaCO3 giving BaSiO3 plus CO2 (gas). In Figure 5.23(a) the linear time dependence of [1 - (1 X B ) I / 3 ] 2 is plotted for several temperatures. The slopes equal to 2 K / R 2 are plotted as a function of R -2 in Figure 5.23(b). Figure 5.23(c) shows the Arrhenius expression, K = K ~ exp - ( R ~ )

(5.60)

that is expected from the dependence of K on the solid state diffusion coefficient. J a n d e r [28] found similar fits for the reactions CaCO3 + SiO2 --~ CaSiO3 + CO2(g)

"~RXNA TZ29S _-- _ 25 kcal/mole (5.61)

CaCO 3 + MoO 3 --~ CoMoO 4 + CO 2(g)

~'~ RXNA TZ29S --_ _ 15 kcal/mole. (5.62)

5.11 Solid-Solid Reactions

175

a 0.03 Xl~3 0"02"

~

870~_

o.olo

0

20

40

60

Time b

100 120

c

-2,100 t . . . . . . . . . . .

~

80

"~ ;Z

80

(min)

o

.j 0

/

1.2 ........

m"" 1.0

I i

, ..... 200 400 600 800 1 / R 2 ( 1 / m m 2)

o.. o0.2

0 0.8s

0.9o

0.95

1000/T (K) F I G U R E 5.23 Solidstate reaction between silica and BaCO3, showing(a) time dependence, (b) particle size dependence, and (c) temperature dependence of reaction. From Jander [28].

There are two important simplifications in this equation: 1. It is valid for small reaction thicknesses; 2. It assumes product and reactants have the same molar volume. When corrections are made for these two simplifications, Carter [29, 30] has shown the following equation to be applicable: 2K [1 + ( Z - 1)(XB)] 2/3 + ( Z - 1)(1 - - X B ) 2/3 = Z + (1 - Z ) - ~ s t (5.63) where Z is the ratio of equivalent volumes product to reactants. Figure 5.24 [31] is a demonstration of the validity of this equation for the reaction of ZnO + A1203--~ ZnA1204

(5.64)

for conditions from zero to complete conversion. This equation is also valid for the oxidation of metal powders (see previous Carter references [29, 30]). In comparing the K values observed for powder reactions with those calculated, difficulties often arise because of the dependence of the reaction rate on the structure of the product layer. In most cases, the product layer is formed incoherently, with m a n y fissures and defects

176

Chapter 5

1.55

Ceramic Powder Synthesis with Solid Phase Reactant

19 m i c r o n s 25 m i c r o n s

-

.0.0 0.2 0.4 0.60

1.51 OO

x! v A

T

"O

1.47 -

-o.8o

1.43

-0.90

N X

T

0 r

e-

o

.u 0 r

N

.39

-0.95

-~

1.35

u.

-1.0

!

0

0:8

1:6 Time

2.4

3:2

3.6

(hr)

FIGURE 5.24 Reaction between ZnO and A1203 to form ZnA1204 at 1400~ in air (two spherical particle sizes). From Schmalzried [31].

t h a t result from the volume change upon reaction. The diffusion coefficient observed may well be a surface diffusion coefficient or a grain boundary diffusion coefficient instead of the diffusion coefficient for a single crystal or a dense polycrystalline body. When new phases are formed, there is a strong tendency for the initial lattice p a r a m e t e r to be a nonequilibrium value corresponding to a coherent interface with the reactant. The diffusion coefficients for such nonequilibrium lattices are normally larger t h a n for equilibrium lattices. As a result, the calculation of K from diffusion data for single crystals or dense polycrystalline bodies sets a lower limit for the actual diffusion coefficient operating in a solid state powder reaction. Reactors for solid-solid reactions are designed in the same way as that for the fluid-solid reactors (see Section 5.10) but with these reactions the mixing of the gas does not need to be considered.

5.12 S U M M A R Y This chapter has discussed fluid-solid and solid-solid reactions as a means to produce a ceramic powder of a particular chemical composition

5.12 Summary

1~

and crystal structure. First, the thermodynamics of these reactions was discussed especially the Gibbs free energy and its change with temperature. Complex reaction schemes can be evaluated as to the most spontaneous using the Gibbs free energy as a criterion. Once the most probable reaction was determined, the kinetics of the reactions were discussed. Fluid-solid reactions have several possible rate determining steps, including surface reaction and mass transfer and heat transfer in the boundary layer and in the product layer. The kinetics of each of the rate limiting steps was discussed for shrinking core and shrinking particle models. Solid-solid reactions take place by solid state diffusion, which may be limited by the diffusion of ions (metal or oxygen) or electrons. Kinetic expressions for the solid state diffusion in ceramic particles was discussed. Reactors for these reactions were also described.

Problems 1. At 25~ and I atm total pressure determine the equilibrium partial pressure of CO2 over CaCO3 particles. (Answer is 1.6 • 10 .23 atm.) 2. Using the enthalpy of the CaCO3 decomposition reaction given in equation (5.4), determine the equilibrium partial pressure of CO2 over CaCO3 at 600~ and 1 atm total pressure. 3. Determine whether W will form an oxide or carbide in air at 900~ Assume that the Pco2 in air is 5 • 10 .4 atm. 4. Does Cr form a oxide or a nitride in air at 700~ 5. Pure metal particles react with a gas of a given composition and at a given temperature to give a ceramic product. What can you say about the kinetics of the reaction if the rate of reaction per gram of solid is (a) proportional to the diameter of the particles, (b) proportional to the square of the particle diameter, (c) independent of the particle size. 6. A batch of spherical monodisperse silicon metal particles is treated in a uniform ammonia gas. The solid is converted to Si3N4 with the same particle morphology, according to the shrinking core model. Conversion is seven-eighths complete after 1 hr and totally complete after 2 hr. What is the rate determining step? 7. Spherical particles of ZnS of size 2.0 t~m are reacted in an 8% oxygen gas stream at 900~ A reaction takes place according to equation (5.10). Assuming that the reaction proceeds by a shrinking core model and that the boundary does not present an important rate resistance,

178

Chapter 5

Ceramic Powder Synthesis with Solid Phase Reactant

(a) Calculate the time needed for complete conversion of a particle and the relative resistance of product layer diffusion during this reaction. (b) Repeat the procedure for particle of size 0.5 ftm. Data: solid density 4.13 gm/cc, reaction rate constant, k r - 2 cm/ sec, gas diffusion in ZnO layer, D A e -- 0.08 cm2/sec.

References 1. Levenspiel, O., "Chemical Reactor Engineering." Wiley, New York, 1972. 2. Castellan, G. W., "Physical Chemistry." Addison-Wesley, Reading, MA, 1969. 3. Kingery, W. D., Bowen, H. K., and Uhlman, D. R., "Introduction to Ceramics," 2nd ed. Wiley (Interscience), New York, 1970. 4. Richardson, F. D., and Jeffes, J. H. E., J. Iron Inst. 160, 261 (1948) 5. Darken, L. S., and Gurry, R. W., "Physical Chemistry of Metals." McGraw-Hill, New York, 1953. 6. Lemaitre, J., Vindick, L., and Delmon, B., J. Catal. 99, 415-427 (1986). 7. Diaz Guemes M. I., Gonzalez Carreno, T., Serna, C. J., and Palacios, J. M., J. Mater. Sci. 24, 1011-1014 (1989). 8. Bowen, P., Highfield, J. G., Mocellin, A., and Ring, T. A., J. Am. Ceram. Soc. 73, 724-728 (1990). 9. Wen, C. Y., and Wang, S. C., Ind. Eng. Chem. 62, 30 (1970). 10. Wen, C. Y., Ind. Eng. Chem. 60, 34 (1968). 11. Kunii, D., and Levenspiel, O., "Fluidization Engineering." John Wiley, New York, 1969. 12. Ishida, M., and Wen, C. Y., Chem. Eng. Sci. 26, 1031 (1971). 13. Ishida, M., Wen, C. Y., and Shirai, T., Chem. Eng. Sci. 26, 1043 (1971). 14. Knudsen, M., "The Kinetic Theory of Gases." Methuen, London, 1934. 15. Shatterfield, C. N., and Frales, F., Aiche J. 5(1) 115 (1959). 16. Hyatt, E. P., Cutler, I. B., and Wadsworth, M. E., J. Am. Ceram. Soc. 41, 79 (1950). 17. Holt, J. B., Cutler, I. B., and Wadsworth, M. E., J. Am. Ceram. Soc. 45, 133 (1962). 18. Gordon, R. S., and Kingery, W. D., J. Am. Ceram. Soc. 50, 8 (1967). 19. Avrami, M., J. Chem. Phys. 7, 1013-1112 (1939). 20. Avrami, M., J. Chem. Phys. 8, 212-224 (1941). 21. Avrami, M., J. Chem. Phys. 9, 177-184 (1941). 22. Tompkins, F. C., in "Treatise on Solid State Chemistry" (N. B. Hanny, ed.), pp. 193-232, Vol. 4. Plenum, New York, 1976. 23. Erofeyev, B. V., C. R. Acad. Sci. URSS 52, 511 (1946). 24. Weimer, A. W., Moore, W. G., Roach, R. P., Hitt, J. E., Dixit, R. S., and Pratsinis, S. E., J. Am. Ceram. Soc. 75(9), 2509-2514 (1992). 25. Wei, G. C., Kennedy, C. R., and Harris, L. A., Bull. Am. Ceram. Soc. 63, 10541061 (1984). 26. Kvorkijan, V., Komac, M., and Kolar, D., in "Ceramic Powder Processing Science" (H. Hausner, G. L. Messing, and S.-I. Hirano, eds.). Dtsch. Keram. Ges., Koln, Germany, 1989. 27. Pettit F. S., Randklev, E. H., and Felten, E. J., J. Am. Ceram. Soc. 49, 199 (1966). 28. Jander, Z., Anorg. Allg. Chem. 163, 1 (1927). 29. Carter, R. E., J. Chem. Phys. 34, 2010 (1961). 30. Carter, R. E., J. Chem. Phys. 35, 1137 (1961). 31. Schmalzried, H., "Solid State Reactions," p. 102. Academic Press, New York, 1974.

6

Liquid Phase Synthesis by Precipitation

6.1 O B J E C T I V E S In this chapter the production of ceramic powders from liquid phase precipitation is discussed. In most cases, ceramic powder precursors (i.e., sulfates, carbonates, oxalates, hydroxides, etc.) are produced by precipitation. These powders must be thermally decomposed to their oxides in a separate step that frequently maintains their precipitated particle morphology although some degree of particle sticking often occurs. This chapter focuses on the fundamental steps of nucleation, growth, and aggregation in precipitation. Control of the kinetics of each of these fundamental steps controls the particle morphology and size distribution during precipitation. The nucleation and growth subjects discussed in this chapter are the same as in gas phase precipitation, to be discussed in Chapter 7. The population balance is used to predict the particle size distribution for idealized batch and constant stirred tank precipitators. 179

180

Chapter 6 Liquid Phase Synthesis by Precipitation

6.2 I N T R O D U C T I O N Using liquid phase reactants, ceramic powders are produced by spray drying, spray roasting, or precipitation. Spray drying and roasting have an initial step of precipitation. They are treated in Chapter 8 since they also include drying and thermal decomposition. Ceramic powders produced by precipitation typically fall into the following categories: metal hydroxides, nitrites, sulfates, oxalates, imides, and so forth. These precursor powders must be thermally decomposed to give the desired ceramic powder as discussed in Chapter 5. One of the most important industrial examples of precipitated ceramic powders is that ofA1203 93H20(gibbsite) precipitated from a sodium aluminate solution that is thermally decomposed to give alumina. Another example is the precipitation of Mg(OH) 2 from a brine solution, which is again calcined to give "dead burnt" magnesia. The main reason precipitation is used to make ceramic powders is that it gives a pure solid product, rejecting to the supernatant most of the impurities. In addition, with precipitation the particle morphology and the particle size distribution can be controlled to some degree. Precipitation has the disadvantage that the powders must be separated from their supernatants and dried, as well as, frequently thermally decomposed to the desired ceramic material. This drying and calcination often leads to aggregates that are cemented together. Special precautions must be used to prevent aggregate formation during drying and calcination (e.g. calcination in very dry atmospheres). The precipitation of powders involves nucleation and growth from a supersaturated solution. There are several ways to cause a solution to become supersaturated to induce nucleation and growth, as illustrated in Figure 6.1. For systems in which the solubility is not a strong function of temperature, evaporation is used to cause supersaturation. For those in which solubility increases with temperature, cooling is used to supersaturate the solution. High pressures are also used to precipitate a particular crystal phase that may not be stable at ambient pressure (e.g., rutile TiO2 instead of Ti(OH)4). The use of high pressure precipitation is referred to as hydrothermal synthesis. Supersaturation can also be produced by adding another component in which the solute is insoluble. Although these methods can be used to generate ceramic powders, the most common method, reactive precipitation, occurs when a chemical reaction produces an insoluble species. Reaction-induced supersaturation is often very high, giving high nucleation rates. With the high number densities of nuclei produced, agglomeration is an important growth mechanism leading to spherical particles that are either polycrystalline or amorphous. The fundamentals discussed in this chapter are useful in understanding melt crystallization in addition

6.2 Introduction

181

~9

>~ .

m

o

Temperature,T FIGURE 6.1 Solubility curves for various types of crystallization systems: Curve A, isothermal solubility; curve B, positive temperature coefficient of solubility; curve C, negative temperature coefficient of solubility.

to precipitation from liquid solution. Melt crystallization is used to make large crystals by slow cooling of molten A1203, for example. When a substrate is transformed from one phase to another, the change in the molar Gibbs free energy, A(~, at constant pressure and temperature is given by AG = (t~2 - t~l)

(6.1)

where t~ is the chemical potential of phase I (solute) and phase 2 (solid). When h(~ < 0, the transition from 1 to 2 is a spontaneous process. When h(~ > 0, it is not thermodynamically possible, on a macroscopic scale. A necessary and sufficient condition for equilibrium is when AG = 0, given by the lines in Figure 6.1; above the lines, AG < 0, and below the lines AG > 0. A supersaturated solution can be called undercooled if dCeq/dT > 0 (curve B in Figure 6.1), or superheated if dCeq/dT < 0 (curve C in Figure 6.1). If T Ois the temperature where the solubility is equal to the actual concentration, then at a temperature T T

AG = - fo AS d T = A I ~ ( T - To)/T o

(6.2)

where AS is the molar entropy and ~ is the molar enthalpy change for the phase transformation. This equation is used for melt crystallization. The molar Gibbs free energy can also be expressed as

AG = - R g T ln(a/ao) = - R ~ T ln(S)

(6.3)

where Rg is the gas constant, T is the absolute temperature, a is the activity of the solute, and a0 is the activity of the pure solute in equilibrium with a macroscopic crystal. Assuming the activity coefficients are one, AG becomes equivalent to - R g T ln(S), where S is the

182

Chapter 6 Liquid Phase Synthesis by Precipitation

saturation ratio given by C S - Ceq

(6.4)

where C is the actual concentration in solution and Ceqis the solubility at the temperature and pressure of the system. For ionic crystal precipitation, the solubility is given by the solubility product as shown in the following example: A +2 + 2B- --, AB2(s) K~p = [A § 2]o[B - ]~

(6.5)

where [A§ o and [B-] o are the ionic concentrations of A and B at equilibrium, respectively. The expression used for the saturation ratio, S, for this example becomes (6.6)

S = [A+2][B-]2/K~p

where [A +2] and [B-] are the actual ionic concentrations of A and B, respectively. In some complex cases, several salts can precipitate from the solution but only one will be the least soluble at the specific pH of the system. Such a complex equilibrium is observed in Figure 6.2,

-2

m t~ 0

o

3

o 0

..,.,I

i

4

5

6

7

8

9

pH

FIGURE 6.2 Influence of temperature on the solubility isotherms of hydroxyapatite, dicalcium phophate (CaHPO4, monetite) and calcite in the system Ca(OH)2H3PO4-KOH-HNO3-CO2-H20([Ca]./[P] = 1, Pco 2 = 10 -3.52 atm, free ionic strength). Taken from Vereecke and Lemaitre [1].

6.3 Nucleation Kinetics

183

where the solubilities [1] of different salts of calcium and phosphate are shown as a function of solution pH. The particle size distribution produced during precipitation is a result of the relative rates of reaction, nucleation, growth, and agglomeration, as well as the degree ofbackmixing in the precipitator. The kinetics of each of these steps will be discussed next.

6.3 N U C L E A T I O N

KINETICS

The material in this section draws heavily from an excellent book by Neilsen [2]. During precipitation new particles are born into the size distribution by nucleation processes. The nucleation rate, which appears as a boundary condition at size L = L* ~ 0 in the population balance, generally has a dominating influence on the particle size distribution. Nucleation is also the least understood of the various rate processes in precipitation. There are three main categories of nucleation: 1. P r i m a r y homogeneous, 2. P r i m a r y heterogeneous, 3. Secondary. Homogeneous nucleation occurs in the absence of a solid interface; heterogeneous nucleation occurs in the presence of a solid interface of a foreign seed; and secondary nucleation occurs in the presence of a solute particle interface. The mechanisms governing the various types of primary and secondary nucleation are different and result in different rate expressions. The relative importance of each type of nucleation varies with the precipitation conditions.

6.3.1 Homogeneous Nucleation In very small quantities of m a t t e r such as clusters of solute molecules, a large fraction of the molecules are at the surface in a state of higher potential energy t h a n the interior molecules (i.e., fewer and weaker bonds). This excess energy is not compensated by an excess of entropy and consequently the free energy for the surface molecules is greater t h a n the free energy for the interior molecules. In a macroscopic body, this excess free energy can be expressed by the surface free energy per unit area, T. In a cluster consisting of a small number (10-100) of molecules or ions, the definition of surface area and surface free energy is r a t h e r ambiguous. None the less, in the theory presented here, we will use the concepts of surface area and surface free energy for convenience. In addition, the surface free energy per unit area has been

184

Chapter 6 Liquid Phase Synthesis by Precipitation

m e a s u r e d for only a few solid materials. Because both the surface a r e a and the surface free energy are ill-defined for a cluster, it is justified to use very simple expressions for them. Classical theories [2-4] of p r i m a r y homogeneous nucleation a s s u m e t h a t in s u p e r s a t u r a t e d solutions solute molecules combine to produce clusters, or "embryos." The overall free energy per cluster, AG, of the aggregates is a result of two terms, the free energy due to the new surface and the free energy due to the formation of new solid:

AG = - ( v / ~ r ) R g T ln(S) + ~/a

(6.7)

where v( = fie r3) is the volume and a ( - [Jar2) is the a r e a of the aggregate, V'is the m o l a r volume of the precipitate, and ~/is the surface free energy per unit area. The a r e a and volume for a n y polyhedron can be used to give a generalized particle radius, r (= 3v/a), which is the exact definition of the r a d i u s of a sphere. This generalized particle radius can be used to calculate the total free energy: hG(r) = -(fivr3/~')RgT In(S) + 3/[Jar2

(6.8)

where flv is the volume conversion factor and ~a is the surface a r e a conversion factor. For a sphere, fie = 4rr/3 and fie = 4~r. W h e n the s u p e r s a t u r a t i o n , S < 1.0, hG(r) is always positive and cluster formation is nonspontaneous. W h e n the supersaturation, S > 1.0, AG(r) has a positive m a x i m u m at the critical size, r*, like t h a t shown in Figure 6.3 [4a]. Clusters larger t h a n the critical size will (_9 <]

-

/

>~ cLd D_ 03 .E) _s

cp o c-

.!

S~

=1 QX

~

~

A

G

_

( 0 ,

S

)

1

Criticol Nuclei Size, r

9

co ..c: (J

Nuclei Size, r

FIGURE 6.3 Classicalnucleation theory dependence of nuclei size on Gibbs free energy at a function of saturation ratio, S. AS > 0 ~ nonspontaneous formation, AG= 0 transient equilibrium, AG < 0 ~ spontaneous formation of a solids phase. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

6.3 Nucleation Kinetics

r*

d

I w

r:t ~

.m

~D

185

In(S)

rr"

q) (9

z 0

r ,

:

I

2

:

I

3

,

I

4

.......

I

5

"

I

6

,

I

7

,

I

8

,

I

9

Saturation Rctio, S

FIGURE 6.4 Critical nuclei size (i.e., AG = AGmax) as a function of saturation ratio. For a given value of S, all r -> r* will grow and all r < r will dissolve. Redrawn with permission from Dirksen and Ring [4a], r*d = 2flayV/(3flvRgT). Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

decrease their free energy by further growth, giving "stable nuclei" that grow to form macroscopic particles. Below the critical size, clusters will decrease their free energy by dissolving. The critical size, r*, is obtained by setting d G ( r ) / d r = 0, giving r* = 2 f l a y V / ( 3 f l v R g T

In S).

(6.9)

Nielsen [2] has adopted the terminology that an embryo is subcritical and a nuclei is supercritical in size. This critical size corresonds to a value of the free energy at the maximum of AGma x = yfla r.2/3

(6.10)

Figure 6.4 shows the critical nuclei size as a function of saturation ratio S. The standard critical size nuclei is given by * = 2flayV/(3flvRgT) rstd

(6.11)

which occurs in the limit of S = e(= 2.718). This standard critical size has the following values for spherical nuclei using the molar volume of 100 cc/mole and a temperature of 300 K.

r*d(/k) y(J/m 2) 802 401 40 20

1.00 0.50 0.10 0.05

For a given value of S, all particles with r > r* will grow and all particles with r < r* will dissolve. This phenomenon, referred to as

186

Chapter 6 Liquid Phase Synthesis by Precipitation

r i p e n i n g [5], will be discussed in more detail later in this chapter. At

high supersaturations, the critical size, r*, approaches the size of an individual molecule (see Figure 6.4), where the theory is invalid. At such large supersaturations, the rate of nucleation is limited by the collision of molecules by diffusion.

6.3.1.1 Embryo Concentrations Using these free energy concepts, the equilibrium number density of embryos of size r is given by Ne(r )

AV(r)]] =N -~-A exp [ - R--~

(6.12)

where NA is Advogadro's number. This embryo size distribution function is shown in Figure 6.5. The rate at which a nuclei of critical size assembled from an embryo and an additional atom is given by the equilibrium reaction X(atom) + X ( e * - a t o m ) ~ X ( r * ) which has the rate d N ( r * ) / d t = kiN(atom)N(r* - atom) - k _ l N ( r * ) t,Q

E (,3 O9 o

200

13..

150

-% z

I O0

5o o

_.J

-50 -I00 -150 o (D O >,, l_

E

-200 0

5

10

15

Nuclei Size,

20 r

25

50

(~,)

LU

FIGURE 6.5 Embryo size distribution function for various values of the saturation

ratio. Plot generated for a sphere of V = 100 cm3/mole, T = 300 K, and T = 0.005 J/m 2. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

6.3 Nucleation Kinetics

187

where k~ is the second-order forward rate constant for the assembly of a critical nuclei from an embryo and one atom (or ion) and k_l is the unimolecular backward rate constant for the splitting off of one atom (or ion) from a nucleus. This rate can be approximated to give

dN(r*) _ k - k exp dt 1--~- 2 --~

RgTI"

(6 13)

"

The term in square brackets can be further simplified by considering this step to be the diffusion of molecules in solution:

[kl-~-k2]

= ~exp

[ - AG~I RgTI

(6.14)

where h is Planck's constant, k B is the Boltzmann's constant, and AG~ is the activation energy for diffusion. (The preceding equation is a weak point in the theory.) This gives a nucleation rate, J, of

j = dN(r*) kBT [ AG~I NA [ G(r*) 1 dt = - ~ exp - RgTI ~ exp RgT J"

(6.15)

The kinetic factors in this equation (i.e., kBT/h exp[-hG~/RgT]Na/~') have a value o f - 1 0 33 cm -3 sec -1 for liquids near their melting points and this is the maximum nucleation rate, Jmax. Alternatively, Einstein's equation [6] for the relation between the root mean square displacement ~, and the diffusion coefficient D,

;~2/2t = D

(6.16)

can be used if we identify t -1 with t-1=

kl~-k2

- d2

(6.17)

and ~, with the molecular diameter, d. This alternative expression gives a nucleation rate

j = dN(r* ) 2D [ G(r* )] dt = d5 exp ~ j.

(6.18)

The kinetic factor in this equation [i.e., 2D/d 5] has a value of ~10 3~ cm -3 sec -~ for liquids and is the maximum nucleation rate, Jma~. A generalized log-log plot of the nucleation rate described by J = Jmax exp(-hG(r*)/RgT) versus supersaturation ratio, S, can be generated as shown in Figure 6.6. The nucleation rate has been made dimensionless by dividing the nucleation rate at S --~ oo (or more precisely as (log S) -2 --. 0) as follows:

log(J/Jmax) = -A(log S) -2

(6.19)

188

Chapter 6 Liquid Phase Synthesis by Precipitation X

o

E "~

0

g. ._J ~

-10

~_~

g g

-2O

Z

-3O

n-:

(1) ~if) c-

E k5

-4O 10

1O0

1000

: E4

Saturation Ratio, S

FIGURE 6.6 Generalized nucleation rate diagram that describes the homogeneous

nucleation rate as a function of the saturation ratio. The number of ions in a critical nucleus, n*, is given by equation (6.21) andA = 4f13aT31z2/{27f12v[kBTln(10)]3}.Experimental nucleation rates, O, are from a BaSO4 precipitation reaction. Redrawn, with permission from Nielsen [2]. where A = 4fi3a T3V2NA/[27f12v(RgT lnl0)3].

(6.20)

From a critical value of S = Sc, given by the x-intercept, the nucleation rate increases with a very steep slope and then asymptotically approaches its m a x i m u m value. The critical value of S depends on A and n*, the critical n u m b e r of atoms, molecules, or ions in the critical nucleus. This critical n u m b e r is obtained from the following equation, d ln(J) = n* = NAflvr*3/(~ ") d lnS

(6.21)

and is related to the critical radius, r*, as shown previously, giving an equation for the nucleation rate as follows: log(J/Jma~) = - n *

(log S ) / 2

(6.22)

This equation is used to give the straight lines in Figure 6.6. Using this type of general diagram Nielsen [2,7] has plotted the data obtained for the nucleation of BaSO4, as shown in Figure 6.6. He found t h a t when S > 1,000, the data followed the generalized plot with a value ofA = 220. The surface energy-, T, of barium sulfate was obtained from this value of A giving 126 erg/cm 2. The value of n* is 18 for this data.

6.3 N u c l e a t i o n K i n e t i c s

189

Below S = 1000, the nucleation rate was essentially constant at 105 cm -3 sec -1. Mullin and Gaska [8] have also verified this theory experimentally with the nucleation of K2SO4 from aqueous solution. The low S data corresponded to heterogeneous nucleation, which will be discussed later in this chapter. When conditions for homogeneous nucleation are first created, an induction time or delay time is required before the steady state nucleation rate, Jo, is established. The nucleation rate has the following transient behavior [9]: J(t) = Jo exp(-t/~)

(6.23)

where the time constant, ~, can be written as [3] -~ 6 d 2 n*/(D lnS).

(6.24)

As S increases, the value o f t decreases. In many liquid systems, the induction time, r, is only several t~sec. In viscous sucrose solution where the diffusion coefficient, D, is large [9], r may be 100 hr. To use the nucleation rate with population balance models, it is necessary to divide the nucleation rate J = dN(r*)/dt by the critical size of the nuclei, r*, giving d~o dt

= J/r* = dN(r*)/(r*dt)

(6.25)

r--*r*

where dVo/dt is the time derivative of the population density by number, Vo, and is measured in m -4 sec -~, which is different than the units of J in m -3 sec -1. Upon further substitution (dVo/dtl~r.) can be defined as = 2D/d 5 exp

d~o dt

- AV(r*)] RgT J/(3AG(r*)/Tfia)I/2"

(6.26)

r-->r*

This expression for the rate of change in the population density of nuclei will be used later in the population balance model as an initial condition. Katz has shown that classical nucleation theory predicts well the dependence of the supersaturation ratio [10] on the nucleation rate.

6.3.2 Heterogeneous Nucleation Most nucleation is in practice likely to be heterogeneous nucleation induced by solid impurite surfaces other than the solute. Nucleation on a foreign surface has a lower surface energy, which leads to a lower critical supersaturation. The rate of heterogeneous nucleation is the same form as that describing homogeneous nucleation in equation (6.13), except that the surface energy, % of the solid-liquid interface is replaced by the surface energy of the solid-seed interface. The only

190

Chapter 6 Liquid Phase Synthesis by Precipitation

difference between the homogeneous and heterogeneous nucleation is that, once the heteronucleii are used up, there are no more of them, limiting the maximum heterogeneous nucleation rate J~a~ n to JJmax

(t - r ) -

J(t)dt/(t

(6.27)

- ~)

where N S is the number of foreign seed nuclei per unit volume, t is time required for heterogeneous nucleation on all of foreign nuclei, and r is the induction time. Thus the total nucleation rate J T is the sum of the homogeneous and the heterogeneous nucleation rates:

JT--Jhomo "+" J h e t e r o

(6.28)

A plot of these two nucleation rates and the total is given in Figure 6.7. Here we see the heterogeneous nucleation (A = 2) taking place at a lower saturation ratio, giving a maximum nucleation rate of 10 -6, which corresponds to all the foreign nuclei. At higher supersaturation, homogeneous nucleation (A = 200) takes place. This type of total nucleation picture was observed by Neilsen [2] for barium sulfate, as shown in Figure 6.8. Attempts to measure primary nucleation kinetics are fraught with difficulties, because the specific mechanisms that give the nucleation rate in any situation are extremely hard to define. Attempts to verify the rate equation and estimate surface energies are complicated by the fact that, before they can be detected, nuclei must grow to a reasonably large size compared to their embryo size. A number of authors [11-14] have attempted to incorporate growth into the analysis 3

CD

~.~

0

O~ ._1

.. I

-51

o

~

-5

t~

/ / ~ u r f o c e Nucleotion - 1021 ~/c

/

~ /

-

-

Heterogeneous

Nucleation

c)

-~ -lO

_,o

/

I /

.--/Ho~ogt~ou~_Nucleob'on -15

~ -2o! I I c(~ 1

E L5

/

10

, 100 Saturation Ratio, S

1oio0

-20 E4

c_ ~c_ L~

FIGURE 6.7 Generalized nucleation rate diagram describing the characteristic differences between homogeneous, heterogeneous, and surface nucleation for this particular example. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

191

6.3 Nucleation Kinetics Ex O o

mE -) O

--) v C~ O _J

q~

HomogeneousNuclection Jrnox = 1030 # / c m 3 s e c

-10

A = 200

0 c 0 -9

0 q)

-20

HeterogeneousNucleation A=2

0

z

co o9 CD C 0 "-(O q~

E

. ~

121

o / ~

3

Jmox -- 105 #/crn sec 9

/~o 9

-

9

9

9

-30

--40

10

1O0

1000

1E4

Saturation Ratio, S

Generalized nucleation rate diagram fit to BaSO4 precipitation data. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

F I G U R E 6.8

of nucleation kinetics, but their assumptions gave ambiguous results. Others have used the nucleation induction time as a means of theory verification [ 15]. Because of the form of the nucleation rate, primary nucleation, either homogeneous or heterogeneous, would be expected to dominate at high supersaturations. Such conditions are characteristic of precipitation processes where relatively insoluble materials are produced by mixing two or more reactant streams, which is a common method of making ceramic particles. This method has problems controlling the distribution of particle sizes produced. The addition of seed particles can be used to stimulate heterogeneous nucleation and thereby control nucleation. Without the seed, homogeneous nucleation will take place that is very sensitive to small variations in supersaturation and thus not easily controlled (see Figure 6.7). With the seed, heterogeneous nucleation takes place at a lower rate than homogeneous nucleation, allowing the control of nucleation and hence particle size distribution. Heterogeneous nucleation is also important in the coating of ceramic particles with another ceramic. Here seed, consisting of the core ceramic particle, is added to the precipitating solution. The supersaturation must be controlled so that only heterogeneous nucleation on the seed takes place. If the supersaturation is too high, homogeneous nucleation will take place and secondary particles of the coating materials will be

192

Chapter 6 Liquid Phase Synthesis by Precipitation

precipitated in addition to the coating itself. For this reason, coating operations require delicate control of the supersaturation.

6.3.3 Secondary Nucleation Secondary nucleation results from the presence of solute particles in solution. Recent reviews [16,17] have classified secondary nucleation into three categories: apparent, true, and contact. Apparent secondary nucleation refers to the small fragments washed from the surface of seeds when they are introduced into the crystallizer. True secondary nucleation occurs due simply to the presence of solute particles in solution. Contact secondary nucleation occurs when a growing particle contacts the walls of the container, the stirrer, the pump impeller, or other particles, producing new nuclei. A review of contact nucleation, frequently the most significant nucleation mechanism, is presented by Garside and Davey [18], who give empirical evidence that the rate of contact nucleation depends on stirrer rotation rate (RPM), particle mass density, MT, and saturation ratio. B 0 ~(S - 1)~ MJT (RPM) h.

(6.29)

Typical values of b lie between 0.5 and 2.5. These values are much lower than the typical ones for primary nucleation by either homogeneous or heterogeneous mechanisms, where b values between 6 and 12 are more common. The importance of M T is first order (i.e., j = 1), suggesting that contact of the crystals with the walls and impeller is the important phenomenon. However, some systems (i.e., K2SO4 [19] and KC1 [20]) have much lower values ofj - 0.4. Typical values of h range from 0 to 8 but most fall between 2 and 4, which are expected from semi-theoretical models [21]. Micro-attrition at a crystal surface was directly observed by Garside and Larson [22]. They found that large numbers of particles were produced in size ranges between 1 and 10 t~m and the size distribution and number density of crystal fragments produced depended on the saturation ratio. At large values of S, more and larger fragments (or nuclei) are produced. A rougher, more fragile crystal surface results from growth at high saturation ratio. It is known [23] that there is a large hydrodynamic shear force just before and another just after contact of a particle with a surface. The influence of these hydrodynamic shear forces on nucleation is unclear, but there is substantial evidence [24-27] that such forces initiate secondary nucleation. After a particle is nucleated, it can grow by various mechanisms. The kinetics of these growth mechanisms are important in determining the resultant particle structure and size distribution. In the next section, we will discuss the more common growth mechanisms.

6.4 Growth Kinetics

193

6.4 G R O W T H K I N E T I C S The material in this section draws from the excellent book by Elwell and Scheel [28]. The process of crystal growth can be described at several size levels: molecular, microscopic, and macroscopic. At the macroscopic level, mass transport limitations control crystal growth. In solution, heat transfer is relatively fast and seldom controls crystal growth. Only when the heat of crystallization is very great (either exothermic or endothermic) might heat transfer play a role. On the other hand, mass transport limitations are frequently important. The macroscopic concentration gradient influences the surface concentration profile, which can lead to the instabilities that cause "step bunching" and dendritic structures. At the microscopic level, step bunches are observed. These consist of hundreds of molecular layers on the surface that have grouped together because of the decreased diffusive flux to each layer in the bunch, compared to the region far away from the bunch. Such step bunches are frequently responsible for trapping solvent inside the crystal structure and other defects. At the molecular level, "growth units" diffuse to the crystal and attach themselves to the surface of the crystal, they diffuse over the surface, and eventually are incorporated into the structure of the crystal or return to the fluid. Of critical importance is the nature of the crystal-solution interface. Both atomically smooth and rough surfaces are shown in Figure 6.9. On the smooth surface, all the atoms represented by cube A are identical. This picture is far from reality. Inside a crystal, an atom will have six neighbors with a binding energy of three times the bond energy (3Ea_a) because each bond is shared by two atoms. For simplicity, only nearest neighbor interactions are considered. If a single new atom is added to the smooth surface, it can form a bond with only one nearest neighbor, so that its binding energy is only one-half the bond energy (1-2Za_a). Other atoms with extra bonds may add to this atom and stabilize a cluster; however, the small binding energy of the first atom is clearly a major barrier to the growth of the crystal. An atom added to a rough surface has several possible sites with different binding energies as follows: Site

Binding energy

A*

1E** a-a

B

Ea_ a

C

~Ea_ a

The site marked * is equivalent to the two-dimensional picture shown in Figure 6.9. The energy marked ** is the same as the addition to atomically smooth surface. Any atom incident on a rough surface has a greater sticking probabil-

194

Chapter 6 Liquid Phase Synthesis by Precipitation A ........

B[-

a b FIGURE 6.9 Simplistic representation of the solid-solution interfaces in a growing crystal: (a) Atomically smooth surface and (b) rough crystal surface. Redrawn, with permission from Elwell and Scheel [28].

ity t h a n one incident on a smooth surface because this sticking probability will depend o n e x p ( ~ a _ a / k s T ) , exp(Ea_a/kBT), and so on. From this simple argument, it can be concluded that the growth rate on a rough surface will be larger t h a n on a smooth surface. Rough surfaces tend to remain rough during growth when adatoms attach at sites that create new corners, which are preferential sites for subsequent adatom additions. On a smooth surface, the rate limiting growth steps is an addition of an atom to the surface, because the subsequent addition to other adatoms at the newly created corners is relatively easy. This quickly completes the new layer, giving rise to a new smooth surface. In the following discussion, layered growth occurs on a smooth surface and continuous growth occurs on a rough surface. Tempkin [29] and Jackson [30] characterized the roughness of a crystal surface with a surface entropy factor, a, defined as

a = 2(Es_s + EW_W- Es_f)/ks T

(6.30)

where Ea_ a is the bond energy. Subscripts f and s correspond to fluid and solid, respectively. The free energy change due to solidification AG s is given in Figure 6.10 as a function of the fractional occupation of a single layer X and the surface entropy factor, a. When a < 2, the minimum hGs occurs at X = 0.5, yielding a rough surface. When a > 2, the minimum hGs occurs n e a r X = 0§ or X = 1-, yielding a smooth surface. Keeping in mind t h a t rough surfaces produce continuous growth and smooth surfaces produce layer growth, estimated values of a can be used to determine the type of crystal growth that will take place, and for this reason, estimates for a are of interest. Bennema and van der Eerden [31] have given the following equation for estimation purposes:

6.4 Growth Kinetics

In

= #(AHs/RgT-

Xeq)

195 (6.31)

where AH s is the enthalpy of the solidification or precipitation, Xeq is the equilibrium mole fraction in solution, and ~: is a crystallographic factor (<1.0) equal to w for a cubic lattice. Bourne and Davey [32-35] have combined estimates of ~ with m e a s u r e m e n t s of growth rates. Generally, these estimates of ~ can be used to successfully predict the observed growth mechanism (layer for smooth surfaces with ~ > 2 and continuous for rough surfaces with ~ < 2). The state of these estimation techniques has been reviewed by Bennema and van der Eerden [31] and Davey [36,37] and Bourne [38]. The observation of flat, light reflecting facets on most crystals grown from solution suggests t h a t these are nearly atomically smooth surfaces. If a crystal is nucleated in a roughly spherical shape, the rough surfaces will grow rapidly and then tend to disappear, leaving flat crystal surfaces. The crystal habit will eventually be bounded by the relatively slow growing crystal faces. These faces are not perfectly flat on the atomic level (note the minima in Figure 6.10 for ~ > 2 do not occur exactly at X = 0 or X = 1), but growth is limited by the nucleation of new layers. Such surfaces are referred to as s i n g u l a r surfaces. The mechanisms and kinetics of surface nucleation will be discussed later in this chapter; however, it may be noted that surface nucleation sites will often be lattice defects, although growth by random two-dimensional surface nucleation on singular surfaces is also possible.

cl, = 1 0

v

v

o_ F<

o_ _u. ._J

o

o 0 el.--' 2 iJJ z w

0

'

1

FRACTION OF SURFACE OCCUPIED, x FIGURE 6.10 Dependenceof fraction of the surface occupied on the Gibbs free energy

for various degrees of surface roughness. Redrawn with the permission of Jackson [30].

196

Chapter 6 Liquid Phase Synthesis by Precipitation

6.4.1 Stages of Crystal Growth Elwell and Scheel [28] describe two types of surface sites where atoms (or growth units) can be integrated into the growing crystal structure: a step and a kink. As shown in Figure 6.10, a step is a location where two nearest neighbor bonds can be made by an adatom with a crystal. A kink site is a location where three nearest neighbor bonds can be made with the crystal. These definitions are necessary to consider the various stages of crystal growth. The process of crystal growth occurs in the following stages: i. Transport of solute from the bulk solution to the crystal surface, ii. Adsorption on the crystal surface, iii. Diffusion over the surface, iv. Attachment to a step, v. Diffusion along a step, vi. Integration into the crystal at a kink, vii. Diffusion of coordination shell of solvent molecules away from crystal surface, viii. Liberation of heat of crystallization and its transport away from crystal. The solute is often an ion that is solvated by a coordination shell of solvent atoms (or other ions). At the crystal surface, desolvation of one or two of the solvent molecules in the coordination shell must occur before the solute can (1) adsorb, (2) attach to a step, and (3) integrate into the crystal at a kink. The diffusion of solvent molecules (or other coordination ions) away from the crystal surface may limit the diffusion of solute toward the crystal surface and thus limit the growth rate. The solute does not become a part of the crystal until the enthalpy of crystallization has been liberated and desolvation is complete. Figure 6.11 schematically presents these processes along with the energetics of each step [35].

6.4.2 Diffusion Controlled Growth Fick's first and second laws describe the diffusion of solute to the surface and the diffusion of solvent and other coordination ions away from the surface of a growing crystal. J = - D VC d C / d t = D V2C

(6.32) (6.33)

In these equations, J is the flux perpendicular to VC and V2C is the Laplacian of the concentration of solute, solvent, or other coordination

6.4 Growth Kinetics

197

The energetics of crystal growth from solution: (a) Movement of the solvated solute molecule and (b) corresponding energy changes for each transformation. Redrawn, with permission from Elwell and Scheel [28].

FIGURE 6.11

ion. Those equations require numerical solutions when the boundary condition is a polyhedron. For a spherical symmetry, however, there is an analytical solution. For spherical coordinates, Fick's first and second laws become J = D dC/dr d C / d t = D [ d 2 C / d r 2 + (2/r) d C / d r ]

(6.34) (6.35)

The spatial boundary conditions are the following: C ( r = R , t) = Ceq C ( r = ~, t >- O) = Ca

(6.36)

corresponding to the concentration C~ far from the crystal and the concentration Ceq at the crystal surface r = R. The initial condition corresponds to the concentration being C~ everywhere, including the surface of the crystal at t = 0. C ( r = R , t = O ) = Ca

(6.37)

1~8

Chapter 6 Liquid Phase Synthesis by Precipitation

These equations have the following error function solution [39]" C -- C e q + (Coo -

Ceq)

{1 - (R/r) erf[(r - R)/(2Dt)I/2]}

(6.38)

When t --, ~ (i.e., steady state), C -

Ceq +(Coc

-

Ceq)[1 -

(R/r)].

(6.39)

In the case of slow crystal growth, it can be assumed that this steady state (in reality, a pseudo-steady state) is set up faster than the crystal grows and

dR/dt-vj-

(TD dC/drlR = ~rD(C~-

Ceq)/R = V D C e q ( S -

1)/R (6.40)

where the crystal radius R has the following time dependance R = [2 (zD(C~ -

Ceq)t]

1/2

-

[2 ~rDCeq(S -

1)t]

1/2

(6.41)

and V is the molar volume of precipitate. Neilsen [40] and Reiss and LaMer [41] solved the same problem with its moving boundary and pseudo-steady state assumption, giving R = [2 VD(C~ -

Ceq)t/q]

1/2

=

[2 T~rDCeq(S

--

1)t/q] 1/e

(6.42)

where q ( < l ) is a function of the volume fraction of precipitate, VCeq (S - 1). Values of q are given in Table 6.1. When VCeq(S - 1) is small, the pseudo-steady state approximation is good. This solution is only good for an isolated sphere. Spheres can be considered in isolation if l?Ceq(S - 1) < 0.001. When many other particles are growing in the

T A B L E 6.1 Full Transient Solution Adjustment Factor, q, in Equation (6.42) as a Function of Volume Precipitate for the Exact Solution to Equation (6.35) as Solved by Nielsen a ~r(C:r - Ceq)

q

I?(C= -- Ceq)

q

0.00001 0.00002 0.00005 0.0001 0.0002 0.0005 0.001 0.002 0.005 0.01 0.02

0.996 0.994 0.991 0.987 0.982 0.972 0.960 0.944 0.912 0.877 0.827

0.005 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

0.730 0.626 0.485 0.382 0.300 0.232 0.172 0.120 0.073 0.034 0.000

a

Redrawn, with permission from A. E. Nielsen, Kinetics of

Precipitation. Pergamon, Oxford, 1964.

6.4 Growth Kinetics

199

neighborhood of this isolated sphere, the effective concentration is reduced, and this will decrease the growth rate. For that reason, these equations can be used only when the distance between neighboring particles is 10 times their diameter [42,43]. For a crystal moving in a supersaturated solution, the diffusion is enhanced by convective mass transport. The flux for convective mass transport can be written as J = Kc(C~

-

Ceq)

(6.43)

where Kc is the mass transfer coefficient given by K c = D / 6 , where 6 is the boundary layer thickness. This gives a growth rate of dR/dt = VD(C~-

Ceq)/~--

YDCeq(S

-

1)/6.

(6.44)

For a sphere, the mass transfer coefficient can be obtained from the Colburn analogy for the Nusselt number: NUAB = K c ( 2 R ) / D = 2.0 + 0.6 Re 1/2 Sc 1/3

(6.45)

where Re [= u(2R)/t~] is the Reynolds number calculated from the slip velocity, u, and Sc [= t~/(pD)] is the Schmidt number (/z = solution viscosity, p = solution density). For other shapes, the boundary layer thickness, 6, can be evaluated, using Carlson [44] analysis with a slip velocity Reynolds number: 6 = {0.463 Re 1/2 Scl/3}-1.

(6.46)

Carlson's theory predicts that 6 ~ u -~/2 so that d R / d t should be u 1/2. This is in reasonable agreement with experiments by Hixon and Knox [45] ( d R / d t ~ u ~ and Mullin and Garside [46] ( d R / d t ~ u ~ Equally useful is the Burton, Prim, and Slichter [47] [BPS] analysis of a crystal rotating at an angular velocity of co in solution giving a boundary layer thickness: 6 ~ 22/3 (~lp)l/~ D1/3co-1/2.

(6.47)

The BPS theory predicts the 6 a co-1/2 so that d R / d t should be a co1/2. At low co < 50 rain -1 Coulson and Richardson [48] and Landise, Linares, and Dearborn [49] found good agreement with the BPS theory. The effect of mixing on particle growth becomes very important when the average power transferred to a fluid is less t h a n that needed for elimination of concentration gradients in the precipitator. When pockets of local supersaturation develop within the reactor, different growth rates will be observed in these pockets. These pockets of local supersaturation can also be generated in systems where the particles are smaller t h a n the mean turbulent eddy size. Therefore, some of the small particles will grow in an effectively stagnant environment. The size at which convection plays a bigger role t h a n Brownian motion as a propelling

200

Chapter 6 Liquid Phase Synthesis by Precipitation

force for the particles can be estimated by considering the average slip velocity of the particles in a precipitator. This value can be estimated by either knowing the energy distribution due to mixing or the circulation time and path in the precipitator. When the crystals are smaller than ~ 10 t~m, the growth rate is typically controlled by diffusion. When the particle number density of these small particles is high, the interparticle distances are comparable to the boundary layer thickness and significantly higher growth rates are observed. Jaganathan and Wey [50] calculated growth in a "crowded environment" and showed higher growth rates for sparingly soluble species.

6.4.2.1 Two C o m p o n e n t Diffusion with Different Concentrations If two components are needed to complete the crystal structure [e.g. Na and C1] then each species has its own diffusive flux: (6.48)

Ji = D i ( C ~ - C i q ) / R .

For a given precipitation a A +~ + b B -~ <-> AaBb (s)

the following solubility product, Ksp, results (6.49)

gsp = [cA] a [CBq] b.

For the crystal A ~ (s) to be formed, the flux of A, J A , and the flux of B, J s , must be related to the crystal stoichiometry, giving J A / a = J s / b = D A ( C A - c A q ) / ( a R ) = D B ( C B - CBq)/(bR).

(6.50)

Here J = R, good for particles smaller than 5-10 tLm [130, 131]. Both CeA and Ces are related through the solubility product. For the case of a 1 : 1 [a :b] stoichiometry with similar diffusion coefficients, (6.51)

D = DA = DB

and the analytical solution is obtained by eliminating Ce~ and C seq yielding 82 1/2} C~)/4]

(6.52)

d R / d t = V J A = V D / R {(CA - C S ) / 2 -- [Ksp + (C A - CB)2/4] 1/2}

(6.53)

JA . J s

. D / R { ( . CA

.C S ) / 2 .

[gsp+(C A

which gives the following growth rate

When CA >> C s this reduces to dR/dt = VDCB/R

a result equivalent to one component diffusion limited growth.

(6.54)

201

6.4 Growth Kinetics

6.4.2.2 C h e m i c a l R e a c t i o n If a chemical reaction is used to generate the insoluble species, the rate of chemical reaction is another possible rate limiting step. To account for the chemical reaction rate, a growth effectiveness factor, ~, is defined as [3,4,51] Growth rate observed = Growth rate when interface is exposed to bulk solution

(6.55)

which is a function of the Damkohler number, D a , 9/ = (1

(6.56)

71Da) n

-

where

Da kr(C =

-

kr[Ceq(S--

Ceq) n-1

-

g c ................

1)]n-1 (6.57)

gc

Ceq

and is the equilibrium concentration, k r is the reaction rate constant, Kc is the mass transfer coefficient, and n is the order of the reaction to form the insoluble species. The effectiveness factor, ~/, varies from 1.0 to 0.0 as D e goes from 0 to = as shown in Figure 6.12 [52]. From this plot we can see the effectiveness factor, ~/, is low for large Damkohler number corresponding to large saturation ratio, S, where growth is limited by diffusion not chemical reaction.

,,Z 0 o o

LL (/1 (/1 ~)

1.00

/9 =

1.0

(Atomistic Growth)

0.80

c

(D > r r

0.60

4--

I,I

.c: 0 L

(.9 o O3

r

>.,

0.40 0.20

0.00 0.01

,

O. 10

1.00

,

10.00

,

100.00

Domkohler Number, Do Crystal growth effectiveness factor as defined by equation (6.56). Redrawn, with permission from Garside [52].

FIGURE 6.12

202

Chapter 6 Liquid Phase Synthesis by Precipitation

6.4.3 Surface Nucleation of Steps The material in this section follows a treatment taken from the excellent book by Nielsen [2]. The generation of a new step on a fiat crystal surface takes place by "two-dimensional" nucleation. In twodimensional nucleation theory [2], it is convenient to treat a cylindrical embryo of radius r (= 2 area/circumference) and height, d, corresponding to the height of one growth unit (i.e., atom or molecule). The total change in Gibbs free energy is given by

AG[otal = --([~A r2 d/V)RgT

ln(S) +

~/e

flL r

where Te is the edge energy per unit length surrounding the embryo. The first term of this equation corresponds to the change in free energy due to the volume of the embryo. The second term corresponds to the free energy due to the length of the edge surrounding the embryo. For a cylindrical embryo f l A - - 7/" and f l L - - 4.* When S > 1.0, AVtStalhas a maximum value

AG~ax = T2e f12 V/[2flA d RgTln(S)] = T2e f12L d2/[2flA ksTln(S)]

(6.58)

occurring at

r~ = "~e [~LV/[2[~A d RgT ln(S)] = ")/e[~L d2/[2flA ks T ln(S)]

(6.59)

The last term of the preceding two equations were obtained by setting = NA d 3, where NA is Avrogado's number. The number of molecules in a critical nucleus is

n* = flA r* 2~de = T2efi2 de/(4flA[ks T ln(S)]e).

(6.60)

The mole fraction of surface embryos, Xr, of size r is given by

Xr = exp[-hGS(r)/R~T].

(6.61)

From which the surface nucleation rate, Js, with units of embryo cm -e sec -1 is given by

Js = D/d4 exp[-AGSax/Rg T]

(6.62)

w h e r e D / d 4 = ( k s T / h ) e x p [ - A G $ / R g T ] as before. Here, AG$ is the activation energy for diffusion in the liquid. A typical value of D / d 4 is 1021 cm -2 sec -1 (i.e., D = 10 -5 cm2/sec and d = 3 x 10 -s cm). Figure 6.6 can also be used for surface nucleation as a function of supersaturation. Because the values of hG~ax are lower than hGmax for homogeneous nucleation, supersaturations lower than the critical supersaturations for homogeneous nucleation will give reasonably high * For a square embyo of edge length, l, and of height, a, the effective radius r = 1/2, ~A = 1, i l L = 4 .

6.4 Growth Kinetics

203

values of the surface nucleation rate. Considering homogeneous, heterogeneous, and surface nucleation at the same time, as shown in Figure 6.7, we have a critical value of the supersaturation ratio, S, for each. These critical supersaturations ratios are arranged in decreasing order, as follows: Sc,homo > Sc,hetero > Sc,surfac e > 1. This order is indicative of the relative magnitudes of the nucleation activation energies that control the different nucleation mechanisms.

6.4.4 Two-Dimensional Growth of Surface Nuclei The growth of a surface nucleus is achieved by either surface or bulk diffusion to the step or kink site at the edge of the growing nucleus. For the liquid phase, bulk diffusion is more important [3], giving a surface growth rate, drs/dt = d 2 D ( S - 1) NACeq. (6.63) This 2-D growth rate must be compared to the surface nucleation rate Js. To make this comparison, the time between two nucleation events, (JsA) -1, on a surface of area, A, is compared to the time necessary to grow a 2-D layer over the whole surface [L/(dr/dt)], where L is the maximum length from the nuclei to the edge of the crystal. When (JsA) -1 > [L/(drs/dt)], each layer, on average, is the result of one nuclei site (Figure 6.13). This gives a macroscopic crystal growth rate for "mononuclear" growth of (dR/dt)MN = flA R2 Dd-3 e x p ( - hGSax/Rg T]

(6.64)

where flA is a crystal shape factor given by area/R 2. When (JsA) -1 > [L/(drs/dt)] , surface nucleation is so fast that each layer is a result of the intergrowth of numerous surface nuclei (Figure 6.13(b)). This gives a macroscopic crystal growth rate for "polynuclear" growth of d [NACeq(S - 1)]2/3 exp[-hG~ax/kBT]. [dR/dt]pN = D --~

(6.65)

When the two mechanisms are equally important (JsA) -~ = 0.6[L/drs/dt].

(6.66)

This corresponds to the crossover point between mononuclear and polynuclear layer growth. At this crossover point, the crystal growth rate is given by d R / d t = 0.2VD Ceq(S - 1)/R (6.67) which is 0.2 times the diffusion controlled growth rate. This indicates that when the growth rate is 20% of the diffusion controlled growth

204

Chapter 6 Liquid Phase Synthesis by Precipitation

R

b FIGURE 6.13 Two-dimensional growth of surface nuclei by (a) mononuclear growth and (b) polynuclear growth models. Redrawn, with permission from Dirksen and Ring [4a].

rate, mononuclear layer growth rate dominates; and between this value and a growth rate equivalent to diffusion controlled growth, polynuclear layer growth dominates.

6.4.5 S c r e w D i s l o c a t i o n G r o w t h At supersaturations less than the critical supersaturation ratio for surface nucleation, Sc, surface~ 1.5, layer growth has been experimentally observed [35]. These new layers are produced by a dislocation that is a continuous source of step and kink sites. The presence of a step associated with an emerging line dislocation at a surface removes the need for surface nucleation. Figure 6.14 shows a crystal face with such a dislocation emerging at point B. Molecules are quickly integrated into the crystal at this mononuclear step. At all points on the line AB, the step moves such that the angular velocity decreases with the distance from point B giving a "screw" or spiral growth pattern to the surface, illustrated in Figure 6.14. The presence of growth spirals has been established on a large variety of crystals grown from vapor [53], from aqueous solution [54], and from melt [35]. The presence of growth spirals gives evidence for Frank's screw dislocation concept [55]; however, the height of the step is typically 50-150 rather than a mononuclear layer as envisaged by Frank. Burton, Cabrera, and Frank [56] and Bennema and Gilmer [57] have developed a theory to predict the crystal growth rate for screw dislocations. The growth rate will depend on the shape of the growth spiral. For an Archimedian spiral, shown in Figure 6.14 [58], the distance between the steps of the spiral Y0 is Y0 = 47rr*

(6.68)

6.4 Growth Kinetics

205

A @

I

\

(a)

Archimedian growth spiral: (top) Schematic presentation of Archimedian growth. Redrawn, with permission from Burton et al. [56]. (bottom) Spiral growth on rare earth crystal. Printed, with permission from Tolkdorf and Welz [58].

FIGURE 6.14

206

Chapter 6 Liquid Phase Synthesis by Precipitation

Jv

Z

Jv

v

FIGURE 6.15 Simple view of the solute flux to a step site. Redrawn, with permission from Elwell and Scheel [28].

where r* = flL~ed2/[2flAks T ln(S)] is the critical radius for the surface nucleation when the supersaturation ratio in solution is S. A more exact t r e a t m e n t [51,59] gives Y0 = 19 r* =

19flL~/ed2 2flAksT ln(S)

(6.69)

which will be used in subsequent development. The base angle, 0, of the growth cone formed by the spiral is given by

0 = tan-l(d/yo).

(6.70)

With each t u r n of the spiral, a new layer of thickness, d, is deposited. The rate of deposition is a result of the flux due to volume diffusion, Jv, and surface diffusion, Js, to the step where integration into the crystal takes place. For simplicity, consider that the curvature of the spiral can be neglected and the rate of movement of the spiral is slow compared to surface diffusion. With these assumptions, the following simple view of the surface can be given as shown in Figure 6.15. At steady state, the two fluxes must balance, giving

djs(y)/dy - Jv = 0.

(6.71)

The surface flux, js, with units of moles cm -1 sec -~ can be expressed in terms of a surface diffusion coefficient, Ds, and a surface concentration, ns"

Js = - D s dns/dy = -Dsnse d(rsldy

(6.72)

where (r~ is the local surface supersaturation ratio. The volumetric flux, Jr, is expressed in terms of a volume diffusion coefficient, De, and a

6.4 Growth Kinetics

207

concentration at the surface Cs = nse (rs/d and that in the bulk solution C = n s e S / d (given in terms of the surface concentration of a hypothetical crystal, nse ) Dvnse[S - O's]

Jv = -

d6

= -

Dv[C- Cs] 6

(6.73)

where d is the thickness of a layer, 6 is the boundary layer thickness, and S is the supersaturation ratio in solution. Substituting these two fluxes into the flux balance equation, we have the following differential equation. y2s d2(rs

+ crs = S

(6.74)

where Ys = [Dsd6/Dv] ~/2 is the mean distance traveled by the solute molecules on the surface, which is much smaller than Yo, the average distance between steps. The boundary conditions for this equation are S-

(rs = fiS crs = S

at y = Yo a t y = 0.

(6.75)

This equation has the following solution: cosh (Y/Ys) S - (rs(y)= fiS cosh (Yo/Ys)

(6.76)

where fl = 1 - Crs/S at the step (i.e., y = Yo). This equation is valid only in the regime where the distance between each kink site is much smaller than the distance between the steps (i.e., x o < Ys) as shown in Figure 6.15. Ifxo > Ys, it is necessary to introduce an extra factor into this equation to account for the nonplanar growth fields. The flux of molecules to the step may now be written as d(rs ] D s n s e f i S tanh(yo/Ys) ! - ~j ys yo

jS[y~ - - D s n s e ~

(6.77)

The crystal growth rate, d R / d t , is calculated from the flux of steps multiplied by their height, d: dR/dt =

Jstepd.

(6.78)

The step flux step, Jstep, is given by

JSbo Jstep -- p d y 0

(6.79)

where p is the crystal density. Making these substitutions, the crystal

208

Chapter 6 Liquid Phase Synthesis by Precipitation

growth rate is given by

dR/dt = DsnsefiS tanh( yo/Ys) yoYspd

(6.80)

Dsnsefl(~l) t a n h ( ~ ) y2p where $1 = S Yo/Ys = 9.5Ted2/ksTys Due to the complications associated with parameter determination, the first term in this equation is usually reduced to an experimentally determined constant. This theory predicts quadratic, dR/dt ~ S 2, behavior when S is less than the critical value (i.e., S < $1) and linear, dR/dt ~ S, behavior when S is greater than the critical value (i.e.,

S>S~). If a number of screw dislocations emerge at the same line dislocation, emerging at the crystal surface, a more complex spacial pattern will be produced. To account for the cooperative effect of multiple spirals, a factor ~ is introduced, giving the crystal growth rate as dR~dr

Ds nse [3~ S 2 Y~ O

$1 tanh

This factor ~: is always less than 1.0 because multiple growth steps close together limit the diffusion rate to each step. In fact, the step height observed experimentally [35] is between 50/~ and 150/~, which corresponds to 10 to 50 steps bunched together. When a two-dimensional diffusion field for the volume diffusion equation is solved [60], the concentration profile associated with each step is cylindrical, as shown in Figure 6.16 [61]. When these diffusion fields overlap significantly, the steps in the middle have a reduced flux and grow more slowly, allowing subsequent steps to catch up, building a step bunch. This step bunch then has its own dynamics and leads to imperfections (vacancies, defects, and occlusions) in the crystal structure.

6.4.6 Summary of Growth Rates All of these crystal growth rates are summarized in Table 6.2. The growth mechanism presented in this table occur in the following order as supersaturation increases. Screw Dislocation: Monosurface nucleation : Polysurface nucleation: Bulk Diffusion. Chemical reaction and heat transfer yield growth mechanisms that are slower than bulk diffusion. The supersaturation driving force, S - 1, given in Table 6.2, can be replaced by S - S(R*) when the

6.4 Growth Kinetics

2{}9

F I G U R E 6.16 Cylindrical solute concentration profile around each kink or step site. Redrawn, with permission from Chernov [61].

particles are small. Here S ( R * ) = exp 3 R * fiv

(6.82)

comes from the Kelvin equation and describes the saturation ratio at which particles of size R* will dissolve. When S - S ( R * ) is positive, particles will precipitate from solution; when S - S ( R * ) is negative, particles smaller than size R will dissolve and particles larger than

TABLE 6.2

Summary of Crystal Growth Rates: dR~dr = K*f(S)*g(R)

Growth mechanism

Diffusion b u l k Monosurface nucleation Polysurface nucleation Screw dislocation Heat conduction Chemical reaction

C ~rDCeq flADd -3 (Dd/3) (NACeq)~3 Dsnsefl/(y2p)

~rkg RgT2/AHf ~VDCeq

f(S)

g(R)

Ref

S - 1 exp[hG,/kBT]w

1/R R2

(S - 1)w3 exp[hG*/ksT] w

1

b

1 1/R 1/R

c

$2/$1 tanh(S1/S) w167 In S w167167 S - 1

a

b

Note. ~r is the molar volume, NAy is Advogadro's number, Ceq is the equilibrium concentration, D is the diffusion coefficient, sub-s surface, L~-/f is the heat of fusion, V is the Damkohler number, k H is the thermal conductivity, flA is the area shape factor for surface nucleii, Yo is the distance between steps, nse is the equilibrium surface concentration, fl = 1 - (rs/S is one minus the maximum surface supersaturation divided by the solution supersaturation, and p is the density. wh V * = fi2~2d2/(4flAksT In S) w167 $1 = (yo/Ys)S w167167 In S = f T AHw/(RT 2) d T a Volmer, M. M., "Kinetik der Phasenbildung," p. 209, Steinkopff, Dresden, Leipzig, 1939. b Nielsen, A. E., "Kinetics of Precipitation." Pergamon, Oxford, 1964. c Elwell, D., and Scheel, H. J., "Crystal Growth from High-Temperature Solution." Academic Press, London, 1975.

210

Chapter 6 Liquid Phase Synthesis by Precipitation

size R* will grow. This dissolution of fines and reprecipitation on larger particles, referred to as Ostwald ripening, occurs in many batch precipitation systems because the supersaturation ratio, S, decreases with time as the batch precipitation proceeds. Initially, at high supersaturation, nucleation produces large numbers of fine particles. This decreases the supersaturation, preventing further nucleation and leading to slow growth, which further decreases the supersaturation ratio. When the saturation ratio falls below the critical value, S(R*), for the fine particles previously precipitated, they will dissolve, holding the supersaturation ratio constant. At this constant supersaturation ratio, only particles larger than R* will grow or ripen at the expense of all smaller particles present in the suspension.

6.5 C R Y S T A L

SHAPE

The material in this section draws from similar material in the excellent book by Ewell and Scheel [28]. The shape of a crystal (i.e., crystal habit) can be controlled by either thermodynamics or kinetics. Only for crystals grown under very, very low supersaturation ratios is a crystal habit established by thermodynamic considerations. These crystals tend to be of mineralological origin. For most other crystal growth conditions, the kinetics of the slowest growing crystal faces give rise to a crystal shape.

6.5.1 Equilibrium Shape Gibbs [62] gave a thermodynamic description of the equilibrium shape of a crystal. The total free energy of a crystal is the sum of free energies associated with its volume, surfaces, edges, and corners. Gibbs shows that the edge and corners have an effect only when the crystal is small and the surface free energy decreases in proportion to the crystal size. For crystals of the same volume, the surface free energy (~ Ti * Ai) will be a minimum for the equilibrium shape, where Ti is the specific surface energy for face i with areaAi. Wulff[63] established that, for a crystal of a fixed weight, one shape has the lowest free energy; where the equilibrium shape is determined by the ratio of h~, the perpendicular distances to a face, to Ti, the specific surface energy of that face, as shown in Figure 6.17. This theorem corresponds to the formula

h_A= h2 T1

T2

hi = --.

(6.83)

Ti

When an equilibrium-shaped crystal grows, the growth rate of each face is proportional to its surface energy. High surface-energy surfaces have the smallest area because of their high growth energies. They are

6.5 Crystal Shape

211

/ hi

FIGURE 6.17 Equilibrium crystal shape as described by Wulff's theorem; in this case,

3'1 < 3"2.Redrawn with permission from Neilsen [2].

also rougher, which leads to higher growth rates. Wulff's theorem has been confirmed by careful experiments by V a l e n t a n [64] for small crystals ~ 1 0 ftm. In Figure 6.18, the flat F faces, stepped S faces, and kinked K faces are shown for a cubic crystal system. The rougher S and K faces grow very quickly and are rarely, if ever, observed. The crystal habit is

F FIGURE 6.18 Hypotheticalthree-dimensional crystal presenting the three main types

of possible faces: flat (F), step (S), kink (K) faces. Redrawn, with permission from Elwell and Scheel [28].

212

Chapter 6 Liquid Phase Synthesis by Precipitation

dominated by the slow growing F faces. From a knowledge of the crystal structure, it is possible to predict the slow growing F faces and therefore the final crystal morphology. Gibbs notes that for macroscopic crystals, the free energy associated with the volume of the crystal will be larger than changes in free energy, due to departures from its equilibrium shape. For these crystals, their shape will depend on kinetic factors, which are affected by crystal defects, surface roughing, and impurities in the solvent.

6.5.2 Kinetic S h a p e The rate determining step for crystal growth of the F faces of a crystal determines its kinetic shape. The S and K faces will almost always grow faster than the F faces. The only exception to this rule is when an impurity is adsorbed on a S or K face, drastically reducing its growth rate to that below an F face. Growth rates of different F faces often exhibit different dependences on the supersaturation ratio. A power law approximation for the growth rate, dR/dtli, of each face has been suggested by many authors [65-68]. This approximation can be expressed as

dR/dtli = ki S mi

(6.84)

which is written in terms of a rate constant ki and a power mi that depend on the growth mechanism and the growth conditions. The relative growth rates for two faces denoted 1 and 2 are

dR/dtll dR/dt]2

kl S ml k 2S m2

(6.85)

If, for example, these growth rates have the form shown in Figure 6.19, face 2 will be the slowest growing and dominate the crystal structure at low supersaturation and face 1 will dominate the crystal structure at high supersaturations. This behavior has been seen in many systems [69]. In one example, potassium iodide changes crystal morphology when precipitated from aqueous solution at a supersaturation ratio of 1.12 [70]. Frequently, during batch precipitation, the supersaturation ratio decreases as crystallization proceeds. This can lead to a change in crystal habit with time. Because the parameter ki depends on temperature, crystal habit will normally change when the growth temperature changes drastically. Crystal defects (dislocations, twins, and inclusions) are also responsible for morphological changes. The flow of solution around a crystal also influences its shape as is discussed in the next section. But the most important factor that can be used to change crystal habit is the addition of impurities to the precipitating system.

6.5 Crystal Shape

213

FACE 2

10 n," qD t~ IZ J::

FACE 1

O L

Saturation Ratio, S

FIGURE 6.19 Comparison of hypothetical growth rates, G1 and G2, for two F faces as a function of the saturation ratio, S. Redrawn, with permission from Elwell and Scheel [28].

6.5.2.1 D i f f u s i o n S h a p e A two-dimensional "square crystal" is shown in Figure 6.20. To the left are drawn curves with the same concentration in solution surrounding the crystal. To the right is drawn the shape of the crystal after different times. The growth rate is ~60% higher at the corner than at

~ C

4-

t~

C1

FIGURE 6.20 Diffusion controled growth of a square crystal. The left-hand side shows a hypothetical concentration gradient around the crystal. The right-hand side shows the growing crystal with time. Redrawn, with permission from Nielsen [2].

214

Chapter 6 Liquid Phase Synthesis by Precipitation

the middle of the face. From this figure we can conclude that the corner of a crystal will tend to grow faster than the center of the faces when diffusion controls the growth rate. In the extreme, this leads to dendrite growth.

6.5.2.2 Surface Nucleation Shape When surface nucleation is so slow that each layer on an F face originated from a single surface nucleus, the shape of the crystal is given by the nucleation rate on the different F faces; those with the lowest nucleation rate are the largest surfaces. At higher supersaturations, several surface nuclei grow together straightening the surface. Since surface nucleation is highly dependent on surface concentration, above a critical supersaturation ratio, diffusion in the bulk liquid will give a nonuniform surface concentration, and the faces will no longer be planar and smooth.

6.5.2.3 Growth Spiral Shape The surface of a screw dislocation will convert an F face to the shape of a pyramid or a cone. The angle, 0, of the cone can be calculated by [51] 0 = tan-l(d/yo)

= tan-l(ks T I n S / ( 1 9 T e d ) ) .

(6.86)

This equation is not accurate at high supersaturation because several spirals operate at once, leading to "step bunching." Non-Archimedian growth spirals occur due to (1) a pair of dislocations of either like or opposite sign, (2) a group of dislocations lying along a line, or (3) "wobbling" at the center of the spiral, giving macroscopic spirals.

6.5.3 Aggregate Shape Particles can aggregate by either Brownian or shear induced aggregation. With Brownian aggregation, diffusion of particles by Brownian motion causes particle collisions. With shear induced aggregation, fluid movement causes particle collisions. The shape of aggregates has been studied by computer modeling in two ways: aggregation particle by particle [71-73] and aggregation cluster by cluster [74,75]. In these models, new particles are added to a growing cluster with different fractal trajectories (i.e., reaction limited with fractal dimension of the trajectory, D t = 0 ; ballistic with D t -- 1 , and diffusion limited with D t - 2 ) . The resulting aggregate shapes are fractal with fractal dimensions given in Table 6.3. When diffusion is responsible for aggregation, the fractal dimensions vary from 2.5 for aggregation particle by particle to 1.8 for aggregation cluster by cluster. Figure 6.21 is a two-dimensional example of a computer generated aggregate grown under conditions of diffusion limited aggregation particle by particle. It has a fractal dimension in 2-D of 1.71. Cluster by cluster aggregates are less dense

215

6.5 Crystal Shape

TABLE 6.3

Fractal Dimension, Df, from Different Aggregation Modelsa

Space dimension

Reaction limited, D t = 0 2 3 Ballistic, D t = 1.0 2 3 Diffusion limited, D t = 2.0 2 3

Particle-cluster

Polydisperse cluster-cluster

2.00 3.00

1.61 2.10

2.00 3.00

1.55 1.95

1.71 2.50

1.45 1.80

a Data taken from Meakin, P., in R a n d o m Fluctuations and Pattern Growth (H. E. Stanley and N. Ostronsky, eds.), pp. 174-191. Kluwer Academic Publishers, London, 1988.

on a v e r a g e t h a n t h a t s h o w n in F i g u r e 6.21. D u e to s h e a r forces, w e a k l y b o n d e d a g g r e g a t e s c a n r e o r g a n i z e to m o r e c o m p a c t s h a p e s (i.e., h i g h e r f r a c t a l d i m e n s i o n ) . M e a k i n [76] h a s c a l c u l a t e d t h e i n c r e a s e in f r a c t a l d i m e n s i o n d u e to b e n d i n g , f o l d i n g a n d t w i s t i n g of c l u s t e r - c l u s t e r a g g r e g a t e s as s h o w n in T a b l e 6.4.

Computer generated aggregate assuming particle-particle aggregation. Fractal dimension of 1.5 in two dimensions made to simulate a fractal dimension of 2.5 in three dimensions. Printed, with permission from Sutherland [72].

F I G U R E 6.21

216

Chapter 6 Liquid Phase Synthesis by Precipitation

T A B L E 6.4 Effective Fractal Dimension Obtained from Reorganization of 3-D Cluster-Cluster Aggregatesa

Trajectory

Original

Bending

+ Folding

+ Twisting

Reaction limited, D t - - 0 D t -- 1 Diffusion limited, D t = 2

2.09 1.95 1.80

2.18 3.13 2.09

2.24 2.18 2.17

2.25 2.19 2.18

Ballistic,

a Data taken from Meakin, P., in Random Fluctuations and Pattern Growth (H. E. Stanley and N. Ostronsky, eds.), pp. 174-191. Kluwer Academic Publishers, London, 1988.

6.5.4 Crystal Habit Modification by Impurities All of these changes in crystal habit caused by kinetic factors are drastically effected by the presence of impurities that adsorb specifically to one or another face of a growing crystal. The first example of crystal habit modification was described in 1783 by Rome de L'Isle [77], in which urine was added to a saturated solution of NaC1 changing the crystal habit from cubes to octahedra. A similar discovery was made by Leblanc [78] in 1788 when alum cubes were changed to octahedra by the addition of urine. Buckley [65] studied the effect of organic impurities on the growth of inorganic crystals from aqueous solution, and in Mullin's book [66] he discusses the industrial importance of this practice. Because crystal growth is a surface phenomena, it is not surprising that impurities that concentrate at crystal faces will affect the growth rate of those faces and hence the crystal shape. With some surface active impurities, small traces, about 0.01%, are all that is required to change crystal habit during crystallization. These impurities can: 1. Reduce the supply of material to the crystal face, 2. Reduce the specific surface energy, 3. Block surface sites and pin the steps of the growing crystal. The impurities that modify crystal habit fall into four categories: 1. 2. 3. 4.

Ions, either anions or cations; Ionic surfactants, either anionic or cationic; Nonionic surfactants like polymers; Chemical binding complexes (e.g., organic dye compounds or enzymes).

Each of these surface active impurities has a propensity to adsorb on a specific crystal surface. The change in the specific surface energy,

6.5 Crystal Shape

21g

d~/, that results from the adsorption of F atoms per unit area is given by Gibbs [62] as -d~/ = F RgT d(ln a2)

(6.87)

where a2 is the activity of the impurity in solution. The adsorbed amount F is frequently related to the impurity activity by the Langmuir adsorption isotherm [79] P :

FM

ba2 1 + ba2

_

or

ba2

F

-- FM-----~F

-

0 1- O

(6.88)

where O is the fractional surface coverage, b(=K/al) is related to the distribution coefficient K and the activity of the solvent al and can also be written as b - b' exp(AH/RgT)

(6.89)

where A H is the enthalpy of adsorption. This Langmuir adsorption isotherm is frequently used to describe the adsorption of ions, chemical binding complexes, and ionic surfactants but not polymers. More than one solvent molecule is displaced by the adsorption of one polymer molecule leading to a slightly modified form to this equation [80], ba2 =

O ~.(1 - O ) ~

(6.90)

where e is the number of solvent molecules displaced per polymer molecule. Ionic surfactants have the possibility of complexation in solution (i.e., micellization) as well as multilayer adsorption, which gives rise to more a complex adsorption isotherm [81] ba2(1/n + k2a~ -1) F = FM1 + ba2( 1 + k2a~_l)

(6.91)

where n is the aggregation number for surface micellization (i.e., number of layers) and k2 is the equilibrium constant for micellization. Ionic surfactants strongly adsorb on oppositely charged crystal surfaces at concentrations much less than the critical micelle concentration (CMC), (i.e., a2 ~- 0.01 9CMC) [82]. On similarly charged surfaces, ionic surfactants will adsorb without selectivity at concentrations near the CMC. Each face of the crystal has a different structure and as a result will be different with respect to adsorption. For example, kaolin platelets have an edge that is predominantly A12Oa and a face that is predominantly SiO2. Consequently, each crystal face will have its own adsorption isotherm, Fi. Kern [70] discusses the adsorption of ions like Cd, in the habit modification of a NaC1 (and Pb in the habit modification of KC1). There is a similarity between the {111} planes of NaC1 and the

218

Chapter 6 Liquid Phase Synthesis by Precipitation

{111} planes of CdC12. Kern postulates that a Cd adsorption layer completely covers that face of a growing NaC1 crystal and its growth is entirely surpressed. The adsorption of impurities at a particular surface decreases the area of the crystal face available for adsorption of solute molecules and therefore the growth of this surface, according to Burrill [83]. Mullin et al. [84] suggest that imputity ions in the vicinity of the surface will (1) reduce the effective surface supersaturation, (2) retard diffusion, and (3) hinder the incorporation of growth units into the crystal. Carbrera and Vermilgeer [85] note that if the mean distance between strongly adsorbed impurities is comparable with the size of the critical surface nuclei(r*), then the step will be "pinned" by the impurities, which decreases the growth rate and traps impurities in the crystal structure. Albon and Dunning [86] have observed step pinning on sucrose crystals caused by raffinose impurities. Decreased growth rates caused by step pinning have been observed with the electrolytic growth of Ag whiskers [87]. Chernov [60] suggests that there are two effects of impurities: reduction of the number of kinks if the impurities are relatively small and mobile, and action as an obstacle for the movement of the steps (i.e., step pinning) if the impurities are large and immobile. Reducing the number of kinks will also decrease the crystal growth. For these conditions, Chernov estimates that an impurity concentration ~10 -3 M will drastically reduce the growth rate. Slavnova [88] has observed qualitative confirmative of Chernov's theory. Sears [89] discussed the effects of poisons on subcritical nucleation kinetics and on the spiral shape. Crystal habit can be drastically changed by the specific adsorption of ionic surfactant impurities at concentrations below their critical micelle concentration. An example of this phenomena is the use of anionic and cationic surfactants to change the habit of adipic acid crystals during precipitation [90]. The addition of cationic surfactant will specifically adsorb on the negatively charged surfaces of adipic acid and limit their growth, yielding platelike particles, as shown in Figure 6.22. Anionic surfactants will adsorb on the positively charged surfaces of the adipic acid crystals and limit their growth rate, yielding needle-like particles. The preferential and strong adsorption of ionic surfactants is frequently used industrially to control crystal habit. Whatever the details of the kinetic mechanism, impurities cause crystal habit modification. Buckley [65] has classified many impurity effects on different crystal habit modifications. In most cases, impurities decrease the growth rate of specific crystal faces, which lead to a change in the crystal habit because the slowest growing faces will dictate the crystal morphology. In some exceptional cases, impurities can increase the growth rate of a particular crystal face. For example, 1% Fe added

6.5 Crystal Shape

219

~" ~..2~~~~ieOn~Cnt

deterg~~! anionic

L-d +LJ ,aces0'is

reta.0ationo,0ro .on

aac~ iS,

needles

FIGURE 6.22 Influence of impurity adsorption on the crystal habit, for the case of adipic acid. Redrawn, with permission from Nielsen [2].

to precipitation of ammonium dihydrogen phosphate gave high-quality, impurity-free crystals that grew at 10 times the rate of the pure solution. Such an increase may be caused by a decrease in the surface energy, reducing the size of the critical nucleus, which increases the surface nucleation rate more than is compensated for by a decrease in step velocity [83]. The effect of impurities, like PPM levels of Pb +2 in the precipitation of NaC1, can improve the quality of the crystals [91] and not enter the crystal lattice. Impurities can also cause the appearance of crystal faces not observed in pure solutions. H a r t m a n [92] has proposed that some impurities will cause step or kink faces to become flat, due to impurity adsorption at the "rough surface." Lateral growth is possible only at steps and growth on this surface is similar to that on an F face [93]. Polymeric surfactants are sometimes used for specific adsorption but more often as steric stabilizing surfactants to control the colloid stability of the suspension during precipitation [94]. A change in the interparticle forces due to polymer adsorption leads to different aggregate structures (i.e., fractal dimensions). These interparticle forces also determine the ease with which the aggregate structure can be reorganized. Jean [95] used the physical adsorption of hydroxy-propyl cellulose onto TiO2 to control its colloid stability during precipitation from alcohol solution. For steric stabilization, the adsorbed polymer must be well solubilized (i.e., better than a theta solvent). The effect of colloid

220

Chapter 6 Liquid Phase Synthesis by Precipitation

stability on aggregate size distribution during precipitation will be discussed in the next section.

6. 6 S I Z E D I S T R I B U T I O N EFFE C T S ~ P OP ULA T I O N BALANCE AND PRECIPITATOR

DESIGN

Coupled with a mass balance, the population balance accounts for all of the particles of each size that are generated in a precipitator. The population balance was first formulated by Randolph [96] and Hulbert and Katz [97]. A general review is provided by Randolph and Larson [98]. The population balance, when performed on a macroscopic scale like that of the whole precipitator, is given by

a~o(R) a(avo(R)) ~k VoKQK § = + B(R) - D(R) at aR V

(6.92)

where Vo(R) is the population of size R, G = dR/dt is the atom by atom growth rate given in Table 6.2, B(R) is the birth function, D(R) is the death function, QK is the flow rate in (+) or out ( - ) with population VOK, and V is the reactor volume. For this macro balance, the terms represent the accumulation of particles in size range R to R + dR due to growth, net particle input from entry and exit streams, and birth and death of particles directly into the size range by nucleation, aggregation, abrasion, or fracture. As written this equation assumes no volume accumulation in the reactor.

6.6.1 C o n t i n u o u s S t i r r e d T a n k R e a c t o r The simplest continuous reactor to consider is that of a constantly stirred tank reactor (CSTR) or precipitator, also called a mixed suspension, mixedproduct removal crystallizer (MSMPR) [98], shown in Figure 6.23. This type of precipitator has a constant volume, V, with an input flow rate equal to its output flow rate, Q. The population ~?0(R) in the precipitator is that which leaves as product. In this case, the population balance is used at steady state (i.e., a~o/at = 0):

Ga~o(R) ~oQ oR V -

(6.93)

where the growth rate, G, is constant and not a function of R (i.e., polynuclear or screw dislocation growth mechanisms that occur at low supersaturations).

6.6 Size Distribution Effects

O] [0

221

Product

Feed .

.

.

.

.

.

.

.

Uniform mixing profile

F I G U R E 6.23

Schematic representation of a continuous stirred tank reactor.

The birth and death functions are assumed for simplicity to be zero. The solution to this differential equation is given by ~?o=~ o exp( - ( R Gr - R*))

(6.94)

where r = V / Q is the mean residence time and 7 ~ is the number density, 7o, at R -- R*. The nucleation rate, B ~ is given by BO = 070 R--~R*

OR = To G o-7 o

(6.95)

The size distribution is solely determined by the mean residence time and the rates of nucleation and growth. In general, the total number of particles present in the system can be calculated by the following integral: N T = fo ~?~

= B~

(6.96)

which is normalization of the distribution function. For determining particle size distributions, it is convenient to use the normalized cumulative distribution function, Fi(R), defined as x ~n ( x ) d x F~(R) =

(6.97) x i n(x)dx

222

Chapter 6 Liquid Phase Synthesis by Precipitation

For example, the cumulative n u m b e r distribution (i.e., i = 0) is given by N(x) = B~ exp(-x)

(6.98)

whereas the cumulative mass distribution (i.e., i = 3) is given by M ( x ) = [1 + x + (1/2)x 2 + (1/6)x 3] exp(-x)

(6.99)

where x ( = ( R - R * ) / G r ) is the dimensionless size. In practice nuclei are produced at some finite size, R*, which is typically several orders of magnitude smaller t h a n the smallest particle size measured to determine the particle size distribution. As a result this critical size is often set to zero. Linear extrapolation of number density to R = 0 implies that the small particles grow in the same m a n n e r as larger particles, which is unlikely, but this is frequently the assumption used. As a result, this type of reactor can be used to obtain information for the average growth rate, G, and the average nucleation rate, B ~ from the slope and intercept of a population plot of equation (6.94). When experiments are performed at different supersaturations, the dependence of nucleation rate and growth rate on supersaturation can be evaluated.

6.6.1.1 C o n t i n u o u s Stirred Tank R e a c t o r with R e c y c l e The width of the size distribution is often measured in terms of the coefficient of variation (c.v.) of the mass distribution. Randolph and Larson [98] have shown that the coefficient of variation of the mass distribution is constant at 50% for this type of precipitator. This coefficient of variation is usually too large for ceramic powders. Attempts to narrow the size distribution of particles generated in a CSTR can be made by classified product removal, as shown in Figure 6.24. The classification function, p(R), is similar to those discussed in Section 4.2 and can be easily added to the population balance as follows: d~o(R) -~o(R)p(R) = dR Gz

(6.100)

Classification functions that fit the classification technique, whether a hydrocyclone or another method discussed in Chapter 4, give rise to specific populations. A convenient classification function shown in Figure 6.25(a) is given by 1 + aR 2 p ( R ) = --------~ 1 + bR

(6.101)

which has the properties that p ( R ) --, 1 as R ~ 0, p ( R ) ~ a/b as R --* ~, and d p ( R ) / d R ---> 0 as R -o 0 and R --~ ~ (Note: a/b must be less

6.6 Size Distribution Effects

Input

223

Reactor no(R)

Stream

Recycle stream

l

Product classifier

p(R) TIo(R)

Output stream 1-p(R) %(R) F I G U R E 6.24

Schematic drawing of a continuous stirred t a n k reactor with a classifica-

tion loop.

than 1.0). Using this classification function, Bourne et al. [99] solved for the population (1 - a/b) tan-l(RbU2)]

(6.102)

which gives narrower particle size distributions than without classification, as shown in Figure 6.25(b). Basically the fine particles are returned to the reactor and are grown to a larger size before they are allowed to leave the system. Narrower particle size distributions are needed for ceramic powders, so this is a useful method of particle synthesis. 6.6.1.2 C a s c a d e of C o n t i n u o u s Stirred T a n k R e a c t o r s Another way to decrease the width of the particle size distribution generated is to use a cascade of CSTRs, where the output from the ith reactor is the input for the (i + 1)th reactor. If nucleation occurs in the first tank of the cascade, followed by growth in all subsequent tanks, then the population leaving the last (Nth) reactor is given by Abegg and Balakrishnan [100] as

TON(x) (Nx) N-1 e-Nx ~o

( N - 1)!

(6.103)

224

Chapter 6 Liquid Phase Synthesis by Precipitation

g Q_ ~

g

1.0

0.8

Total Recycle: p(R) = 1.0 No Product Classification: p(R) = 0.0

0:6

LI_ 0.4

Fines Destruction

0

"~

a=t

0.2

" b=5

o=

121

1

9 b=

10

0.0

Particle

Size,

R

b o gEE R"

1.0

d

o -~ D

0.8

C~

lassification

.....,

.~ a

0.6

nesOo _, u ,on

r

n Z CO ~D

0.4

=1

0.2

i

b = 10 9 b=5

C

o "~ C

E

0.0 Dimensionless

Particle

Size,

x (=R/TO)

(a) Classification function as described in equation (6.101). (b) Particle size distribution effects with classified product removal (i.e., equation (6.102). Both redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

FIGURE 6.25

where N is the number of tanks in the cascade and x[=(R - R * ) / G r , N i r = F~i=l r ] is the dimensionless size. The particle size distribution as a function of the number of tanks, N, is given in Figure 6.26(a) in terms of the cumulative weight distribution. From this figure, it is evident that the size distribution becomes narrower as the number of tanks in the cascade, N, increases. If nucleation occurs in each tank in the cascade, the population leaving the last tank has the following form:

a

1.0

c

._o o D

0.8

D_

_c

0.6

(b (D

.> _o a E (D

N=IO N = 20

0.4 0.2 0.0

I

).1

1.0

Dimensionless b c 0 ._

u 0

LL 4-,

_c ._~ (b

_o

E

Particle

10.0

Size, x (=R-R*/~-G)

1.0 0.8 0.6

~J

.>

........7

N = 100

,

,

,

0.4 0.2

LP

0.0 O.

1.0

Dimensionless C

,..--.,,, v

E

Particle

10.0

Size, x (=R-R*/'rG)

60 5O

0

o

40

C3

>

3O Multi-Point Nucleation

0

cq)

2O

~(1) 0 Q)

10 0

Single-Point Nucleation

'

2i0

'

Number

4t0

...........

of Tanks

601

,

in S e r i e s ,

8'0

'

......O0

N

F I G U R E 6.26 The effect of the number of tanks in series, N, on the cumulative weight distribution of a precipitated powder. (a) This plot is for the case of nucleation in the first tank only and further particle growth takes place in the remaining cascade. Recalculated from Equation (6.103) in terms of a weight distribution. (b) This plot is for the case of nucleation in all N tanks of the cascade. Recalculated from equation (6.104) in terms of a weight distribution. (c) Coefficient of variation = 100[(x16% x84%)/(2xs0%)] for both single- and multipoint nucleation. The coefficient of variation for this graph is calculated from panels (a) and (b). All panels redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK. -

226

Chapter 6 Liquid Phase Synthesis by Precipitation

VON(x) 7o

~~ [(Nx) ~ (Nx) g-~ ]

- 1 + -= [ i!

+ ~ ( N - 1)! e-yx

(6.104)

which is plotted in Figure 6.26(b) in terms of the cumulative weight distribution. Again, as the number of tanks increases, the particle size distribution becomes narrower. The coefficient of variation of the particle size distribution, defined as c.v. = 100(x16~ - xs4~)/(2Xso~), is plotted in Fig. 6.26(c) as a function of the number of tanks in series for both single- and multipoint nucleation models. For all numbers of tanks in a series, the multipoint nucleation model gives a larger c.v. than the single point nucleation model. For a large number of tanks in a series, the value of c.v. continuously decreases for single point and multipoint nucleation models. Increasing the number of tanks in the series decreases the width of the particle size distribution, thus giving more uniform ceramic powders.

6.6.2 Batch Precipitation Batch precipitation reactors are used through out the industry on a variety of size scales to produce ceramic powders. Examples include Cr203 for magnetic recording media, ferrites for magnets, and AgI for photographic emulsions. By far the most frequent utilization of batch precipitation is for small scale precipitation. Another example of a batch precipitation is during the initial stage of spray drying, when a droplet may be considered a small evaporative batch precipitation reactor. The population balance for the batch reactor in the absence of breakage and agglomeration is given by

a~o O(G*~o) +~ = 0 Ot OR

(6.105)

where 7o is the population, which is a function of time, t, and size R. The growth rate, G = dR/dt, is elaborated in Table 6.2 (i.e., dR/dt = K*f(S)*g(R)). The mass balance including the chemical reaction kinetics (i.e., k R CeqS) provides the time dependance of S, the saturation ratio, and thus the time dependence of f(S); for example,

dS dR Ps Ceq --~ = kR CeqS - [4/37rR .3 B ~R* + 47rR2 -~- Vo(R, t) R] ~

(6.106)

where B ~ is the nucleation rate. A particular solution to the population balance gives the population, To(R, t), as a function of time. The preceding equation is separable if v0(R, t) is a product of a time function, T(t), and a size function, ~(R), i.e., To(R, t) = T(t)*~(R)), giving a solution [4a] as follows:* * Note g'(R) g - ~ dR = d In g(R).

227

6.6 Size Distribution Effects

=

<,,

(6,107) where X.i is the eigenvalue and a i is the eigenvalue coefficient for this equation. The eigenvalues and coefficients are obtained from the initial condition, Vo(R, t = O) = i=1 ~ a i exp -

L g(R)

g*-ff(R)

dR

= ~~ (6.108)

and from this initial condition the eigenvalue coefficients can be determined: ai

= ~=~ E V~

exp

-

[g(R)

(6.109)

K*-ff(R) d R .

For a constant supersaturation (i.e., S # function of time) and a growth rate that is not a function of size (i.e., g(R) = 0 corresponds to polynuclear and screw dislocation growth), the size distribution at any time is simply a shifted version of the initial size distribution, T~ after the nucleation is completed: ~~

= R*, t = O) = ~~

(6.110)

- R* - Gt, t).

For different values of n in g(R) = R n, other kinetic expressions can be developed, if all the nuclei are the same size R*'. Integration of the growth rate (i.e., dR~dr = K * f ( S ) * g ( R ) ) gives: ('Kf(S)dt Jo

= Kf(S)t = ~ R-ndR

= [--nR(1-n)]R,

JR*

or =

ln[RR---g]

forn for n = 1.

This equation assumes that all the nuclei are the same size, R*. The equation gives an expression for the increase in particle size with time for various growth rate mechanisms. This expression is similar to that used by Nielsen [2] in his chronomal analysis. The only difference is that chronomal analysis uses the fraction precipitated, a(t), as a variable instead of the mean particle size, R(t). The fraction precipitated is defined as a(t) = C o - C(t)= ~ R(t! .~3 C 0 -

Ceq

\ R ( t = ~)}

where Co is the initial concentration of the precipitating species. Linear plots of the various chronomals, given in Table 6.5, versus time are used to verify the various growth mechanisms.

TABLE

6.5

G r o w t h Chronomalsa: I i ( a ( t ) ]

Growth mechanism Diffusion l i m i t e d

=

Kit Ii(a)

Ki

l~nI~l1--OL ~ 0~,3~] - ~tao ~I1 +~2 a l / 3 /1

R.2 3VDCeQ(S - 1)

-2

0.55 R* P o l y n u c l e a r surface M o n o n u c l e a r surface

x -2/3 (1 - x) -p dx,

p = (m + 2)/3

{kdTD2[Ceq(S - 1)](m+2~}1/a w h e r e "-Is = k[Ceq(S - 1)]m

- 3 a 1/3

d 3 e x p [ A G * / k s T]

3DR* a

D a t a t a k e n f r o m Nielsen, A. E., Kinetics of Precipitation, P e r g a m o n , Oxford, 1964.

229

6.6 S i z e D i s t r i b u t i o n Effects

MONO- NUCLEAR/POLY- NUCLEAR GROWTH n=+l ..... ~

E 0

D .s t ~

4

#

= 0.5

#m

a

=

0.2

pm

For all five distributions, are I = 2 0 %

O3

c~

3

# 0.82 pxn a = 0.17/zm =

_0

E

2

z

/ r ~ /~ = 1.36 pm f i a = 0.27 # m

-(3

' 2 I ,/"~

L / a \

~N 0

E

L

0

Z

0

0

J

1

,u = 2.24 w'rJ

2

.-.o.., 5

4

5

6

Particle Size, R (~m)

F I G U R E 6 . 2 7 M o n o n u c l e a r a n d p o l y n u c l e a r g r o w t h c u r v e s , b a s e d on t h e g r o w t h m o d e l of d R / d t ~ r n ( w h e r e n = 1). C u r v e 1 initial d i s t r i b u t i o n (t = 0), ft = 0 . 5 / ~ m , G = 0.2 /~m, O're1 - - 20%; c u r v e 2 , d i s t r i b u t i o n at t = 0.5 t i m e u n i t s , / ~ = 0.82 ftm, cr = 0 . 1 7 ftm, OreI - - 20%; c u r v e 3 , d i s t r i b u t i o n at t = 1 t i m e u n i t , ft = 1.36 ftm, o- = 0 . 2 7 / ~ m , O're1 - 20%; c u r v e 4, d i s t r i b u t i o n at t = 1.5 t i m e u n i t s , ft = 2 . 2 4 / ~ m , G = 0 . 4 5 / ~ m , Gre] = 20%; a n d c u r v e 5, d i s t r i b u t i o n at t - 2 t i m e u n i t s , ft = 3 . 7 0 ftm, o- = 0 . 7 4 / ~ m , O're1 - - 200~. R e d r a w n w i t h p e r m i s s i o n f r o m D i r k s e n a n d R i n g [4a]. R e p r i n t e d from [4a], c o p y r i g h t 1 9 9 1 , w i t h k i n d p e r m i s s i o n f r o m E l s e v i e r S c i e n c e Ltd., T h e B o u l e v a r d , L a n g f o r d L a n e , Kidlington 0X5 1GB, UK.

For different values of n in g(R) = R n, the standard deviation of the particle size distribution [101] will (1) increase with time for n > 0, (2) remain constant for n = 0, and (3) decrease for n < 0. These three cases are illustrated in Figure 6.27 for n = 1, Figure 6.28 for n = 0, and Figure 6.29 for n = - 1 (i.e., diffusion limited growth). During diffusion limited growth, the standard deviation of the particle size distribution decreases as the particles grow. Diffusion limited growth can be obtained relatively easily using higher supersaturations, as discussed in conjunction with Table 6.2. For these reasons, batch precipitation with diffusion limited growth is frequently used for the precipitation of ceramic powders where a narrow size distribution is required.

6.6.3 Effect of Aggregation on the Particle Size Distribution In this section, the population balance will be used to model batch and CSTR precipitators where aggregation is a competing growth mechanism. Figure 6.30 is an example of the aggregate microstructure in

230

Chapter 6 L i q u i d Phase Synthesis by Precipitation

DIFFUSION CONTROLLED GROWTH c" 0 ..Q

~

L. .,4,-/

03 o~ C3

(D Xb

E

........

20 18

a = 0.03 # = 1.50/Jm

14 f/. --- 1 . 1 2 / ~ m .

IT

Crre I = 1.2

#m

= 1.6

==

O're I = 2 . 2

i

.

~

8

# = 0.5 urn a=0.1 #m

6

9

~

a = 0.03 pm

12 10

/z = 2 . 0 6 # m o = 0.02 #m

/z= 1.81 p r n

16

Z -o

I"1----1

22

4

N

o

2

E t-

5i

0 0.0

O

z

0.5

1.0

1.5

Porticle Size, R (/~m)

:

2.0

2.5

6.28 Polynuclear growth curves, based on the growth model of d R I d t ~ r n (where n = 0). C u r v e 1, initial distribution (t = 0), tL = 0 . 5 t~m, o" = 0 . 1 t~m, O're] = 2 0 % ; curve 2, distribution at t = 0 . 5 time units, t~ = 1 . 0 t~m, (r = 0 . 1 t~m, O'rel 1 0 % ; curve 3, distribution at t = i time unit, t~ = 1.5 t~m, o- = 0 . 1 t~m, O're1 6 . 7 % ; c u r v e 4, distribution at t = 1 . 5 time units, tL = 2 . 0 tLm, (r = 0 . 1 t~m, (rre] = 5 . 0 % ; and curve 5, distribution at t = 2 time units, t~ = 2 . 5 tLm, o- = 0 . 1 t~m, Grel 4 . 0 % . Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0 X 5 1 G B , U K .

FIGURE

- -

- -

- "

which we can see the individual nuclei 50/~ in diameter building up the structure of the 0.3 t~m particle produced by the precipitation of CeO2 by forced hydrolysis [102]. To begin this development, the macroscopic population balance is rewritten: O~qo(R______~+) Ot

O(G~qo(R)) OR

~ ~oKQK + k = B(R) - D(R) V

(6.111)

with birth, B(R), and death, D(R), by aggregation included. In this form of the population balance, we assume a negligible volume accumulation and no particle breakage. For a solution to this population balance, a mathematical relationship must first be developed for the birth and death functions. The birth and death functions predict the importance of particle aggregation on the final particle size distribution. The aggregation rate was first developed by von Smoluchoski [103] in his rapid coagulation theory. His equation predicts the time change in the number of particles per unit volume consisting of k primary particles"

ONk Ot

--

I i=~-1i=k-1 ~., ~ 47rR~jDijN~Nj-Nk ~ 4~rRikDikN~ 2 i= l j = k - i i= l

(6.112)

231

6.6 Size D i s t r i b u t i o n Effects

P O L Y - N U C L E A R GROWTH o~

A

r o :+~

6

.

.

.

.

.

.

.

.

.

.

n=O .

.

.

.

F o r oil five d i s t r i b u t i o n s ,

c~ "E

#=o.5~m

."+-' Or)

O're1 ~ 2 0 %

~ = 1.O~m ~ = 1.5~m

or

.~10~

or

= 6.7~

.

.

.

.

.

.

.

.

.

.

.

.

cr = 0.1 /zm

~=2.0~'n

are I ~ 5 . 0 ~

Cb

~-2.5~m

are1.4.0

L.

(D ..0

E

:3 Z "0

._N o

E

L 0 Z

O! 0.0

2

3

1:0

1;5

=

" 0.5

2[0

2.5

9 3.0

Particle Size, R (um~ F I G U R E 6.29 Diffusion controlled g r o w t h curves, b a s e d on t h e g r o w t h model of d R / d t = r n ( w h e r e n - - 1 ) . C u r v e 1, initial d i s t r i b u t i o n (t = 0), ft = 0.5/~m, r = 0.1 fern, r 1 = 20%; curve 2, d i s t r i b u t i o n at t = 0.5 t i m e units, ~ = 1.12 ram, cr = 0.04 ram, O're1 3.9%; curve 3, d i s t r i b u t i o n at t = 1 t i m e unit, tL = 1.5 t~m, (r = 0.03 t~m, O're1 2.2%; curve 4, d i s t r i b u t i o n a t t = 1.5 time units, t~ = 1.81 t~m, (r = 0.03 t~m, O'rel 1.6%, a n d curve 5, d i s t r i b u t i o n at t = 2 t i m e units, t~ = 2.06 t~m, (r = 0.02 tLm, (rre] = 1.2%. R e d r a w n w i t h p e r m i s s i o n from D i r k s e n a n d Ring [4a]. R e p r i n t e d from [4a], c o p y r i g h t 1991, w i t h k i n d p e r m i s s i o n from E l s e v i e r Science Ltd., T h e Boulevard, L a n g f o r d Lane, K i d l i n g t o n 0X5 1GB, UK. -

-

-

"

-

-

F I G U R E 6.30 F o r m a t i o n of CeO2 p a r t i c l e s by t h e forced h y d r o l y s i s of a n acidic Ce(804)2 solution at 90~ ( a - c ) P a r t i c l e f o r m a t i o n over a 6 h r period a n d (d) final p r o d u c t a f t e r 48 h r of a g i n g (with p e r m i s s i o n [102]).

~~

Chapter 6 Liquid Phase Synthesis by Precipitation

where Rij( = R i + Rj) is the sum of radius i and j, Do(= kBT/6rrl~ *(1/R i + 1/Rj)) is the relative Brownian diffusion coefficient for two particles, i and j. The two summation terms on the right-hand side of this equation are the birth and death functions, respectively, but they do not have the same units as the population balance. To make this formulation compatible, it must be divided by Rk, giving B(R, t) - D(R, t) 10Nk Rk Ot

k B T i=~-l J=~-I (R i -F Rj )2 2kB ~ (Ri + Rk) 2 31-r i=l j=k-i R i Rj g i g j - 3t ~Tgk~i=l RiRk gi" (6.113)

The birth and death functions now have the same units as the population balance. To attempt a solution, an integral or continuous approach will be used in place of this discrete summation. This suggests that there is a continuous distribution of particle sizes (i.e., the sizes of interest for the population balance are much larger than that of singlets, doublets, etc.). Some key substitutions for this integration are necessary: Ni = x Vo(X)

Ri = x

Nj = (R - x) ~?o(R - x)

Rj = R - x

Nk = R Vo(R)

Rk = R

i=i=~_k-l, i=j=k-lk-i_E__l =

dx

= =

lfo dx .

Substitution of these relationships into the discrete form of the birth and death functions yields R

(6.115)

B ( R ) = Ka fo Vo(X)Vo(R - x) dx

D(R) = 2

~R2

~ (x + R)2Vo(X) dx = 2 Ka Vo(R) NT

~ + -~ + 1

)

(6.116) where NT is the total number of particles per unit volume, Ka[ = kBT/ 3/zW] is the aggregation rate constant, W is the colloid stability factor averaged over all particle sizes, and (~i) is the ith moment of the distribution, given by ~oc

fo x,o(X)dx = J0 xi,o(X) (~i) = fo ~o(xl d x

NT

(6.117)

6.6 Size Distribution Effects

233

The governing differential equation for particulate growth now becomes O~o(R) ot

t

= Ka

0[avo(R)] oR

s:

~k VoKQK V

~Oo(x)rlo(R - x ) d x - 2Ka~o(R)NT

R-5 + ~- + 1 .

(6.118)

An analytical solution to this integro-partial differential equation is not possible without some simplifying assumptions. In the sections that follow, analytical solutions are presented for particle growth in a CSTR and batch precipitation reactors. For systems in which shear is the dominant collision mechanism and not Brownian diffusion, the birth and death functions can be rewritten in terms of the mean shear rate, ~/, as follows [104]. B(R) = -~R 4/

x(R - x) ~)o(x) rlo(R - x) dx

4 ~o(R) D(R) - -~ R ~/ f0~ x(R + x) 3 ~o(X) dx

(6.119) (6.120)

Shear aggregation brings an added complexity to the modeling of aggregation kinetics. This complication due to shear is not discussed further.

6.6.3.1 C o n t i n u o u s Stirred Tank R e a c t o r with Aggregation The role agglomeration plays in a CSTR is explained by considering the macroscopic population balance at a steady state: V O~o(R) + ~o(R___~) OR T = Ka ~jo rio(x) ~oo(R - x) dx - 2Ka ~o(R)NT

+

+ 1

(6.121)

where G is the atomistic growth rate, which is assumed to be not a function of size R (i.e., either polynuclear or screw dislocation growth). Making the assumption used by Dirksen and Ring [4a] that the term (x2/R 2 + 2~/R + 1) is equal to 4 and putting equation (6.121) in dimensionless terms, we find L

OH(L____~)OL + H(L) = B fo H(x) H ( L - x) dx - A 9H(L)

(6.122)

where L = R/G~ and H(L) = ~?o(R)/v ~ where vo(= Vo(R = R* ~ 0)) is the nuclei population assumed to be at zero size, which corresponds to

234

Chapter 6 Liquid Phase Synthesis by Precipitation

the boundary condition for the differential equation. The two dimensionless coefficients in this equation are B = Ka r *GT * 7 ~ and A = 8 * NT * Ka r. Their ratio, B / A , is the nuclei number density divided by eight times the total number density (Note: G 9~o is the nucleation rate [105]). This B / A ratio is less than 1.0 for all conditions. This equation has the following Laplace transform, sF(s ) - 1 + F (s ) = BF2(s ) - AF(s )

which can be solved for F(s) as follows: (s + A + 1)_+ X/(s + A + 1)2- 4B F(s) =

2B

"

Taking the minus sign, the inverse Laplace transform is given by [129] H ( L ) = - - - - L I ~ ( 2 V B L ) e x p [ - (A + 1)L]

(6.123)

X/ *L

where I~(x) is the modified Bessel function of order 1. The minus sign must be taken for a finite answer. Taking the plus sign gives rise to poles in the solution. When L ~ 0, this solution has the characteristic that H(L) ~ 1.0 as it should since I~(L) ~ L/2. When L ~ ~, the solution converges to zero when A+ 1>2V~ because II(L) diverges exponentially (i.e., exp(L)/~v/2rrL). This constraint is consistent with the physical system (i.e., B / A < 1.0) noted earlier. The total number density can be determined by noting that the following integral is equivalent to taking the Laplace transform, L { }, of the solution without the exponential term (where the Laplace transform coordinate s = A + i) as follows [129]: NT = G~ ~

So

H(L) d L

=

-~

L

~

1

* II(2V~-B * t)

}

(6.124)

_ Vr vo [(A + 1 ) - ~/(A + 1) 2 - 4B]. 2

Noting the preceding inequality, the square root in the expression gives a real number. This solution has the characteristics shown in Figure 6.31. When Ka is zero, A and B are zero and the solution reduces a simple exponential ~o(L) = ~o exp(-L) which is the result expected for the population without aggregation [105]. Increasing the value of A while keeping constant the ratio B / A , the amount of aggregation increases as is shown in Figures 6.31 and

6.6 Size Distribution Effects

235

F I G U R E 6.31 D i m e n s i o n l e s s p o p u l a t i o n as a f u n c t i o n of size for v a r i o u s d e g r e e s of a g g r e g a t i o n for t h e r a t i o B / A = 0.5: C u r v e A, no a g g r e g a t i o n , A = B = 0; c u r v e B, A = 0.1, B = 0.05; c u r v e C, A = 1.0, B = 0.5; c u r v e D, A = 10, B = 5; c u r v e E, A = 100, B = 50.

6.32. In all cases, as A increases for a fixed ratio B/A, the population decreases in the small paticle sizes and increases in the larger particles sizes. The decrease in the small particle sizes is more drastic when A is larger. This increase is slight on a n u m b e r basis, as seen in Figure 6.31, but in all cases the curve with aggregation becomes larger t h a n that for the curve without aggregation at some large size. This crossover is seen for some curves is shown in the expanded view of these two figures. Numerical solutions [106-109] also show similar results. This crossover is excentuated by the dimensionless mass distribution plotted in Figure 6.32. Here we see the mode of the mass distribution increases to larger sizes as the value of A is increased. In all cases, both with and without aggregation, the solution is monomodal. This can be verified by noting the zeros of the first derivative:

oH(L) 0L

2

- ~II(2V~L)

(A + 1) e x p [ - (A + 1)L] (6.125)

2V/-B I2(2VBL) e x p [ - (A + 1)L] + V~L which can have only a single zero. Therefore this solution has only one extreme (i.e., monomodal). Unimodal particle size distributions for all supersaturations have also been observed for the precipitation of ammonium polyuranate [110] in accordance with this theory.

236

Chapter 6 Liquid Phase Synthesis by Precipitation

FIGURE 6.32 Dimensionless mass distribution for the condition of Figure 6.31.

Figure 6.33 shows the size distribution for TiO2 precipitated by hydrolysis of metal alkoxide in alcohol in a CSTR [111]. Upon examination of the large TiO2 particles by a transmission electron microscope they show a grape cluster morphology [112], which also suggests an aggregation growth mechanism. Other experimental systems [113-115] have also shown bimodal particle size distributions and aggregate particle morphology. If we consider that the most frequent collision is between the small nucllei and the large aggregates, equation (6.121) simplifies to G

a~0(R) ~ vo(R) oR

- Ka R ~o ~o(R) - 2 K a ~o(R) NT

~ + -~ + 1 .

(6.122) This simplification is realized when the population of nuclei is much larger than the population of aggregates, which is the case at high levels of saturation where nucleation plays a dominant role. This simplification is also justified by the fact that the colloidal stability ratio, W, is large for aggreate-aggregate collisions (i.e., K a is small) but small for nuclei-aggregate collisions (i.e., Ka is large). With this simplification, another analytical solution to the macroscopic population balance becomes possible [4a]. The solution to this population balance for a CSTR

6.6 Size Distribution Effects

237

Distribution of agglomerated TiO2 precipitated in a CSTR: (a) Scanning electron micrograph of agglomerated TiO2 produced in a CSTR and (b) bimodal number distribution of the same agglomerated sample. Printed with permission from Lamey and Ring [111]. Reprinted from [111], copyright 1986, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

F I G U R E 6.33

238

Chapter 6 Liquid Phase Synthesis by Precipitation

o g-

KNTRo

A

rY

6

.

o N

1.50

. . . . .

1.25

K_ (D

E z O3 O9

1.00

"r/~ o

~,-

0.25

Rmox

= 17.1

-I

NT

0.75 0.50

- 1

KNT'r

D XD 03

,e = o.:3

1

E =

1

!~,~

]~

~

/3 = 0.1

fl = 0.2

N

E

o b9 E q)

0.00,

'

1 2

,

3

,

,

4 5

E

',

"i

6

7

I

8

! .....

6 -1.50

6

p-

O

:3 X3 o03 _

r (1) X3

E 03 03 (D E

~o 03 f: (D

E

I

;--'T--l"-

i '

..',

:

KNTRo

1

KNTT

0.2

G

1.00

Eb

z

i

Particle Size Rotio, R ( = R / R o ' )

O rY

i

9 10 11 12 13 14 15 16 17 18 19 20

-

~~ o NT

-

1

e=10 Rmo x = 35.3

0.50

tk9

0.00, 1 3

j

, ; J ,. , , . . . . . . . . . 5 7 9 1113151'71'92:12'52'52'72'93'13'5353'73'9

Particle Size Rc:io, R (=R/Ro) FIGURE 6.34 Solution to the differential equation for a CSTR with aggregation (i.e., equation (6.123). (a) Variation in s for constant fl and ~; (b) variation in s for constant /3 and ~; (c) variation in ~ for constant fl and ~. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

239

6.6 Size Distribution Effects o

w ~" ~-.

o

"r]~ o

r

NT

1.0

KNTR o #

0.8

0.2

-

G

r = 0.5

L.

.-~ C3 (1) ..(3

F

0.6

Rmo x = 34.7

1

~=

-1

KNT'r

0.4

:3

z

r

O3

0.2

O

"~ c

r

r

___~ c-

Rmo x = 5.40.0

1

3

:

5

7

E

P3

Rmo x = 16.4

I i , , ~ : 9 11 13 15 17 1 9 2 1 2 3 2 5 2 7 2 9 3 1

33353739

Porticle Size Rotio, R (=R/Ro) FIGURE 6.34

(Continued)

can be written as

ex {

+

]1 (6.123)

+

2(x 2)

+ - ~ - In

where fl =

KaNrR* G '

1

~

Ka NT f,

~

v~R* NT .

(6.124)

For any population, a physical constraint m u s t be placed on the system in which conservation of mass accounts for the m a x i m u m particle size present in the reaction environment. Mathematically, this means t h a t the following population balance relationship must hold:

Mr =

IRmaxx3 ~o(X) dx. JR*

(6.125)

Therefore, the new variable, Rmax, constrains the population balance so t h a t it agrees with the mass balance. This means that for fixed values of fl, s, and ~b, the final d i s t r i b u t i o n ~ i n c l u d i n g the m a x i m u m particle s i z e ~ c a n be predicted. The qualitative effect of fl, e, and ~ on the particle size distribution is realized by varying these parameters as shown in Figs. 6.34. For most cases shown in Figure 6.34, a bimodal size distribution with one mode corresponding to nuclei that have grown

240

Chapter 6 Liquid Phase Synthesis by Precipitation

atomistically and the other mode to aggregates. In the other cases shown in Figure 6.34, a single mode distribution function is observed, corresponding to an exponential decay typical of a CSTR without aggregation. In all of these cases, the aggregation rate is small compared with the atomistic growth rate. Other theories [111,116] give bimodal sized distributions. Due to the many approximations made in this derivation, only qualitative agreement of experiment with theory have been made to date. The size distribution can be easily classified to keep the aggregates and recycle the nuclei. This gives a narrow size distribution for the production of ceramic powders.

6.6.3.2 Batch Reactor with Aggregation Particle growth in a batch environment is more difficult to predict because the steady state assumption previously used for the CSTR case is no longer applicable. For a batch precipitator, the simplified population balance becomes O~o(R, t) + G 0~~ _t_____) Ot OR = K~Rrl ~ "oo(R, t) - 2Xa~o(R, t) NT(t)

)

~ + ~- + 1 .

(6.126)

Again the assumption on the aggregation rate are that the most frequent collisions are between the larger particle and the small pattides. This partial differential equation can be approximated by an ordinary one by creating a new characteristic time variable, t'(= t R / G = R*/G], which is constant. With this variable change the population balance becomes 2 G\

= K~R'o ~ ~oo(R, ) - 2KarlO(R, t') N~(t')

+

+ 1 .

(6.127) By application of this transformation under conditions of constant t', the dimensionless solution to the characteristic population balance for a batch reactor can be found to be ~ ((~,)2.1)

~ ~,o

+

(x2

(6.128) + -~ln

where fl=

KaNTR* G '

~~ R* 4)=

NT

(6.129)

241

6.6 Size Distribution Effects

This solution to the population balance is only good under conditions of constant t'. This means that, for every particle of size R, a corresponding time is needed for its formation. This assumption is less in error for narrow distributions t h a n for broad ones. If we make the further approximation that the aggregates are narrow in their size distribution, a delta function can be used to describe them. The delta function has a first moment, R---~, and a second moment, R---~+ (R-~)2 ~ (~-~)2. This simplifies the death function as follows: D ( R , t')

= 2ga~0(R

,

t) NT(t)

-~ + - ~ + 1

= 8Ka~Qo(R ,

t') NT(t').

(6.130) The work of Smoluchowski also gives a relationship for the decrease in the total n u m b e r of particle per unit volume, N T , as a function of time" GNotl/2 NT(t') = G(t~/2 + t ' ) - R

(6.131)

where No is the initial n u m b e r density of singlet particles at t = 0. Using these further simplifications, the following solution can be obtained:

-1+ where X = KaNoR*. 4G '

~ _ Gtl/2 _ tl/2 R* t' "

(6.133)

This solution to the population balance is good only under conditions of constant t'. This means that, for every particle of size R, a corresponding time is needed for its formation (i.e., t = (R - R*)/G). For this population also, a physical constraint m u s t be placed on the system in which conservation of mass accounts for the m a x i m u m particle size present in the reaction environment. Mathematically, this means that the following population balance relationship m u s t hold: rRmax

M T = JR|* X3~o(X) dx.

(6.134)

Therefore, the new variable, Rmax,constrains the population balance so t h a t it agrees with the mass balance. This means that for fixed values of X and ~, the final d i s t r i b u t i o n ~ i n c l u d i n g the m a x i m u m particle size--can be predicted. The qualitative effects of X and 9 on the particle size distribution is shown in Figures 6.35(a) and (b). In

Chapter 6 LiquidPhase Synthesis by Precipitati~

242

o

~/,= 0 2 5

131 g..

2.001

.

/

E

.

.

.

.

=

.

KN R " o o 4G

t

0.5

•

I

O

D .O t_

/

. w

~

/i

x=~

03

. m

C3 t_ (D

E

1.00~ i~

X = 1.0

Rmax=5"l

/i

/

/ !

/

X = 2.O

z

O3 03 q0 C ~O

03 C

0.50

0.00

E i5

1

2

3

4

Particle

5

-6

~ 7

i 8

Size Ratio, R (=R/R

: 9

o)

b o~ ~, ~.

ti/2

X = I - O = ~

1.00

,/, = 0.s I

..2

max = 3"21 ii

.o_

/i

a

,-

~

E

o')

= 5.1

/hA

".~

z

~ = 0.25 Rmax

/1

lk = O. I

i

/I

A

~

KN~176

,G

.:O.O

Rma x =

/l

/

0.25

C

.9 (-

q)

E a

0.00

1

3

5

!

7

9

11

13

15

~

Particle Size Ratio, R ( = R / R o ) Solution to the differential equation for a batch reactor with aggregation (i.e., equation (6.132)). (a) Variation in ~ for constant X; (b) variation in X for constant ~. Redrawn with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

F I G U R E 6.35

243

6.6 Size Distribution Effects E i,< Q; N

Or) (1)

(l) (:T, Cr~

ou -0 (D

1.8 1.6

I

:

CuOX 2

1.4

A = Ti02

1.2

9

= ZrO 2

1.0 0.8 0.6 0.4 0.2 0.0 0.0

,

I

0.5

,

I

1.0

..........,

........... I

1.5

,

I

,

2.0

2.5

Dimensionless Time, In(1 +t/t1/2) Characteristicanalysis on typical ceramic powder formation reactions. Data plotted accordingto equation (6.135). Redrawn, with permission from Dirksen, S. Benjelloun, and Ring [120].

FIGURE 6.36

these figures, we also see a bimodal size distribution with one mode corresponding to nuclei that have grown to R = R * + Gt and the other mode corresponding to the aggregates. Using this theory, a kinetic expression for the mean size of the aggregates, h(t), after the initial burst of nuclei can be derived: h ( t ) - h 0 = h01n(1 + t~/2) = ~ m R * ln(1 + t~/2)

(6.135)

where the initial aggregate size, ho, is composed of m nuclei of size R*, corresponding to the conservation of hDF = m(R*)D~

(6.136)

where DR is the fractal dimension [117, 118] of the aggregates. Figure 6.36 is a plot of the mean aggregate size as a function of time for several experimental systems [119-121]. In all cases the experimental results follow the linearized fit with the fit parameters given in Table 6.6. The value of h0 (intercept) is reasonable compared to the initial aggregate size observed in the electron micrographs of the particles [120]. In other experimental cases, notably FePO4 precipitation [122] and CeO2 precipitation by forced hydrolysis [102], aggregated particle structures are produced during batch precipitation. Unfortunately in these experiments, the growth of aggregates was not studied, so a comparison with the preceding simplified theory cannot be made.

244 TABLE 6.6

Chapter 6 Liquid Phase Synthesis by Precipitation Results from Characteristic Analysisa

System

ko(s lope ) (tLm)

ko(i n te rcep t ) (t~m)

tl/2 (min)

Diffusion controlled*

Copper oxalate b Zirconiac Titania d

0.109 1.334 0.146

0.097 0.692 0.109

43 291 12

Accept Accept Reject

* The null hypothesis is that the reaction is diffusion controlled at a 95% confidence level. a Redrawn, with permission from J. A. Dirksen and T. A. Ring, Chem Eng. Sci. 46(10), 2389-2427 (1991). b Dirksen, J. A. Benjelloun, S., and Ring, T. A., Colloid Polymer Sci. 268, 864 (1990). c Ogihara, T., Mizutani, N., and Kato, M., J. Am. Ceram. Soc. 72, 421 (1989). d Jean, J.-H., and Ring, T. A., Langmuir 2, 251 (1986).

6.7 C O P R E C I P I T A T I O N OF C E R A M I C P O W D E R S The recent ceramic literature is filled with examples of ceramic powders produced by "coprecipitation" of a precursor salt then thermal decomposition to the oxide. The benefit of coprecipitation most often stated in this literature is "atomistic mixing" in the resulting ceramic powders. In the ceramic literature the coprecipitation process is described as follows. A solution of metal salts, often nitrates, is mixed with a precipitating agent common for all the metals. Oxalic, citric, or other organic acid or ammonium hydroxide are frequently used as the precipitating agent because many metal oxalates or citrates or hydroxides have a low solubility. What happens during this coprecipitation process? Basically there are two possibilities: (1) precipitation of a mixed metal precursor crystal (i.e., a double, or triple, salt) with a specific stoichiometric ratio of the metals, which will be called true coprecipitation; or (2) segregative precipitation of individual particles of the different metal oxalates, for example, that are colloidally unstable and hetero-coagulate together into a mixed aggregate particle, which will be called simultaneous precipitation and coaggregation. Which possibility takes place depends on the thermodynamics of the solution.

6.7.1 True Coprecipitation Consider a solution with two metals A and B in solution with a precipitation agent X. With true coprecipitation, only one equilibria is

6.7 Coprecipitation of Ceramic Powders

245

responsible for the formation of a solid phase as follows: A +2 + nB +a + mX--->

nBnXm(s)

with only one solubility product K~nX~. Examples of this type of precipitation are hydrated metal arsenate, selenates, permanganates, chromates, and dichromates, and ferrocynates (e.g., Co3(AsO4)2 98H20, CdSeO4 92H20, LiMnO4 93H20, ZnCr204 92H20, Co2Fe(CN)6 9xH20). These examples can be referred to as double metal hydroxides or double salts. There is little information in the chemical literature on the subject of multimetal carboxylate solubility products. The only mixed metal carboxalate example that the author could find in the literature is that of copper acetate meta arsenate [123], Cu(C2H302)2 93Cu(AsO2)2, which is an insoluble pigment called Paris green. Even more complicated examples of three mixed metal precipitates exist for example CsGa(SeO4)2 92H20. In this case and in the case of double salt precipitation, a mixed metal complex is likely to form in solution that is then precipitated out of the solution. For that reason, knowing what mixed metal complexes form in solution is a way of predicting what mixed metal precipitates can be produced. A number of mixed metal alkoxides can be produced in alcohol solution. These mixed metal alkoxides can be hydrolyzed to give a mixed metal hydroxide that may be insoluble or may further react via condensation polymerization to form the mixed metal oxide. A large list of the possible mixed metal alkoxide complexes in alcohol solutions is given in Bradley's book [124] on metal alkoxides. A SrTi(O-i-Pr)~* complex in solution is responsible for the production of SrTiO3 after hydrolysis and calcination and a ZrTi(O-i-Pr)~ complex is responsible for the production of Z r T i Q . Some thorough experiments with mixed metal alkoxide complexes have been performed by a group at the University of New Mexico at Albuquerque lead by Mark Hampden-Smith. His group has produced MTi(O-i-Pr)5 complexes [126] (where M = Li, Na, or K) and MSn(OEt)~ complexes [127] where M = T12 or Zn. Although these mixed metal alkoxide complexes are a means of obtaining atomic mixing of the various metals, the stoichiometric ratio of metals may or may not be that desired for the ceramic powder. A different stoichiometric ratio, n, for the same two metals is not likely to be precipitated out just because the initial solution contains a different stoichiometric ratio. Also for many electronic ceramic compositions it is desirable to have many other metals incorporated at the ppm level into the ceramic powder as sintering aids, grain growth inhibitors, and crystal phase stabilizers. Adding these other metals is very difficult * ( O - i - P r ) is the isopropoxide group.

246

Chapter 6 Liquid Phase Synthesis by Precipitation

with true coprecipitation because only the one double salt, A B n Z m ( s ) , is insoluble.

6.7.2 Simultaneous Precipitation and Coaggregation The other precipitation possibility is the simultaneous precipitation of two insoluble species controlled by two separate precipitation reactions as follows: A § + 2 X - ~ AX2(s) B +3 + 3X- --* BX3(s)

withKs~AZ2 withKSZ3.

To precipitate a particular bulk stoichiometric ratio, it is necessary to note the different solubility of the two salts and compensate for the extra solubility of one of the salts with a higher initial concentration of the metal ion. Once this is done, the solid produced will have the desired bulk stoichiometry. But how well mixed will these metals in the solid be? Certainly not at an atomic level. The scale of chemical segregation depends on the relative rates of nucleation, growth, and aggregation in precipitation. Specifically the rate of hetero-aggregation is the most important factor that will determine if the individual particles are of a single solid AX2(s) or BX3(s) or a mixture of the two. The rates of nucleation, growth, and homo-aggregation of each solid will determine the size of the particles either AX2(s) or BX3(s) separately. The extremes of this process are shown schematically in Figure 6.37. In this figure, we see nucleation and atomistic growth taking place

// Nucleation + Growth

Aggregation

Homo-

Nucleation + Growth

HeteroAggregation

FIGURE 6.37 Segregation of AX2(s) (black squares) and BX3(s) (open circles) during simultaneous precipitation and coaggregation.

6.7 Coprecipitation of Ceramic Powders

247

separately and then either homo-aggregation or hetero-aggregation. With homo-aggregation, we have chemical segregation on the aggregate size scale. With hetero-aggregation, we have chemical segregation on the individual particle size scale. The size of the individual particles is controlled by the rates of nucleation and atomistic growth of each of the solids AX2(s) and BX3(s). To decrease the size of the chemical segregation, the individual particles should be the smallest possible. This can be done by using precipitation conditions of high supersaturation, above the critical value for homogeneous nucleation. This gives high rates of nucleation, which will deplete the supersaturation in solution faster than the similarly high rate of atomistic growth. To promote hetero-aggregation in preference to homo-aggregation the system must be completely colloidally unstable. This is frequently done with a large concentration of inert salt left in solution after the precipitation reaction, which decreases the double layer thickness and causes fast electrostatic coagulation of all particles in the suspension. In other cases, polymeric flocculants can be used for this purpose. This type of behavior can be seen in the coprecipitation of BaY2Cu3 (oxalate) produced by quickly mixing equal volumes of two solutions one 0.4 M in oxalic acid and the other 4.8 • 10 .2 M in Ba(NO3)2, 2.4 • 10 .2 M in Y(NO3)3, and 6.8 • 10 .2 M in Cu(NQ)2. Five minutes after mixing the two solutions three types of particles (i.e., yttrium, barium, and copper oxalate) are observed to be hetero-aggregated together see Figure 6.38(a). EDAX examination of the larger spherical particles shows that they are copper oxalate and the very fine particles are barium oxalate and yttrium oxalate. Due to their morphology the copper oxalate spheres appear to be formed by homo-aggregation. Figure 6.38(b) shows the same precipitation product after 2 days in the mother liquor. The barium and yttrium oxalate particles have been reorganized by dissolution and reprecipitation into a square sheet-like crystal of a double salt of yttrium and barium oxalate 30 t~m on an edge and spheres of copper oxalate remain unchanged. At this time we know very little about this double salt of yttrium and barium oxalate [128]. Were it not for the fact that 5 min after mixing a hetero-aggregated precipitation product was observed, the segregation shown in Figure 6.38(b) could have been interpreted to be the result of the separate nucleation, growth of YBan(oxalate), and homo-aggregation with copper oxalate particles. The segregation observed in these precursor powders is often lost by calcining the powders where solid state interdiffusion and reaction of the different metals takes place. If the segregation is large scale, then it will take a long time for this interdiffusion to take place. For this reason an understanding of the segregation processes in coprecipitation is important.

FIGURE 6.38 Precipitation of YBa2Cu 3 oxalate, a precursor of the superconducting YBa2Cu307. (a) Sample taken 5 rain after mixing two solutions as described in the text. By EDAX large spheres are copper oxalate and the small particles cementing the spheres together are a yttrium bartium oxalate mixture. Bar = 10 ftm. (b) Sample taken 2 days after mixing two solutions as described in the text. By EDAX spheres are copper oxalate and thecrystal in the center of the picture is a yttrium bartium oxalate mixture coaggreg, ated with the copper oxalate spheres. Bar = 20 ftm. Printed with permission from Dirksen and Ring [4a]. Reprinted from [4a], copyright 1991, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

6.8 Summary

249

6.8 S U M M A R Y This chapter has developed the fundamental concepts for the precipitation of ceramic powder precursors. Classical nucleation theory was discussed to explain homogeneous nucleation, the process by which particles are created initially. In addition, heterogeneous nucleation and secondary nucleation were discussed. Particle growth was discussed next, giving several common growth mechanisms that operate at different supersaturations and have different size and supersaturation dependences. With batch reactors, the supersaturation changes during precipitation and as a result the growth mechanism may also change during the course of batch precipitation. The slowest growing crystal faces determine the morphology of the precipitated particles. Impurities adsorb on each crystal face differently and effect their growth rates and therefore the crystal morphology. The addition of impurities to precipitating systems is a common industrial practice to control crystal morphology. The overall particle size distribution for two idealized precipitators, batch and CSTR, was discussed. Aggregation plays an important role in the governing population balance as another important particle growth mechanism. Control of the kinetics of nucleation, growth, and aggregation allows the precipitation of ceramic powder precursors with desired stoichiometries, morphologies and size distributions.

Problems 1. In the precipitation of Mg(OH)2, we find the crystalites to have a hexagonal platelet morphology. Assuming that we mix equal volumes of a 0.5 M solution of MgC12 with one 1.1 M in NH4OH at 100~ (a). What is the final pH of the solution after precipitation? (b). What is the induction time for nucleation? (c). What is the critical nuclei size? What is the number of molecules in a critical nuclei? (d). What is the initial nucleation rate? Data for Mg(OH)2: Solubility in water at 100~ = 0.04 gm/liter, M w = 58.33, p = 2.36 gm/cc, T S L " - 120 erg/cm 2, molecular diffusion coefficient D = 1.2 • 10 .5 cm2/sec. 2. In the batch precipitation of Mg(OH)2, we have calculated the initial nuclei size. After many hours the saturation ratio decreases 100fold. What is the stable particle size after many hours? Will the initial nuclei that have growth to twice their initial size remain or will they dissolve and a new particle size reprecipitated? If so, what is this new particle size?

250

Chapter 6 Liquid Phase Synthesis by Precipitation

3. To develop a new lower sintering temperature alumina chip substrate, a new type of powder is required consisting of an aluminum oxide core and a magnesia coating. Note: Magnesia at >0.05% weight is a liquid phase sintering aid for alumina allowing sintering at -1600~ (instead of-1900~ for pure alumina) and also preventing exaggerated grain growth. To produce this powder, it is necessary to coat an alumina powder (mean size of 0.5 ftm, geometric standard deviation of 1.3) with a precipitate of Mg(OH)2 using solutions of MgC12 and NH4OH. Because the composite powder will be tape cast, it will need a large amount of polymeric binder in the paste formulation, and for that reason the coating must be uniform and no unseeded homogeneous nucleation of Mg(OH)2 should occur, because these fine particles will produce a green body with much finer pores and greater difficulty removing the binder. Determine the window for the saturation ratio to cause seeded heterogeneous nucleation but essentially prevent homogeneous nucleation of the Mg(OH)2. Data: Mg(OH)2-AI(OH)3Tss = 10 erg/cm 2. 4. For the preceding coating precipitation, what is initially the diffusion controlled growth rate of the particles if the initial saturation ratio is S = (Sc,homo- Sc,hetero)/2? 5. In the electro-fusion process, alumina is heated by an electric arc furnace to its melting point of 2045~ Then it is allowed to cool and crystalize very slowly, giving rise to large crystals ( - 5 cm). The three principle exposed crystal surfaces have the following surface energies, ~/~ = 125.0, ~/2 = 276.3, ~/3 = 277.2 ergs/cm 2. Determine the equilibrium crystal morphology of this electro-fused A1203. Is this likely to be the crystal morphology? 6. For the Mg(OH)2 precipitation described in problem 1, determine the particle size distribution produced in a CSTR precipitator without aggregation if the supersaturation ratio in the tank is 100 and the growth is mononuclear. The volume of the CSTR is 10 liters and the total flow rate is 1 liter per min. 7. How many tanks in series are needed if the desired particle size distribution should have a coefficient of variation less than 35%? Assume multipoint nucleation. 8. For the CSTR in problem 6 the average colloid stability ratio is 1.0 due to the high concentration of NH4C1 in the solution. Determine the particle size distribution produced by the precipitator if aggregation is also considered. Use the viscosity of water 1 cp.

References 1. Vereecke, G., and Lemaitre, J. Cryst. Growth 104, 820-832 (1990). 2. Nielsen, A. E., "Kinetics of Precipitation." Pergamon, Oxford, 1964.

References

251

3. Becker, R., and Doring, W., Ann. Phys. (Leipzig) [5] 24, 719-752 (1935). 4. Volmer, M. M., and Weber, A., Z. Phys. Chem. 119, 227 (1926). 4a. Dirksen, J. A., and Ring, T. A., Chem. Eng. Sci. 46(10) 2389-2427 (1991). 5. Garside, J., J. Chem. Eng. Sci. 40, 1 (1985). 6. Einstein, A., Ann. Phys. (Leipzig) [4] 17, 549 (1905). 7. Nielsen, A. E., Acta Chem. Scand. 15, 441-442 (1961). 8. Mullin, J. W., and Gaska, G., Can. J. Chem. Eng. 47, 483 (1969). 9. Dunning, W. J., in "Chemistry of the Solid State" (W. E. Garner, ed.), p. 159. Butterworth, London, 1955. 10. Hung, C.-H., Krasnopoler, M. J., and Katz, J. L., J. Chem. Phys. 90(3), 1856-1865 (1989). 11. Schonel, O., and Nyvlt, J., Collect. Czech. Chem. Commun. 40, 511 (1975). 12. Hulbert, H. M., Chem. Ing. Techn. 47, 375 (1975). 13. Janse, A. H., and de Jong, E. J., Trans. Inst. Chem. Eng. 56, 187 (1978). 14. Mullin, J. W., and Janci~, S. J., Trans. Inst. Chem. Eng. 57, 188 (1979). 15. Ilievshi, D., Zheng, S. G., and White, E. T., CHEMECA 89, Technol. Third Century Broadbeach, Queensland, Australia, 1989, Paper 29b (1989). 16. Botsaris, G. D., in "Industrial Crystallization" (J. W. Mullin, ed.), p. 3 Plenum, New York, 1976. 17. Estrin, J., in "Preparation and Properties of Solid State Materials" (W. R. Wilcox, ed.), Vol. 2. Dekker, New York, 1976. 18. Garside, J., and Davey, R. J., Chem. Eng. Commun. 4, 393 (1980). 19. Randolph, A. D., Beckman, J. R., and Kraljevich, K., AIChE J. 23, 500 (1977). 20. Shaw, B. C., McCabe, W. L., and Rousseau, R. W., AIChE J. 19, 194 (1973). 21. Garside, J., and Shah, M. B., Ind. Eng. Chem. Process Des. Dev. 19, 509 (1980). 22. Garside, J., and Larson, M. A., J. Cryst. Growth 43, 694 (1978). 23. Happel, J., and Brenner, H., "Low Reynolds Number Hydrodynamics." PrenticeHall, New York, 1965. 24. Powers, H. E. C., Ind. Chem. 39, 351 (1963). 25. Sung, C. Y., Estrin, J., and Youngquist, G. R., AIChE J. 19, 957 (1973). 26. Jagannathan, R., Sung, C. Y., Estrin, J., and Youngquist, G. R., AIChE Symp Ser. 193, 76, 90 (1980). 27. Wang, M. L., Huang, H. T., and Estrin, J. AIChE J. 27, 312 (1981). 28. Elwell, D., and Scheel, H. J., "Crystal Growth from High-Temperature Solution." Academic Press, London, 1975. 29. Tempkin, D. E., "Crystallization Processes," p. 15. Consultants Bureau, New York, 1964. 30. Jackson, K. A., "Liquid Metals and Solidification," p. 174. Am. Soc. Metals, Cleveland, OH, 1958. 31. Bennema, P., and van der Eerden, J. P., J. Cryst. Growth 42, 201 (1977). 32. Bourne, J. R., and Davey, R. J., J. Crystal. Growth 36, 287 (1976). 33. Bourne, J. R., Davey, R. J., and Hunger Buhler, K., J. Cryst. Growth 34, 221 (1976). 34. Bourne, J. R., and Davey, R. J., J. Cryst. Growth 39, 267 (1977). 35. Bourne, J. R., Davey, R. J., and McCullock, J., Chem. Eng. Sci. 33, 199 (1978). 36. Davey, R. J., in "Industrial Crystallization 78" (E. J. Jong and S. J. Janci~, eds.), p. 169. North-Holland Publ., Amsterdam, 1982. 37. Davey, R. J., Curr. Top. Mat. Sci. 8, 249 (1982). 38. Bourne, J. R., AIChE Symp. Ser. 193(76), 59 (1980). 39. Volmer, M. M., "Kinetik der Phasenbildung," p. 209. Steinkopff, Dresden, Leipzig, 1939. 40. Neilsen, A. E., J. Phys. Chem. 65, 46 (1961). 41. Reiss, H., and LaMer, V. K., J. Chem Phys. 18, 1 (1950). 42. Reiss, H., J. Chem. Phys. 19, 482 (1951). 43. Ham, F. S., J. Phys. Chem. Solids 6, 335 (1958).

252

Chapter 6 Liquid Phase Synthesis by Precipitation

44. Carlson, A. E., in "Growth and Perfection of Crystals" (R. H. Doremus, B. W. Roberts, and D. Turnbull, eds.), p. 421. Wiley, New York and Chapman & Hall, London, 1953. 45. Hixon, A. W., and Knox, K. L., Ind. Eng. Chem. 43, 2144 (1951). 46. Mullin, J. W., and Garside, J., Trans. Inst. Chem. Eng. 45, 1285 (1967). 47. Burton, J. A., Prim, R. C., and Slichter, W. P., J. Chem. Phys. 21, 1987 (1953). 48. Coulson, J. M., and Richardson, J. F., "Chemical Engineering 2." Pergamon, Oxford, 1956. 49. Landise, R. A., Linares, R. C., and Dearborn, E. F., J. Appl. Phys. 338, 1362 (1962). 50. Jaganathan, R., and Wey, J. S., J. Cryst. Growth 51, 601 (1981). 51. Garside, J., and Tavare, N. S., Chem. Eng. Sci. 36, 836 (1981). 52. Garside, J., Chem. Eng. Sci. 26, 1425 (1971). 53. Verma, A. R., "Crystal Growth and Dislocations." Butterworth, London, 1953. 54. Forty, A. J., Philos. Mag. [7] 42, 670 (1951). 55. Frank, F. C., Discuss. Faraday Soc. 5, 48 (1949). 56. Burton, W. K., Cabrera, N., and Frank, F. C., Philos. Trans. R. Soc. London, Ser. A 243, 299-358 (1951). 57. Bennema, P., and Gilmer, G. H., in "Crystal Growth" (P. Hartmann, ed.). NorthHolland Publ., Amsterdam, 1973. 58. Tolkdorf, W., and Welz, A., J. Cryst. Growth 13/14, 566 (1972). 59. Carbrera, N., and Levine, M. M., Philos. Mag. [7] 1, 450 (1956). 60. Chernov, A. A., Sov. Phys.--Usp. (Engl. Transl.) 4, 129 (1961). 61. Chernov, A. A., Sov. Phys.--Cryst. (Engl. Transl.) 8, 63 (1963). 62. Gibbs, J. W., Trans. Connec. Acad. Arts Sci. 3 (1875); "Collected Works." Longmans, Green, New York, 1928. 63. Wulff, G., Z. Kristallogr. Mineral. 34, 449-530 (1901). 64. Valetan, J. J. P., Ber. Dtsch. Math.-Phys. Kl. K. Sach. Ges. Wiss. (Leipzig) 67,1 (1915). 65. Buckley, H. E., "Crystal Growth." Wiley, New York, 1951. 66. Mullin, J. W., "Crystallization," 2nd ed. Butterworth, London, 1972. 67. Van Hook, A., "Crystallization." Reinhold, New York, 1961. 68. Alexandru, H. V., J. Cryst. Growth 5, 115 (1969). 69. Belyutsin, A. V., and Dvorikin, V. F., in "Growth of Crystals" (A. V. Shubnikov and N. N. Sheftal, eds.), Vol. 1, p. 139. North Holland, Amsterdam, (1958). 70. Kern, R., Growth Cryst. 8, 3 (1969). 71. Vold, M. J., J. Colloid Sci. 18, 684 (1963). 72. Sutherland, D. N., J. Colloid. Sci. 25, 373 (1967). 73. Vold, M. J., J. Phys. Chem. 63, 1608 (1959). 74. Jullien, R., J. Phys. Rev. A 29, 997 (1984). 75. Ball, R. C., and Jullien, R., J. Phys. Lett. (Orsay, Fr.) 45, L103 (1984). 76. Meakin, P., in "Random Fluctuations and Pattern Growth" (H. E. Stanley and N. Ostronsky, eds.) p. 174-191. Kluwer Academic Publishers, London, 1988. 77. Rome de L'Isle, "Crystallographie, 2nd. ed., p. 379. Paris, 1783. 78. Leblanc, N., Journ. Phys. 33 (1788); Ann. Phys. (Paris) 23, 375 (1788). 79. Langmuir, I., J. Am. Chem. Soc. 40, 1361 (1918). 80. Adamson, A. W., "Physical Chemistry of Surfaces," 4th ed., p. 396. Wiley (Interscience), New York, 1982. 81. Zhu, B. Y., and Gu, T., J. Chem. Soc., Faraday Trans. 1 85(11), 3813-3817 (1989). 82. Novich, B., and Ring, T. A., Langmuir 1, 701 (1985). 83. Burrill, K. A., J. Cryst. Growth 12, 239 (1972). 84. Mullin, J. W., and Leci, C. L., Chem Eng. Prog., Symp. Ser., Paper 39a (1970). 85. Carbrera, N., and Vermilyea, D. A., in "Growth and Perfection of Crystals" (R. H. Doremus, B. W. Roberts, and D. Turnbull, eds.), p. 393. Wiley, New York and Chapman & Hall, London, 1958.

References

253

86. Albon, N., and Dunning, W. A., Acta Crystallogr. 15, 115 (1962). 87. Price, P. B., Vermilyea, D. A., and Webb, M. B., Acta MetaU. 6, 524 (1958). 88. Price, P. B., Vermilyea, D. A., and Webb, M. B., Acta Metall. 6, 524 (1958). 88. Slavnova, E. N., Growth Cryst. 1, 117; 2, 166 (1958). 89. Sears, G. W., J. Chem. Phys. 29, 104-105 (1958). 90. Michaels, A. S., and Tausch, F. W., Jr., J. Phys. Chem. 65, 1730-1737 (1961). 91. Egli, P. H., and Zerfoss, S., Discuss. Faraday Soc. 5, 61 (1949). 92. Hartman, P., Growth Cryst. 7, 3 (1969). 93. Ledfirset, M., and Monier, J. C., Colloq. Int. C.N.R.S. 152, 537 (1965). 94. Mates, T. E., and Ring, T. A., Colloids Surf. 24, 299-313 (1987). 95. Jean, J. H., and Ring, T. A., Proc. Br. Ceram Soc. 38, 399 (1986). 96. Randolph, A. D., Can. J. Chem. Eng. 42, 280 (1964). 97. Hulburt, H. M., and Katz, S., Chem. Eng. Sci. 19, 555 (1964). 98. Randolph, A. D., and Larson, M. A., "Theory of Particulate Processes," 2nd ed. Academic Press, San Diego, CA, 1988. 99. Bourne, J. R., Hunger Buehler, K., and Zabelka, M., in "Industrial Crystallization" (J. W. Mullin, ed.), p. 283. Plenum, New York, 1976. 100. Abegg, C. F., and Balakrishnan, N. S., Chem. Eng. Prog., Symp. Ser. 110(67), 88 (1971). 101. Overbeek, J. T. G., Adv. Colloid Interface Sci. 15, 251 (1982). 102. Matijevic, E., Pure Appl. Chem. 60(10), 1479-1491 (1988). 103. von Smoluchowski, M., Z. Phys. Chem. 92, 129 (1917). 104. Ives, K. J., in "The Scientific Basis of Flocculation" (K. J. Ives, ed.), p. 37. Sijthoff and Noordhoff, The Netherlands, 1978. 105. Randolph, A. D., and Larson, M. A., "Theory of Particulate Processes." Academic Press, New York, 1971. 106. Lui, Y.-M. R., and Thompson, R. W., Chem. Eng. Sci. 47, 1897-1901 (1992). 107. Hounslow, M. J., AIChE. J. 36, 106-116 (1990). 108. Tavare, N. S., and Patwardhan, A. V., AICHE. J. 38, 377-384 (1992). 109. Saleeby, E. G., and Lee, H. W., Chem. Eng. Sci. 49(12), 1879-1884 (1994). 110. Hoyt, R. H., Ph.D Thesis IS-T-811, Iowa State University, Ames (1978). 111. Lamey, M., and Ring, T. A., Chem. Eng. Sci. 41, 1213-1219 (1986). 112. Edelson, L. H., and Glazer, A. M., J. Am. Ceram. Soc. 71(4), 225-235 (1988). 113. Berry, C. R., Photogr. Sci. Eng. 20, 1 (1976). 114. Margolis, G., and Gutoff, E. B., AIChE J. 20, 467 (1974). 115. Gutoff, E. B., Cottrel, F. R., and Denk, E. G., Photogr. Sci. Eng. 22, 325 (1978). 116. Delpech De Saint Guilhem, X., and Ring, T. A., Chem. Eng. Sci. 42, 1247-1249 (1987). 117. Mandelbrot, B. B., "The Fractal Geometry of Nature." Freeman, San Francisco, CA, 1982. 118. Witten, T. A., and Cates, M. E., Science 232, 1607-1612 (1986). 119. Ogihara, T., Mizutani, N., and Kato, M., J. Am. Ceram. Soc. 72, 421 (1989). 120. Dirksen, J. A., Benjelloun, S., and Ring, T. A., Colloid Polym. Sci. 268, 864-876 (1990). 121. Jean, J. H., and Ring, R. A., Langmuir 2, 251 (1986). 122. Wilhelmy, R. B., and Matijevid, E., Colloids Surf. 22, 111-131 (1987). 123. Weast, R. C., and Selby, S. M., "Handbook of Chemistry and Physics," 47th ed. Chem. Rubber Publ. Co., Cleveland, OH, 1966. 124. Bradley, D. C., Mehrotra, R. C., and Gaur, P. D., "Metal Alkoxides." Academic Press, London, 1970. 125. Smith, J. S., Dolloff, R. T., and Mazdiyasni, K. S., J. Am. Ceram. Soc. 53, 91 (1970).

254

Chapter 6 Liquid Phase Synthesis by Precipitation

126. Hampden-Smith, M. J., Williams, D. S., and Rheingold, A. L., Inorg. Chem. 29, 4076 (1990). 127. Hampden-Smith, M. J., Smith, D. E., and Duesler, E. N., Inorg. Chem. 28, 3399 (1989). 128. Benjelloun, S., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1993). 129. Barouch, E., and Ring, T. A., unpublished material. 130. Christoffersen, J., Christoffersen, M. R., and Kjaergaard, N., Kinetics of dissolution of calcium hydroxyapatite in water at constant pH, J. Cryst. Growth 43, 501-511 (1978). 131. Nielsen, A. E., Electrolyte crystal growth mechanisms, J. Cryst. Growth, 67, 289310 (1984).

7

P o w d e r Synthesis with Gas P h a s e Reactants

7.1 O B J E C T I V E S This chapter discusses four methods of gas phase ceramic powder synthesis: by flames, furnaces, lasers, and plasmas. In each case, the reaction thermodynamics and kinetics are similar, but the reactor design is different. To account for the particle size distribution produced in a gas phase synthesis reactor, the population balance must account for nucleation, atomistic growth (also called vapor condensation) and particle-particle aggregation. These gas phase reactors are real life examples of idealized plug flow reactors that are modeled by the dispersion model for plug flow. To obtain narrow size distribution ceramic powders by gas phase synthesis, dispersion must be minimized because it leads to a broadening of the particle size distribution. Finally the gas must be quickly quenched or cooled to freeze the ceramic particles, which are often liquid at the reaction temperature, and thus prevent further aggregation. 255

256

Chapter 7 Powder Synthesis with Gas Phase Reactants

7.2 INTRODUCTION

Gas phase reactors are used to produce the purest ceramic powders because it is relatively easy to obtain purified reactant gases with impurities at the ppm to ppb level. Dopants can be easily added to the ceramic powder in a controlled way by simply mixing in another reactant gas. Depending on the reaction kinetics these dopants may not be homogenously distributed within the resulting ceramic powder. In addition, gas phase reactors are typically continuous reactors that can be controlled very precisely by inexpensive methods. Gas phase reactors produce, for the most part, very fine ceramic powders and narrow particle size distributions, which is an advantage; however, these particles can also be fractal structures that are strongly agglomerated together. In process sintering of these fractal aggregates can sinter to give a roughly spherical particle. Important disadvantages are to be considered also. Gas phase reactors produce very fine ceramic particles (<0.5 t~m) at low volume fraction. These particles are difficult to separate from the large volume of gas, and as a result, a significant fraction of the cost of producing gas phase ceramic particles is spent on particle-gas separation, specifically the energy lost during the pressure drop in bag filtration. Several gas phase reactions produce ceramic powders on a commercial scale. Flame synthesis of titania by reaction of titanium tetrachloride with oxygen or steam to produce a submicron spherical pigment grade titania (albeit sintered aggregates) is the largest single gas phase powder synthesis reaction operated in industry today.* The second largest is the reaction with silicon tetrachloride, which is burned in air to produce silica.** Carbon black is also commercially produced by burning a hydrocarbon feedstock with insufficient oxygen.*** All of these reactions are exothermic and require no external energy source to keep them going after the reaction has been initiated. For endothermic reactions, gas phase powder synthesis can be performed with an external energy source to provide the necessary heat for reaction. This external energy source can be provided by a furnace, laser, or plasma (direct current, dc arc, or radio frequency, RF). In a furnace, conduction radiation and convection provide the energy. The ions and electrons of the plasma provide the energy in plasma synthesis. The wavelength of the laser must be coupled with a molecular vibration of one of the reactants or the carrier gas to provide the energy necessary for reaction in laser synthesis. On an industrial scale, only flame and * Operated by Dupont, Tioxide, and others. ** Operated by W. R. Grace Co. *** Operated by Cabort Corporation.

7.2 Introduction T A B L E 7.1

257

Gas Phase Powder Synthesis Routes

Reaction energy F l a m e synthesis Furnace synthesis

Exo Endo a

P l a s m a synthesis Laser synthesis

Endo a Endo a

Heat transfer in by

Radiation Convection Ions and e -b Laser ~, = molecule vibration b

Maximum temperature 2500 K 2300 K 10,000 K 1800 K

a Exothermic reactions are also possible. b S t a n d a r d methods of heat transfer are also possible.

furnace reactors are used. Much research is being done to perfect laser and plasma reactors for future ceramic powder synthesis. Gas phase ceramic synthesis is the subject of several review papers. The treatment here is analogous to that in Flagan [1], Friedlander [2], and Pratsinis and Kodas [3] but instead of using the traditional aerosol nomenclature, this chapter uses the nomenclature developed in Chapter 3 on population balances for educational continuity. Each of the gas phase powder synthesis methods is summarized in Table 7.1. The maximum temperatures are also listed. The adiabatic flame temperature is the maximum possible temperature achieved in flame synthesis and will depend on the concentration of reactants in the feed. Powder synthesis in a furnace uses conduction, convection, and radiation, giving a maximum temperature o f - 2 3 0 0 K. A plasma is an ionized gas. High velocity electrons remove other electrons from the neutral gas molecules present in the plasma, thereby producing ions and electrons that sustain the plasma. There are two types of plasma: (1) thermal (e.g., dc arc and radio frequency induction plasmas), where the temperature of electrons and ions is equal; and (2) low temperature (e.g., glow discharge), where the temperature of the electrons is much greater than that of the ions. Glow discharges are not used extensively for powder synthesis [4]. The electrons and ions move in the electric field of the plasma, reacting with the reactants to form the ceramic powder and other electrons. The maximum temperature of a thermal plasma can reach as high as ~15,000 K (e.g., 200 kHz-20 MHz argon plasma). As a result, ceramic materials synthesized in a plasma will be vaporized or molten and only upon leaving the plasma will they cool to form a solid ceramic particle. If the cooling rate is fast, amorphous material can be produced during this quench. Laser synthesis of ceramic powders can provide a maximum temperature of ~1800 K, but this temperature depends on the

258

Chapter 7 Powder Synthesis with Gas Phase Reactants

power of the laser used. The laser energy must be effectively coupled to a molecular vibration of a reactant or the carrier gas in the reaction cell. The energy absorbed by the gas in the reaction cell is highly dependent upon the pressure within the reaction cell and its energy coupling constant. Typical energy use efficiencies are 15% [3]. Figure 7.1 shows general reactor configurations for flame, furnace, plasma, and laser synthesis reactors for ceramic powders. In each case, there is a reaction zone that has a high axial velocity. The radial velocity component depends on the type of reactor. Some reactors have swirling flow to induce turbulence and mixing. In the reaction zone, gas phase reactants react to form a product that must nucleate a second phase, either liquid or solid, and then condense it from the vapor phase. This nucleation process is analogous to that discussed in Chapter 6, having a maximum free energy associated with a critical nuclei size. The typical size of these primary particles are between 1 and 500 nm. This primary particle can then grow by the incorporation of other product molecules one by one or by agglomeration with other particles. Because most of these vapor phase reactors have a very high reaction rate, the saturation in the vapor phase increases rapidly until the critical supersaturation for nucleation is reached, producing a very high number of nuclei per unit volume. After this burst of nucleation, the particles grow atom by atom from the remaining supersaturation and agglomerate as they move axially down the reactor. Figure 7.2 is a schematic of the number density of particles as a function of axial position in the flame front. Also shown in Figure 7.2 is the average diameter as a function of axial position. As the initial burst of nucleation takes place, the average particle diameter remains nearly constant. As soon as agglomeration starts, the average diameter increases. As the particles move down the reactor, the rate of particle growth decreases until a steady state diameter is reached. This decrease in growth rate is caused by the depletion of reactant or cooling of the reactant gases to decrease the reaction rate or quenching of the particles to decrease the agglomeration rate. When the particles are cooled sufficiently (i.e., below their melting point), they no longer stick together on collision, effectively stopping particle growth by aggregation. In this chapter each of the steps, reaction, nucleation, growth, and aggregation, as well as quenching will be discussed.

FIGURE 7.1 Reactor configuration for gas phase reactors: (a) Flame reactor, (b)furnace reactor, (c) laser reactor, (d) radio frequency (RF) plasma reactor, (e) direct current (dc) plasma reactor.

260

Chapter 7 Powder Synthesis with Gas Phase Reactants

F I G U R E 7.2 Schematic diagram of gas phase reactor with (a) number density of particles, (b) average diameter of particles, and (c) temperature of flame.

7.3 G A S P H A S E R E A C T I O N S Several g a s phase reactions can be utilized for flame, furnace, plasma, and laser synthesis.

7.3.1 F l a m e As described previously, flame synthesis reactions include the oxidation of silicon chloride to produce silica the oxidation of titanium chloride to produce titania and the oxidation of other metal chlorides (see Table 7.2 also). SiC14(g) + O2(g)--~ SiO2(s) + 2C12(g) TiC14(g) + 2H20(g)~ TiO2(s)+ 4HCI(g).

7.3 Gas Phase Reactions

261

TABLE 7.2 Flame Synthesis of Ceramic Powders ,,,

Ceramic powder

A1203 SiO2 TiO2 C Fe203, Cr203, A1203, V203, TiO2, SnO2, SiO2, ZrO2 GeO2, SiO2, POs, B203

Reactant

Reference

Al(C3H70)3 {acetylacetonate} SIC14 TIC14 Alkanes Metal Chlorides

a b c d e

GeC14, SIC14, SiBr4, POC13, BC13

f

a Sokolowski, M., Sokolowska, A., Michalski, A., and Gokieli, B. J., Aerosol. Sci. 8, 219-230 (1977). b Ulrich, G. D., Chem. Eng. News 6, 22-29 (1984). c George, A. P., Murley, R. D., and Place, E. R., Garaday Symp. Chem. Soc. 7, 63-77 (1973). d Dannenberg, E. M., J. IRI, 190-195 (1971). e Formenti, M., Juillet, F., Mereaudeau, P., Techner, S., and Vergnon, P. In "Aerosols and Atmospheric Chemistry" (G. Hidly, Ed.), Academic Press, New York, 1972. f French, W. G., Pace, L. J., and Foertmeyer, V. A., J. Physics Chem. 82, 21912194 (1978).

Other metal chlorides are sometimes added as trace gasses in this combustion process to alter the crystal phase and particle morphology of the product. Each of these reactions is highly exothermic, thus allowing the reaction to sustain itself by the energy produced by the reaction. The adiabatic flame temperature, TA, is given by T A : Tin +

~HRxN

Cp(products)

(7.1)

where Tin is the reactor feed temperature, AHRxN is the enthalpy of reaction at Tin and Cp(products) is the heat capacity at constant pressure of the products of combustion. In another example of flame synthesis, H2 (or other fuel) and 02 are used for combustion, and droplets of an aqueous salt solution are entrained in one of the streams. In a particular example an aqueous salt solution of yttrium, barium, and copper nitrates was used to create the aerosol entrained in the dry 02 stream of a hydrogen-oxygen coannular diffusion flame with the oxidant as the inner stream. The result was an unagglomerated YBa2Cu3Ox powder with a critical superconducting temperature of 92 K [5] confirming its high parity. The advantages of flame reactors are these: high purity gases can be used to produce a high purity solid product; simplicity of reactors; scale up has been demonstrated for carbon black, silica, and titania; a large range of particle diameters has been demonstrated. The disadvan-

262

Chapter 7 Powder Synthesis with Gas Phase Reactants

tages are that flame reactors produce hard agglomerates that under some conditions are low in density and fractal in shape (see Figures 2.6 and 7.22).

7.3.2 Furnace Decomposition Decomposition of complex metal organic molecules are typically endothermic reactions performed in furnace reactors. An example of a furnace decomposition is the thermal decomposition of dimethylchloro-silane to give silicon carbide [6]: (CHa)2SiC12(g) ~ SiC(s) + 2HCI(g) + H2C = CH2(g). Energy is provided to this endothermic reaction by the conduction, convection, and radiation from the furnace walls. This type of decomposition can also be caused on a hot substrate to produce a ceramic film. This process is called chemical vapor decomposition (CVD).

7.3.3 P l a s m a Plasma reactors provide very large quantities of energy for highly endothermic reactions. As a result of the high heating rate, the reactions are very fast, taking place by ionic intermediates. Cooling can also be relatively fast as the particles leave the plasma. With high cooling rates, nonequilibrium materials (e.g., amorphous materials) are sometimes produced. An example of an endothermic plasma synthesis reaction is the reaction of silica and aluminum nitride to produce a sialon [6]: SiO2(s) + A1N(s)~ SiA1ON(s) In this case, silica particles and alumina nitride particles are vaporized and recondensed as a composite particle with an amorphous structure. Table 7.3 lists other possible plasma synthesis reactions. The list includes carbides, nitrides, and oxides. Each of the reactions given is an endothermic reaction requiring heat from the plasma.

7.3.4 Laser Laser synthesis is a method whereby the energy in the laser is absorbed by one of the reactants of the chemical reaction or the carrier gas. This absorption of energy is enough to initiate a chemical reaction. By choosing the wavelength, ~,, of the laser, reaction pathways can be enhanced to give a desired product mix. To sustain an endothermic chemical reaction, larger amounts of laser energy are required. Often large power CO2 lasers are used that have several turnable wavelengths in the infrared spectrum. Examples of laser synthesis reactions are

7.4 Reaction Kinetics

263

TABLE 7.3 Plasma Synthesis of Ceramic Powders

Compound

Starting materials

Plasma type

SiC SiC SiC WC WC TiC TaC TaC B4C

CH3SiC13 SiOx + CH4 Sill4 + CH4 W + C/W + CH4 W30 + CH4 TIC14 + CH4 + He Ta + CH4 TaC15 + CH4 + He BC13 + CH4 + H2

RF/arc Arc Rf Arc Arc Arc RF Arc RF

Si3N4 Si3N4 Si3N4 A1N TiN TiN ZrN TaN MgN NbN VN HfN BN

SIC14 + NH3 + H2 Si3H4 + NH3 Si + N2/NH3 AINH3 TIC14 + N2 + H2 Ti + N2 Zr + N2 Ta + N2 Mg + N2 Nb + N2 V + Ne HfC14 + N2 + H2 BC13 + N2 + He

RF/arc RF RF/Arc RF RF RF RF/arc Arc Arc Arc Arc RF RF

TiB2

TIC14 + BC13

Plasma

A1203 A]203/Cr203 SiO2 SiO2/A1203 TiO2, TiO2/Cr203 ZnO, Sb203, BaO SiO2, MgO MgO ZrO2, ZrO2/A1203 ZrO2/SiO2

A1/A1C13 + O2 A1 halide + 02 + CRO2C12 SIC14 + 02 Si + A1 + 02 TiCla + 02 + CrO2C12 Oxides Mg(NO3)2(aq) Zr(NO3)2(aq)+ Al(NO3)3(aq) Zr(NO3)2(aq) + silicone oil

RF/arc RF RF RF RF Arc RF RF RF

Carbides

Nitrides

Borides Oxides

Source: Los Alamos National Laboratory [3] with additions by this author, from [6].

listed in Table 7.4. Carbides, nitrides, and borides have been produced by laser synthesis.

7.4 R E A C T I O N KINETICS The kinetics of each of these chemical reactions is highly dependent upon reactant concentration and temperature. Consider for a moment

264

Chapter 7 Powder Synthesis with Gas Phase Reactants T A B L E 7.4

Laser Synthesis Reactions

Oxides SiHt a + 202---> SiO2(s) + 2H20 TIC14 + O2 ~ TiO2(s) + 2C12 AICI3 + ~O2 --) A1203(s) + 2~C12 Carbides SiHt a + CHt --> SiC(s) + 4H 2 Sill4 a + ~1 C2H a4 --> SiC(s) + 3H2 CH3SiH3 a --~ SiC(s) + 3H2 SIC14 + CH4 ~ SiC(s) + 4HC1 Nitrides 3Sill4 a + 4NH3--~ Si3N4(s) + 12H 2 Borides TIC14 + B2H6a---~TiB2(s) + 4HC1 + H2 _

a Indicates absorption at ;~ = 10.6 tLm of CO2 laser, ~(CO2) = 9 . 2 - 1 1 / z m tunable.

the general reaction of (a) moles of A and (b) moles of B to produce (d) moles of D. aA(g) + bB(g)--~ d D ( s or 1). For this reaction, assumed to be nonequilibrium, the rate of reaction is given by Rate = - 1 / a d[A]/dt = - 1 / b d [ B ] / d t = 1 / d d[D]/dt ~ kl[A]~[B]~ (7.2) where [A] is the concentration of A, k~ is the reaction rate constant, and a and fl are the orders of reaction with respect to A and B, respectively. The rate constant k~ has temperature dependence according the Arrhenius law [7] as follows: kl = ko e x p ( - E A / R g T )

(7.3)

where EA is the activation energy for the reaction and k0 is the preexponential factor that has been shown to be either proportional to temperature, considering transition state theory [8], or proportional to the square root of temperature, considering collision theory [9]. In this formulation, the reverse reaction is neglected because it is often not important at the high operating temperatures of these gas phase reactions. However, if the back reaction is important the rate expression will be given by Rate = kl[A]~[B] ~ - k2[D] 8 where the reverse reaction rate constant is k2.

(7.4)

7.4 Reaction Kinetics

265

At equilibrium, the rate is zero and the following simplification can be made [D]~/[A]~[B] ~ = k l / k 2

= Kequilibriu m

(7.5)

where K e q u i l i b r i u m is the equilibrium constant for the reaction at the temperature of the reaction. A simplified reaction rate expression for equilibrium reaction is given by Rate = kl([A]

-

[A]e)a([B] - [S]e )fl

(7.6)

where the equilibrium concentrations of A and B are written a s [A] e and [B]e. This generalized reaction rate expression will be used to discuss the various types of reactions used in gas phase synthesis of ceramic powders. There are two general types of reactions: thermal decompositions and chemical combinations.

7.4.1 C o m b i n a t i o n R e a c t i o n s The reaction between two reactants occurs when the two reactant molecules collide with the proper orientation and energy to form an activated complex. This activated complex then decomposes into the products of the reaction. The rate limiting step of the reaction mechanism will determine the reaction rate expression. Depending on the overall reaction mechanism, the overall order of reaction, (~ + fl), will generally vary from 0 to 2 and sometimes involve rate laws that are different from that generalized previously. Many of the reactions with two reactants listed in Tables 7.2, 7.3, and 7.4 are combination reactions. Above 1000~ the reaction kinetics for combination reactions are fast, taking place in less t h a n a second in most cases. This speed coupled with the heat of reaction means that either a large amount of heat is absorbed quickly with an endothermic reaction or given off quickly with an exothermic reaction. A classic chemical combination reaction is that of the oxidation of TIC14: TiC14(g) + O2(g)--* TiO2(s) + 2C12(g). This reaction was studied [10-13] in a heated furnace (both horizontal and vertical) and a magnetically rotated dc plasma reactor. In the magnetically rotated dc plasma reactor Mahawili and Weinberg [10] found the reassociation of C1 radicals affects the global reaction rate at low oxygen concentration. Under these conditions, the reaction was zero order with respect to TIC14 and oxygen _ d[TiCl4] = k' e x p ( - E a / R g T ) dt

(7.7)

with a rate constant given as a preexponential factor, k', equal to a constant 0.36 • 1 0 - 4 moles/(sec cm 3) and an activation energy, EA,

266

Chapter 7 Powder Synthesis with Gas Phase Reactants

equal to 42 kJ/mole. In the furnace reactor with the reactants diluted with argon, Pratsinis et al. [13] found an oxidation rate expression given by _ d[TiC14______]]= (k a + kb[O2]l/2)[TiC14]

dt

(7.8)

where = 8.26 x 104 exp(-Ea/RgT)sec-1 kb = 1.4 x 105 exp(-Ea/RgT) (liters/mol) 1/2 sec -1 with Ea = 88.8 -+ 3.2 kJ/mol.

ka

This rate expression is valid over the range of oxygen to TIC14 ratio from near 0 to 10 and a t e m p e r a t u r e range from 700 to 1000~ These results are different from those observed in a dc plasma reactor due to the increased concentration of C1 radicals in the plasma, which gives a different reaction mechanism, where either the elementary reactions are different in the reaction pathway or there are different rate determining steps in the reaction pathway or both. Powers [14], French et al. [15], and T a n a k a and Kato [16] have studied the kinetics of the oxidation of SIC14. T a n a k a and Kato found the reaction to be pseudo-first order in SiC14 concentration. French et al. have found the reaction to be pseudo-first order with a rate constant given as a preexponential factor, k0, equal to a constant 8.0 • 1014 sec -1 and an activation energy, EA, equal to 98 kcal/mole. Powers found a two part reaction pathway for the rate determining step given by SiC14(g) ---> SiC13(g) + 1/2C12(g) O2(g) + SiC14(g)--* SiO2C13(g) + 1/2C12(g) with subsequent oxidation to SiO2 and C12, giving the overall reaction SiC14(g) + O2(g)--* SiO2(s) + 2C12(g) with a reaction rate obtained from the two rate determining steps given by ,.tr,~i~l 1 "~L~'v'4"

dt

:

(k a +

kb[O2])[SiC14]

(7.9)

where = 1.7 • 1 0 1 4 exp(-Ea/RgT) sec -1 kb = 3.1 • 1 0 1 6 e x p ( - E a / R g T ) (liters/mol) sec -1 with E a - - 96 kcal/mol.

ka

This reaction mechanism suggests that the removal of the first chlorine atom is the rate determining step. Suyama and Kato [12] and French et al. have measured the rate of

267

7.4 Reaction Kinetics

oxidation of several metal halides and found the reactions to be pseudofirst order under conditions of excess oxygen. The activation energies and preexponentials are as follows: Halide SiC14 SiBr 4 A1C13 TiC14 A1Br 3 FeC13 GeBr 4 POBr 3 BC13

Ea, k c a l / m o l

[12]

k0 [15]

90 -25 17 11 <11 ---

8 X 1014 5 • 1011 m --

--

Not pseudo-first order

-2 • 101~ 3 • 1011

Ea, k c a l / m o l

[15]

98 67

64 62

These activation energies are closely associated with the energy of dissociation of the first halogen atom [6], which appears to be the rate determining step in the mechanism of oxidation.

7.4.2 T h e r m a l Decomposition Reactions Thermal decompositions occur for a single reactant, A: aA(g)--~ dD(s or 1). The second reactant (B, in the previous general reaction scheme) does not contribute to the reaction kinetics. The energy needed to break the bonds in the reactant molecule is provided in major part by the kinetic energy of the gas molecules. This kinetic energy is converted into vibrational energy within the molecule, breaking the chemical bonds between the atoms in the molecule. The kinetic energy (K.E.) of a reactant gas molecule is proportional to temperature (i.e., K.E. = 3/2 ksT). Therefore, the higher the temperature the more kinetic energy is available for its transfer into vibrational energy and the faster the thermal decomposition reactions will take place. A1NH3, (CH3)2 SIC12, CH3SiH3, and CH3SiC13 are classic thermal decomposition reactants. The heat for these thermal decompositions reactions can be provided by a furnace, laser, plasma, or flame. The kinetics of thermal decomposition reactions are either zero order (i.e., a = 0) or first order (i.e., = 1). At normal operating temperatures (i.e., >1000~ the reverse reaction that establishes equilibrium is very slow in comparison to the forward reaction. Under these conditions, the reaction will continue until nearly all of the reactant is consumed. The zero order TIC14 decomposition in the magnetically rotated dc plasma reactor of Mahawili and Weinberg [10], discussed in the preceding section, is an example of a thermal decomposition reaction kinetics (see equation (7.7)).

268

Chapter 7 Powder Synthesis with Gas Phase Reactants

7.4.3 L a s e r R e a c t i o n s When a laser is used as a source of heat, the bonds in the molecules that absorb the radiation increase their vibrational energy. The intensity of radiation absorbed follows the Beer Lambert law: /absorbed

--

I0 - I =/o[1 - exp(-aCL)]

(7.10)

where I 0 is the intensity of the incident radiation, a is the molar absorption coefficient t h a t depends upon wavelength, C is the concentration of absorbing species, and L is the path length of the radiation. The resulting increased vibrational energy is (1) used to activate a bond for reaction as in the reaction

A-~A*

d[A*] _ k~Iab~orbed dt

(7.11)

(2) converted to kinetic energy, increasing the temperature of the gas mixture; or (3) allowed to lose its energy by releasing fluorescent radiation, hv', A*--* A + hv'

d[A*] dt - k2[A*].

(7.12)

When the absorption of radiation is coupled to the reactant gas, the reaction kinetics are related to the radiation profile within the reactor, which in most cases is very complex and yields a complicated reaction profile. When the reactant molecules are the molecules that absorb the radiation, the molecular vibrations lead to (1) a decomposition t h a t is analogous to a thermal decomposition, written as A*(g)--~ dD(s or l)

d[A*] dt - k3[A*]

(7.13)

assuming a first order reaction, the products of this laser decomposition can be either ions or free radicals that react with other gas molecules to give a product species, or (2) a combination reaction, written as aA*(g) + bB(g) --~ dD(s or l)

-a1 d[A*] dt - k4[A*]~[B]~.

(7.14)

For the case of laser induced decomposition reactions, production rate of D is given by

I d[D]_ k3[A*] = kllab~orbed d dt k2 + k3

(7.15)

which gives the rate of photochemical reaction. To calculate the quanturn efficiency, the photochemical reaction rate given in equation (7.15) must be divided by the intensity of absorbed radiation, Iab~orbed" An

7.4 Reaction Kinetics

269

analogous expression can be developed for the combination reaction in a laser reactor. Q u a n t u m yields for a particular reaction pathway caused by a given wavelength of light can be experimentally measured; and these are tabulated in the literature, one example of which is F r a n k and H a n r a h a n [17].

7.4.4 P l a s m a Reactions In plasmas, the gas is ionized giving ions and electrons. Both the ions and the electrons are accelerated by the electric field either ac, RF, or dc arc of the plasma. The average kinetic energy of the gas, a measure of its temperature, has three contributions; one from the gas molecules, one from the ions, and one from the electrons. Due to their low mass, electrons accelerate to very high velocities in the electric field and can dominate the average kinetic energy of the system. For this reason, temperatures of-15,000~ are possible with plasma reactors. As a result of the ionization of gases, the reactions that can take place are different than those in a nonionized gas. The reaction kinetics are also enhanced in an ionized gas because the transition states of reactions are not difficult to achieve with ions as reactants.

7.4.5 Complex R e a c t i o n M e c h a n i s m s In some cases two step reactions can take place. An example [18] of such a two step reaction is that observed with the laser synthesis of SiC using the reactants Sill4 and CH 4. In this reaction, silane undergoes thermal decomposition giving silicon as follows: SiH4(g)--~ Si(1) + 2H2(g). The silicon atoms nucleate a liquid droplet of silicon, and this silicon surface is a catalytic surface for the thermal decomposition of methane, as follows: CH4(g)---> C(s) + 2H2(g). The carbon diffuses into the molten silicon droplet and reacts to produces SiC: Si(1) + C ( s ) ~ SiC(s). This reaction sequence may be further complicated by the presence of disilane, Si2H ~ in the reactants and the stepwise removal mechanisms of hydrogen atoms from both the silane and the methane molecules. Another example of a complex reaction pathway is the reaction of carbon with liquid B203 ( M P = 733 K, B P = 2,133 K) droplets produced

270

Chapter 7 Powder Synthesis with Gas Phase Reactants

in a plasma reactor. Above 1733 K, liquid boron oxide is reduced to gaseous boron suboxide as follows [19]: C(s) + B203(1)--* B202(g) + CO(g) which is subsequently reduced to boron carbide [20] 5C(s) + 2B202(g)--. B4C(s) + 4CO(g) Such gaseous reaction intermediates are common and are also seen with SiO2. The suboxide SiO is a stable gas at relatively low temperatures (i.e., >2153 K).

7.5 H O M O G E N E O U S

NUCLEATION

Homogeneous nucleation theory was developed by Volmer [21], Bradley [22], and Becker and Doring [23] to explain the mist formation from a supersaturated vapor phase. Vapor to liquid and liquid to solid phase transitions in the absence of heterogeneous nucleation sites needs an activation energy that results from an increase in surface free energy resulting from an embryo of the condensed phase. The free energy of the embryo or nuclei, AGembryo, is given by

4

(,1)

AG~mb~yo= 4Irr2o" - -~TrrapmRgTln ~

(7.16)

4 7rr aOmRgTln(S) = 4~'r2o" - -~ where Rg is the gas constant, Pm is the molar density of the condensed phase and o- is the specific surface energy of the liquid. This free energy results from two terms: one from the formation of the new surface and one from the formation of a condensed phase. The partial pressure, P~, divided by the equilibrium partial pressure, po, of the condensing species is equivalent to the saturation ratio, S, used in Chapter 6 for liquid phase crystallization. The partial pressure of the condensing species is a function of the type of transformation (i.e., vapor to liquid condensation or vapor to solid sublimation) and the temperature. The partial pressure of the condensing species, po, as a function of temperature, T, is given by

P~

[M-/tra~ _ T)] 0 ( 1 = [1 atm] exp l_ Re T~--ATM

(7.17)

where ~-/trans 0 is the enthalpy of the transformation and TI_ATM is the temperature where the vapor pressure is 1 atmosphere. When the saturation ratio is less than 1, the AGembryo increases monotonically as shown in Figure 7.3. When the saturation ratio is

7.5 Homogeneous Nucleation

271

S<1

,~G ZSGmax

r* S>l

$'IGURF, 7.3 Gibbs free energy as a function of embryo size, r: S = P/Po > 1.0, S = P/Po < 1.0.

greater t h a n 1, the hGembryo exhibits a maximum, also shown in Figure 7.3. This m a x i m u m occurs at a critical radius given by 2(r

r* =

(P1)

pmRgT ln ~11

=

2(r PmRgT ln(S)

(7.18)

at a free energy given by AG* =

16zr~ 3

[

3 mR Tln

=

16zrcr 3

(PI)] 2 3[pmRgTln(S)]2

(7.19)

The critical radius, r*, is a radius above which an embryo will grow spontaneously. A plot of the critical radius as a function of the saturation ratio, S, is given in Figure 7.4. As the partial pressure ratio increases the critical size decreases for each value of the surface energy. A spherical embryo of radius r is bombarded by vapor molecules at a rate 4~rr2NO/4where N (= P1/kBT) is the molecular n u m b e r density of the vapor and 0 (= X/8kBT/Trm) is the m e a n molecular speed. As a result of using kinetic theory of gasses for the m e a n molecular speed we can write the condensation rate, CR, as CR=

4zrr2qp1 X/-2zrmkBT

(7.20)

where q is a sticking coefficient, k B is Boltzmann's constant, and m is the molecular mass. In addition, the concentration of embryos, Nr, with radius r is described by a Boltzmann relation: P1

Nr = ~S T e x p ( - A V r / k s T ) ~

(7.21)

272

Chapter 7 Powder Synthesis with Gas Phase Reactants

F I G U R E Z 4 Critical radius as a function of S = P/Po" r*std = 2fla~'V/(3fl~RGT), a plot of equation (7.18). V is the molar volume.

where P1/kBT is the molecular density and A G r is the free energy for the embryo of radius r. The fraction of molecules with energy greater than A G r is given by the exponential term in equation (7.21). The nucleation rate, J, which corresponds to the production rate of embryos of critical radius r* is expressed as the product of the condensation rate on the critical embryo times the population of critical embryos: P1 P1 (-~G*~ J = CR* Nr. = q47rr .2 (27rmkBT)l/2 k s T exp k s T ]

(7.22)

Several modifications of this nucleation equation have been suggested by Zeldovitch [24] that allow Figure 7.5 [25] to be drawn, which shows the nucleation rate as a function of the saturation ratio and the number of atoms in the critical embryo (see Section 6.3 for details). The nucleation rate is very low for small values of the saturation ratio. But, at a critical saturation ratio, the nucleation rate increases drastically and saturates at a maximum rate corresponding to the condensation rate, times the molecular density, PJkBT. For the laser induced thermo-decomposition of silane, SiH4(g)--* Si(g) + 2H2(g) the critical nucleus size is given as a function of reaction temperature in Figure 7.6. Here, we see that the critical nucleus size is less than the atomic radius of the product silicon for temperatures of interest between 1000 and 1500~ which have been measured in the reaction zone [18]. Every silicon atom resulting from the silane pyrolysis reaction can be regarded as a nucleus of critical size. This result poses several problems. First, the bulk properties of the solid, silicon, used in the free energy cannot be used. Second, this theory is based on the clustering of

273

7.5 Homogeneous Nucleation

A= =

A=20

__.

-10

n*=2 A = 100

~-----

A = 200

=" -20

..J

n*= 1(:

-30

~ -40

I

i

i

,

I I iiZl

1

i

10

i

i

I I liWl

100

I

I

I

I I II!

I

1000

I

I

I

~0

I I III

1E4

Saturation Ratio, S = PI/P1~ FIGURE 7.5 Nucleation rate as a function of P/Po and also as a function of critical nucleus size. From Nielsen [25].

Homogeneous nucleation

0.18

E

e~

0.16-

Ii~

Atomic radius of silicon

"

~t~ 0.14

0.2 atm

L.

Ii~ 0.12 0.7 atm

0 :3 0.10C 0 ~ "~.

0

O.O8|it ~

" 0 . 0 6 " --" 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(Thousands) Temperature (K) FIGURE 7.6 Critical nucleus radius of silicon calculated by homogeneous nucleation theory for 0.2 atm and 0.7 atm. Atomic radius of silicon is 0.146 nm. From Sawano [18].

274

Chapter 7 Powder Synthesis with Gas Phase Reactants

0

35

Homogeneous nucleation

...............

9

/

. -

~: 3 0 " ~

7 t

A

A

i

i

_-

~

I

i

~

.

0 I~ ~-

C

-

25

0 2o :3 C 0 --

15" 0.6

!

I

0.8

!

I'

i

1.0

I

1.2

I

I'

1.4

i

1.6

1.8

2.0

(Thousands) Temperature (K) FIGURE 7.7 Nucleation rate of silicon particles calculated by homogeneous nucleation theory for 0.2 atm and 0.7 atm. From Sawano [18].

1.0

1600 K

O r

0.8

--"

m

=.,..

O) 0 . 6 -

q.-

O

C :3 O

0.4-

E ,~

_

0.2 0.1

l' 0

t

'

1000 K ,

I 0.02

i

i 0.04

I

I 0.06

/

I 0.08

! ....

0.1

Time (msec)

FIGURE 7.8 Silane decomposition versus time at various temperatures based on Coltrin's mechanism [26]. From Sawano [18].

7.6 Collisional Growth Theory

275

atoms from a gas having a concentration defined by its partial pressure. If we ignore these problems, the nucleation rate predicted for these critical nuclei sizes are in the neighborhood of 1030 number/m3/sec for flame t e m p e r a t u r e s above 1400~ as shown in Figure 7.7. For these conditions, there is no barrier to nucleation, and the nucleation rate shown in Figure 7.7 is simply equivalent to the silane pyrolysis rate. Figure 7.8 [26] shows the amount of silane reacted as a function of time for different temperatures. At 1000 K, essentially no reaction is observed in 0.1 msec. At 1300 K, 80% of the silane is reacted after 0.1 msec. At 1600 K, essentially all of the silane is reacted in much less t h a n ~ msec. Because the critical nucleus radius is about equal to the atomic radius of a silicon atom for the temperatures experimentally observed, it can be considered that the rate at which the silicon atoms collided is an aggregation process. Under these conditions atomistic growth and aggregation are the same process. When the critical nuclei are larger t h a n an atom, atomistic growth occurs as an atom collides with a particle, and aggregate growth occurs as two particles consisting of many atoms each collide to form an aggregate. Critical nuclei of atomic dimensions are frequently observed with gas phase reactions of all types. This puts into question some of the assumptions used in the previous nucleation theory (e.g., surface free energy and volumetric free energy for only one atom or one molecule).

7. 6 C O L L I S I O N A L GR 0 W T H T H E OR Y Particle growth will be derived in terms of a mass flux from the gas phase to the surface. Transport mechanisms vary with conditions and particle size. Differences in the mechanisms are characterized by the Knudsen [27] number, KN = h/r, where the m e a n free path of the gas molecules is ~[= (2Ir d2(P1/ksT)) -1, where d is the molecular diameter] and r is the radius of the particles. For the m e a n free path small compared to the particle size, KN < 1, a diffusion limited process is applicable. When the m e a n free path is larger t h a n the particles, KN > 1, particle growth is controlled by collision of gas molecules onto the particle surface. During gas phase synthesis under high temperature conditions, the latter, KN > 1, is often more appropriate. In the discussion of growth of micron size particles, we will use an example of laser decomposition of silane to form silicon particles as an example. Assuming pure silane decomposes to atomic silicon and hydrogen gas, the Knudsen n u m b e r for a silicon "particle" one atom in size is 65. For this reason, there is no difference between the mass flux to the surface of a silane atom and simple collision of gas molecules. Collision theory

276

Chapter 7 Powder Synthesis with Gas Phase Reactants

was originally developed by Smoluckowski [28]. Hidy [29] expanded this collision theory to coagulation, and Ulrich [30] applied it to particle growth using the kinetic theory of gases. The average velocity of an atom or molecule is described by the kinetic theory of gases:

(8kBTtI/2

0 = \ 7rm /

(7.23)

where m is the molecular mass. Assuming each particle is the same diameter, the collision frequency, Z, for one particle with its neighbors is given by

where d is the molecular diameter and N(= P1/kBT) is the molecular number density of molecules. The factor 89prevents duplicate counting. As each collision reduces the number of particles by 1 the change in number density is expressed as

d N - qZ = qd2N2 (4~kmBT)l/2 - dt

(7.25)

where q is a sticking coefficient. Using mass conservation, this equation is rewritten as follows [30]:

dt

t1 o 1 o 11 o \ Pm ] \4~rpm/

where Mw is the molecular weight, Co is the initial concentration, and Pm is the molar density of the condensed phase (or product species). This differential equation has the following solution for the initial condition N = No at t = 0:

N(t)=No 1 +

(CoMN~ 1/6 q\ ~ ] (ksT) ~/2

at

.

(7.27)

This equation can be used to calculate the average particle radius, using

4/37r73= Co N(t)"

(7.28)

Figure 7.9 is a plot of the average particle radius both calculated and measured experimentally by light scattering as a function of reaction time in the reaction zone. The measured values are always about 60% of the calculated values, using collision theory based on 100%

7.6 Collisional Growth Theory

277

60

E" 50 ~ ~ '~ '~_~O~o(~eoOtttV

"-"

301

(1) 20 "6

I1.

~c~o~*~~ ~"~"

//i

10 0 0

0.002

0.004

0.006

0.008

0.010

Time (sec) b 5O

--1

40

30L_

~

/

20

/

~ lO 0

1"

0

i

I

0.002

'

I

0.004

i

I

0.006

i

I

0.008

i

0.010

Time (sec) FIGURE 7.9

Partical r a d i u s versus time in reaction zone at 0.2 atm: (a) calculated by collision t h e o r y at 10% and 100% conversion, (b) m e a s u r e d by light scattering. F r o m Sawano [18].

decomposition [18]. This is because the laser reaction zone is diluted by the coaxial gas inside the chamber, and some fraction of the gas will, therefore, not react. The experimentally measured results have a shape similar to that of the calculated value, which demonstrates that the concept of nucleation followed by collisional growth gives a reasonable explanation of the particle formation in gas phase laser synthesis.

278

Chapter 7 Powder Synthesis with Gas Phase Reactants

7.7 POPULATION

BALANCE

FOR

GAS

PHASE SYNTHESIS As seen in Figure 7.1, all of the gas phase reactors have a gas inlet point after which gasses start to heat up in a reaction zone. Then the gas cools down as it moves axially down the reaction zone into the quench zone. Figure 7.2 shows a schematic of this axial reaction zone. Methods used to model this axial reaction zone have been developed in chemical engineering. These methods use a plug flow model of the reaction zone. This plug flow model assumes that an axial slice of gas has a uniform composition that changes with distance (or time of flight) from the gas inlet. The transient population balance for a plug flow reactor in the absence of breakage and agglomeration is given by aV ~ a ( G v ) + u ~aT = 0

at

Or

0x

(7.29)

where G is the growth rate as a function of particle radius r and u is the superficial velocity in the x direction in the plug flow reactor. The population balance coupled with the mass balance, reaction kinetics, initial, and boundary conditions provides a particular solution to the population balance for the final population. At a steady state, d v / d t = 0 and for a growth rate, G, that is not a function of size or not a function of time and the population balance becomes G aT(r, x ) + u ~a~?(r, x ) = O.

Or

0x

(7.30)

If the chemical reaction is fast, as with a high powered laser beam, the nucleation will take place instantaneously, giving an initial burst in the population resulting in the boundary condition ~7(r = r o , x = O ) = N o = 7o

ro

(7.31)

where ~7ois a constant equal to the n u m b e r density, No, of silicon atoms just after reaction divided by their size, r o. Using a characteristic time r = x / u - r G , the partial differential equation (7.19) is given by (O'(r'x)) 2G \ Ox

=O

(7.32)

which has the initial condition given by equation (7.31). This initial condition gives a value for the characteristic time ~ = - r o / G . The solution to equation (7.30) with this initial condition is given as ~?(r, x)l~ = constant.

(7.33)

7.7 Population Balance for Gas Phase Synthesis

q(r)

dr/dx

279

= G/u

r

ro

FIGURE 7.10 Schematic of population, ~7(r), in a plug flow reactor.

The solution to the population balance equation is shown in Figure 7.10. With a single nucleation event, the population that nucleates at t = 0 has an invariant shape or s e l f - p r e s e r v i n g s i z e d i s t r i b u t i o n (i.e., a delta function in this case) with respect to r = x / u - r / G = - r o / G . As the particles move down the flame front, they increase in size according to the following equation: (7.34)

r - r o = Gx/u.

This model is an idealized simplification of gas phase reactions. In a more general case with a distribution of nuclei sizes, the narrower is the nucleus size distribution the narrower is the product size distribution. Under conditions in which growth is limited by diffusion, the growth rate is not constant with size, and either a narrowing of the size distribution takes place as the particles grow, as shown in Figure 6.28, or a broadening of the size distribution takes place as the particles grow, as shown in Figure 6.27, depending on the size functionality of the growth rate. To account for these nonconstant growth rates, equation (7.30) is replaced by the following integration over the time in the reaction zone: f ~ 1

(x/u

x

ro --~ d r = ~o d t -u- -

(7.35)

One case that is common in gas phase powder synthesis occurs when, after nucleation, the growth is controlled by the chemical reaction. In this case, the mass flux due to chemical reaction can be related to the growth rate, giving 47rr2G = 47rr2 d r _ MwD d [ D ] _ MwD dkl[A]~[B]~ dt pN o dt pN o

(7.36)

280

Chapter 7 Powder Synthesis with Gas Phase Reactants

for the generalized reaction aA(g) + b B ( g ) ~ d D ( s or 1) The time coordinate in this expression can be related to the axial position, x, and the superficial velocity, u, as t = x/u. Multiplying by d t and integrating gives 4~r

r2dr = ro

i:

[>3 _ r]] =

foWO

dkl[A]~[B] ~ d t

(7.37)

PNo

where the concentration of [A] and [B] are functions of time or axial position as t = xu. This expression shows how the mean size, ~, increases with position x down the reaction zone after the nucleation of No particles per unit volume of size r o. This expression does not predict the broadening or narrowing of the particle size distribution with growth because the nuclei are assumed to be all of the same size and grow at the same rate as that of the chemical reaction.

7.8 DISPERSION MODEL SYNTHESIS REACTORS

FOR

GAS

Real reactors have an additional complication associated with the dispersion of the flow profile due to a nonuniform radial velocity profile and due to fluctuations in the velocity profile caused by eddy, molecular, and Brownian diffusion as shown in Figure 7.11. This flow dispersion causes the particles to be in the reaction zone for a distribution of growth times, giving a distribution of particle sizes even if the nuclei are all of the same size and growth rate is not a function of particle size. Deviations from plug flow can be measured by monitoring a tracer input, as shown in Figure 7.12, and watching how the tracer is dispersed as it flows down the reactor. Considering the flow and dispersion of molecules according to their axial diffusion coefficients we can calculate this dispersion of a concentration profile by using Fick's law, given by aC _ D a2C aC at 0-~-y - u a--x-

(7.38)

FIGURE 7.11 Dispersed plug flow, Fluctuations due to different flow velocities, molecular diffusion, and turbulent flow.

7.8 Dispersion Model for Gas Synthesis Reactors

281

FIGURE 7.12 The dispersion model predicts a symmetrical distribution of tracer at any instant. From Levenspiel [9]. Copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

where D is the diffusion coefficient responsible for the dispersion of flow and u is the axial velocity of the flow. This partial differential equation can be put into dimensionless form:

OC (D)02C

OC

0-0 =

OX

~

OX2

(7.39)

where X and 0 are given by the following relations: X -

ut+x L

t

tu

0 = _-- = - t L

(7.40) (7.41)

where L is the length of the reactor and (D/uL) is the dimensionless dispersion number. The dispersion number is a measure of the extent of deviation from plug flow. For an initial population corresponding to a delta function, this differential equation (7.39) has the following solution:

2 l__ex [ which is now a function of only 0 because the time is related to the position resulting from the movement of the concentration front. The solution of this differential equation for different conditions is shown in Figure 7.13. The width of the concentration profile at the exit of the reactor is dependent on the value of (D/uL). When it goes to zero, plug flow results. When it goes to infinity complete back mixing results as shown in Figure 7.13.

282

Chapter 7 Powder Synthesis with Gas Phase Reactants 2.0

I

D

r l

-

!

Plug flow, ~.L=O_~ I I /~ / S m a l l amount I / I\ o, dispersion,

i

1.5

I

/

11~~. :

/ M i x e d flow, / O _oo

o.1

u-L-

'

:

=:

I

/ i t,n,eroeO,a,e '

~

/

=o.oo~ ,

|

V

amount of

II ~ d i s p e r s i o n ,

/

|

/

I

~

D-0

. ~ uL-,"

025

I 1 \ ,aroeaoun,

o

-J

0

0.5

1.0

1.5

2~.0

0"=t/t FIGURE 7.13 Residence time distribution for various extents of back mixing as predicted by the dispersion model. From Levenspiel [9]. Copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

Levenspiel [9] gives the variance of residence time distribution, ~r2, as

~

(r 2 = 2 ~ - ~ - 2

~-~

1-exp

(~ ~

.

(7.43)

At small values of the dispersion n u m b e r the variance of the residence time distribution decreases and approaches plug flow, where the following approximation can be applied:* (r 2 -

2D uL"

(7.44)

This dispersion number, (D/uL), for fluid flow in a cylinder can be obtained from a chemical engineering correlation by Levenspiel [9] noting that the intensity of dispersion D / u d t , (where d t is the diameter of the cylinder) is plotted as a function of Reynolds's number Re = upgdt/lZg; pg is the gas density and t~g is the gas viscosity. (Please note that the Reynolds's number of the flow is altered by the presence of particles. Particles increase the gas density and reduce the effective kinematic viscosity. The net result is to accentuate turbulence and * Note: This expression is valid only for D/uL < 0.1.

7.8 Dispersion Model for Gas Synthesis Reactors

283

intensify mixing [31].) Multiplying the intensity of dispersion, D / u d t , by the axial ratio of the reaction d t / L , gives the dispersion number, (D/uL): D _ D dt u L - u d t " "L

(7.45)

which completely characterizes the degree of back mixing in the reactor. The dispersion number influences the residence time distribution, as shown in Figure 7.13. Because a particle being synthesized by a gas phase reactor is in the reactor for various periods of time, it will grow by either condensation or aggregation for various periods of time, giving various sizes. As a result, the particle size distribution depends upon the residence time in the reactor and, thus, the dispersion number, which is in turn a function of the Reynolds number and the Schmidt number for the gas flow as seen in Figure 7.14 [32]. For gases in laminar

Correlation for the dispersion of fluids flowing in pipes. Adapted from Levenspiel [32]. Copyright 9 1972 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

FIGURE 7.14

284

Chapter 7 Powder Synthesis with Gas Phase Reactants

flow (i.e., Reynolds number < 2100), we find the dispersion intensity,

D/udt, to be a function of the Schmidt number. The Schmidt number is defined as the Sc = t~g/pgDAs, where DAB is the molecular diffusion coefficient, which is different than the axial dispersion coefficient. In the transition regime, the dispersion intensity is a very weak function of the Schmidt number. When the Reynolds number is high (i.e., turbulent flow), the dispersion intensity is essentially constant and not a function of Schmidt number. The following treatment considers the effect of the residence time distribution on the size distribution of particles produced in a gas phase reactor. To do this we have to assume that the particles are produced by nucleation, either single point at the inlet of the reactor or multipoint through out the reactor, and particle growth is atom by atom with a growth rate G. Using the residence time distribution, the particle size distribution can be calculated for these two cases of nucleation [33].

7.8.1 S i n g l e - P o i n t N u c l e a t i o n With a reactor in which single-point nucleation, giving a population with particles all the same size, is followed by growth, the population of particles has the following form [33], plotted in Figure 7.15 as cumulative distribution: 9 8 X

m

7

1

(9 N ~9 6 u}

_.e

l

5-

(...

.O

4-!

r

~i

~)

3-

E

= m

I~

2-

7 oo

01

I

,

'

,

I

5 1'o 2'o 3'0 40 5oe'o Cumulative

,

,'

,

I

80 9'0 9's 98 99

weight

%

FIGURE 7.15 Cumulative weight distribution versus dimensionless size for a cascade of reactors with nucleation only in the first tank or for a dispersed plug flow reactor with nucleation only at the entrance to the reactor. Data from Abegg and Balakrishnan [33].

7.8 Dispersion Model for Gas Synthesis Reactors

285

9 8

X

7

N ~9

6-

~

N=I

l

5-

C oI~ 4 -~, C

~i

E

m ~

Ol

,

1 2

,

5

,

10

',

,

,

I,

, . '

20 30 40 50 60 70 80

,

,

90 95

,

I

98 99

Cumulative weight % FIGURE 7.16 Cumulative weight distribution versus dimensionless size for a cascade of reactors with equal nucleation rates in all the t a n k s or for a dispersed plug flow reactor with equal nucleation rate all along the reactor. Data from Abegg and B a l a k r i s h n a n [aa].

(Nx )N- 1e -Yx

~ON(X) :

( N - 1)!

(7.46)

where x is the dimensionless size, r/Gt, and the integer, N,* is given by N=~ 7.8.2 Multipoint

l(D) -1 ~-~ .

(7.47)

Nucleation

With a reactor in which nucleation and growth occur all along the reactor, the population of particles has the following form [33] plotted in Figure 7.16 as cumulative weight distribution:

)i (Nx 1 ~ON(X) -- 1 + l ~ z (Nx i=1 i! + ( N - 1)-------~.e-Yx

(7.48)

For a plug flow powder synthesis reactor that has nucleation and growth occurring at the same time, Ring [34] has developed a model that predicts the geometric standard deviation of the log-normal particle size * Note t h a t N is an integer corresponding to the n u m b e r of t a n k s in series, which is an analogous model for accounting for back mixing in a plug flow reactor as N ~ ~ a plug flow design results.

286

Chapter 7 Powder Synthesis with Gas Phase Reactants

10-1

10-2

10.3

10 -4

1.0

1.1

1.2

1.3

1.4

1.5

1.6

~Jg FIGURE Z17 Dispersion number versus geometric standard deviation: A, single point nucleation model; B, multipoint nucleation model. Data from Ring [34].

distribution as a function of the dispersion intensity. This model is shown in Figure 7.17 for both single-point nucleation and multipoint nucleation. As the dispersion intensity decreases, the geometric standard deviation approaches that of a monosize system, ~g = 1.0. Thus, the fluctuations due to turbulent and molecular diffusion in plug flow can be accounted for with this simple model of nucleation and growth in real rather than ideal flow reactors, as seen in the following problem.

P r o b l e m 7.1. Gas P h a s e R e a c t o r M e a n Size a n d Size D i s t r i b u t i o n In the production of ceramic TiO2 powder in a flame reactor, a gas after mixing consisting of 1.0 moles/sec TIC14, 8.0 moles/sec N2, and 2.0 moles/sec 02 is fed to the tubular reactor at 100~ with a diameter of 3.88 cm so that the superficial velocity is 100 m/sec. If there is an instantaneous nucleation of particles 10/~ in diameter (19 TiO2 molecules) corresponding to 0.001% of the reactants and the subsequent growth rate is identical to that produced by the chemical reaction: TiC14(g) + Q(g)--> TiO2(s, amorphous) + 2C12(g) + AHRxN = 24.35 kcal/mole with the kinetics _ d[TiC14_______~]~ ka[TiC14] dt

7.8 Dispersion Model for Gas Synthesis Reactors

287

k a = 8 . 2 6 • 104 exp(-Ea/RgT) s e c -1 a n d E a - 8 8 . 8 +- 3 . 2 k J / tool., determine the mean particle rize, F, and the geometric standard deviation, %, of the particle size distribution. Data" TiO2 M/fusion = 11.4 kcal/mole, Tf = 1825~ Cp = 20.51 cal/ mole/~ p = 3.8 gm/cc, Mw = 64 gm/mole. where

N2, 02, C12: Cp = 10 cal/mole/~ Product gas viscosity = 0.05 centipoise, density = 0.173 gm/liter, Schmidt number = 0.25 For the solution, after the complete reaction, the gas flow rates are 1.0 moles/sec TiO2, 2.0 moles/sec C12, 8.0 moles/sec N2, and 1.0 moles/sec 02 . At I atmosphere pressure the adiabatic flame temperature is 2500~ (using equation (7.1), which results when the product is TiO2 solid and the gases are fed at 100~ When the product is TiO2 liquid the adiabatic flame temperature is 1350~ These two temperatures are above and below the fusion temperature for TiO2 of 1825~ and for this reason the TiO2 produced is for the most part solid with a fraction liquid. But this fixes the flame temperature at the fusion temperature of 1825~ At 1 atmosphere pressure and 1825 + 273 K, the initial 11 moles of reactants and final 11 moles of gaseous products will occupy 1892 liters of volume giving the initial reactant concentration: 0.00053 moles/liter TIC14, 0.00423 moles/liter N2, and 0.00106 moles/liter 02. If 0.001% of the TIC14 has reacted to nuclei, we have 3.2 • 10 ~s nuclei/liter and new initial concentration that is essentially unchanged. These particles grow according to a growth rate, G = dr/dt, which is related to the reaction kinetics 47rr2G _ M w d[TiO2] _ M w ka[TiC14] = M w ka[TiC14]o exp(_kat ) pNo dt pNo ~o The time coordinate can be related to the axial position, x, and the superficial velocity, u, as t = x/u. Multiplying by dt and integrating gives 47r

r2dr = r0

7o

ka[TiC14] o [exp(-kat)] dt

which can be rearranged to give the mean size as a function of time as Fa - ro3 =

47rNo------~ ka[TiC14]0 exp(-kat) dt

Mw -47rNop [TiC14]o [ 1 - exp ( - k a X ) ] which is plotted in Figure 7.18. In the figure, distance x is given in meters down the reaction zone.

288

Chapter 7 Powder Synthesis with Gas Phase Reactants 10 9Ro .

_

R (x)

=

,,

..,_----~

m

...~

!

I Ro 0

FIGURE 7.18

x

0.5

D i s t a n c e x g i v e n i n m e t e r s d o w n t h e r e a c t i o n zone.

The asymptotic mean size is 59~ reached at 0.5 m, assuming that the reactor is an ideal plug flow reactor where all the particles are the same size. To further this analysis, we can add dispersion into this reactor analysis and correct for the nonideal nature of this reactor. The dispersion analysis allows the prediction of the geometric standard deviation of the partice size distribution due to variations in the residence time distribution. The geometric standard deviation of the particles size distribution is given by the dispersion number in Figure 7.17. The dispersion number is given by D _ D D__.d t uL u d t L" To calculate the dispersion intensity, D / u d t , the Reynolds number and the Schmidt n u m b e r are needed: Re

- 100 m/sec 93.88 cm 0.173 gm/liter = 13,400 t~ 0.05 poise Sc = 0.25 from data -

udtp

where d t is the tube diameter of the reactor, t~ is the product gas viscosity, and p is the product gas density. The Reynolds number and Schmidt n u m b e r give a value of D/udt = 2. Thus D _ 19 9d_t_= 2 , ~ =3.88 0 . 0 1cm 55 uL - u d t L 500 cm

"

In this equation the reactor length, L, is that for the steady state particle size to be reached, not the total reactor tube length. This dispersion n u m b e r gives a geometric standard deviation, cry, of 1.2 using the single-point nucleation model of Figure 7.17. This model assumes that the particles do not aggregate during growth by coalescence.

7.9 Population Balance with Aggregation

7.9 P O P U L A T I O N

289

BALANCE

WITH A G G R E G A T I O N When the particles are formed at high temperatures the particles are often liquid droplets. These droplets stick together when they collide, altering the particle size distribution produced. Accounting for aggregation in the population balance in gas phase reactors is performed in the following way: O~l(r, t) + G O~(r, t) + u ~~i(r, t) = B(r, t) - D(r, t). at ar Ox

(7.49)

The first term in this partial differential equation describes the temporal change of the population; ~/; the second term describes the atomistic growth of the particles (which assumes that G is independent of particle size r), and finally the last two terms account for the birth and death of particles of size r by an aggregation mechanism. The birth function describes the rate at which particles enter a particle size range r to r + hr, and the death function describes the rate at which the particles leave this size range. In the case of continuous nucleation, an additional birth rate term is used for the production of atoms (or molecules) of product by chemical reaction. In this case, the size of the nuclei are the size of a single atom (or molecule) and the rate of their production is identical to the rate of chemical reaction, ktC, where C is the reactant concentration, giving B(ro, t) - d~l____~o= l d N o _ 1 ktCNAv dt rodt ro

(7.50)

where NAV is Advogadro's number. When nucleation is discontinuous, there is an initial burst of nuclei. As a result, the preceding equation is not used because the reactant concentration is essentially zero for most of the flow. In this case, the initial burst of nucleation is accounted for in the initial condition for the population balance, which corresponds to ,/(r = ro, t = 0 ) = ~/o(t = 0) = ~/o

(7.51)

where the population of nuclei, ~/0(t), decreases with time due to aggregation. In this example, we will consider only a single burst of nuclei generated at time zero (~/o = constant). Aggregation of these nuclei gives the following time dependence with respect to the nuclei population, as described by von Smoluchowski [28] and Ives [35]:

~l(r=r~176

~o t-~/2)2

(7.52)

290

Chapter 7 Powder Synthesis with Gas Phase Reactants

where tl/2 = (1/Kv~ is the aggregation half life. Therefore, the population balance, as described in Equation (7.49), is useful in that it predicts the number distribution of particles grown under kinetically competing conditions, that of atomistic growth and particle aggregation. For a solution to the population balance to be possible, mathematical relationships must be developed for the birth and death functions accounting for particle aggregation.

7.9.1 Rapid Flocculation Theory The birth and death functions predict the importance of particulate aggregation on the final particle size distribution. The key concepts for the development of these two functions come from Smoluchowski's rapid flocculation theory, which was derived in Section 6.6.3:

f2

B(r, t)

= K a

D(r, t)

= 2KaT~(r ,

77(x ,

(7.53)

t) ~?(r - x, t) dx

i f ~ (x + r) 2 ~(x, t) dx

t) -~

(7.54) = 2Ka77(r , t ) N T [ - - ~ + --r + 1

where N T is the total number density of particles and the aggregation rate constant is Ka-

k, Tq 3t~g

(7.55)

Here, t~g is the viscosity of the gas and q is the sticking factor. The ith moment is described as follows: XiT~(X) d x (xi)=

.~

IV(x)

xiT~(X) d x :

dx

o

NT

"

(7.56)

o

Applying the new birth and death functions given in equations (7.53) and (7.55), respectively, the governing differential equation for the population of particles, v(r, t), becomes Ov(r, t) + G Or(r, t) + u O~(r,t) Ot = K~

Or

(7.57)

Ox

rl(x, t)~(r - x, t) dx - 2Karl(r, t)Nr(t)

--7 + - - + 1 . F

An analytical solution to this integro-differential equation is impossible without some simplifying assumptions. The birth function can be

7.9 Population Balance with Aggregation

291

simplified by assuming that aggregation takes place by the collision of nuclei with the larger aggregates and not by the collision of particles of similar sizes, either nuclei-nuclei or aggregate-aggregate. By limiting the type of growth mechanism by which particles can be born to a collision between a nuclei and a large particle, the convolution integral describing the birth function can be simplified: (7.58)

B ( r , t) = Ka~lo(t)r~l(r , t).

Certainly this assumption is not valid for all times. But, for much of the time after the initial burst of nucleation, this is probably the case for the following two reasons. First, because there is a high number density of nuclei compared to the aggregates, the probability of a collision between a large aggregate particle and a nuclei is higher than the collision between two larger aggregates. The sticking probability (q) is different for each type of collision. For a collision between two nuclei or two aggregates, it is low, but the sticking probability for the collision between a nuclei and an aggregate is much higher--approaching 1.0. With this simplifying assumption in mind (i.e., equation (7.57)), the new population balance becomes O~l(r, t) + G 0,/(r, t) + u 0~/(r, t)

0t

Or

0x

i(x )

]

(7.59)

= Ka~or~o(r, t) - 2Ka~O(r, t ) N T ( t ) [ - - ~ + --r + 1 .

This partial differential equation can be approximated by an ordinary one by creating a new characteristic variable; that is, ~" = t - r~ G - x / u . Applying the initial condition for this problem, the characteristic can be shown to be a constant: ~" = - r o / G . Making this variable transformation, the new characteristic population balance becomes 2G

O~(r,or t)

9 = K a ~ o r ~ ( r , t) - 2Kar~(r, t ) N T ( t ) [ r2 + --r + 1 .

(7.60)

By application of this transformation under conditions of constant r, the dimensionless solution to the characteristic population balance for a batch reactor is

[

~/(r,t)] = exp { _ f l [ ( ~ _ 1 ) _ 4 6 _ [ ( ~ 0 ) 2 - 1 ] ~/0 (7.61) +

(X2)(1~)r--~

+ 2 x} In r( ~r) ]

where fi =

KaNT(t)ro G '

r

~lo(t)ro NT(t ) 9

(7.62)

292

Chapter 7 Powder Synthesis with Gas Phase Reactants

The dimensionless group/3 corresponds to the ratio of the aggregation rate to the atomistic growth rate. Large values of fl suggest that aggregation is fast compared to atomistic growth, whereas small values of fl suggest that atomistic growth dominates. The dimensionless number, 6, corresponds to the fraction of the total number of particles that are nuclei. Both fl and 6 are functions of time; as the time increases both of these dimensionless groups decrease. Recall that this solution to the population balance is valid only under conditions of constant r, which means that for every particle of size, r, there is a corresponding time needed for its formation.

7.9.2 A P h y s i c a l C o n s t r a i n t on the Population Balance A physical constraint must be placed upon the population balance to ensure that the conservation of mass is upheld. Mathematically, this means that the following population balance relationship must hold:

MT(t) = fiv

f

rmax

ra,l(x, t) dx

(7.63)

r0

where flv is the coefficient for the calculation for the volume of a particle (e.g., 4/37r for a sphere, 1.0 for a cube). The maximum particle size present, rmax, is the constraint in this population balance. This means that, for fixed values of j3, 6, and time, the final distribution~including the maximum particle s i z e ~ c a n be predicted. The qualitative effect of fl and 6 on the particle size distribution is shown in Figures 7.19 and 7.20 [36]. When fl and 6 are equal to 0.01 (Figure 7.20b), we find a bimodal size distribution consisting of a small population of nuclei and a narrow size distribution of aggregates with nothing between the two modes. This plot is characteristic of the final stages of aggregation. When 6 is increased to 0.1 and/3 remains the same (Figure 7.20a), we find a larger population of nuclei. This aggregate size is smaller, and the relative standard deviation of the aggregates is larger than in Figure 7.20b. Figure 7.20a is also characteristic of the final stages of aggregation, but earlier in time than the example in Figure 7.20b. When fl is 0.01 and 6 is 0.5 (Figure 7.19b), a broad and continuous distribution of particle sizes between the nuclei and the maximum sized aggregates is found. Increasing the value of 6 to 0.667 (Figure 7.19a), it is found that the maximum size is decreased drastically and the shape of the size distribution is changed by putting more emphasis on the population of nuclei than on the aggregate population, which is typical of an earlier snapshot in time and corresponds to an increase in ~b. This model shows that the agglomeration of both

7.9 Population Balance with Aggregation

293

F I G U R E 7.19

N u m b e r d e n s i t y of p a r t i c l e s v e r s u s size for v a r i o u s d e g r e e s of a g g l o m e r a t i o n [36]: (a) fl = 0.01, ~ = 0.667, t = L/u = 0.5tu2; (b) fl = 0.01, ~ = 0.5, t = L/u = 0.5tl/2.

small and large particles will drastically increase the width of the particle size distribution produced. In addition, turbulent and molecular diffusion will broaden the size distribution of the particles produced in gaseous reactors. These effects can be minimized by short reaction zones and short reaction times, giving fairly narrow particle size distributions.

294

Chapter 7

Powder Synthesis with Gas Phase Reactants

F I G U R E 7.20 N u m b e r d e n s i t y of p a r t i c l e s v e r s u s size for v a r i o u s d e g r e e s of a g g l o m e r a t i o n [36]: (a) fl = 0.01, ~ = 0.1, t = L / u = tl/2, (b) fl = 0.01, ~ = 0.05, t = L / u = tl/2.

Only on cooling the particles as they leave the hot zone of the reactor do the processes of agglomeration and growth stop. As a result, the final particle size distribution produced by a gaseous reactor is highly dependent on the geometry of the reactor, its degree of dispersion, and the rate of cooling provided as the particle leaves the flame front. Quick cooling provides the narrowest of size distributions. The analysis of

7.9 Population Balance with Aggregation

295

reaction, nucleation, growth, and agglomeration is applicable to all types of gaseous reactions, including flame, furnace, plasma, and laser reactors. Attention to these details can improve the quality of the powder produced by such techniques.

7.9.3 O t h e r N u m e r i c a l M o d e l s A number of numerical approaches have been applied to the solution of the general population balance equation: 0~/(r, t) + G 0~/(r, t) + u ~0~/(r, t) = B(r, t) - D(r, t) Ot

Or

Ox

(7.64)

with B(r, t) = Ka

f

r

V(x, t)~(r - x, t) dx + B(ro, t)

ro

fr

= Ka

__

(7.65)

~(x, t)~(r - x, t) dx + 1 koCNAv ro

FO

1

D(r, t) = 2Ka~l(r , t ) - ~ fro (x + r)2~/(x, t) dx

(7.66)

where the aggregation rate constant is Ka _ kBTq 3t~g"

(7.67)

Here,/xg is the viscosity of the gas and q is the sticking factor. This aggregation rate constant is due to Brownian motion. The shear aggregation rate constant can be added to the Brownian aggregation rate constant for turbulent flow conditions [37, 38]. These numerical approaches to a solution differ primarily in their representation of the particle size distribution. Solution for a generalized population balance has a high computational cost due to the stiffness of the general population balance equation with high reaction rates [1]. Assuming a general particle size distribution (e.g., monodisperse, self-preserving size distribution [2] or one or more log-normals [39, 40] greatly simplifies the calculation, but these solutions tend to overestimate the aggregation rate [41]. Models based on determining moments of the particle size distribution [42-44] (e.g., total number, surface area, and volume of particles per unit volume of gas) have their own shortcomings, because they describe only a single mode of a population and require a large number of moments to describe bi- or multimodal particle size distributions. An intermediate level of approximation, sectional representation [45, 46], which represents the population as a histogram with sectors

296

Chapter 7 Powder Synthesis with Gas Phase Reactants

of the same mass concentration, results in a substantial decrease in computational cost. Which of these models is appropriate depends on the required accuracy and the nature of the problem being examined (i.e., growth by aggregation or atomistic growth). In most gas phase synthesis reactors, the reaction is very fast, generating large supersaturations and high nucleation rates. Nucleii are formed and have little time to grow by atomistic growth before they aggregate, decreasing the number and increasing the mean particle size. Soon all the information about the initial nucleii is lost by this rapid coagulation. Because most models can describe aggregation quite well, the dynamics of aggregating systems is easily represented. On the other hand, when atomistic growth dominates (i.e., nucleation is controlled to a low rate), the choice of a model is more critical [1]. A weakness of all these models is the inability to predict the dynamics of small cluster sizes away from the major mode of the particle size distribution. A number of hybrid models [47] have been developed that account for large particles with either (1) a continuous representation or (2) a sectional representation and the small clusters descretely. With one of these models, the discretes e c t i o n a l model [47-49], it is possible to predict the entire evolution of the particle size distribution accurately. This model assymptotically approaches the self-preserving size distribution after a few milliseconds of reaction. The discrete-sectional model has been used by Landgrebe and Pratsinis [50, 51] to develop numerical solution nomographs for vapor synthesis of ceramic powders (e.g., TiO2) when the atomistic growth rate is neglected. The nomographs predict the particle number concentration, geometric mean diameter, and geometric standard deviation of the particle size distribution as a function of residence time and chemical reaction time (both dimensionless when actual times are divided by the aggregation time, tl/2 = 1/K~?~ The chemical reaction time is a function of temperature. Experimental data from both furnace and flame reactors were compared to these nomographs with reasonable agreement [45].

7.10 QUENCHING THE A GGRE GA TION As the reaction consumes more and more of the reactants, the particles grow by atomistic and collisional growth. When the reaction is complete, atomistic growth stops but collisional growth continues as long as the particles are still at a temperature in which they are molten or sticky. To control the size of the particles produced by a gas phase reactor it is necessary to suddenly cool the particles to well below their melting point at the point at which they are the desired size. This step

297

7.10 Quenching the Aggregation a DOUBLE PIPE QUENCH

FLUID INLET HOT IGASI

INLETL

GE~.AS

HOT GASSES

I

COLD FLUID OUTLET ! !

z

_-.,~

-'dz~ Vl

k

I-"'

!11 II

"~1

I

Tc2

.~1 Pe2 tt

DILUTION QUENCH l l l

HOT GAS INLET

T2,pZ,v2,S2

II

:

S =S3 = I = I

MIXED GAS OUTLET T3,p3,v3

= = =

b

DILUTION GAS

FIGURE 7.21 Quenching: (a) Heat transfer quenching by double pipe heat exchanger in an aerosol reactor. (b) Dilution quenching by the addition of a cold gas.

is referred to as quenching the reaction. Quenching can take place by passing the hot gasses through a heat transfer section as shown in Figure 7.21(a) or by dilution of the hot gasses with cold gasses as shown in Figure 7.21(b). Both practices are used on large scale gas phase reactors in the industry. The speed with which the quenching is applied is an important factor in determining the size distribution of the product. If quenching takes place rather slowly and from the wall to the interior of the flowing aerosol, then the particles at the wall will be

298

Chapter 7 Powder Synthesis with Gas Phase Reactants

quenched quickly, stopping the collisional growth process and producing small particles, but those at the center of the flow will be quenched slowly, continuing the collisional growth process for a longer time and producing larger particles. As a result, the total population will have a broader size distribution as a result of quenching slowly from the wall of the reaction. For a highly turbulent flow, the broadening of the distribution is minimized. With dilution quenching, a uniform mixing of the hot and cold gasses gives a uniform quench. But uniform mixing is difficult with large volumes of gas flowing at high flow rates. In addition, the heat capacity of the cold gas is generally less than that of the particle ladened reactor gas. Therefore, larger quantities of cold gas than the initial reactor gas are needed. This leads to a major dilution of the product aerosol, which can cause difficulty when the particles are to be separated from the product gasses after the quench. For this reason, heat transfer quenching is the most frequent form of quenching, albeit problematic with respect to the broadening of the particle size distribution as just discussed.

7.10.1 Heat Transfer Quench Referring to Figure 7.21(a), the double pipe quench, we will consider the heat transfer between the hot gases and the cold fluid circulated in the jacket. At a steady state with no heat losses, the energy balance for the hot gas (h) and the cold fluid (c) is given by

Qh = whAHh = -Qc = -wcAHc

(7.68)

where Q is the heat transferred, w is the fluid flow rate, and AH is the change of enthalpy of the fluid. For incompressible liquids and ideal gasses the relation AH = CpAT is a valid approximation, allowing the above equation to be rewritten as

Qh = whCph(Th2- Thl)= - Q c -

-wcCpc(Tc2- Tel)

(7.69)

where Tel and Tee are the initial and final temperatures of the cold fluid stream, respectively; and Thl and The are the initial and final temperatures of the hot gas stream, respectively. When the hot gas is an aerosol, the heat capacity, Cph , is an effective heat capacity, which accounts for the particles and the conveying gas. When the particles are initially liquid and undergo a phase transition at their melting point upon cooling, the effective heat capacity also accounts for the heat of fusion of the solid phase. The macroscopic energy balance for the hot fluid can also be applied in differential form, giving

dQh = whCphdTh.

(7.70)

7.10 Quenching the Aggregation

299

This equation can be applied to a segment dz long where dQh is also given by

dQh = Uo(27rrodz)(Tc- Th)

(7.71)

where the temperatures of the hot and cold stream in the segment dz are Th and Tc, Uo is the overall heat transfer coefficient based on radius r o (inside or outside radius of the pipe).

Uo = (1~hi + 8/kp + 1/ho) -1

(7.72)

where h is the heat transfer coefficient (i is inside, o is outside) and kp is the thermal conductivity of the pipe of wall thickness, 8. Combining the previous two equations, we have

dT~ _ 27rrodz ( T c - T h ) - Uo whCph"

(7.73)

An analogous expression can be written for the cold stream as follows:

dTc

_

(Tc-

27rrodz Vo WcC.c "

(7.74)

By adding together the previous two equations, we find

L iT--hh~- T~) J :

(1

1)

U~ WhCph + w:C,c (27rr~

(7.75)

Integrating this equation, we find

[(Thl-Tcl)

]:

c2)J

Uo

( 1

1 )(27rr0L).

+ WcG

(7.76)

This expression shows the change of the average temperature in the hot gasses with the length of the cooling zone, L. Coupled with the average velocity, ~, of the hot gasses, the time, t, to cool to the melting point of the solid material can be obtained from the preceding equation, noting that L = ~/t. For highly turbulent gas flow with water as the cold fluid, the distance (or time) required to quench the hot gasses to the melting point of the solid from the flame temperature can be calculated.

Problem 7.2. Quenching Heat Transfer for a Plasma Reactor Determine the distance to the point of fusion for A12Q(1) particles in a gas at 2500~ flowing inside a tube of diameter 10 cm. The gas is, for the most part, air flowing at a rate of 10 kg/min ladened with 5% by weight particles. The cooling liquid is water flowing at 100 kg/min with an initial temperature of 25~ flowing in the annular space around the hot gas.

300

Chapter 7 Powder Synthesis with Gas Phase Reactants

Data: A1203Tf = 2045~ Cph = 0.249 cal/gm/~ Cpc = 1.0 cal/gm/ Uo = 50 kcal/m2/hr/~ For a solution, the heat balance for the hot gas and the water is given by ~

whCph(Th2- Thl) = -wcCpc(Tc2- Tel) (10 + 0.5)kg/min * 0.001 cal/gm/~ 9(2500~176 = - 1 0 0 kg/min 91.0 cal/gm/~ 9(Tee - 25~ from which Tc2 can be calculated to be 36.89~ lnL(-~h;

Using the equation,

Cl)1 (1 ~c2)J = Uo whCph + wcCpc

we can solve for L, which is 2.035 m.

7.10.2 Gas Mixing Quench Another form of quenching involves the addition of a cool gas to the reaction stream as shown in Figure 7.21(b). Often this gas has the same composition as the reaction mixture because it is just recycled gas t a k e n from the bag filter downstream. As a result, for this gas, the heat capacity, Cp, and the molecular weight, Mw, are the same as that of the gas in the reaction mixture. (Please note that, due to differences in particle loading, there is a difference in the heat capacity of these streams, but not accounted for in this problem.) Choosing two reference planes a and b as shown in Figure 7.21(b) we find mass, momentum, and energy balances as follows. Mass" W 3 ---- W 1 -+- W 2 ---- W Momentum (or force)" w3v3 + p3S3 = WlVl + plS1 zv W2V 2 + P282 = F

Energy:

w3 [Cp3(T3 - To) + V----~]= Wl [ C p l ( T 1 -

(7.77)

T0) + ~ ]

where wi is the mass flow rate, Pi is the pressure, Si is the crosssectional area of the pipe for that flow, and v~ is the velocity. Subscripts: are 0 for reference, 1 for stream 1, 2 for stream 2, and 3 for stream 3; no subscript (bold), total mass (w), rate of momentum transfer (or force, F), and energy (E) flow rate using the equation of state

P3 = p3RGT3

(7.78)

where P3 is the molar volume after mixing. Note: This analysis does not account for the ceramic particles specifi-

7.11 Particle Shape

301

cally. The average heat capacity, Cp, for streams 1 and 3 contain the heat capacity of the ceramic powder and its heat of fusion if the temperature is less than its melting point. The following analysis, however, neglects the heat capacity of the ceramic particles and their heat of fusion and assumes that Cpl ~ Cp2 ~ Cp3 ~ Cp (i.e., the very low particle loading case) and assumes the pressure times cross-sectional are terms are negligible. From these equations, assuming adiabatic mixing, the velocity after mixing can be determined as follows: va=-w

~/+1

1+ -

1-2(~/2+1 \ ~/2

wE ~

(7.79)

where ~/is the specific heat ratio CJCv. When the quantity in square brackets is unity, the velocity of the final stream is sonic. Therefore, in general, one of the solutions for v3 is supersonic and one is subsonic. Only the subsonic solution can be obtained under experimental conditions, because the supersonic solution is unstable [52]. Using the mass and energy balances, the temperature after mixing is given by

v ).wv

.

(7.80)

The temperature T 3 should be lower than that of fusion of the ceramic powder produced in the reactor to stop sticking aggregation of the particles. The energy of fusion must be accounted for in this energy balance for detailed calculations for temperatures below the fusion point of the ceramic powder when particle loading is high. When the contribution of the particles to the heat capacity of the gas stream is important (i.e., the high particle loading case), the mass, momentum and energy balance equations in (7.77) must be solved simultaneously. Typically, the details of streams 1 and 2 are known, and we need to calculate the outlet velocity and temperature (i.e., stream 3). Using the mass balance, we can calculate w3. With w3, we can use the momentum balance to calculate the outlet velocity v3, assuming that the pressure times cross-sectional area terms are negligible. With w3 and v3 we can use the energy balance to calculate the outlet temperature, T3, which completes the solution. When the pressure times cross-sectional area terms are important, the problem is more difficult and simultaneous solution of the mass, momentum, and energy balance equations (7.77) must be performed.

7.11 P A R T I C L E

SHAPE

The ceramic particles produced by gas phase reactions exhibit several different shapes depending on the conditions under which they were made. If the flame temperature is much higher than the melting

302

Chapter 7 Powder Synthesis with Gas Phase Reactants

F I G U R E 7.22 Particle morphologies of ceramic powders produced by gas phase synthesis: (a) Solidified coal flyash identified as Fe304 (please note the crystallites due to slow crystallization during quenching; (b) SiO2 (amorphous) from a flame reactor Bar - 5000 /k (please note the fractal nature of this aggregataed cluster; (c) TiO2 produced by thermal decomposition of Ti(OC3HT)4 at low concentration (0.025%) in a furnace reactor (please note aggregation similar to that of SiO2 in b); (d) TiO2 produced by thermal decomposition of Ti(OC3H7)4 at higher concentration (0.4%) in a furnace reactor. Figure (a) taken from McCrone and Delly [53], (b) taken from Phys. Rev. Lett. 52126], 2371-2374 (1984), (c and d) taken from Oshima et al. [54].

point of the ceramic powder, then the liquid droplets formed are spherical. If the quenching is very fast (i.e., ~ 10+~~ these liquid droplets form spherical amorphous ceramic particles, see Figure 7.22(d). If the quenching is not that fast, these liquid droplets form polycrystaline ceramic particles, see Figure 7.22(a) [53a]. The faster the cooling rate during quenching, the smaller and the more numerous are the crystals per liquid droplet. If the cooling is very fast amorphous particles are produced. If the flame temperature is slightly above the melting point of the ceramic, initially liquid droplets are formed but within the flame they cool to below their melting point. At these temperatures, the

7.12 Summary

303

particles are sticky and form fractal aggregates upon collision. Diffusion limited particle-cluster aggregation of this form gives a fractal dimension of 1.7-1.8. This type of structure is observed with flame synthesized silica (see Figure 7.22(b)), titania (see Figure 7.22(c)) [54], and carbon black. Cluster-cluster aggregation is another form of aggregation that can occur with a fractal dimension of 2.2-2.5, depending on the interparticle forces acting during aggregation. If the particles are held near their melting point for a long time before they are quenched, they will sinter together. Xiong and Pratsinis [55] have developed a population balance model that accounts for the nucleation, aggregation, and sintering of particles in a flame reactor. The result is a two-dimensional population defined by particle area and particle volume after sintering reaction at a particular temperature for a particular reaction time. For particles synthesized at flame temperatures below the melting point of the ceramic, the first particles formed are crystaline. These particles, typically grow atom by atom, maintain their crystal habit in much the same way as crystals grow in solution (see Chapter 6 for details). If the particles are sticky, they will aggregate, giving aggregates of crystaline particles.

7.12 S U M M A R Y This chapter gives a description of the four methods of gas phase ceramic powder synthesis: flame, furnace, laser, and plasma. Different types of reactors are chosen due to considerations based on the enthalpy of reaction. A flame reactor must have an exothermic reaction, whereas in principle the other reactors can operate with either exothermic or endothermic reactions. In each case, the reaction thermodynamics and kinetics are similar but the reactor design is different. The presence of ions sometimes changes the reaction pathways and kinetics for plasma reactors. To obtain narrow size distribution ceramic powders by gas phase synthesis, an idealized plug flow reactor design is the best. Dispersion of the gas flow, always present in real reactors, leads to a broadening of the particle size distribution. In addition, slow quenching of the reaction mixture gives broader particle size distributions.

Problems 1. A CO2 laser reactor producing SiC particles is operated in the laminar flow regime. You are required to quadruple the production rate (at fixed ~). There are two alternatives: quadruple the length of the reactor leaving the tube diameter the same, or double the diameter of the reactor leaving the reactor length unchanged. Compare the

304

Chapter 7 Powder Synthesis with Gas Phase Reactants

deviation from plug flow or both of these larger reactors with that of the present reactor. Which scale-up would you recommend? Also consider the changes that are necessary in the power of the laser used to effect the reaction. Data: Assume that the reactors are long enough for the dispersion model to be applied and that laminar flow prevails at all points. The Beer Lambert law of light intensity, I, is applicable I / I o = exp(aCL), where a is absorptivity of the reactant gas mixture at a concentration, C, which absorbs the light of the CO2 laser and L is the path length. 2. For a furnace reactor with a diameter of 2 cm and a length of 10 cm from the gas mixing point to the point of quencing, the flow rate is 1 liter per minute at a temperature of 2600~ Determine the geometric standard deviation of the particle size distribution assuming the multipoint nucleation model. What can you do to make a narrow particle size distribution? Data: Properties of the gas, density = 0.173 gm/liter, viscosity - 0.05 centipoise, Schmidt number = 0.25. 3. Given the TIC14 decomposition data in Section 7.4.1 determine the time for 99% complete reaction of 0.01 moles per liter TIC14 in argon at 1500~ 4. At 1500~ what is the equilibrium partial pressure of TiO2. Is TiO2 a liquid or a solid at this temperature. 5. Calculate the saturation ratio of T i Q as a function of the reaction time for the conditions given in problem 3. 6. For 3 t~sec after the start of reaction, what is the nucleation rate for the saturation ratio given in the previous problem. 7. Assuming that the nucleation rate is constant at 1030 particles per cm 3, determine the mean particle size as a function of time for an initial concentration of 0.01 moles per liter TIC14 in argon at 1500~ Assume the collision theory of aerosols is applicable and the sticking coefficient is 1.0.

References 1. Flagan, R. C., Ceram. Trans. I(A), 229 (1988). 2. Friedlander, S. K., "Smoke, Dust and Haze." Wiley, New York, 1977. 3. Pratsinis, S. E., and Kodas, T. T., in "Aerosol Measurement" (K. Willeke and P. Baron, eds.), Chapter 33, p. 724. Van Nostrand-Reinhold, New York, 1992. 4. Singh, R., and Doherty, R., Mater. Lett. 9, 87-89 (1990). 5. Journal of Research of the National Institute of Standards and Technology 9{}(4) (1991). 6. Sheppard, L. M., Metal Progress 4, 53 (1987).

References

305

7. Glasstone, S., Laidler, K. J., and Eyring, H., "The Theory of Rate Processes." McGrawHill, New York, 1941. 8. Bishop, D. M., and Laidler, K. J., J. Chem. Phys. 42, 1688 (1965). 9. Levenspeil, O., "Chemical Reaction Engineering." Wiley, New York, 1972. 10. Mahawili, I., and Weinberg, F. J., AIChE Symp. Ser. 186(75), 11 (1979). 11. Antipov, I. V., Koshunov, B. G., and Gofman, L. M., J. Appl. Chem. USSR (Engl. Trans.) 40, 11-15 (1967). 12. Suyama, Y., and Kato, A., J. Am. Ceram. Soc. 59(3-4), 146-149 (1976). 13. Pratsinis, S. E., Bai, H., and Biswas, P., J. Am. Ceram. Soc. 73(7), 2158-2162 (1990). 14. Powers, D. R., J. Am. Ceram. Soc. 61, 295-297 (1978). 15. French, W. G., Pace, L. J., and Foertmeyer, V. A., J. Phys. Chem. 82, 2191-2194 (1978). 16. Tanaka, J., and Kato, A., Yogyo Kyokaishi 81(5), 179-183 (1973). 17. Frank, A. J., and Hanrahan, R. J., J. Phys. Chem. 82(20), 2194-2199 (1972). 18. Sawano, K., Ph.D Thesis, MIT, Cambridge, MA, 1985. 19. Lamoreaux, R. H., Hildenbrand, D. L., and Brewerk, L., J. Phys. Chem. Ref. Data 16(3), 419 (1987). 20. Xiong, Y., Pratsinis, S. E., and Weimer, A. W., AIChE J. 38(11), 1685-1692 (1992). 21. Volmer, M., "Kinetik der Phasenbildung." Edwards Brothers, Ann Arbor, MI, 1945. 22. Bradley, R. S., Q. Rev., Chem. Soc. 5, 315 (1951). 23. Becker, R., and Doring, W., Ann. Phys. (Leipzig) [5] 24, 719 (1935). 24. Zeldovich, J., Soy. Phys.--JETP (Engl. Transl.) 12, 525 (1942). 25. Nielsen, A. E., "Kinetics of Precipitation." Pergamon, Oxford, 1964. 26. Coltrin, M. E., Kee, R. J., and Miller, J. A., J. Electrochem. Soc. 131 (2), 425-434 (1984). 27. Knudsen, J. G., and Katz, D. L., "Fluid Dynamics and Heat Transfer. McGraw-Hill, New York, 1958. 28. von Smoluchowski, M., Z. Phyz. Chem. (Leipzig) 92, 9 (1917). 29. Hidy, G. M., J. Colloid Sci. 20, 123-144 (1965). 30. Ulrich, G. D., Combust. Sci. Technol. 4, 47-57 (1971); Ulrich, G. D., and Reihl, J. W., J. Colloid Interface Sci. 87, 257-265 (1971). 31. Field, M. A., Gill, D. W., Morgan, B. B., and Hawksley, P. G. W., "Combustion of Pulverised Coal," p. 71. British Coal Utilization Research Association, Leatherhead, UK, 1967. 32. Levenspiel, O., Ind. Eng. Chem. 50, 343 (1958). 33. Abegg, C. F., and Balakrishnan, N. S., Chem. Eng. Prog. Symp. Ser. 110(67), 88 (1971). 34. Ring, T. A., "Large-Scale Generation of Narrowly Sized Ceramic Powders--An Assessment," l l t h Conf. Product. Res. & Technol. "Comput.-Based Factory Conf. Proc., Autom. Carnegie-Mellon University, Pittsburgh, 1984. 35. Ives, K. J., in "The Scientific Basis of Flocculation" (K. J. Ives, ed.), p. 37. Sijthoff & Noordhoff, The Netherlands. 36. Dirksen, J. A., and Ring, T. A., in "High-Tech Ceramics, Views and Perspectives" (G. Kostors, ed.), Chapter 3. Academic Press, San Diego, CA, 1989. 37. Swift, D. L., and Friedlander, S. K., J. Colloid Sci. 19, 621 (1964). 38. Wang, C. S., and Friedlander, S. K., J. Colloid Interface Sci. 24, 170 (1967). 39. Whitby, K. T., J. Aerosol Sci. 12, 173 (1981). 40. Lee, K. W., and Chen, H., Aerosol Sci. Technol. 3, 327 (1984). 41. Seigneur, C., Hudischewskyj, A. B., Seinfeld, J. H., Whitby, K. T., Whitby, E. R., Brock, J. R., and Barnes, H. M., Aerosol Sci. Technol. 5, 205-222 (1986). 42. Xiong, Y., and Pratsinis, S. E., J. Aerosol Sci. 22(5), 637-655 (1991). 43. Pratsinis, S. E., J. Colloid Interface Sci. 124, 416 (1988). 44. Pratsinis, S. E., J. Aerosol Sci. 20, 1461 (1989). 45. Gelbard, F., Tambour, Y., and Seinfeld, J. H., J. Colloid Interface Sci. 76, 541 (1980). 46. Crump, J. G., and Seinfeld, J. H., Aerosol Sci. Technol. 1, 15-34 (1982).

306

Chapter 7 Powder Synthesis with Gas Phase Reactants

47. Wu, J. J., and Flagan, R. C., J. Colloid Interface Sci. 124, 416 (1988). 48. Gelbard, F., and Seinfeld, J. H., J. Comput. Phys. 28, 357-375 (1979). 49. Landgrebe, J. D., and Pratsinis, S. E., J. Colloid Interface Sci. 139(1), 63-86 (1990). 50. Landgrebe, J. D., Pratsinis, S. E., and Mastrangelo, S. V. R., Proc. 2nd World Cong. Part. Technol., Kyoto, Japan, 1990, pp. 352-359. 51. Landgrebe, J. D., Pratsinis, S. E., and Mastrangelo, S. V. R., Chem. Eng. Sci. 45(9), 2932-2941 (1990). 52. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena," p. 471. Wiley, New York, 1960. 53. McCrone, W. C., and Delly, J. G., "The Particle Atlas." Ann Arbor Sci. Publ., Ann Arbor, MI, 1973. 54. Oshima, K., Torimoto, Y., Tsuto, K., Okuyama, K., Ushio, R., and Kousaka, U., Kona (Hirakata, Jpn.) 9, 59-71 (1991). 55. Xiong, Y., and Pratsinis, S. E., J. Aerosol Sci. 22(8), s199-s202 (1991).

8

Other Ceramic Powder Fabrication Processes

8.1 O B J E C T I V E S This chapter is devoted to the other ceramic powder fabrication processes not easily classified in the solid, liquid, and gas phase synthesis schemes of Chapters 5, 6, and 7. These methods include spray drying, spray roasting, freeze drying, metalorganic decomposition, sol-gel synthesis, and melt and flux solidification. Each of these techniques is described in various levels of detail.

8.2 S P R A Y D R Y I N G Spray drying is an industrial process used very often in the generation of dry powders from liquids or suspensions. A basic description of the preferred techniques can be found in Spray Drying Handbook by Masters [1]. A guide to spray dryer performance evaluation is given

307

308

Chapter 8 Other Ceramic Powder Fabrication Processes

by AIChE [2]. Spray drying is also a method of ceramic powder production that uses either a concentrated metal salt solution, a powder dispersion (a sol) with a polymeric binder, or a gel solution [3] as a feed material. The feed material is introduced into the top of a dryer (Figure 8.1) by some method of atomization. The atomizer may be a high speed rotating disc, a two-fluid (pneumatic) nozzle, or a single fluid (pressure) nozzle. The function of the atomizer is to (1) make many droplets of the feed material, increasing its surface to volume ratio to aid in drying, and (2) form droplets that will give the desired particle size distribution for the dried product. Once the droplets are atomized, they flow by gravity through the spray drying chamber until they reach the bottom where they exit with a gas stream through the product outlet. Because of the small droplet size created by atomization the actual drying time is measured in seconds. The total residence time for a droplet/particle is on average no more than 30 sec. The heated gas may be fed into the dryer either at the top, cocurrent to the liquid droplets, or at the bottom, countercurrent to the liquid droplets. The gas stream, usually large in volume compared to that of the liquid, may be heated directly or indi-

FIGURE 8.1

Schematic of spray drying process.

8.2 Spray Drying

309

rectly by any convenient means before entering the dryer. This gas provides the energy needed to 1. Heat and evaporate the liquid and heat the vapor to the exit temperature, 2. Provide the energy required for the heat of the reaction or crystallization, if applicable, 3. Heat the solid material to the exit temperature, 4. Compensate for heat losses from the chamber by conduction, convection, and radiation, 5. Maintain the exit gas at a sufficiently high temperature so that vapor does not condense in the spray drying and particle separation equipment. After leaving the drying chamber, all gases (and vapors) from drying pass through product recovery equipment before being released to the atmosphere. Cyclonic separators, followed by dry fabric filters or wet scrubbers, are used to separate the particles from the gases. The principal use of spray dryers is for ordinary drying of water solutions (also organic solutions) and aqueous slurries. In the spray drying of slurries, a flowable powder (10-100 ftm diameter) useful for dry pressing or plasma spraying is produced. Spray roasting is a variant of spray drying in which the spray dried material is subject to drying temperatures up to 1000~ where thermal decomposition, discussed in Chapter 5, takes place. Several types of thermal decomposition are possible, including dehydrating hydrous salts, decomposition of metal salts by decomposition of the salt anions (e.g., CO3-, SO42), and phase transitions in the solid material. Spray drying involves three fundamental steps: (1) atomization, (2) droplet drying, and (3) gas-droplet mixing. Each of these steps will be discussed in the following sections. The particles produced by spray drying are sometimes spherical and other times in the shape of a punctured spherical shell. The distribution of internal structure and chemistry including binder are not uniform inside a spray dried particle because it dries from the outside, first bringing impurities from the center of the droplet to the surface where they are crystallized out or left behind when the solvent evaporates.

8.2.1 A t o m i z a t i o n The object of atomization is to produce a large number of small droplets from a liquid stream so that the droplets can be dried into particles. Atomization is accomplished usually by one of three types of devices: (1) a high-pressure nozzle, (2) a two-fluid nozzle, or (3) highspeed centrifugal discs. These atomizers are low in cost, produce broad

~1~

Chapter 8 Other Ceramic Powder Fabrication Processes

Jet breakup: (a) Neckingin a liquid stream withL > 1.5R. (b) Disturbances of the circumference of a liquid jet of diameter D. Breakup occurs [4] when the amplitude of the disturbance is equal to D/2, which occurs first at a wavelength of 4.51 x D. (c) 1-2 t~m filament forms between two drops as liquid jet nears the breakup point. Drawn from a photo by Castleman [6].

FIGURE 8.2

droplet size distributions, a n d are of relatively low efficiency. The energy r e q u i r e d to increase the surface a r e a of the s t r e a m into droplets is typically less t h a n 1% of the total e n e r g y consumption. The other 99% is lost as h e a t to the system. With these atomizers, low viscosity solutions m a y be atomized into droplets as small as 2 t~m. The largest droplet d i a m e t e r (used for d e t e r m i n i n g drying time) is r a r e l y larger t h a n 500 t~m. All these atomizers* rely on Rayleigh instabilities shown in F i g u r e 8.2(a). W h e n the length, L, of a cylindrical liquid filament is g r e a t e r t h a n 1.5 x radius, the filament is u n s t a b l e and b r e a k s up into droplets. These instabilities are induced by n a t u r a l (or artificially induced) vibrations in the liquid. For a cylinder of liquid, these vibrations will grow exponentially and r u p t u r e the cylinder if the wavelength, X, is either a multiple of its length (Rayleigh breakup: X = n L j ) * Centrifugal discs produce sheets of liquid that perforate, forming filaments that break up by Rayleigh instabilities. See Figure 8.4.

8.2 Spray Drying

311

or a multiple of its diameter (radial breakup: ~ = nDj) as shown in Figure 8.2(b) [4]. The Rayleigh breakup mechanism leads to spherical droplets that have a diameter 1.89 times the diameter of the cylinder, Dd = 1.89 Dj. For high-viscosity liquids, droplet size does not follow this rule but yields a larger [5] droplet diameter, Dd, given by

Dd=l.89Dj(l+

3t~1

)1/6

(~/plDjg)U2

(8.2)

where Dj is the jet diameter, t~ is the liquid viscosity, p~ is the liquid density, ~/is the surface tension, and g is the acceleration due to gravity. For radial breakup, the m a x i m u m disturbance occurs at a wave length, ~ 4.5Dj, which leads to a droplet with a diameter smaller t h a n that of Raleigh breakup. Filaments 1-2 tLm in diameter trailing behind a just formed droplet, as shown in Figure 8.2(c) [6], can also break up giving submicron satellite droplets shown in Figure 8.2(a). As the interfacial velocity, u l, between the gas and the liquid increases, the break up process changes, because this velocity reinforces standing surface waves, which leads to a more complex jet break up, which occurs as a result of transverse oscillations. When the interfacial velocity, u l, is higher still, the droplets shatter into very fine droplets. The most important droplet stability criteria is the ratio of aerodynamic forces to surface tension forces defined by the Weber number [7, pp. 18-60], Nwe.

18-60],Nwe.

h2_ nmax lPllJd Nwe -

~/g

.

(8.3)

For low viscosity liquids when the dimensionless viscosity group [7],

Nv(= ~l/((rpl) u2] is less t h a n 1, a droplet will be stable below a m a x i m u m size, Dy ax, defined by the critical Weber n u m b e r and the gas liquid contact time. For long gas-liquid contact times, two large droplets are produced. For short contact times, many small droplets are produced. This type of droplet breakup yields a very broad droplet size distribution. For high-viscosity fluids, the critical Weber n u m b e r [3] is corrected as follows" Nwe

= xvATcritwe+

14Nv1"6

(8.4)

where the critical Weber number, 2-u ATcrit is given for various conditions of breakup in Table 8.1. Other types of atomizers for finer droplets use one or more of the following forms of energy to aid in droplet breakup" (1) sonic energy from the gas stream, (2) ultrasonic energy, (3) electrostatic energy, or (4) violent eruption of vapor within the jet when an additive liquid is injected below its vapor pressure.

312 T A B L E 8.1

Critical NWe 4 10-20

Chapter 8 Other Ceramic Powder Fabrication Processes Critical Weber Numbers for Droplet Breakup [7]

Breakup mode

Contact time

Nv

Vibrational into two large drops and several smaller drops Shattering into many small drops

>10 sec

<1.0

10 -3 sec

<1.0

8.2.1.1 P r e s s u r e N o z z l e s

In a pressure nozzle, liquid is forced at high pressure (50-300 kPa) with a high degree of spin through a small orifice, 0.1-5.0 mm in diameter. As the liquid leaves, it achieves its sonic velocity. Sonic velocities for several liquids are given in Table 8.2 [8]. The average velocity at the exit of the orifice, v2, is given by the Bernoulli equation [9] as 2

AP pl

(8.5)

where AP is the pressure drop across the orifice and Pl is the liquid density. These nozzles tend to operate at their sonic velocities. For this reason, the capacity of a spray dryer is controlled by the number of the orifices used, their size, and the sonic velocity. Too small an orifice can plug with foreign m a t t e r or become encrusted. The orifice will be subject to wear at high pressures or when solids are suspended in the feed liquid, changing the spray characteristics with equipment age. The mean droplet diameter from a single fluid nozzle is estimated with the following rule of thumb: Dd = 500 t~m" (psi) 1/3 where AP is the pressure drop across the orifice in lb/in2(psi).

TABLE 8.2 Sonic Velocities of Common Liquids [8]

Fluid

Ysoni c (m/sec)

Air Water Methanol Ethanol

331 1498 1103 1207

(8.6)

8.2 Spray Drying

313

FIGURE 8.3 Atomization nozzle designs for internal mixing and external mixing

8.2.1.2 T w o - F l u i d N o z z l e s

In a two-fluid nozzle, a liquid at relatively low pressure (0-60 psi) and an atomizing gas (either air or stream) at relatively high pressure ( - 1 0 0 psi) are forced through a double orifice. There are many types of nozzle design, but all types fall into two categories: internal mixing and external mixing, as shown in Figure 8.3. Many designs attempt to add swirl to the gas or liquid. These two designs of gas injection lead to high interfacial velocities, which produce smaller droplets with broader size distribution than pressure nozzles. The mean droplet size from a two-fluid nozzle is a complex function of the superficial velocity and fluid properties [10]. Some nine different engineering correlations

314

Chapter 8

Other Ceramic Powder Fabrication Processes

[3] have been developed. Two-fluid nozzles have been employed for dispersion of thick ceramic pastes not capable of being atomized by pressure nozzles.

8.2.1.3 High-Speed Disc Atomizers This type of atomizer discharges liquids in sheets from the periphery of a high rpm disc. These sheets of liquid perforate, forming filaments that then break up into droplets, as shown in Figure 8.4 [11, 12]. The principle design objectives are to ensure bringing the liquid up to disc speed and to obtain a uniform drop size distribution. Disc diameters vary from 5 cm to 21 cm with speeds from 3000 to 50,000 rpm, with higher speed usually reserved for small disc diameters. The degree of atomization is a function of the tangential velocity (Ddisc/2~) rather than the angular velocity (~). The mean droplet diameter for a disc atomizer is a complex function [7] of the tangential velocity, density,

Sheet breakup: (a) by perforation [11], (b) by sinusoidal growth [12]. Redrawn from a figure in Perry and Chilton [7, p. 18-60]. Reproduced with permission of McGraw-Hill, Inc.

FIGURE 8.4

8.2 Spray Drying

315

viscosity and surface tension. Disc atomizers have the advantage of being able to atomize suspensions and thick pastes that erode and plug nozzle atomizers. Discs are able to operate over a wide range of feed rates and disc speeds without large changes in the droplet size distribution.

8.2.2 Droplet Drying One of the principle advantages of spray drying is that spherical aggregates are produced as shown in Figure 8.5. These powders flow easily and for that reason are used for dry pressing. This spherical particle may be solid or hollow, depending on drying conditions. Generally materials that form a gelatinous skin or crust during drying will form hollow spheres. This skin slows down the drying from the surface. Continued heat transfer vaporizes the central part of the droplet, cracking the shell. Sudden vaporization can create a large hole in the shell as shown in Figures 8.5 and 8.6. There are four formation mechanisms for hollow particles [13]. (1) For drying gel solutions, a low permeability gel film forms around the droplet, reducing the evaporation rate. (2) For drying salt solutions, the liquid flows to the droplet surface, where evaporation takes place increasing the concentration near the surface. If the evaporation rate exceeds the rate of diffusion of solute back into the center of the droplet, then crystallization will take place at the surface of the droplet, creating a crust that slows the drying from the surface. (3) For drying slurries, as the liquid flows to the surface of the droplet, particles are carried along, thus forming a filter cake at the surface of the droplet. If the cake is sufficiently impermeable then it will slow the evaporation rate at the surface of the droplet. (4) Air entrained in a droplet can also cause hollow particles after drying. For materials that dry without crust formation into solid spherical particles, the final powder mean diameter,_ Dp, can be determined by the initial mean droplet diameter, Dd, and the solid content of the liquid as follows [14]"

Dp

(f)dCdt1/3

where Cd and Cp are the weight fraction of solids in the droplet and the dry particle, respectively; and Pd and pp are the bulk density of the droplet and the dry particle, respectively. For a molecule to be vaporized and diffuse away from the surface of the droplet, the latent heat of vaporization must be transferred to the droplet surface. A plot of the temperature around an evaporating water droplet is shown in Figure 8.7 [15]. Here we see the droplet is 15.1~ cooler than the surrounding

316

Chapter 8

Other Ceramic Powder Fabrication Processes

F I G U R E 8.5 Spherical aggregates produced by spray drying a suspension of A1203 particles. (a) Note that the particles are roughly a spherical shape with diameter - 1 5 0 /~m and also a generation of smaller spheres that result from satalite droplets. As a result of the spherical shape (and size), this powder flows easily into a mold for pressing. (b) Most of the particles show hollow cores, this dimble is the point where the core gas escaped during the last stages of drying. Photo courtesy, I. Adamou, Vice President for R+ D, Maret, SA, CH-2014 Bole, Switzerland.

8.2 Spray Drying

317

FIGURE 8.6

Frames of a motion picture of an evaporating drop of NaC1 illustrating crust formation and final structure. Drawn from a photo from Marshall [10].

FIGURE 8.7 Temperature around an evaporating drop. Redrawn from Ranz and Marshall [15 ].

318

Chapter 8 Other Ceramic Powder Fabrication Processes

gas due to the cooling caused by evaporation of the water solvent. The rate of mass transfer, J1, is given by

\RgTs RgTB]

(8.7)

where Rd is the droplet radius, Kc is the mass transfer coefficient, P1s is the solvent partial pressure at the drop surface, pS is the solvent partial pressure in the bulk gas, Ts is the surface temperature, TB is the bulk gas temperature, and Rg is the gas constant. The mass transfer coefficient, Kc, for a sphere can be determined from the Sherwood number, Sh (- Kc2Rd/DAB, where DAB is the molecular diffusion coefficient of the solvent, A, in drying gas, B,* typical values are 10-1-10 cme/sec) and the following engineering correlation [15] N S h ---- 2 . 0

+ 0.'::~T~/2~rl/3-~, Re x" Sc

(8.8)

where NRe (= UDOg2Rd/lUbg)is the Reynolds number (based on the interfacial velocity, u l;/zg, the gas viscosity; and pg, the gas density) and Nsc (= tZJpgDAB)is the Schmidt number. The partial pressure of a pure liquid at the surface varies as a function of the surface temperature, Ts, according to the Clausius Clapeyron equation [16]: pS=Poexp[

(1To Ts1)]

Rg

(8.9)

where AHvap is the enthalpy of vaporization andPo is the vapor pressure at To. With salt solutions, the solvent vapor pressure is given by ps = Ts(1 _

Xsalt)Po

exp [

Rg

(1To Ts1)]

(8.10)

where Ts is the activity coefficient for the solvent in solution and Zsalt is the mole fraction of the salt in solution. For dilute solutions the activity coefficient, T,, has a value of 1.0. A similar expression can be used for the vapor pressure of polymer solutions. The mass flux, J1, is related to the heat transfer flux, Q1, required to evaporate those molecules Q1 = z~kHvap$ J1.

+ -~mB p (dA

1 i] 2

(8.11)

8.2 Spray Drying

319

The heat transfer flux is given by

Q ~ - 4~R~h(T ~ - T s)

(8.12)

where h is the heat transfer coefficient. The heat transfer coefficient for a sphere is given by the Nusselt number NNu (= h2Rd/k, where k is the thermal conductivity of the drying gas) and the following engineering correlation NNu = 2.0 + 0.u,~I~T1/Re 3I~T2/ "" Pr 3,

(8.13)

where Ypr (-- Cp[~tg/k)is the Prandt number and Cp is the heat capacity of the drying gas at constant pressure. For gases, typical values of Npr are 0.7 to 1.0. The temperature at the surface of the droplet is determined by the evaporation rate obtained from simultaneous mass and heat transfer equation (8.10). (This temperature is the same as the wind chill factor discussed in weather forecasts, which can be calculated with equation (8.10) assuming the form of a moist sphere the size of the average human head.) Under some drying conditions, the heat transfer is the slow step, limiting evaporation, and in others the mass transfer is the slow step. In either case the evaporation rate, dRd/dt, can be written as

Pi 47rRd2 dRd Qi Mw ~ = J~ = ~/vap

(8.14)

where Mw is the molecular weight of the evaporating liquid and p~ is its density. This expression can be rewritten as

dR d_M_wKc ~ PSi_dt p~ \RgT S

pS ~ = Mwh RgTB]

tOl~/va------~ (TB -

Ts)

(8.15)

for mass transfer and heat transfer limited evaporation. Both of these expressions show a constant rate (i.e., dRd/dt - constant, not a function of size) that is frequently observed for a certain period of droplet drying. During this period, the surface of the droplet is always wet by the flow of fluid from within the droplet to the surface, and the temperature at the surface is essentially constant. Only when a crust is formed is the evaporation rate decreased and the temperature of the surface increased. When the droplet decreases in size, either Kc or h increases according to either the mass or heat transfer correlation (equation (8.8) or (8.13)). When the slip velocity is small, the correlations giving either Nsh or NNu are equal to 2.0 (Stokes regime). Leading to the following after substitution"

dRd_MwDA B ~ pS dt Rppi \RgTs

pB ~ : R~TB/

Mwk

(V B - VS).

RdPlAHvap

(8.16)

320

Chapter 8

Other Ceramic Powder Fabrication Processes

T A B L E 8.3 E v a p o r a t i v e Conversion, XB, versus Time for S h r inking Liquid Drop Rate Controlling Step: Sphere Conversion XB = 1 - (Rd/R~) 3

Boundary layer mass transfer

Boundary layer heat transfer

t = 1 - (1 - XB)V3

t = 1 - (1 - XB)2/3

T

T

02

plRd

plAHvapR ~ T

T-"

2MwDAB R-gTs

----

R--~B

An integration of this equation from R~ to R d gives the time, t, needed to dry a droplet from one size to another. The results of this integration are given in Table 8.3. These results are in the form t/T = f(Xs), where is the maximum time to dry the droplet for a particular rate determining step. In the equations given in Table 8.3, the surface temperature of the solvent is not known. The surface temperature is obtained by making the heat and mass transfer fluxes equal, as given in equation (8.16). This is equivalent to equating the expressions for the values of T for heat and mass transfer given in this table. The equations in this table are good only when the surface of the droplet is wet. When the droplet forms a crust, pore diffusion will be important. But before we discuss drying a crusted-over droplet, we must first discuss the precipitation that causes the crust to form when a salt solution is dried.

8.2.2.1 Precipitation Inside the Droplet For salt solution droplets, evaporation increases the salt concentration in the droplet causing precipitation, as shown in Figure 8.8. Experimental measurements on a wall drop verify this behavior, as seen in Figure 8.9. For salt solution droplets, the mass balance provides the time dependence of S, the saturation ratio; for example,

Ceq ~

= J1 -

[

4/37rR*aB~

+ 47rR2 --~ Vo(R, t)R ~

(8.17)

where B ~ is the nucleation rate and dR/dt is the crystal growth rate. The population balance for this droplet in the absence of breakage and agglomeration and assuming good mixing in the droplet is given by 0 -~- * ~?0 0~o ~ ot

oR

= 0

(8.18)

8.2 Spray Drying

321

m

8

Ceq

Time

FIGURE 8.8 Partial pressure of H20 at the droplet surface and salt concentration within the droplet as a function of time.

where To is the population based on number which is a function of time, t, and size, R. The growth rate d R / d t is elaborated in Table 6.2 (i.e., d R / d t = K*f(S)*g(R)). The assumption of good mixing is most accurate when the evaporation rate is slow. A particular solution to the population balance gives the population, To(R, t), as a function of time. As discussed in Chapter 6, the preceding equation has the following solution [17]: ~?0(R, t ) - i=1 ~ ai exp

- f hif(S)dt

.exp

- f Lg(R)

K*g(R

I (8.19)

/

J I

~o,.. I

Air temp.

First

crystals

~

200

~

/I

Drop

400

20.4"~1

!

600

I

/

800

20

~c

1000

sec

FIGURE 8.9 Variation of drop temperature and drop diameter in the drying of a drop of NaC1. Data taken from Marshall [10].

322

Chapter 8

Other Ceramic Powder Fabrication Processes

where h i a r e the eigenvalues and a i are the eigenvalue coefficients for this equation. This solution is a series solution that is the product of an exponential size dependent function and an exponential time dependent function. The eigenvalues and coefficients are obtained from the initial condition: rl(R t = O) = '

.=

ai e x p

-

~ -

[g(R)

K * ghi( R ) ] d R }

= v~

(8.20)

From this initial condition the eigenvalue coefficients can be determined: ai = i=1 ~ ~?~

-

[g(R)

g*g(R)

dR.

(8.21)

For a constant supersaturation (i.e., S ~ function of time) and a growth rate, G = d R / d t = K * f ( S ) * g ( R ) , which is not a function of size (i.e., g ( R ) = 0 corresponding to polynuclear and screw dislocation growth), the size distribution at any time is simpy a shifted version of the initial size distribution, ~?~ after the nucleation is completed: v~

t = O) = v ~

- Gt, t).

(8.22)

For different values of n in g ( R ) = R n, other kinetic expressions can be developed. Figure 8.10 [18] shows the type of powder produced on spray drying a solution that consists of metal salts of barium and iron in the ratio 1:12 (i.e., barium ferrite). Here we see the remains of the spherical droplets with a surface that consists of the metal salt precipitates, which form a narrow size distribution of platelet crystals (see Figure 8.10(a) and (b)). This narrow crystal size distribution is predicted by the population balance model if nucleation takes place over a short period of time. When these particles are spray roasted (in a plasma gun), the particles are highly sintered into spherical particles (see Figure 8.10(c)).

8.2.2.2

Droplet

Drying

with

a Crust

When an impervious skin forms as shown in Figure 8.6 on the droplet due to the solids in the suspension or the crystallization of the soluble species, the constant rate drying period ends and new shrinking core drying kinetics are applicable. For this case, we have the boundary layer mass and heat transfer described previously and pore diffusion and heat transfer, which can become the rate determining step. This type of drying is observed in Figure 8.9 as a nearly constant droplet diameter at drying times of 800/200 sec. The pore diffusive flux of A

Ba1Fe12 metal salt solution spray dried into a barium ferrite precursor powder, showing in (a) a broad size distribution of aggregated particles that result from the dried droplets, in (b) a higher magnification of a particle's surface. The platelet crystallites of metal salts are a narrow size distribution as predicted by the population balance for crystallization in evaporating. When the powder, shown in (a), is spray roasted, with a plasma gun in this case, it forms the highly sintered spherical particles shown in (b). Smaller particles show signs of melting and subsequent solidification. An exposed view of the interior of a larger spherical particle in (c) shows slightly sintered barium ferrite grains. Taken from Herman [18].

F I G U R E 8.10

324

Chapter 8

O t h e r C e r a m i c P o w d e r F a b r i c a t i o n Processes

through the product layer, J2, is given by P1

J2 = 47r r2DAE

(8.23)

- constant

dr

where r is the radius where the pores are filled with liquid, DAE is the effective diffusion coefficient of the solvent A in the porous layer. The effective diffusion coefficient for a porous layer is given by

[1

DAE = ~

+

_S~

(8.24)

where D g (= a~/18RgT/TrMw) is the Knudsen [19] diffusion coefficient (a is the pore radius), DAB is the molecular diffusion coefficient through the gas in the pores, s is the void fraction of product layer, and ~ is the tortuosity of the pores. The heat flux in the porous layer is given by

Q2 = 4~'r2Ke-~r

r =

constant

(8.25)

where Ke is the effective thermal conductivity of the product layer. If the product layer is porous, the effective thermal conductivity is given by

K e = [ 1 - sks

+ -~f] -1

(8.26)

where s is the void fraction in the porous product layer, ks is the thermal conductivity of the solid, and kf is the thermal conductivity of the gas in the pores. To complete the picture, the flux of solvent vapor is maintained by the transfer of heat to the liquid surface in the pores of the dry product:

01 4~.r 2 dr

Mw

=J2 _

Q2 .

(8.27)

The evaporation rate for pore diffusion and pore heat transfer gives a size dependent evaporation rate given by

,11

dr MwDAE RgT d--[ = Pl * dr

_

Mwg

e

r -- Pl~Hva-------~*

dT ~

I r

(8.28)

8.2 S p r a y D r y i n g

325

T A B L E 8.4 Evaporative Conversion, XB, versus Time for S h r i n k i n g Liquid Core Model: Sphere XB = 1 -- (r/RCd~ a B o u n d a r y layer

Pore d i f f u s i o n or c o n d u c t i o n

Rate Controlling Step: Mass T r a n s f e r (pore diffusion) t/r = XB t/r = 1 -- 3(1 --XB) ~3 + 2 ( 1 - X B) plR~ ~ ~1 (RCO)2 d T ~-

3M~c(R~s R~B)

T--

6MwDe(~gT)r

Rate Controlling Step: H e a t T r a n s f e r (pore conduction) t/r = X B t/r = 1 - 3(1 - XB) ~3 + 2(1 -- XB) plAHvapR~~ plAHvap(R~~ T= t = 3Mwh(Ts-

Ts)

6 M w K e ( T B - Tr)

for mass transfer and heat transfer limited evaporation. Both of these expressions show a rate dependent on the thickness of the porous crust, (R~~ - r), giving a decreasing evaporation rate with time, as shown in Figure 8.9 for the drying of a droplet of NaC1 solution. Initially the droplet dries from its surface according to rate expressions given in Table 8.3. After some degree of drying, a crust is formed at a size, R~~ The crust becomes thicker and becomes rate controlling. For this reason, it is possible that more than one resistance controls the evaporation rate throughout droplet drying. To account for the simultaneous action of these resistances is straightforward. A sum of the time values for each step is a means to obtain the total drying time: Ttota 1 :

(TMT -~- THW)constantrate -~- (TMT -~- TpD -~- THT -~- THC)crus t

(8.29)

where the r values are those given in Table 8.3 for the constant rate period and Table 8.4 for the crusted period, where the subscripts correspond to mass transfer, MT; pore diffusion, PD; heat transfer, HT; and porous heat conduction, HC. In the equations given in Table 8.4 the surface t e m p e r a t u r e of the water is not known. The surface temperature is obtained by making the heat and mass transfer fluxes equal as given in equation (8.27). This is equivalent to equating the expressions for the values of r for heat and mass transfer given in this table. The largest r value is that of the rate determining steps. There is always one rate determining step for heat transfer and another for mass transfer.

8.2.2.3 Problem: Droplet Drying Time A countercurrent spray dryer is operating with air at a superficial gas velocity of 1 m/sec at 150~ with a spray of droplets 50 ftm in diameter consisting of an aqueous suspension of 0.5 ~m A120 a particles.

326

Chapter 8 Other Ceramic Powder Fabrication Processes

Determine the maximum time to dry these droplets if we assume that the pore radius between the A1203 particles is 0.1 t~m. Assume that the droplet, initially at a A1203 volume fraction of 0.6, is the same size for the whole drying period. Data for water: M w = 18 gm/mole, Cp = 1 cal/gm/~ ~ - / v a p ~-~ 539.55 cal/gm, ~kHfusion -- 79.71 cal/gm, TBp - - 1 0 0 ~ p = 1.0 g m / c m 3, /~ - 0.01 poise. Data for air: M w = 29 gm/mole, Cp = 0.25 cal/gm/~ p = P M w / R ~ T , tL = 2 ( M w k s T / N A v ) ~ kf = t~Cp/Pr, P r - 0.73, diffusiOnwater,ai r -- 2 • l O - 3 T 3/2 ~ / 1 / M w a + 1 / M w ~ 1 / ( P [ ( d a + d~)/2] 2) cm2/sec, d e = 3.617/~, d0 = 2.655 A, P in atmospheres, and T in K. Data for A1203: p = 4.2 gm/cc, thermal conductivity, ks = 0.002 cal/cm/sec. To solve this problem, we will use the equations in Table 8.4 to determine the maximum drying time, r, for the various rate determining steps. In these equations the surface temperature of the water is not known. This problem shows a way in which the surface temperature can be determined. Using the terminal settling velocity, V t , V t :

g 4R~[0.6 94.2 gm/cm 3 - pg] 18 t~g

16.3 cm/sec. The Reynolds number, Re, can be determined to be 0.032, which is well within the laminar regime. From this Reynolds number, the mass and heat transfer coefficients can be determined if the temperature is assumed to be the bulk temperature, TB = 150~ Vt =

[2.0 + 0.6 Re 1/2 Sc 1/3]DAB 2Rd K c = 232.703 cm/sec

Kc =

h = [2.0 + 0.6 Re~/2Pr ~/3] kf 2R~ h = 0.031 cal/cm2/sec. Using equation (8.22) the effective diffusion coefficient, consisting of the Knudsen and molecular diffusion, can be determined as a function of temperature. Using equation (8.26) the effective thermal conductivity of the porous A1203 network, consisting of solid and gas conductivity, can be determined as a function of temperature. Using equation (8.9) the partial pressure of water can be given as a function of temperature. As a result of these substitutions the maximum times for drying for the various rate determining steps, ~, can be written as a function of t e m p e r a t u r e ~ t h e surface temperature of the water. A plot of these drying times as a function of the surface temperature is given in Figure 8.11.

8.2 Spray Drying

FIGURE 8.11

327

Drying time as function of surface temperature.

In this figure we see several intersections between r curves for mass transfer (negative slope) and heat transfer (positive slope) steps. The intersection between the boundary layer mass transfer and boundary layer heat transfer of the constant rate period gives a surface temperature of 300 K at a m a x i m u m drying time of 0.11 sec. This is, however, not the longest drying time. Pore diffusion has a longer drying time at 1.05 sec at 300 K. The intersection with the longest m a x i m u m time, ~, is for mass transfer by pore diffusion and boundary layer heat transfer occurring at a surface temperature, Ts, of 338.4 K or 65.4~ and a time of 0.176 sec. All the other r values at this temperature are at shorter times as follows: rBL(T s) rPD(T s) ~HTB L(Ts) rHTPD(Ts)

= = = =

0.022 0.176 0.176 0.043

sec sec sec sec.

Thus the surface temperature is fixed at 65.6~ with pore diffusion the rate determining step. At all other surface temperatures, pore diffusion gives the largest r value for the mass transfer steps, and boundary layer heat transfer gives the largest r value for the heat transfer steps.

8.2.3 Gas-Droplet Mixing G a s - d r o p l e t contact can be either cocurrent or countercurrent (and sometimes crosscurrent). Up until now we have considered t h a t the gas has a large volume with constant t e m p e r a t u r e and partial pressure. In a spray dryer the gas in contact with a particle constantly changes due to the g a s - d r o p l e t contact pattern. These different dryer configu-

~8

Chapter 8

Other Ceramic Powder Fabrication Processes

FIGURE 8.12

Spray dryer flow patterns: (a) Cocurrent flow, (b) countercurrent flow.

rations are shown in Figure 8.12. The various methods of contacting fluids include countercurrent, crosscurrent, and cocurrent plug flow. Consideration of the residence time distribution for each type of fluid-solid contact is necessary to understand its effect on the drying conversion, XB. As a result of a given residence time distribution, E(t), the average conversion of B, XB, is given by [20]

1 - XB =

-

fo

[1 -- XB(t)]E(t) dt

(8.30)

where Xs(t) is the conversion function given in Tables 8.3 and 8.4, depending on the kinetics applicable. The preceding equation assumes that the gas is of uniform composition throughout the reactor at all times. If the gas composition changes with the time or position within the reactor, a more complex equation must be used. The residence time distribution can take any form. However, two simplified residence time distributions are frequently used: back mixed flow [20], -t/o

E(t) - e

0

where 0 is the mean residence time; and plug flow [20], =

where 6 is centered at t = 0.

(8.31)

8.2 Spray Drying

329

To account for the effect of particle size distribution in addition to the residence time distribution is difficult, because different size particles can remain in the reactor for different periods of time. To account for these effects completely, a population balance must be performed, where the conversion, XB, is an internal variable (see Chapter 3). This type of t r e a t m e n t is beyond the scope of this chapter. A simplified method of accounting for the effects of a particle size distribution, W(r), on the mean conversion, XB, is by [20, p. 381] 1 - XB=

-

[1 - XB(r)]W(r) d r

f0

(8.33)

where W(r) is the normalized population weight distribution. This equation assumes that the particles, regardless of size, have the same residence time within the reactor. This is not always a good assumption because fine particles follow the gas stream lines much better t h a n large particles, which tend to settle. For single size particles, the mean conversion for a mixed flow reactor is given in Figure 5.14. The mean residence time, 0, must be much larger t h a n the maximum time for reaction, r, for the average conversion, XB, to be complete (i.e., X~ - 1.0) for all the rate determining steps. In a plug flow reactor, however, the mean residence time, 0, needs to be larger than only the maximum time for reaction, r, for complete reaction. A comparison of the time for a certain conversion in a mixed and a plug flow reactor is given in Figure 5.15. For a specific conversion the mixed flow reactor time is always larger than the plug flow reaction time.

8.2.4 Spray Dryer Design The time, t, spent by a droplet in the spray tower of height, z, is given by vector addition of the gas velocity, V~, and the terminal settling velocity of the droplet, vt (i.e., v = z / t = Vg + vt). With countercurrent flow, the gas velocity and the terminal settling velocity are in different directions and therefore have different signs. The terminal settling velocity is given by vt =

when the Reynolds number,

D2(ps - P~)g

18t~g

NRe is

Vt -- ~]

(8.34)

less t h a n 1 and

4D(p s - p~)g 3CDPg

(8.35)

at other Reynolds numbers corresponding to various drag coefficients, CD. In the Newton regime, where the drag coefficient CD is 0.44 (i.e.,

330

Chapter 8

Other Ceramic Powder Fabrication Processes

NRe is greater than 100 and less than 100,000). When the droplets are small, the terminal settling velocity of the droplet is small compared to the gas velocity and the droplets are carried along with the gas. The height of a spray dryer column, z, necessary to take a liquid to dryness is given by [7] (assuming that boundary layer mass transfer is the rate determining step) z - HoGNoG where HOG (= ~ut

VJKca)

(8.36)

is the height of transfer unit, and NOG

( - fp~o dPi/(P~ - Pi)) is the number of transfer units. In the preceding equations, Vg is the superficial gas velocity, a is the total droplet area per unit volume (assumed to be a constant), and Kc is the mass transfer coefficient for a sphere, which can be determined from the Sherwood number, Sh (= Kc2Rd/DAB,where DAB is the molecular diffusion coefficient of the solvent, A, in drying gas, B), and the following engineering correlation [21]: Nsh =

2.0 + 0.6 Rei/2Sc i/3

(8.37)

Depending on the contacting pattern (cocurrent or countercurrent) the equilibrium partial pressure P~ will depend on the surface temperature, which depends on the heat transfer at each location in the column. For the constant rate period, the surface temperature is a constant and the integrand needed to calculate NOG is greatly simplified:

NoG = ln [ (Pf - P1)~

(8.38)

where [P~ - P 1 ] i n is the solvent partial pressure difference between the surface of the droplets, PT, and the gas at the spray dryer inlet, and [P~' - Pi]out is the same partial pressure difference at the spray dryer outlet. This expression is good for boundary layer mass transfer as the rate determining step in drying. For other rate determining steps, the definition of NOG will account for the heat transfer, pore conduction, and pore diffusion giving a much more complicated expression.

8.3 S P R A Y R O A S T I N G Spray roasting is simply an extension of spray drying to higher temperatures, where thermal decomposition of the sprayed salt decomposes. The following steps are involved:

8.3 Spray Roasting

1. 2. 3. 4.

331

Atomization of a salt solution, Mixture of droplets with a heated gas, Evaporation of the solvent (often water), Thermal decomposition of the dried salt particles in the heated part of the equipment.

Figure 8.13 shows a common spray roasting apparatus. It consists of a atomization head, a gas mixing zone, and a zone heated by a furnace. A tube furnace is commonly used as a heat source [22-24], but a plasma may also be used. RF plasmas have been used in a similar cocurrent flow pattern [25-28], and a dc plasma has been used in a countercurrent flow configuration [29]. Plasmas are used because they have high temperatures and steep temperature gradients. In general, any soluble metal salt can be used, and for this reason metal nitrates and metal acetates are commonly used. Table 8.5 gives the decomposition temperatures of several metal salts. The morphology of spray

FIGURE 8.13 Schematic of spray roasting apparatus.

~32

Chapter 8 Other Ceramic Powder Fabrication Processes

TABLE 8.5 DecompositionTemperatures for Metal Salts [8], Lower Temperatures Are for Dehydration Salt

Zn(NO3) 96 H 2 0 Ni(NO)3)2 96H20 AI(NO3)3 99H20 Cu(NO3)2- 6H20 Pb(NO3)2

Temperature (~ )

105-131-360 400 (m.p. 57) 150 (m.p. 73) 264-170 470

Salt

Temperature (~ )

Zn(C2H302)292H20 Ni(C2H302)294H20 Co(C2H302)3 Cu(C2H302)2" H20 Y(C2H302)3" 4H20

100-400(m.p 260) 350 100 240 500

plasma roasted powder is shown in Figure 8.10(b & c). It shows a classic spherical shape caused by either sintering or melting followed by solidification in the high temperature of the plasma. The size distribution is essentially the same as the droplet size distribution and is therefore controlled by atomization. These decomposition temperatures are typically less than 500~ and often follow complex mechanisms for example. Zn(NO3) 96H20(s) 1~176176> Zn(NO3)(s) + 6H20(g) -55~176ZnO(s) + NO2(g). (8.39) This complex reaction sequence leads to equally complex decomposition kinetics. As discussed in Chapter 5, the shrinking core model is applicable to simple one-step thermal decompositions of the type A(g) + b B ( s ) ~ rR(g) + sS(s). Fluid-solid reactions can be kinetically limited by several steps: 1. 2. 3. 4. 5.

Surface reaction, Mass transfer in the boundary layer surrounding the particle, Diffusion in the product layer, Heat transfer in the boundary layer surrounding the particle, Heat conduction in the product layer.

These rate determining steps are discussed in detail in Chapter 5. Using the relation between the flux J and the change in core radius given in equations (5.36) and (5.37) with the definitions of the fluxes for mass and heat transfer equations (5.27), (5.28), (5.31), (5.32) (with equations (5.33) and (5.34)), it is possible to determine the time dependence of conversion, XB:

for the shrinking core model for a sphere as given in Table 8.6.

8.3 S p r a y R o a s t i n g

333

T A B L E 8.6 C o n v e r s i o n , XB, v e r s u s T i m e for S h r i n k i n g C o r e Model: bB(s) --* A(g) + C(s) or S p h e r e XB = 1 -- (r/R) 3

B o u n d a r y layer

Pore diffusion

Surface reaction

R a t e c o n t r o l l i n g s t e p : m a s s t r a n s f e r (pore d i f f u s i o n or s u r f a c e r e a c t i o n ) t/r = XB t/r = 1 -- 3(1 -- XB) 2/3 + 2(1 -- XB) t/r = 1 -- (1 -- XB) 1/3

pBR r = 3bKg--------~

pBR2 T = 6bDe--------~

R a t e c o n t r o l l i n g step: h e a t t r a n s f e r (pore c o n d u c t i o n ) t/T = XB t/~ = 1 -- 3(1 XB) 2/3 + 2(1 -

pBR r = 3 A H ~ x N b h ( T B - Ts)

r =

pBR T - bKsCAB

-

X B)

PBR2 6 A H ~ N b k e ( T B - TS)

The conversion time expressions in Table 8.6 assume that a single rate resistance controls the reaction of the particle. Initially, the product layer provides no rate resistance because it is very thin. After some degree of reaction, however, the product layer is thicker and often becomes rate controlling. For this reason, it is not reasonable that just one resistance controls the reaction rate throughout the whole reaction. To account for the simultaneous action of these resistances is straightforward because all are linear in reactant concentration. Thus a sum of the time values for each step is a means to obtain the total reaction time [20, p. 371]: Ttotal-

TMT + TpD + TSR + THT + THC

(8.41)

where the r values are those given in Table 8.6 when the subscripts correspond to mass transfer, MT; pore diffusion, PD; surface reaction, SR; heat transfer, HT; and heat conduction, HC. The various methods of contacting fluids include countercurrent, crosscurrent, and cocurrent plug flow. Consideration of the residence time distribution for each type of fluid-solid contact is necessary to understand its effect on the conversion. As a result of a given residence time distribution, E(t), the average conversion ofB,Xs, is given by [20] oo

1 - XB = fo [1 - XB(t)]E(t) dt

(8.42)

where Xs(t) is the conversion function given in Table 8.6, depending on the kinetics applicable. This equation assumes that the gas is of a uniform composition throughout the reactor at all times. If the gas composition changes with the time or position within the reactor, a more complex equation must be used. The residence time distribution can take any form of two simplified residence time distributions, back

334

Chapter 8

Other Ceramic Powder Fabrication Processes

mixed flow or plug flow, as described in equations (8.31) and (8.32). To account for the effect of particle size distribution in addition to the residence time distribution is difficult because different size particles can remain in the reactor for different periods of time. To account for these effects completely, a population balance must be performed where the conversion is an internal variable (see Chapter 3). This type of treatment is beyond the scope of this chapter. A simplified method of accounting for the effects of a particle size distribution, W(r), on the mean conversion, XB, is by [20, p. 381]

f0

1 - XB =

[1 - XB(r)]W(r) dr

(8.43)

where W(r) is the normalized population weight distribution. This equation assumes that all particles, regardless of size, have the same residence time within the reactor. This is not always a good assumption because fine particles follow the gas stream lines much better than the large particles, which tend to settle. For single size particles, the mean conversion for a mixed flow reactor is given in Figure 5.14. The mean residence time, 0, must be much larger than the maximum time for reaction, r, for the average conversion, XB, to be complete (i.e., XB = 1.0) for all of the rate determining steps. In a plug flow reactor, however, the mean residence time, 0, only needs to be larger than the maximum time for reaction, ~, for complete reaction. A comparison of the time for a certain conversion in a mixed flow reactor and a plug flow reactor is given in Figure 5.15. For a specific conversion, the mixed flow reactor time is always larger than the plug flow reaction time. Using a plug flow furnace configuration with the ultrasonic nebulation of a solution of zirconium oxy-chloride and yttrium nitrate, Dubois et al. [23] have produce slightly porous spherical yttrium stabilized zirconia with particles between 0.2 and 1.8 t~m in diameter and geometric standard deviations between 1.2 and 1.6. Using a countercurrent dc plasma configuration with the atomization of yttrium, barium and copper nitrates in the mole ratio 1" 2"3, Kong and Pfender [29] have produced slightly porous spherical YBa2Cu3Ox with particles between 0.5 and 2.0 t~m in diameter and geometric standard deviations between 1.3 and 2.0. These two examples show one of the important advantages of this powder synthesis method. Each droplet can be considered a separate micro-reactor, and as a result segregation of the two metal salts, which occurs during precipitation, is limited to the size of the final particle, which is determined by the initial droplet size. Thus mixed oxides or doped oxides can be prepared by this technique without large scale chemical segregation.

335

8.4 Metal Organic Decomposition for Ceramic Films

8.4 M E T A L O R G A N I C D E C O M P O S I T I O N FOR CERAMIC FILMS Solutions of metal organics can also be spray roasted to give ceramic powders as mentioned earlier. But by far the most interesting use of metal organic decomposition is for the manufacture of ceramic thin films. In this case, metal organic solutions can be dip coated or spin coated onto a substrate. After drying, a metal organic coating is obtained, which can then be thermally decomposed to a ceramic. These metal organic solutions can also be used to put coatings on other ceramic powders to make a layered composite powder. Readers interested in thin films by metal organic decomposition are referred to the book Sol-Gel Science [30] and other review papers [31-33]. The thermodynamics and kinetics of metalorganic decomposition can be discussed in terms of the fluid-solid reactions presented in Chapter 5. An example of this type of metal organic thermal decomposition is shown in Figure 8.14. Here we can see the weight loss of a Ba- and Cu-ethylhexanoates and Y-napthenate solution in chloroform. Each of the metal carboxylates has a different decomposition profile when measured separately. When the weight loss of each component is summed in the appropriate proportions for the composite solution (i.e., lOO .

90

.

~

80 70

~

_ ........ _ ~

:

YBaz Cu Oz Cu 2 - E t h y l h e x a n o a t e Ba 2 - E t h y l h e x a n o a t e Y Naphtenate Combined Powders

---.... --'-....

~

60

';

..... "

30

20

L..

.~ ,

..

...........

o

~ .................

~-~. |

'~

"'~ ~

oo

9

9 oo~oo

L------M

o

o o o o

o o o

go

o

e g o

o

go

. . . . . . . . . . . .

~INNlilNliPI N

l

/

I

I

I

N

I

N

I

I

',oo

700

U

oo

l

l

o o o o

O

U

l

/ !

900

"

,ooo

Temperature (~

FIGURE 8.14

Thermogravimetric analysis of superconducting YBa2Cu3OT-x film, solution, and the raw materials used to form the precursor solution: Y naphtenate, Ba 2ethylhexanoate, and Cu 2-ethylhexanoate. The raw materials curve is a calculated from proportions of raw materials in the composite film solution. From Gallagher and Ring [31].

336

Chapter 8

Other Ceramic Powder Fabrication Processes

stochiometric ratio Y :Ba: Cu 1 : 2 : 3), the profile is different than that observed experimentally for the solution. Interactions between the different metal carboxylates in solution cause slight changes in decomposition temperatures of the components, and there is also an overlap of the individual decompositions. This pyrolysis results in an amorphous and homogeneous solid consisting of CuO, BaCO3, and Y203. Decomposition of BaCO3 occurs at 850~ after which the oxides can react by solid-solid reactions to form the YBa2Cu307_x phase. Crystallization of this mixed oxide takes place at temperatures near 900~

8.5 F R E E Z E D R Y I N G Freeze drying is used to dry either salt solutions or ceramic suspensions in a gentle way, avoiding thermal decomposition of the metal salts and aggregation of the particles. There are four steps in freeze drying: 1. Either a mixture of soluble salts containing the desired ratio of metal ions is dissolved in a solvent or a ceramic powder suspension is made. 2. The solution is sprayed into droplets between 0.1 and 0.5 mm in diameter and rapidly frozen by a cold fluid (liquid or gas) so that no compositional segregation or aggregation can occur and the solvent crystals that nucleate are very small. Very high cooling rates can give a solidified solvent that is amorphous. 3. The solvent is removed by vacuum sublimation being careful not to form any liquid phase that could cause chemical segregation. 4. The resulting powder is either used directly or the metal salts are calcined at a temperature that decomposes the crystallized salts and converts them to very fine crystalites of the desired ceramic powder. The first step of freeze drying is atomization, which was discussed in Section 8.2. The next step is solidification, which is discussed in the book Transport Phenomena in Metallurgy [9]. The solidification or freezing time for a droplet of volume, V, and surface area, A, is given by the time, t, required to transfer the heat of fusion from the fluid to the freezing droplet: f t

Q= ~ J0

Aqsurfacedt = p'VH}

(8.44)

where p' is the density of the solution in the droplet, H} is the effective heat of fusion that represents the sum of the latent heat of fusion, Hw,

8.5 Freeze Drying

337

and the liquid's super heat, CpATsuperhea t . Cp is the heat capacity of the solution in the droplet. Considering the solution to the transient heat conduction problem for an infinite slab, the heat flux into the freezing liquid, qsurface, is given by qsurface-

-k (Tf- To) k/~at

(8.45)

where Tf is the freezing point of the liquid in the droplet, T O is the t e m p e r a t u r e of the fluid used for freezing, a[= k/pCj is the thermal diffusivity, and k is the thermal conductivity of the freezing fluid. This approach neglects the curvature of the freezing droplet. Integrating, we find that the time to freeze a shape of volume V and a r e a A is given by t = C

(8.4:6)

where (8.47) This expression is valid for nearly flat shapes but inaccurate for a sphere. In the case of a sphere, Adams and Taylor [34] have developed an exact solution using two dimensionless parameters, ~ and T: -

V~at

(8.48)

where t is the time to freeze a volume V with a surface area A (i.e., (V/A) = (D/6), D is the diameter of a sphere) and

~/= [ (Tf- T~ ]j With these parameters, the freezing times for a true sphere can be calculated from the expression r,,,

+

- 1

(8.50)

This expression is plotted in Figure 8.15.

8.5.1 Problem: Freezing Time for a Drop Determine the freezing time for drops of water initially at 25~ of size 1 tLm, 10 t~m, 100 t~m, and 0.1 cm suspended in air at -50~ and

33~

Chapter 8

Other Ceramic Powder Fabrication Processes

0 0

2

F I G U R E 8.15 Dimensionless sphere solidification time, r(= k/at/(V/A), versus T(= [(Tw - To)/p'H~]pCp) , corresponding to a plot of the equation r~(2/V/-~ + T/3) = 1, where t is the time to freeze a spherical droplet of volume V and surface area A, note VIA = D/6, where D is the sphere diameter.

1 atm. Assume t h a t there is no evaporation or sublimation and t h a t the air is s t a g n e n t and transfers heat by conduction only. D a t a for ice: Cp = 0.5 cal/gm/~ hHvap = 539.55 cal/gm,/~kHfusion 79.71 cal/gm, Tw = 0~ p = 1.0 gm/cm 3, k1 = 0.0015 cal/sec/cm/~ Data for air: Mw = 29 gm/mole, Cp = 0.25 cal/gm/~ p = PMw/RgT, t~-- 2(MwkBT/NAv)~ 2, kg-- 5.6 • 10 .6 cal/sec/cm/~ To resolve this problem we have to determine % defined in equation (8.49), and t h e n use it in equation (8.50) to determine r, from which the actual time can be determined, using equation (8.48): :

=[

p'H}

J

1.0 gm/cm a * [79.71 cal/gm + 0.5 cal/grn/~ 9(25-0)~ 9 1.586 • 10 .4 gm/cc 90.25 cal/gm/~ = 2.15 • 10 .5

]

Solving for r, using equation (8.50), we find t h a t r = 390, from which freezing time, t, can be calculated using equation (8.48) as follows:

t

5.6 x 10 .6 cal/sec/cm/~ 1.586 ~ io-~-~/-~c . ~ c--a~gm/~ ] for all the different droplet diameters. These values follow"

8.5 Freeze Drying Diameter

F r e e z i n g time

1 t~m 10 tLm 100 t~m 0.01 cm

3.0 x 10 .5 0.003 0.3 30

~~

The freezing time is short (i.e., less than a second) for all droplets smaller than 100 t~m. As we have noted in Chapter 6, some salts have a solubility that decreases with decreasing temperature. In this case these salts will crystallize during the freezing process, which can lead to chemical segregation into different salt crystals within the droplet. When the solubility increases with decreasing temperature, a more homogeneous mixing of the various metal salts is found. After freezing, the time to sublimate the solvent is given by the drying expressions in Tables 8.3 and 8.4, where the enthalpy of vaporization for drying is replaced by the enthalpy of sublimation. The enthalpy of sublimation is often equal to the sum of the heats of fusion and vaporization [16]. The enthalpy of sublimation is also substituted for the enthalpy of vaporization in the Clausius Clapeyron equation (8.9) required for the calculation of the solvent partial pressure. The same rate determining steps of boundary layer mass transfer and heat transfer as well as pore diffusion and porous heat conduction are applicable in sublimation. After sublimation, a dry powder consisting of either unaggregated powder or metal salts (sometimes hydrated) is obtained. During the subsequent calcination of the metal salts, thermal decomposition takes place the same way as discussed in Chapter 5. We can have decomposition of two general types: (1) fluid-solid reaction, where the fluid is a gas; or (2) solid-solid reaction. Fluid-solid reactions can be represented by A(gas) + bB(solid)--~ dD(solid) A(gas) + bB(solid)--* dD(solid) + eE(gas) bB(solid)--~ dD(solid) + eE(gas). With each of these reactions, a solid of one type (B) is the reactant and a solid of another type (D) is the product. A gas is also a reactant or a product of the reaction. In some cases, the solid product (D) forms a shell on the outside of the particle, B, producing a diffusion barrier for further reaction, giving a shrinking unreacted core. In other cases, the product D flakes off the surface of particle B, giving a shrinking sphere. The reaction kinetics for all of the various rate determining steps in fluid-solid reactions were discussed for two general models, shrinking core and shrinking particle, in Chapter 5.

340

Chapter 8

Other Ceramic Powder Fabrication Processes

After a mixture of oxides is produced by the thermal decomposition of the freeze dried and calcined powder, further solid-solid reactions can take place. Solid-solid reactions operate by the mechanisms discussed in Chapter 5, which include solid state diffusion and chemical reaction. Diffusion in ceramic solids is always ionic in nature and depends on defect or hole diffusivity, as well as electron conductivity. Once the ionic reactants are in close association, chemical reactions can take place, giving mixed oxides like B a T i Q and PbZrO3. A useful laboratory freeze drying apparatus is described by Rigterink [35]. This apparatus uses a dry ice (CO2(s)) plus an acetone bath to freeze the water droplets.

8.6 S O L - G E L

SYNTHESIS

The earliest routes for forming ceramics from sol-gel solutions involved the precipitation of metal oxide particles from solutions. These form a true colloidal suspension: a sol. Upon destabilization of this sol, aggregation takes place and a rigid network is formed: a gel. A gel is intermediate between a solid and a liquid. The term sol-gel has since been used by the materials science community to describe, albeit erroneously, virtually all chemical processing of ceramics from solutions (e.g., metal oxide particle precipitation or metalorganic decomposition). This discussion focuses on the gel aspects of sol-gel synthesis and not on the sol aspects, which are treated separately in this book. The schematic diagram in Figure 8.16 and the micrographs in Figure 8.17 demonstrate the variations in sol-gel processing. The starting solutions may be made from a variety of chemical systems. These include metallo-organic, polymeric, and ionic solutions. The chemistry chosen dictates the physical characteristics of the gel produced. These characteristics include its structure and, specifically, the density distribution in the gel network. The networks formed may be described as follows:

1. Particle networks give high and nonuniform density distributions formed by the aggregation of particles. The surface chemistry of the particles control their suspension properties and aggregate structure. Aggregate structure is also controlled by the number density of particles in solution, as this effects aggregation rate and the fractal dimension of the resulting aggregate. Virtually any fine particulate sol may be destabilized to form a particle network. 2. Aggregate networks give low and nonuniform density distributions formed by the clustering of aggregates. In this case, a solution is prepared under conditions where a solid phase is precipitated out in a

8.6 Sol-Gel Synthesis

341

FIGURE 8.16 Schematicofroutes for structural evolutionofmetallorganics in solution.

From Gallager and Ring [31].

very small particle size (i.e., high supersaturation). During this precipitation, the particles aggregate simultaneously. The result is the formation of a rigid network. A qualitative discussion of both is presented here. 3. P o l y m e r n e t w o r k s give low and uniform density distributions formed by crosslinking polymers. The polymers used contain the metal desired in the final ceramic. In some cases, the polymer backbone is a series of metal oxygen bonds making an inorganic polymer. The various stages of sol-gel synthesis are discussed next.

8.6.1 Precursor Solution Chemistry Ceramic materials can be made from the oxides, nitrides, carbides, or sulfides of metals. The precursor solution must contain both the metal and the requisite O, N, C, or S in close liaison. Three broad

342

Chapter 8

Other Ceramic Powder Fabrication Processes

8.6 Sol-Gel Synthesis

343

classifications based on the physical properties of the precursor solution will be used: (1) sols that are colloidally destabilized to form an aggregate network, (2) true solutions, and (3) solutions that polymerize to form a network. The network formed in the true solution case results from simultaneous precipitation and aggregation. The precursors most often used in the sol-gel process are hydrolyzed alkoxides in alcohol solutions. A short discussion of the alkoxide chemistry is useful to explain their gelling characteristics. The alkoxide precurors are commonly formed as one of a series of homoleptic alkoxides M(OR)n, where n = 1-6. Organic molecules such a s alcohols tend to be strong rr electron donors and thus stabilize the highest oxidation state of the metal [36]. The specific synthesis route of an alkoxide is dictated by the electronegativity of the metal. The most common routes for metal alkoxide formation are [37] 1. Reduction of alcohols for electropositive metals, 2. Substitution on metal halides for electronegative metals, 3. Catalyzed reaction with labile M-NR2 or M-C bonds for less active metals. The electronegative metals usually form unstable alkoxides that tend to polymerize rapidly to form [-M(OR)2-O-] n . Alkoxides are easily solubilized in alcohols. Alkoxide precursors must be kept free of water to avoid hydrolysis. Hydrolysis is the first step in the reaction of alkoxides to form gel networks. This is difficult because alkoxide solutions easily absorb water from the atmosphere. An example of an alkoxide solution prepared as a YSa2Cu307_ x superconducting precursor solution is given by Zheng et al. [38]. They used Y, Ba, and Cu isopropoxides in propanol, which was further modified by the addition of an organic acid chelating agent. The chelating agents stabilize the alkoxides with respect to hydrolysis. Polymer network solutions can also be formed from aqueous chemical systems. Aqueous metal chelates that have at least one additional carboxyl group as a reaction site can undergo polyesterification with a polyhydroxyl alcohol to form a network [39, 40]. Aqueous metal ions can also react with polyacrylic acid and be precipitated as a crosslinked polymer [41]. The polymerization mechanism and its rate are important factors in determining the molecular weight of the polymers and the density distribution of the microstructure formed.

FIGURE 8.17 (a) Transmission electron micrograph of aqueous alumina gel (x125,000). Versal alumina gel, photo courtesy of Kaiser Aluminum and Chemical Company. (b) Scanning electron micrograph of sol-gel 0.2065/xm average diameter silica spheres. From the work of Kovats and Ring, DC-EPFL, 1992. TM

344

Chapter 8 Other Ceramic Powder Fabrication Processes

The next stage in processing ceramic films from solutions is the development of structure. There are two types of structure development to consider: (1) the structure of the primary aggregate or polymer, and (2) the structure that develops upon gelation. It is possible to control both types of structure development through the chemistry of the precursor solutions. The control of structure is essential to the final microstructure of the dried gel. For aggregating systems, the aggregate structure is controlled by the rate of aggregation (see Table 6.4). Fast aggregation obtained with sols with high particle concentrations gives higher fractal dimension (i.e., 2.5) because aggregation is controlled by the incorporation of a particle into the aggregate structure. For slower aggregation, Brownian diffusion of an aggregate to another aggregate controls the aggregation rate giving lower fractal dimension (i.e., 1.8). Aggregation under shear lowers the fractal dimension of the aggregate due to bending and folding as shown in Table 6.4. For polymerizing systems, the polymer structure depends on the polymerization and crosslinking reaction kinetics. Metal alkoxides are easily hydrolyzed. Once hydrolyzed, they polymerize by polycondensation mechanisms. The hydrolysis and polycondensation mechanisms may be represented as [42] follows: Hydrolysis as M ( O R ) n + H20---) M ( O R ) n - I O H + R O H

Polycondensation as 2 M ( O R ) n _ I O H ----> ( R O ) n _ I M - O - M ( O R ) n _ 2 0 H + R O H 2 M ( O R ) n _ I O H ----> (RO)n_IM-O-M(OR)n_I + H20

where the R is an alkyl group. Because alkoxides are immiscible in water, it is necessary to put the alkoxides and water into a mutual solvent, such as an alcohol, for the hydrolysis reactions to occur. The relative rates of reaction for the hydrolysis and condensation dictate the structure and properties of an alkoxide gel. These reaction rates are schematically described in Figure 8.18 [43] for the example of a silicon ethoxide. In acidic solutions, hydrolysis is achieved by a bimolecular displacement mechanism that substitutes a hydronium ion (H § for an alkyl [44]. Under these conditions the hydrolysis is rapid compared to the condensation of the hydrolyzed monomers and promotes the development of larger and more linear molecules, as is described in Figure 8.19. Under basic conditions, hydrolysis occurs by nucleophilic substitution of hydroxyl ions (OH-) for alkyl groups [45]. Here the condensation is rapid relative to hydrolysis, promoting the precipitation of three-dimensional colloidal particles as shown in Figure 8.17(b) and 8.19.

345

8.6 Sol-Gel Synthesis FAST

Condensation

Hydrolysis

SLOW

0

2

4

6

8

10

Log[H +]

F I G U R E 8.18 Schematic comparison between the rates of hydrolysis and condensation for silicon ethoxide solutions. Redrawn from Schaeffer [43].

In addition to adjusting pH, the reaction rates may be varied by the temperature, water concentration, and the choice of alkoxide ligands. The steric interactions of the alcohol solvent chosen also influences the reaction rates. If the alcohol solvent is different from that of the alkoxide

~-,~~

i

L~~-,

j

It_

J

~

":." 9

":

9

9 . 9

9

": ee ~176

a,,o~,.IJ,"

,,, o~O;~-~ linear

polymers

random a g g r e g a t e s

ACID C A T A L Y Z E D

uniform p a r t i c l e s

BASE C A T A L Y Z E D

F I G U R E 8.19 Schematic diagram of the effect of pH on the growth and gelation of a silicon ethoxid. Redrawn from Gallagher and Ring [31].

346

Chapter 8

Other Ceramic Powder Fabrication Processes

ligand, transesterification will modify the reaction rates. For example [36], M(OEt) n + mPrOH--.

M(OEt)n_m(OPr) m + mEtOH.

Heterometallic complexes that contain an M ' - O - M " linkage can also be formed in solution, allowing multimetal ion gel networks to be formed; for example, a P b - T i alkoxide has been prepared by the reaction of titanium isopropoxide [Ti(OC3HT)4] with lead acetate [Pb(C2H302)2 92H20] in methoxyethanol [46]. This alkoxide complex is used in forming PbTiO3 precursor. Solutions may also be partially prehydrolyzed and mixed to control the reaction kinetics and avoid segregative precipitation. Prehydrolysis can help circumvent the problems of different reaction rates in multialkoxide systems. Another important sol-gel solution chemistry is that of the Pechini process [39]. Here metals are complexed in aqueous solution by citric acid. Then a polyalcohollike ethylene glycol is added and the solution heated to 150 to 250~ causing polyesterification of the metal chelates, which results in a gel. The gel is then thermally decomposed to produce a mixed oxide with very fine crystallites, typically 20 to 50 nm, clustered into aggregates. Over 100 different mixed oxide compositions have been prepared by this method. Gelation is initiated by (1) chemical reactions among precursor molecules, (2) changes in solution chemistry, or (3) evaporation of the solvent. But irrespective of the process by which it is attained, the gelation of a precursor solution begins with the formation of fractal agagregates of the discrete units present in solution (monomers, oligomers, polymers, or particles). Brinker and Scherer [30] have described the gelation process as one in which these aggregates grow to the point at which they impinge on one another and link, forming a continuous network. Though reactions may not be completed, the point at which a single aggregate cluster appears to encompass the entire solution is defined as the gel point. A model for gelation comes from the mathematics of percolation theory, where molecular groups are represented as points on a lattice, and bonds are formed randomly with probability P between nearest neighbor lattice sites [47]. This eventually leads to the formation of a random cluster that spans the lattice. The percolation threshold is analogous to the onset of gelation. At gelation, the chemical reactions may not be complete. The continuation of reactions, referred to as aging, usually reinforce~ of the network. The sol-gel transition in alkoxides is not reversible, as it often is with particulate gels. After the formation of the network, many characteristics of the structure are set but may still be altered during film or fiber deposition, drying, and thermal decomposition. Gels can be used for film formation, spinning ceramic

8.6 Sol-Gel Synthesis

347

fibers, atomizing into droplets that are thermally decomposed into powders, or dried into large monoliths.

8.6.2 F i l m F o r m a t i o n Gels are frequently used for ceramic film formation. Many aspects of the processing of films are common to all the deposition techniques. Schroeder [48] has outlined the conditions necessary for thin film formation. The solution must wet the substrate, it must remain stable with aging, it should have some tendency toward crystallization into a stable high-temperature phase, and for multiple layers the previous layers must be either insoluble or heat treated to make them insoluble before subsequent depositions. For a solution to wet a substrate the contact angle 0 between the surface of a drop of solution and the substrate must be less than 90 ~ The conditions for this are described by Young's equation: TLV cos 0 - Tsv -- TSL

(8.49)

where T is the surface tension for the liquid-vapor, LV, solid-vapor, SV, and solid-liquid, SL, interfaces, respectively. For wetting, the easiest variable to control is the surface tension of the solution. Alcohol solutions used with metal alkoxides wet better than aqueous solutions because they have lower surface tensions. The method of depositing a solution onto a substrate also affects the microstructure of the film. Flow due to gravitational and centrifugal draining of the solvent before drying causes shear, which deforms the network. The capillary forces at the meniscus between substrate and solution in dip coating also act to shear the network. The effect of shear is to restructure clusters and induce interpenetration or alignment of polymer [49]. Whether the network formation occurs in the early stages of deposition or later dictates how much of an effect the deposition technique has on the structure. The effect of the structure on the character of the final film is summarized in Figure 8.20. A solution of linear and randomly branched polymers, as shown in Figure 8.20, panel A, forms a highly stable and concentrated solution that gives a high and uniform density film. The linear polymers align themselves with the applied shear stress from spin or dip coating and pack into an even higher density structure. The fine pore structure created in gelation increases the capillary forces during solvent removal, which increases the density and reduces pore volume. These are important characteristics for protective and electronic coatings. Figure 8.20, panel B, shows the character of a film formed from aggregates in solution. Aggregates have a low and nonuniform density

348

Chapter 8

Other Ceramic Powder Fabrication Processes

FIGURE 8.20 Schematic diagram showing the relationship between gel structure and film structure. Redrawn from Gallagher and Ring [31]. A similar figure is given in Brinker [49].

distribution. To prevent gelation they must be used at a high dilution. The higher dilution decreases the amount of material being deposited at the substrate surface. During solvent evaporation aggregation occurs. This leads to larger pores, higher pore volumes, and lower densities. These are desirable film characteristics for catalysts, antireflective coatings, and sensor materials such as surface acoustic wave devices that respond to the absorptive nature of the film. In the gelled sols shown in Figure 8.20, panel C, the voids of the film are very large, because the packing of the gelled aggregates is improved only slightly by the shear forces during deposition. These films sinter well with low stress, giving high-density films. These are useful properties for making thick and protective films. Polymerizable systems, like the metal alkoxides, are interesting because it is possible to form all of these different film structures by simply manipulating the solution chemistry. Properties such as structure, viscosity, and concentration are easily controlled with polymers,

8.6 Sol-Gel Synthesis

349

and there are fewer problems with the precipitation or segregation of phases. One of the most important coating variables is solution rheology, which affects the film's thickness and evenness during deposition.

8.6.3 Gel Drying The drying of gels is best described as a function of the solvent weight loss in the following stages [30], which are the same as those for green body drying.

1. Constant rate period. Liquid flows to the surface to replace that lost to evaporation. This is the stage where most of the shrinkage and warping occurs, as a result of the capillary pressure differentials within the film. 2. Critical point. The gel matrix reaches a point where it can no longer shrink to release the solvent necessary to reduce the capillary pressure. At this point the liquid meniscus enters the matrix and drying from within the film begins. This is when a crack is most likely to appear in the material, especially those produced with alkoxides, where the M - O - M network is stiff. Cracking occurs more with thicker films than with thin films. 3. First falling rate period. The last layers of solvent are removed by flow along the pores walls to the surface. 4. Second falling rate period. The volatilized solvent is removed by the diffusion of vapor to the surface. The timing of these stages is intimately linked to the structure of the gel. There are several ways to reduce cracking during drying of gels. Chemicals may be added to the solutions that reduce drying stress. These are known as drying control chemical additives (DCCA) [50]. In general, these additives either reduce capillary pressure at the gel surface, reduce the solvent vapor pressure, or modify the pore size distribution. Another technique is to use hypercritical evacuation [51]. Above the critical temperature and pressure of the solvent, there is no longer a liquid-vapor interface and therefore no capillary pressure to cause shrinkage. Gels formed under these conditions are called aerogels. With the film constrained by a substrate, all shrinkage during drying is normal to the surface of the substrate. If thin enough (<0.5 micron), films tend not to crack upon drying. Lange [52] hypothesizes that, because crack growth is a function of the stress relieved per unit volume, a low volume of material, as in films, makes crack growth an unfavorable process [30].

350

Chapter 8

Other Ceramic Powder Fabrication Processes

8.6.4 Thermal Decomposition of Gels As the gel increases in temperature, weight is lost from the degradation of the gel. At low temperatures, up to about 150~ there is the loss of the solvent, physically adsorbed water, and weakly bound ligand molecules. The fine porosity of the matrix slows down the liberation of these molecules past the temperature when they normally volatilize. Above 250~ pyrolysis of the molecular network begins. The volatilization of the decomposition products of ligand groups is responsible for the weight loss. The structure decomposes first by cleavage of the weakest bonds [53]. The decomposition temperature varies with bond strength, crosslinking, and structure of the groups affected, as well as the atmosphere in which it takes place. Different chemical structures have different decomposition behavior. In the absence of oxygen, or presence of difficult to decompose molecular structures, residual organic groups form chars that remain in the structure until high temperatures. Above 800~ this residual organic is removed by decarbonation. Carbon is difficult to remove competely and the presence of carbon, and other chemical residues in the final piece is normal.

8.6.5 Gel Sintering With decomposition, the collapse of the skeleton begins and this eventually leads to the complete consolidation by sintering. Because gels are amorphous, the dominant sintering mechanism is viscous flow. Viscous flow is a much faster mechanism than the diffusion in crystals, and this contributes to lower sintering temperatures in gels as compared to sintering of mineral powders. Scherer [54] has studied viscous sintering of porous gel networks and concluded that the rate of contraction for the network E varies with 1

~ ~,svN~

(8.50)

where ~/sv is surface energy of the gel, V is the viscosity, and N is the number of pores per unit volume. As one decreases the pore size, keeping the density constant, the number per unit volume increases, and as a result the gel sinters faster. Another observation is that viscosity decreases with increasing temperature and increases with the degree of crosslinking. But if one heats the sample fast, the effects of crosslinking are minimized and sintering is enhanced. Though rapid heating may be detrimental in bulk materials because of trapped pores, this is not a problem of the same magnitude in thin films. It is also important to sinter to full density before crystallization occurs, which reduces the sintering rate.

8.7 Melt Solidification

8.7 MELT

351

SOLIDIFICATION

Large grained ceramic powders can be produced by the melt solidifcation processes. The most common example is that of electric arc melted A1203 ,* which is slowly cooled to crystallize the melt. To melt the A1203, a graphite electrode is placed in a steel container (volume ~ 1 m 3) filled with powdered A1203. An electric arc is passed between the electrode and the ceramic powder. After a sufficient current has passed the ceramic powder melts. The steel vessel is protected from the melt by a layer of solidified ceramic powder. After melting the electrode is removed, and the vessel is set aside to cool. Solidification then begins. In some cases the cooling rate is increased by passing cooling water over the surface of the steel vessel. Even under these conditions, the melt is slowly cooled, crystallizing rather large crystals (e.g., 0.1-5 cm). Ceramic and abrasive powders are produced from these crystals by grinding and classification. Molten solutions can also be used for crystallization. Fluxes are selected that have a high solubility of the to-be crystallized material. Borate fluxes are used for some oxide systems, sodium sulfide fluxes are used for sulfide systems, and molten metal fluxes are used for carbide and nitride systems. In both the melt and flux systems, the solubility is highly temperature dependent. The solubility in flux systems is not particularly well known except for a few systems. The general solubility behavior is like that in liquid phase precipitation, which was discussed in Chapter 6. In melt solidification, the supersaturation, S, is given by T

AG = - ~ AS d T = AI~ ( T - T o)

To

%

(8.51)

=

where T Ois the temperature where the solubility is equal to the actual concentration, T is the absoluted temperature, AS is the molar entropy, AH is the molar enthalpy change for the phase transformation, and Rg is the gas constant (a is the activity of the solute and a0 is the activity of the pure solute in a macroscopic crystal for flux systems). Solidification is discussed in the book Transport Phenomena in Metallurgy [9]. The freezing time for a volume, V, with a surface area, A, exposed to cooling is given by the time, t, required to transfer the heat of fusion: Q

=

Aqsurface

dt = p ' V H }

*A process practiced by Alu Swiss-Lonza, SA.

(8.52)

352

Chapter 8

Other Ceramic Powder Fabrication Processes

where p' is the density of the molten liquid. H} is the effective heat of fusion that represents the sum of the latent heat of fusion, HW, and the liquid's super heat, CpATsuperhea t . Cp is the heat capacity of the molten liquid. Considering the solution to the transient heat conduction problem for an infinite slab, the heat flux into an infinite mold, qsurface, is given by qsurface = - k ( T f -

T o)

k/rr~t

(8.53)

where TWis the fusion point of the molten liquid, T Ois the temperature of the mold, ~(= k/pCp) is the thermal diffusivity, and k is the thermal conductivity of the mold. Integrating we find that the solidification time for a shape of volume V and with an external surface area A is given by I__\o

t = C (v/~ \~/ where

c =

(TT=

1(1)

o)J koc

(8.54)

"

(8.55)

The size of the crystalites produced will depend on the nucleation rate and crystal growth rate at the solidification front. Both are controlled by the local supersaturation. The rate expressions for homogeneous nucleation (equation (6.15)) and heterogeneous nucleation (equation (6.27) and crystal growth (Table 6.2) are given in Chapter 6. For the flux systems, the melt is obtained by mixing powdered materials of the flux material and the to-be crystallized solute. This mixture is melted in a crucible and slowly cooled. The solubility of the solute decreases as the melt is cooled. This causes supersaturation of the solute, which nucleates and grows. The rate of cooling effects the supersaturation at the point of crystallization and dictates the nucleation and growth rates throughout crystallization. With slow cooling, the lowest nucleation and growth rates are observed, giving the largest crystals. For slowest cooling rates, the solubility in the flux will determine the crystal size. The more soluble the solute is the more material there is for crystallization into the few crystals nucleated.

8.8 S U M M A R Y This chapter has given information on several other methods of ceramic powder synthesis not easily classified into the chapters on solid, liquid, and gas phase synthesis. These methods include spray

8.8 Summary

353

drying, spray roasting, freeze drying, metalorganic decomposition, sol-gel synthesis, and melt and flux solidification. There are many similarities between spray drying, spray roasting, and freeze drying, which start with atomization of a liquid followed by drying (and thermal decomposition) or freezing. Sol-gel synthesis is particularly different than the precipitation because an aggregate (or polymer) network is formed. This gel can be dried (or spray dried) to form a ceramic powder. Use of sol-gel synthesis to make films and fibers were also superficially discussed. Melt and flux synthesis is used to produce the largest ceramic powders, which are often used for abrasives.

Problems 1. Determine the spray dried particle diameter resulting from the Rayleigh breakup of a jet 1 mm in diameter operating on a 0.001 molar doped iron nitrate solution of N i - A 1 - G a spinel ferrite (i.e., MFe204 M = Nio.5-Alo.3-Gao.2). Assume that the drying conditions are sufficient to evaporate the water but not thermally decompose the hydrated metal nitrates. 2. Determine the spray roasted particle diameter resulting from the Rayleigh breakup of a jet 0.1 mm in diameter operating on a 0.01 molar doped iron nitrate solution of N i - A 1 - G a spinel ferrite (i.e., MFe204M = Nio.5-A10.3-Gao.2). Assume that the roasting conditions are sufficient to thermally decompose the hydrated metal nitrates that result from spray drying to their respective individual oxides. 3. Determine the maximum drying time for an aqueous 1.5% volume solid A1203 gel droplet 100 t~m in diameter at 20~ in air at 100~ if the pores are like that shown in Figure 8.17(a). Assume that the size of the droplet does not change during drying. Data for water: Cp - 1 cal/gm/~ AHvap = 539.55 cal/gm, ~-/fusion - - 79.71 cal/gm, TBp 100~ p = 1.0 gm/cm 3, t~ = 0.01 poise. Data for air: M w = 29 gm/mole, Cp = 0.25 cal/gm/~ p = P M w / R g T , t~ = 2 ( M w k s T / N A v ) ~ 2, d i f f u s i ~ = 0.076 cm2/sec. 4. An aqueous gel (1 liter per min) is spray dried at 25~ in a countercurrent spray dryer. The droplets produced by atomization are 50 t~m in diameter and remain that size throughout drying. If the drying rate is limited by mass transfer in the boundary layer, what height of the spray dryer of cross-sectional area 1 m 2 is required if the drying air (100 liter per min) enters at 150~ and ~ 0% relative humidity. See problem 3 for the data. =

5. Determine the time to freeze a 10 t~m diameter droplet of water in dry air at 200 K. See problem 3 for the data.

354

Chapter 8

Other Ceramic Powder Fabrication Processes

6. Determine the time to dry a frozen water droplet 10 t~m diameter in dry air at 200 K. Use the shrinking core model. Is heat or mass transfer the rate determining step? See the data in problem 3. 7. Spherical particles of ZnSO4 of size 2.0 ftm are reacted in SO3 free air at 900~ A reaction takes place according to ZnSO 4 ~

ZnO(s) + SO3(g)

Assuming that the reaction proceeds by a shrinking core model and that the boundary layer does not present an important rate resistance, (a) Calculate the time needed for complete conversion of a particle and the relative resistance of product layer diffusion during this reaction. (b) Repeat the calculation for a particle of size 0.5 t~m. Data: Solid density 3.54 gm/cc, reaction rate constant, k r = 2 cm/sec, gas diffusion in ZnO layer, DAE = 0.08 cm2/sec.

References 1. Masters, K., "Spray Drying Handbook." Pitman Press, Bath, UK, 1979. 2. "AIChE Equipment Testing Procedure: Spray Dryers." AIChE, New York, 1988. 3. Marsh, G. B., Fenelli, A. J., Armor, J. N., and Zambri, P. M., Eur. Pat. Appl. 86 102830.6 (1986). 4. Beddow, J. K., "Particulate Science and Technology," p. 85. Chem. Publ. Co., New York, 1980. 5. Brodkey, R. S., "The Phenomena of Fluid Motions." Addison-Wesley, Reading, MA, 1967. 6. Castleman, R. A., J. Res. Natl. Bur. Stand. (U.S.) 6, 281,369 (1931). 7. Perry, R. H., and Chilton, C. H., "Perry's Chemical Engineering Handbook," 5th ed., McGraw-Hill, New York, 1973. 8. Weast, R. C., and Selby, S. M., "Handbook of Chemistry and Physics," 4th ed. CRC Press, Cleveland, OH, 1966. 9. Geiger, G. H., and Poirier, D. R., "Transport Phenomena in Metallurgy." AddisonWesley, Reading, MA, 1973. 10. Marshall, W. R., Jr., Chem. Eng. Prog., Monogr. Ser. 50, 2 (1954). 11. Fraser, et al., AIChE J. 8(5), 672 (1962). 12. Dombrowski, N. and Johns, W. R. Chem. Eng. Sci. 18, 203 (1963). 13. Duffee, J. A., and Marshall, W. R., Jr., Chem. Eng. Prog. 49(9), 480-486 (1953). 14. Lukasiewicz, S. J., J. Am. Ceram. Soc. 72(4), 617-624 (1989). 15. Ranz, W. E., and Marshall, W. R., Chem. Eng. Prog. 48, 173 (1952). 16. Castellan, G. W., "Physical Chemistry," p. 322. Addison-Wesley, Reading, MA, 1964. 17. Dirksen, J. A., and Ring, T. A., Chem. Eng. Sci. 46(10), 2389-2427 (1991). 18. Herman, H., Kona (Hirakata, Jpn.) 9, 187-199 (1991). 19. Knudsen, M., "The Kinetic Theory of Gases." Methuen, London, 1934. 20. Levenspiel, O., "Chemical Reactor Engineering," p. 384. Wiley, New York, 1972. 21. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena," p. 647. Wiley, New York, 1960. 22. Sroson, D. W., and Messing, G. L., Adv. Ceram. 21, 99-108 (1987).

8.8 Summary

355

23. Dubois, B., Ruffler, D., and Odier, P., J. Am. Ceram. Soc. (in press). 24. Ishizawa, H., Sakurai, O., Mizutani, N., and Kato, M., J. Am. Ceram. Soc. 65(10), 1399-1404 (1986). 25. Kagawa, M., Kikuchi, M., Syono, Y., and Nagae, T., J. Am. Ceram. Soc. 66(11), 751 (1983). 26. Kagawa, M., and Nagae, T., Sci. Report of the Res. Inst., Tohoku University, Sr. A, 3112], 216 (1983). 27. Kagawa, M., Ohta, H., Komatsu, H., and Syono, Y., Jap. J. App. Phys., 24[4], 477 (1985). 28. Ono, T., Kagawa, M., and Syono, Y., J. Plasma Chem. and Plasma Processing 712], 201 (1987). 29. Kong, P. C., and Pfender, E., J. Plasma Chem. Plasma Process. 9(1), (1986). 30. Brinker, C. J., and Scherer, G., "Sol-gel Science: The Physics and Chemistry of Solgel Processing." Academic Press, San Diego, CA, 1990. 31. Gallagher, D., and Ring, T. A., Chimia 43, 298-304 (1989). 32. Barbe, C. Gallagher, D., and Ring, T. A., Int. Symp. Mol. Level Des. Ceram., Nagoya, Japan, 1991, pp. 29-38 (1991). 33. Gallagher, D., Scanlan, F., and Ring, T. A., Proc. "Euromat '91," Cambridge, UK, European Materials Research Society, 731 (1991). 34. Adams, C. M., and Taylor, H. F., Trans. Am. Foundrymen's Soc. 65, 170-176 (1957). 35. Rigterink, M. D., Am. Ceram. Soc. Bull. 51, 158-161 (1972). 36. Colomban, P., Ceram. Int. 15, 23 (1989). 37. Okamura, H., and Bowen, H. K., Ceram. Int. 12, 161 (1986). 38. Zheng, H., Colby, M. A., and Mackenzie, J. D., Mater. Res. Soc. Symp. Proc. 121, 537 (1988). 39. Pechini, M. P., U.S. Pat. 3,330,697 (1967). 40. Lessing, P. A., Am. Ceram. Soc. Bull. 68, 1002 (1989). 41. Micheli, A. L., Ceram. Int. 15, 131 (1989). 42. Livage, J., and Henry, M., "Ultrastructure Processing of Advanced Ceramics" (J. D. Mackenzie and D. R. Ulrich, eds.), p. 183. Wiley, New York, 1988. 43. Schaeffer, D. W., Science 243, 1023 (1989). 44. Pohl, E. R., and Osterholtz, F. D., "Molecular Characterization of Composite Interfaces" (H. Ishida and G. Kumar, eds.), p. 157. Plenum, New York, 1985. 45. Pope, E. J. A., and Mackenzie, J. D., J. Non-Cryst. Solids 87, 185 (1986). 46. Blum, J. B., and Gurkovich, S. R., J. Mater. Sci. 20, 4479 (1985). 47. Stauffer, D., Coniglio, A., and Adams, M., Adv. Polym. Sci.: Polym. Networks 44, 103 (1982). 48. Schroeder, H., Phys. Thin Films 5, 87 (1969). 49. Brinker, C. J., Ceram. Eng. Sci. Proc. 9, 1103 (1988). 50. Hench, L. L., in "Science of Ceramic Chemical Processing" (L. L. Hench and D. R. Ulrich, eds.), p. 52. Wiley, New York, 1986. 51. Kistler, S. S., J. Phys. Chem. 36, 52 (1932). 52. Lange, F. F., in "Fracture Mechanics of Ceramics" (R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds.), Vol. 2, p. 599. Plenum, New York, 1974. 53. Burns, G. T., Angelotti, T. P., Hanneman, L. F., Chandra, G., and Moore, J. A., J. Mater. Sci. 22, 2609 (1987). 54. Scherer, G. W., J. Am. Ceram. Soc. 67, 709 (1984).

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PART

III CERAMIC PASTE FORMATIONM I S E - E N PATE The third part of this book describes the formation of a ceramic paste. This process involves the following steps: 1. wetting the ceramic powder surface by a solvent, 2. deagglomeration of the ceramic particles, 3. adsorption of something to prevent reagglomeration, and 4. colloidal stabilization. The first three topics are usually performed simultaneously in a single unit operation~powder mixing and blending with a solvent system. These three topics are discussed in Chapter 9. In Chapter 10, colloidal stabilization is discussed. There are two methods of colloid stabilization, electrostatic and steric. Electrostatic stabilization involves the adsorption of ions onto the surface of the ceramic powder, which forms a mobile double layer of counterions. During a collision between particles this charge cloud prevents coagulation of the ceramic powder. Steric stabilization involves the adsorption of a polymer onto the particles' surface, which prevents flocculation of the ceramic powders. The topics discussed in Chapter 10 effect the "shelf life" of a ceramic suspension; that is, the time that can elapse between when it is prepared and when it is used without deleterious effects due to agglomeration. In Chapter 11, colloidal properties of ceramic suspensions are discussed, which include settling, Brownian motion, and colligative properties of the suspension. Be-

358

Part III

Ceramic Paste Formation--Mise-En P(tte

cause ceramic suspensions are a mixture of powder, polymer, and solvent, very complex colloidal properties are observed. One of the most interesting colligative properties of a ceramic suspension is osmotic pressure. The osmotic pressure of a ceramic suspension results from the packing of the particles in the ceramic suspension and the concentration of salts, polymers, and surfactants in the liquid phase of the suspension. Suspension rheology, another colloidal property of a ceramic powder suspension, is discussed in Part IV, Chapter 12. Suspension rheology is dependent on the interparticle forces in the ceramic suspension and the packing of the particles. Suspension rheology is important for green body formation and is therefore in the next part of this book, "Green Body Formation."

9

Wetting, D e a g g l o m e r a t i o n, and Adsorption

9.1 O B J E C T I V E S To form a paste from a ceramic powder and a fluid, the following steps must be performed: 1. Wetting of the powder by liquid. 2. Deagglomeration of the powder. 3. Adsorption onto the powder surfaces to prevent reagglomeration. Wetting is controlled by the thermodynamics of the work of dispersion. Deagglomeration is typically performed by either ultrasonification or milling, although high-shear mixing is sometimes used. Milling was discussed in Chapter 4. Ultrasonification is discussed in this chapter. Adsorption of specific molecules or ions is usually necessary to prevent reagglomeration. The adsorbate molecules can include solvent molecules, ions, surfactant molecules, and polymers. The various laws governing the adsorption of these different soluble species are discussed. 359

360

Chapter 9

Wetting, Deagglomeration, and Adsorption

9.2 WETTING OF A P O W D E R B Y A LIQUID The material s u m m a r i z e d in this section was taken from excellent books by P a t t o n [1] and A d a m s o n [2]. The surface tension of the liquid is in great part responsible for wetting. The concept of a surface tension results from the unbalanced forces between molecules at the surface of a liquid. The surface tension of a liquid can be m e a s u r e d when a liquid (soap) film is attached to a wire frame as shown in Figure 9.1. The force required to increase the area of the film by an area 2Lx is equivalent to 2L times the surface tension, T. F = 2LT

(9.1)

The factor 2 in this equation is a result of the two surface layers being formed on either side of the liquid film. The work required to increase the area by 2Lx is given by the product of the force, 2LT, times the distance, x. W(work input) = F . x

= 2 L T "x

(9.2)

The free energy stored per unit area, E, is given by the ratio of the work, W, to the area, A, which is identically equal to the surface tension: W E-A-

2LT "x 2L-x - ~

(9.3)

For this reason, the surface tension has units of either energy per unit area or force per unit length. The surface tension of a liquid is m e a s u r e d in one of several ways: capillary rise, ring detachment, or drop weight. Each method of surface tension m e a s u r e m e n t is outlined in Figure 9.2. With capillary rise, the fluid is suspended by the surface

Contracting soap film on two sides, back and front, of a wire frame. The force necessary to create a new area, A, depends on the surface tension, %

F I G U R E 9.1

9.2 Wetting of a Powder by a Liquid

361

tension of the curved liquid interface. With the ring detachment and drop weight methods, the force at detachment or the drop weight at detachment is proportional to the surface tension of the liquid. Surface tensions of various organic liquids are tabulated in the appendix of this book. The surface tension of an organic liquid can be calculated [1, p. 234; 3, 4] from its chemical structure as follows: ~/LV =

P~ * M W ~ * ~ N j 9 j=l

(9.4)

Schematic diagram indicating various methods to measure the surface tension of a liquid: (a) drop weight, (b) ring detachment, and (c) capillary rise.

FIGURE 9.2

~62

Chapter 9 Wetting, Deagglomeration, and Adsorption TABLE 9.1

Parachor Values a for Organic Structures

Elements C O (OH or ether) 02 (ester or acids) N (nitrile) P F Br

4.8 20.0 60.05 29.15 37.7 25.7 68.0

H (to C) H (to O) O (carboxyl) N (amines) S C1 I

17.1 11.3 43.25 12.55 48.2 54.3 91.0

Organic structures Bonds Double Triple

Ring structures 23.2 46.6

Three member Four member Five member Six member

16.7 11.6 8.5 6.1

a Patton, T. C., "Paint Flow and Pigment Dispersion," 2nd ed., p. 234. Wiley (Interscience), New York, 1979. Other values are available in Hertzog, E. S., Ind. Eng. Chem. 36, 998 (1944) and Meissner, P., Chem. Eng. Prog. 45, 151 (1949 which are summarized in Perry, R. H., and Chilton, C. H., "Perry's Chemical Engineering Handbook," 5th ed., pp. 3-240. McGraw-Hill, New York, 1973. 5 Includes double (or triple) bond.

where Pl is the density, Mw is the molecular weight of the organic liquid, and Pj are the so-called parachor numbers, given in Table 9.1, of each part of the chemical structure, j, with Nj equal to the number of parts j in the chemical structure. If there are, for example, three double bonds in the chemical structure of a solvent, then Nj is 3 and Pj is 23.2. The surface tensions predicted by this parachor method are typically accurate to _ 1 erg/cm 2 for many organic liquids. At the boiling point, T~, of a liquid the surface tension can be estimated from Walden's rule [5]: AHvap(Tb)Pl(Tb)

~/LV =

364

(9.5)

where the enthalpy of vaporation, AHvap(Tb)is given in cal/mole and the liquid density, pl(Tb), is given in gm/cc for a surface tension in dynes/cm. For mixtures of liquids the surface tension may be estimated from known pure-component surface tensions, YLy-i, as follows:

9.2 Wetting of a Powder by a Liquid

~/LV-mix---- E "YLv-iXi i

363

(9.6)

where X i is the mole fraction of the various liquids in the mixture. This equation is most applicable for hydrocarbons at T < boiling point. The surface tension (or surface energy) of a solid is determined by the contact angle of the solid with various nonsolvating liquids. Upon extrapolation of the contact angle, 0 (given by Young's equation, which follows) to either 0 = 90 ~ where [6] ~/SL = 1 ~/LV or 0 = 0 ~ where [1, p. 222] ~/SL ~ ~/LV, the surface energy of the solid can be obtained. It should be noted that the surface energy of a solid will be effected by the adsorption of surfactants and equilibrium versus nonequilibrium conditions, the vapor phase and vapor adsorption, advancing versus receding contact angles, and the spreading of monomolecular films [7]. The wetting of the solid by a liquid is a contact angle phenomenon as shown in Figure 9.3. For a drop of liquid in contact with a solid, the vectorial balance of the surface tensions of the solid-vapor, solid-liquid, and liquid-vapor gives the well-known Young's equation [8, 9]: (9.7)

COS 0 ---- ~ / S Y - ~/SL ~/LV

where 0 is the contact angle between the solid and the liquid and ~/ij is the surface free energy or surface tension between i and j, where S = solid, L = liquid, and V = vapor. For wetting, the contact angle

........

Liquid

Solid Water

Oxide Water

Nonoxide Schematicof force balance for Young's equation with example of water wetting on an oxide ceramic and nonwetting on a nonoxide ceramic (i.e., carbides and nitrides).

FIGURE 9.3

364

Chapter 9

Wetting, Deagglomeration, a n d Adsorption

must approach 0, so that the liquid spreads over the solid surface easily. Nonwetting means the contact angle 0 is >90 ~ so that the liquid tends to ball up on the surface easily. For the case of a finite contact angle, the spreading coefficient, SL/S, is given by [2, p. 340]: S L / S -- 'YSV -

'YLV -

(9.8)

'YSL

When the spreading coefficient is positive, the liquid spreads over the surface of the solid. When the spreading coefficient is negative, the liquid does not spread.

P r o b l e m 9.1. S p r e a d i n g H 2 0 on S i O 2 Silica with a saturated layer o f - O H groups on its surface has a surface tension measured [10] by solubility considerations of 46 erg/ cm 2. Without this layer o f - O H groups on its surface it has a surface energy [7] of 880 erg/cm 2. Will water with its surface tension [11] of 72 erg/cm 2 spread on a flat surface of silica? S o l u t i o n The spreading coefficient, SL/S = T s v - TLV 880 - 72 - 46 = 762 erg/cm 2, is positive so spreading is spontaneous. The contact angle cannot be calculated because cos 0 > 1.0, but the value of cos 0 is still a valid estimate of the work of adhesion, penetration and spreading as discussed next. -

TSL

--

The spreading coefficient can also be used to distinguish whether one liquid, for instance, water, will spread on another liquid, for instance, a hydrocarbon. Here, values for the denser liquid replaces those of the solid in equation 9.8. Figure 9.4 shows [1, p. 216] the surface tension as a function of the number of carbon atoms in the hydrocarbon chain, the surface tension of water, and the spreading coefficient. For alkanes smaller than octane, we find a positive spreading coefficient on water, but for molecules larger t h a n octane, we find a negative spreading coefficient. Therefore, spreading occurs only for alkanes smaller than octane. For spreading on solids and liquids, the values of TSL and TLV should be made as small as possible. From the practical point of view, this is best done by adding a surfactant to the liquid phase so that it will absorb at both the solid-liquid and the liquid-vapor interfaces, lowering both their surface tensions. The surface energetics of wetting can be broken down into three different processes [12], adhesion, penetration, and spreading. These steps are schematically shown in Table 9.2. For a smooth solid surface with complete liquid solid contact, the work of adhesion, penetration, and spreading are given by the following equations [13]"

9.2 Wetting of a Powder by a Liquid

36~

F I G U R E 9.4

Conditions for the surface tension which give s p r e a d i n g of aliphatic hydrocarbons on water. D a t a from P a t t o n [1, p. 216].

T A B L E 9.2 S c h e m a t i c I l l u s t r a t i o n of the T h r e e I n t e r m e d i a t e Steps in the Dispersion of a Rough Cube a

Wetting process

Initial state

Final state

[]

W o r k of adhesion (one face)

. . . . . . . [] . . . . . . . . . .

WA = --i(~/sy- ~'SL) -- ~/LV ........

Liquid . . . . . .

. . . . . . Liquid . . . . . . . .

W o r k of p e n e t r a t i o n (four faces)

Wp = - 4 i ( ~ / s y - ~SL)

. . . . . . . [ ]. . . . . . . . . .

.......

. . . . . . . Liquid . . . . . . .

. . . . . Liquid . . . . . . . . .

.........

[] . . . . . . . .

........

Liquid . . . . . .

. . . . . . . Liquid . . . . . . . . . . . . . . . [] . . . . . . . . .

[] . . . . . . . . . .

W o r k of s p r e a d i n g (one face)

Ws = - i ( ~ / s v - ~/SL) + ~/LV

[]

Total: W o r k of dispersion (six faces)

w , = w~ + we+ ws = - 6 i ( ~ / s y - ~/SL)

........

Liquid . . . . . .

. . . . . . . Liquid . . . . . . . . . . . . . . . [] . . . . . . . . .

a P a t t o n , T. C., " P a i n t Flow a n d P i g m e n t Dispersion," p. 240. Wiley (Interscience), New York, 1979. Note different sign convention used here. h G (or W) negative corresponds to spontaneous.

366

Chapter 9

T A B L E 9.3

Wetting, D e a g g l o m e r a t i o n , a n d A d s o r p t i o n

Spontaneous Wetting Conditions a

Surface tension

Tsv > TLV

Contact angle for smooth surface

All 0

N a t u r e of surface: Adhesion Penetration Spreading

Smooth yes yes b

Tsv < TLV

0 < 90 ~

Rough yes yes c

Smooth yes yes no

Rough yes yes d

0 > 90 ~

Smooth yes no no

Rough e

no no

a Taken from Patton, T. C., "Paint Flow and Pigment Dispersion," 2nd ed., p. 222. Wiley (Interscience), New York, 1979. Copyright 9 1979 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc. b Yes for i cos 0 < - 1 . 0 ; no for i cos 0 > - 1 . 0 . c Yes for (Ysy - TSL) > TLV; no for (TSV -- TSL) < TLV. d Yes for i(Tsy - TSL) > TLV; no for i(Ysy - TSL) < TLV. e Yes for i cos 0 > 1.0; no for i cos 0 < 1.0.

Adhesion (one face contact) Penetration Spreading (over one face)

WA = - ( T s v - TSL) -- TLV Wp = - ( T s v - TSL) Ws = - S L / s = - ( T s v - TSL) + TLV

(9.9) (9.10) (9.11)

Each of these fundamental steps is described by a thermodynamic expression for the work associated with that step and must be negative for wetting to occur. When W is positive then work must be done on the system for the process to take place. As a result, spontaneous wetting will occur under two conditions, Tsy > TLV and Tsy < TLV. For Tsv > TLV, TSL must also be less than Tsv. This means that the term in parentheses, ( T s v - TSL), in the predeeding expressions will always be positive, indicating that the processes of adhesion and penetration are spontaneous although spreading may or may not occur. When Tsv < TLV, no such generalizations can be made. In the past, much confusion has occurred as a result of the failure to differentiate these two wetting regimes. These two wetting regimes merge when Tsv = TLV. Different concepts must be applied to obtain wetting on either side of this critical point, as shown in Table 9.3.

9.2.1 Smooth versus Rough Surfaces The degree of roughness or irregularity, i, is conventionally measured by dividing the actual true or contour area, A~, of the surface by its projected area or outside geometric area, Ag" Ai

i = --

(9.12)

9.2 Wetting of a Powder by a L i q u i d

367

In general, for liquids, which are normally smooth, the value of i is equal to 1. For solids, i is always greater t h a n 1. The effect of surface roughness on adhesion, penetration and spreading for a rough cube is shown in Table 9.3. The total work for the sum of all of these steps is referred to as the work of dispersion. When the work of dispersion is negative, dispersion is spontaneous. For Tsv > ~ L V , ~/SL must again be less t h a n Tsv. This means that the term in parentheses, (Tsv - TSL), in the previous expressions will always be positive, indicating that the processes of adhesion and penetration are spontaneous (i.e., WA and Wp are negative) although spreading may or may not occur. Whether spreading occurs depends on whether i ( T s y - ~/SL) is larger than ~/LV" Thus, rougher surfaces (i.e., i > 1.0) promote spreading. For the second wetting regime, when Tsy < TLV, no such generalizations can be made. However by using the liquid-solid contact angle [1, p. 218], TLV COS 0 i = ( ~ / S V -

TSL) i

(9.13)

for a rough surface with complete contact between the solid and the liquid, it is possible to calculate wetting behavior. Comparing this equation to Young's equation for a smooth surface, assuming complete wetting, gives a simple ratio for the surface roughness, i, i-

cos Oi cos 0

(9.14)

When the contact angle is 0 ~ the cosine function has a maximum value of 1.00. Considering the preceding equation, it appears that a seemingly unreal situation arises when cos 0 i takes on a value greater t h a n 1.0 (cos 0~ = i cos 0 > 1.0). However, when this situation occurs, it should be interpreted as a point where the solid surface, by virtue of its roughness, has gone beyond the zero contact angle and the wetting regime has been shifted from Tsv < T i v to Tsv > TLV. Angles where cos 0 > 1.0 do not exist but the value of i cos 0 is still valid for estimating the work of adhesion, penetration, and spreading as follows [1, p. 219]: A d h e s i o n (one face contact) Penetration S p r e a d i n g (over one face)

WA = -T/v(COS Oi + 1) = --TLV(i cos 0 + 1) Wp - --~Lv(cos 0i) -- --Tiv(i cos 0) Ws = -TLy(cos Oi + 1) = --TLV(i cos 0 -- 1)

(9.15) (9.16) (9.17)

These equations are most useful for predicting wetting for the regime Tsy < TLV. When the contact angle is less t h a n 90 ~spontaneous adhesion

and penetration always occur. For contact angles more t h a n 90 ~, spontaneous penetration does not occur and adhesion becomes more probable as roughness increases. For contact angles more t h a n 90 ~, retraction (dewetting) occurs. For contact angles less t h a n 90 ~ spontaneous spreading occurs especially for small contact angles where spreading

368

Chapter 9

Wetting, Deagglomeration, and Adsorption

is enhanced by surface roughness. Table 9.3 is a summary of the conditions that lead to spontaneous wetting.

Problem 9.2. Wetting of a Rough Solid Surface A 45 ~ contact angle is given for a liquid on a smooth solid surface. What roughness, i, must the solid surface have to induce spontaneous spreading? S o l u t i o n Setting Ws = --SL/s = TLV(i COS 0 -- 1) = 0 and solving for i gives a value of 1.41.

9.2.2 P a r t i a l Wetting of a Solid Sometimes only a fraction of the total surface area is wet by the liquid. The actual area wet is given by the symbol a. Thus the ratio (a/i) must be multiplied by various terms in Table 9.2. For rough surfaces with partial liquid-solid contact, the three steps of dispersion are shown in Figure 9.5. For this partially wet surface, the contact angle is given by [1, p. 218]

TLV (COS 0i + (i --~a)) = (Tsv -

TSL) a

(9.18)

Partial contact of the liquid to the solid surface further promotes spreading when Tsy > T L V "

9.2.3 Internal Wetting With particles that are formed by an agglomeration process, there is internal surface area which can be wet by the solvent. A simplified case of internal porosity is that of a horizontal capillary where gravity can be neglected. The rate of penetration of a liquid into the tube, dl/ dt, is given by the Washburn equation [14]:

dl dt

:/r[Tsy- TSL] ~/

4tv

(9.19)

where l is the depth of penetration in time t, r is the radius of the capillary, and V is the viscosity of the liquid. For the internal wetting of a porous agglomerated particle, the radius of the capillary may be replaced by the mean radius of curvature, k, for the pores in the particle [15]. Young's equation can be used in conjunction with this equation to simplify the term [Tsv - T S L ] : T L V COS 0.

9.2 Wetting of a Powder by a Liquid

369

FIGURE 9.5 Adhesion, penetration, and spreading of a liquid on a rough surface with partial solid-liquid contact. The left side and the right side of each diagram represents the conditions before and after wetting takes place, respectively. This figure is similar to one from Patton [1, p. 211] but uses the opposite sign convention corresponding to spontaneous being equivalent to AG (or W) negative.

370

Chapter 9 Wetting, Deagglomeration, and Adsorption

9.2.4 Heat of Wetting When a powder is immersed in a liquid and wetting takes place, heat is given off. The heat of wetting for various powder surfaces is given in Table 9.4. Let us start by analyzing the heat of mixing between two liquids and then alter the resulting expressions for a solid and a liquid. Drago [16, 17] studied the theory behind the heat of mixing between two liquids, where this heat arises from Lewis acid-base interactions. Drago tabulated E and C constants for a large number of organic liquids which correspond to the covalent, C, and electrostatic, E, components of the Lewis acid-base interactions. Drago E and C values for many solvents are listed in Table 9.5 and in the appendix of this book. The Lewis interactional energy between an acidic liquid and a basic liquid, A H ab, c a n be calculated by (9.20)

_ A H a b -- EAEB + GAGB

where EA and EB are constants representing the electrostatic part of the Lewis acid solvent and the electrostatic part of the Lewis base solvent, respectively; and CA and CB are constants representing the covalent part of the Lewis acid solvent and the covalent part of the Lewis base solvent, respectively. Table 9.5 is a listing of the E and C values given by Drago and others [18, 19] for a series of solvents common to the ceramic literature. These E and C constants can predict within 10% the heat of mixing of two solvents if Lewis acid-base interactions are the sole interactions operating.

TABLE 9.4

Heat of Wetting (cal/cm 2) at 25~ a

Solid

H20

C2H50H

n-Butylamine

CCl4

TiO2 AI203 SiO2 BaSO4 Graphon Teflon 6

5505 400-600 d 400-600 490 32 6

400

330

240 c 270 220

110

106

n-CGH14 135 100 100 103 47 e

a Data taken from Adamson, A. W., "Physical Chemistry of Surfaces," 3rd ed., p. 335. Wiley (Interscience), New York, 1976, if not otherwise referenced. 5 Zettlemoyer, A. C., Ind. Eng. Chem. 57, 27 (1965). c Harkins, W. D., "The Physical Chemistry of Surfaces." Reinhold, New York, 1952. d Wade, W. H., and Hackerman, N., Adv. Chem. Ser. 43, 222 (1964). e Whalen, J. W., and Wade, W. H., J. Colloid Interface Sci. 24, 372 (1967).

9.2 Wetting of a Powder by a Liquid TABLE 9.5

3 71

Drago E and C Values for Various Sovlents a

Drago E and C values (kcal/mole) 1/2 Solvent Hydrocarbons 1. hexane 2. toluene Chlorides 3. methylene chloride 4. chloroform 5. carbon tetrachloride 6. 1,2-dichloroethane 7. trichloroethylene 8. chlorobenzene Cyanide 9. acetonitrile Ethers 10. isopenthyl ether 11. tetrahydrofuran 12. dioxane Ketones 13. acetone 14. 2-butanone 15.2-heptanone Esters 16. ethyl formate 17. ethyl acetate Aldehyde 18. benzaldehyde Alcohols 19. methyl alcohol 20. ethyl alcohol 21. n-propyl alcohol 22.2-propyl alcohol 23.2-furfuryl alcohol 24. benzyl alcohol 25. n-octyl alcohol 26. ethylene glycol Amine 27. propylamine Carboxylic acid 28. propionic acid 29. n-octanoic acid 30. oleic acid

CA

CB

EA

1.91

EB

0.087 3.40 3.31

0.02 0.150

1.34

0.886

3.19 4.27 2.38

1.11 0.978 1.09

2.33 2.38

0.987 1.09

1.74

0.975

1.12

0.975

0.451

3.88

0.446

4.17

6.1

1.4

a E and C data taken from several sources as tabulated by Okuyama, M., Garvey, G., Ring, T. A., and Haggerty, J. S., J. Am. Ceram. Soc. 72(10), 1918-1924 (1989). Reprinted by permission of the American Ceramic Socity.

372

Chapter 9

Wetting, Deagglomeration, and Adsorption

Fowkes [20] extended the Drago theory to the heat of wetting of a solid surface. Because Lewis acid and base adducts form between solid surfaces and solvents in much the same way as between two liquids in a solution, Fowkes contends that surfaces could be treated as a continuum with unique interactional characteristics described in a way similar to that of Drago. By selecting nonassociating solvents that do not participate in covalent reactions or hydrogen bonding with the surface, Fowkes contends that it is possible to isolate the Lewis acid-base interactions between the solvent and the surface. He states that the contribution to the heat of wetting, h H w , as measured by calorimetry are limited to dispersional energy, A H d, and the energy of the Lewis adduct formation, A H ab. AH w -

(9.21)

A H ab + A H d

where A H d is calculated from the solvent solid-vapor and liquid-vapor surface tensions as follows: (9.22)

A H d = 2(TSVTLV) 1/2

Combining equations 9.20 and 9.21 and 9.22, we arrive at an expression for calculating the Lewis interaction energy. Using heats of wetting measured by microcalorimetry Okuyama et al. [21], characterized the powder surfaces for silica, silicon carbide, and oxidized silicon carbide surfaces. The results of their analysis are given in Table 9.6 in terms of Drago E and C values and the values of the solid surface free energy used for the calculation. In Table 9.6 we see that the pure silicon carbide surface is basic, although the oxidized silicon carbide surface is acidic like that of silica. With these E and C values Okuyama et al. [21] were able to predict that the silica and oxidized silicon carbide surfaces are best wet by

TABLE 9.6

Drago E and C Values (kcal/mole) 1/2 for Ceramic Powders

Drago E and C values a (kcal/mole) 1/2 b

TSV

Powder

EA

CA

Es

CB

(erg/cm 2)

SiO2(Hi--Silm233) Laser SiC OxuSiC

4.65 m 1.45

0.66 m 0.43

m 1.85 ~

m 1.47 ~

300 399 330

a Okuyama, M., Garvey, G., Ring, T. A., and Haggerty, J. S., J. Am. Ceram. Soc. 72(10), 1918-1924 (1989). Reprinted by permission of the American Ceramic Society. b Girifalco, L. A., and Good, R. J., J. Phys. Chem. 61, 904-909 (1957); Good, R. J., Surf. Colloid Sci. 2, 1 (1978); J. Colloid Interface Sci. 59(3), 398-419 (1977).

9.2 Wetting of a Powder by a Liquid

373

basic solvents and the pure silicon carbide was best wet by acidic solvents. With this good wetting, higher density ceramic green bodies were produced [21] by gravity settling, centrifugal settling, and colloidal pressing.

P r o b l e m 9.3. S o l v e n t S e l e c t i o n Using the Drago E B and CB values for laser SiC, chose two solvents from Table 9.5 which will give a large enthalpy of wetting. S o l u t i o n The laser SiC surface is basic and will give the highest enthalpy of wetting if acidic solvents are used with high values of EA and CA. The solvents with the largest EA and CA values in Table 9.5 are two of the alcohols listed: ethyl alcohol and benzyl alcohol. Unfortunately microcalorimetry experiments for the heat of wetting are difficult to perform due to the care that must be taken to keep the powder surface free from adsorbed impurities. As a result an approximate method based on an infrared band shift caused by the interaction of the solvent with the solid surface has been developed. Drago often used spectroscopic shifts, hV~H, of the OH stretching frequency of phenol to determine heats of mixing of bases with phenol. His equation is given as - A H ab-- 3.08 kcal/mole + (0.0103 kcal/mole cm-1)Av~

(9.23)

In these studies he used concentrations of phenol less than 0.02M to avoid association and added an excess of base to obtain the acid-base complex. Fowkes [20] has focused on the spectral shifts of the carbonyl ( C ~ O ) stretch frequency of esters and ketones adsorbed from polymers onto silica fillers. He has found that the spectral shifts of carbonyl stretch has two contributions, one due to dispersion interactions, hvd=o and the other due to acid-base interactions, hV~bo . The two shifts result in the following equations for the heat of mixing, A H ab, and the surface tensions, T" A H ab = (0.2381 kcal/mole c m -1) A r cab _ O

-AT d = (1.429 mJ/m2/cm -1) hvd=o

(9.24) (9.25)

For systems in which hydrogen bonding occurs, the enthalpy of solvent mixing given by Drago and the enthalpy of wetting given by Fowkes is often in error. The hydrogen bond index (HBI) was developed by Gordy [22] as a method to put on a relative scale the importance of hydrogen bonding in the mixing of solvents. Gordy's hydrogen bond index is a scale between 0 (benzene) and 39 (water). For a particular solvent, the tendency to

374

Chapter 9 Wetting, Deagglomeration, and Adsorption

hydrogen bond is measured by the shift of the infrared vibrational frequency of the oxygen deuterium band (wavelength 3.6-4.35 ftm) in a test molecule (CH3OD) dissolved in the solvent. A 10 wave number shift equals 1 hydrogen bond index unit. Gordy showed that organic solvents largely proton acceptors follow the order: Amines > Ethers > Keytones > Esters > Nitro compounds Values of the hydrogen bond index for many common organic solvents are given in the appendix of this book. The hydrogen bond index has been used to determine if a particular solvent is good for wetting and dispersing a particular solid by Davies and Karuhn [23], Gallager and Aksay [24], Okuyama et al. [21], and Itoh et al. [25]. In addition, other techniques have been used to determine if a particular solvent is good for wetting and dispersing a particular solid. Solid isoelectric point, IEP, minus solvent pKa has been used by Bolger [18] to correlate good solvents for powder dispersion. Dipole moment of solvents has been used by Okuyama et al. [21] to correlate good solvents with powder dispersion. Dielectric constant of the solvent has been found by Mizuta et al. [26] to correlate well with dispersion stability. This approach suggests that matching the Hamaker constants for the solvent and the solid is responsible for colloidal stability of the dispersion as will be discussed in Chapter 10. As a result the discussion in the last two paragraphs, selection of a solvent for wetting and dispersion of a ceramic powder is a problem that does not have one unique solution or even one approach. A particular approach in choosing an appropriate solvent for wetting and dispersion will be more successful than another depending on how well the approach accounts for all the interactions between solid and solvent present in the various experimental systems.

9.3 D E A G G L O M E R A T I O N Deagglomeration can take place by two methods: comminution and ultrasonification. Comminution is the subject of Chapter 4, so it will not be discussed in detail here. In the comminution of aggregates, the population balance can be used to predict the size distribution as a function of time in a batch mill or as a function of mean residence time in a continuous mill. Aggregates have the same type of birth and death functions for particle breakage as polycrystalline particles but the rate constants are much higher and the size selectivities for aggregates are different than those for the comminution of crystalline materials. The

9.3 Deagglomeration

375

use of surfactants in the liquid medium of the grinding medium has been shown to increase the comminution rate constant drastically. Rehbinder [27] has demonstrated that the fine grinding of solids to create new interfaces is facilitated considerably by the adsorption of surface-active agents at structural defects in the surface. DiBenedetto [28] has suggested that wetting lowers the fracture energy of the solid. A theory of such spontaneous dispersion has been proposed by Rehbinder [29] and the weakening of a solid by the adsorbed materials including surface-active agents is referred to as the Rehbinder effect.

9.3.1 Ultrasonification Ultrasonics are most often utilized for powder deagglomeration in the laboratory. This technique is frequently used with submicron-sized powders which are hard to disperse by other methods. When a liquid is exposed to progressively higher ultrasonic powder, small vapor bubbles start to appear above a certain threshold energy [30]. These bubbles, called cavities, have a strong influence on the neighboring material. Beyond the threshold, the number of cavities increases rapidly with increased ultrasonic power. Cavities form everywhere within the ultrasonic field and collapse violently after a short lifetime, on the order of 2 tzsec. When cavities collapse, they produce locally very high velocity jets in the neighborhood of 100 m/sec and pressure gradients [31] of 20 GPa/cm. This collapse has been monitored in silicone oil by sonoluminescence spectra and shown to give an effective cavitation temperature [32] of 5075 +- 156 K. The resulting mechanical forces on the aggregated particles are extremely strong and continue as long as the ultrasonic power is above the threshold value for cavitation. These hydrodynamic forces are strong enough to break [33] apart weakly bonded particles, such as those joined by Van der Waals (VdW) forces. The distances over which these VdW forces are effective are fairly short [33], on the order of 10 nm. Figure 9.6 [34] shows the mean diameter of SiC powder after 5 min exposure to an ultrasonic field at different power levels. The threshold value for cavitation at 25 W is easily seen. The fundamental description of the effects of ultrasonication on dispersed agglomerated particles is extraordinarily complex [35]. The forces on the particles result from these short-lived jets having dimensions that are typically larger than the agglomerated particles and cavity number densities that are generally much lower than the number densities of agglomerated particles. In addition, the strength of the agglomerates can be expected to vary between individual particles and with time, as the primary particles are broken from the initial agglomerates and reattached to form new, weaker agglomerates. A simple heu-

376

Chapter 9 Wetting, Deagglomeration, and Adsorption

300

250 E e--

rr" hA

hA 200123 Z < hA

150u

I00

0

I

20

I

40

I

60

,

80

POWER L E V E L ( W )

FIGURE 9.6 Aggregate mean diameter after 5 min ultrasonic irradiation as a function of input powder level. The powder type is L 0 0 6 S i C and the solvent is 2-propyl alcohol. Taken from Aoki et al. [34]. Reprinted by permission of the American Ceramic Society.

ristic model of the ultrasonic dispersion process was developed by Aoki et al. [34], based on the interaction probability between agglomerates of primary particles and ultrasonically induced cavities. This model has two major components: 1. The probability of deagglomeration (klNcNp) 2. The probability of agglomeration (k2N~). The rate at which the aggregate number density, Np, changes can be expressed as dNp - k l N c N p - k2N~ dt

(9.26)

Using a highly simplified agglomerate particle shape, the relation between the number of agglomerated particles formed by a combination of primary particle of diameter, dR, is given by d~Np = d~No

(9.27)

9.3 Deagglomeration

377

where do is the diameter of the initial agglomerate particle and No is the initial number density. Values of n ranging from 1 to 3 are possible depending on the structure of the agglomerate as follows: n = 1

corresponding to unbranched linear agglomerates with the conservation of total length, corresponding to arbitrarily shaped particles that fracture with a conservation of total area, and corresponding to arbitrarily shaped particles that fracture with a conservation of total volume.

n = 2 n = 3

As shown in Figure 9.7, Aoki et al. [34] were able to explain why the mean diameter of the agglomerates first decreased and then increased, with increased exposure time to the ultrasonic radiation at power levels above that necessary for cavity formation. These results show the competitive nature of the two processes (deagglomeration and agglomeration) taking place in the ultrasonic bath. This increase in particle size is real in that observations of the particle size without

400 --'0--"

E

Theoretical behavior with ultrasonic

3O0

Experimental data

E:

without u l t r a s o n i c

rr LI.I I--LLI

< a

Experimental data with ultrasonic

[]

200

o

go

z

o

w 0

I00

0

0

0

0

I

30

0

I

60

90 TIME (rain) FIGURE 9.7 Agglomerate mean diameter as a function of ultrasonic irradiation time. The primary particle size of the type L006 SiC powder is 100 nm. The input power is 40 W and the solvent is 2-propyl alcohol. Taken from Aoki et al. [34]. Reprinted by permission of the American Ceramic Society.

78

Chapter 9 Wetting, Deagglomeration, and Adsorption 35

o 50-

o v

>" 2 5 I-(Jr) Z W IZ 2O w

/

w

,/

o G009 o

v B082 o

v

I0

0

I

I

I

20

40

60

POWER

13010

,~ B 0 6 4

80

LEVEL (W)

FIGURE 9.8 Centrifuged sediment densities (3000 G for I hr) of ultrasonically treated powders as a function of power level. Dispersions were treated for 5 min. The powders were dispersed in Oloa-hexane. The Oloa content was 20% of the powder weight. All SiC powders B064, B082, and G009 are similar to one another. Taken from Aoki et al. [34]. Reprinted by permission of the American Ceramic Society.

ultrasonication remain constant, but with the ultrasonication the size increases at long exposure times. The settled density of SiC powders treated with 5 min of ultrasonic energy at various power levels is as shown in Figure 9.8. Above the critical power level for cavitation, the settled density increases to a maximum value then decreases in correspondence to the mean agglomerate diameter decreasing and increasing. Therefore, it is necessary with each type of agglomerated powder and each power setting on the ultrasonic equipment to establish the time where the minimum diameter is observed for best deagglomeration. This corresponds to the optimum time for ultrasonication, allowing the production of the greatest number of deagglomerated particles. In the liquid medium during deagglomeration, it is important to have something that will adsorb on the freshly broken surface to prevent reagglomeration. Adsorption is the next subject to be discussed. Once the particles are deagglomerated they will undergo Brownian diffusion and sedimentation, which are discussed in Chapter 10.

9.4 Adsorption onto Powder Surfaces 9.4 A D S O R P T I O N

ONTO

POWDER

379

SURFACES

During adsorption the local concentration of molecules in the neighborhood of a surface differs from that of the bulk phase. Figure 9.9 shows this enhanced concentration at the solid-liquid interface. At this interface, there is a surface excess concentration of surface active molecules. This surface excess corresponds to the shaded area in Figure 9.9. Dividing this excess number of moles of i at the surface, n [, by the area of the surface, we obtain the surface excess concentration, F~, of component i given by F~ = n___~ i A

(9.28)

Historically, the first derivation (by Gibbs) of an adsorption isotherm was that for the liquid-vapor interface. This derivation is presented next to put in place the nomenclature used in adsorption on both liquid-vapor and solid-liquid interfaces. A derivation (by Langmuir) for adsorption at the solid-liquid interface is presented after that for the liquid-vapor interface. Adsorption at the liquid-vapor interface is

FIGURE 9.9 Concentrationprofile in a liquid mixture in the vicinity of the solid-liquid interface. Z is the direction perpendicular to the interface. The shaded area is the surface excess concentration.

380

Chapter 9 Wetting, Deagglomeration, and Adsorption

important in the processing of ceramics because this adsorption controls the surface tension of the liquid. The removal of bubbles from a liquid and from a ceramic slip is also controlled by the surface tension of the liquid. In addition, the wetting of ceramic powders by a liquid is controlled by the liquid surface tension as we have seen in Section 9.2.

9.4.1 Gibb's Adsorption Isotherm for the Liquid-Vapor Interface With a liquid-vapor interface, Gibbs [36] has developed a thermodynamic treatment of the variation of surface tension with composition. This derivation comes from the book Physical Chemistry of Surfaces by Adamson [2, p. 340]. This derivation sets the stage for adsorption at the solid-liquid interface, which will be discussed next. In two-component liquid the change in the liquid-vapor surface tension is given by

dT = - F [ dt.t i - F~ d~2

(9.29)

where tL~ is the chemical potential of the i component, and F[ is the surface excess concentration at any arbitrary plane near the surface. When the two phases, gas and liquid, are in contact, component 1, the solvent, is present in large excess compared to component 2, the surfactant. In accordance with the Gibb's assumption, we choose a plane where the surface excess concentration of the solvent is equal to zero (F~ = 0) so that the changing surface tension is given only by the second term in the preceding equation. Because the chemical potential of the surfactant is given by

P~2 = ~o + RT In a2

(9.30)

where a2 is the solution activity of the surfactant and tL0 is the chemical potential of the component 2, surfactant, at infinite dilution. We can take the derivative with respect to the chemical potential and insert the result into the proceeding equation to obtain d T = -F(21) RT d (ln a2)

(9.31)

here F~ has been replaced by F(2~)to indicate that the Gibbs assumption F~) = 0) has been made. The above equation can be arranged to give [36]: F(21)=-1( dT )SA R-T d (~na2)

(9.32)

where SA is the liquid-gas surface area, which is held constant.

381

9.4 Adsorption onto Powder Surfaces

For dilute solutions, we can replace the activity of the surfactant a2 with its concentration, thereby giving us F~I)=

RT

For the liquid vapor interface, the surface tension is easily measured as a function of the concentration as shown in Figure 9.10. The preceding equation can be used to determine the surface excess concentration of surfactant as a function of the surfactant concentration if the surface tension of the solution as a function of surfactant concentration is known. For dilute aqueous solutions of organic substrates, the semiempirical equation for the surface tension, 7, of a solution of concentration C2,

---7=1- B In (1 + ~__~2)

(9.34)

To

has been used [37], where 7o is the surface tension of the solvent (water in this case), A and B are constants. This equation is equivalent to the

lOO

80 A

E

i

O

-

Na dodecyl (lauryl) sulfate

--

Nonyl phenoxypolyoxyethylene

-

Na dioctylsulfosuccinate

--

Fluoro surfactant (Zonyl FSC)

m o

60 0 w o

40

2o "1

ot 0.00

I

I

0.02

0.04

9

I

0.06

Concentration

9

I

0.08

9

I

0.10

9

012

(%W)

FIGURE 9.10 Surface tension for aqueous solutions of four surfactants. Data taken from Patton [1, p. 285] and Perry and Chilton [4].

382

Chapter 9 Wetting, Deagglomeration, and Adsorption

Langmuir adsorption isotherm corresponding to F(21) =

aC2 1 + bC2

(9.35)

where a and b are different constants. The Langmuir adsorption isotherm is now derived for adsorption at the solid-liquid interface.

9.4.2 Adsorption Isotherms for the Solid-Liquid Interface This derivation comes from the book Physical Chemistry of Surfaces by Adamson [2, p. 340]. The moles of a solute species adsorbed per gram of adsorbent is given experimentally by h C2Vsol/m, where h C2 is the changing concentration of the solute following the adsorption, Vsol is the total volume of the solution, and m is the grams of the adsorbent. It is convenient in the following development to suppose that mole numbers and other extensive quantities are on a per gram of adsorbent basis, so that n ~ the moles of solute adsorb per gram is given by

n~ = VAC2 = n o AX S

(9.36)

where n o is the total moles of solution per gram of adsorbent and AX s is the change in mole fraction of the solute following adsorption. In dilute solutions, both forms of this equation are equivalent. The quantity n ~ is generally a function of C2, the equilibrium solute concentration, and temperature for a given system. At a constant temperature, n ~ is a unique function of C2. This function is called the adsorption isotherm. The usual experimental approach is to determine this function, that is, to measure adsorption as a function of concentration at a specific temperature. Several forms for this function have been proposed. An important example of a specific model for adsorption is known as the Langmuir [38] equation. The Langmuir model assumes that the surface is composed of many adsorption sites. All adsorbed species interact only with their adsorption site and not with each other laterally. Adsorption is therefore limited to a monolayer. Adsorption process can be written as A (solute in solution, a2) + B (adsorbed solvent, X~) A (adsorbed solute, X~) + B (solvent in solution, al)

(9.37)

with an equilibrium constant for this process given by

K =X~al X~a2

(9.38)

where al and a2 are the solvent and solute activities in solution, respec-

9.4 Adsorption onto Powder Surfaces

383

tively. The activities in the adsorbed layer are given by the respective mole fractions X~ and X~. Because the t r e a t m e n t is restricted to dilute solutions, a l is constant and X~ + X~ = 1 so that the preceding equation becomes ba 2

(9.39)

X ~ = 1 + ba2

where b = K / a l . This equation can be rewritten, in terms of the fraction surface coverage 0 = F(21)/F2(1) m ~ where F 2(1) m is the surface excess concentration at monolayer coverage, 0=

ba2

(9.40)

1 + ba 2

where 0 = n ~/n s = X ~ and n s is the number of moles of adsorption sites per gram. Equation 9.40 is often written in the following linearized form: 0 1 - 0 - ba2 ~- bC2

(9.41)

where C2 is the concentration of solute or adsorbate. The equilibrium constant, K, can be written as K = e -AG~

e AS~

e -~H~

(9.42)

where AH ~ is the net enthalpy of adsorption often denoted by - Q , where Q is the heat of adsorption. Thus the constant b = K / a l can be written by b = b ' e Q/RT

(9.43)

where b' is (e~S~ This Langmuir adsorption isotherm is frequently followed by dilute solutions and at least qualitatively by some concentrated solutions. Compliance with the form of the Langmuir isotherm does not, however, give a sensitive test of the model. There are several reasons why real systems deviate from the theoretical model: 1. Adsorption is complex involving several types of interactions, solvent-solute, solvent-adsorbent, and solute-adsorbent. 2. Few solid surfaces are homogeneous at the molecular level. 3. Few monolayers are ideal because they are usually very concentrated. 4. Solutions are often not dilute. For this reason caution should be exercised in the interpretation of the b' values and heats of adsorption, values of Q, obtained from the best fits of the Langmuir adsorption isotherm. Microcalorimetric studies

384

Chapter 9 Wetting, Deagglomeration, and Adsorption

should be performed separately to measure the heat of adsorption. Heats of adsorption have been measured by Fowks [19, 20] for polymethylmethacrylate and chlorinated poly(vinly chloride)onto silica and calcium carbonate particles from solutions of various acid, neutral, and basic solvents and triethylamine onto a-Fe203 from cyclohexane.

9.4.3 Binary Solvent Adsorption Figure 9.11 [39] shows composite isotherms resulting from various combinations of individual ones. Type I isotherms result where there Type II

Type l r-

.o

A

._o 0

Concentration

Concentration Type III

--]

Type IV

C 0

2

0 c/) "0

r

Type V

.

C O m

f:L t_ O

Concentration

FIGURE 9.11 Classification of excess adsorption isotherms for binary solutions with solubility. After Kipling [39]. Reproduced by permission from Quantitative Review by Ripling Royal Society of Chemistry, Cambridge, UK, 1951.

9.4 Adsorption onto Powder Surfaces

385

is no affinity by component 1 for the surface that component 2 must go through a maximum. The reason for this m a x i m u m is the term X~ in the following equation for the surface excess concentration: F~ = (nS/~)(X~

- X~2) = n o AX~2/~, = (n~X~

-

n

sxX2)/E l

(9.44)

where n si and n li are the moles of component i in adsorbed surface layer and in the solution, X s and X~ are the mole fractions of component i in the adsorbed surface layer and in solution at equilibrium and E is the surface area per gram. Because n si + n l i = n o, the total n u m b e r of moles of component i in the system, and nS + n l = n 0 the total n u m b e r of moles in the system, the last two terms in equation 9.44 can be written. It is important to note that the experimentally defined apparent adsorption n 0 AXI2/E, which is also F ~, does not give the amount of component 2 in the adsorbed layer, n~. Only in a dilute solution, where Xl2 --* 0 andX~ --* 1.0, is this true. The adsorption isotherm, F~ plotted against Xl2, is thus an isotherm of compositional change. Again referring to Figure 9.11, in all other cases where component 1, the solvent, has an increasing affinity for the surface as in isotherms type II to type IV, the apparent adsorption of component 2 will be negative in concentrated solution. Type I isotherms result where there is essentially no affinity for the surface by the two molecules. Depending upon the distribution coefficient, the adsorption isotherms for ideal binary solutions can be shown in Figure 9.12. Here the distribution coefficient, K, as defined in equation 9.42 determines the shape of these isotherms.

FIGURE 9.12 Excess adsorption isotherms for ideal bindary solution with infinite solubility for different values of K the adsorption distribution constant given by equation 9.44.

386

Chapter 9

Wetting, Deagglomeration, and Adsorption

FIGURE 9.13 Adsorption isotherms for a solute with a finite solubility and a Lanqmuir adsorption isotherm, equation 9.40, w i t h X = C2/C~, where C~ is the maximum concentration (i.e., solubility or critical micelle concentration) of the solute and b is K/a i (where K is the distribution constant and a i is the activity of the solvent). This figure assumes that the solution is dilute where the activity a2 is equal to the concentration C2.

When one of the solvents has a limited and low solubility, C~, we find the classic Langmuir absorption isotherm is obtained with a slight modification to the activity axis as shown in Figure 9.13. Such a solubility limit can be obtained by precipitation or micellization in the case of surfactants. Micellization is the association of ionized surfactant molecules into structures where the hydrophobic parts of the surfactant molecules expel water. Micelles have different structures (i.e., spheres, cylinders, and lamellar structures) depending on the surfactant molecule and its concentration of surfactant in solution. Each structure has a different maximum concentration, C~, which limits adsorption. In Figure 9.13, the activity is replaced by concentration in the dilute solution case and the concentration axis C2 becomes dimensionless by division by the solubility limit C2/C~ when the constant b is replaced by bC~ in equation 9.40.

9.4.4 Adsorption of Ions The interaction of an electrolyte with an adsorbent may take one of several forms. The electrolyte may adsorb in total, in which case the situation is similar to that for molecular adsorption described earlier. It is more often observed that ions of one sign (+ or - ) are held more

9.4 Adsorption onto Powder Surfaces

387

F I G U R E 9.14

Schematic representation of molecular arrangement close to a solid surface showing the inner (IHP) and outer (OHP) Helmholtz planes, the stern layer, diffuse double layer, also called the Gouy layer, and the slip plane where the zeta potential is measured. Also shown is the potential for various distances from the surface.

strongly than those with the opposite sign, forming a charged adsorbed layer which is compensated by an oppositely charged ionic cloud as shown in Figure 9.14. When ions adsorb at the solid-liquid interface (inner Helmoltz plane, or IHP), they displace solvent molecules and are referred to as potential determining ions (PDI). Associated with the IHP is a semi-ordered layer of solvent molecules mixed with hydrated cations forming the outer Helmholtz plane (OHP). The potential at the OHP which is a result of the complex adsorption of ions and is difficult to measure. The potential at the slip plane, the zeta potential, ~, is easily measured by electrophoresis; however, this plane is some distance away from the OHP in the diffuse double layer or Gouy layer. The PDIs for oxides in water are generally the OH- and H + ions. As a result of the

388

Chapter 9

Wetting, Deagglomeration, and Adsorption

water equilibrium, H20

gw > H + + OH-

(9.45)

with Kw = 10 -14~ at 24~ the solution pH is ideally suited to monitor the change in the proportion of H § and OH- ions at the surface and therefore the charge of the adsorbed layer. The point where there is no charge at the surface is called the point of zero charge (PZC). The PZC can be determined from the following equilibria [40, 41] at the surface of a hydrated metal oxide particle M-OH~ M-OH

gl

~M - O H + H §

(9.46)

K2 ~M - O - + H §

(9.47)

and the equation PZC = [PK1 + pK2] 2

(9.48)

This equation works for most simple metal oxides dispersed in HC1 or KOH or KC1. Table 9.7 gives the values of pK1, pK2, and PZC for various oxides. For some oxides other PDIs are important. For example, when a small amount of barium ion (i.e., 10-3M BaC12) is added to a dispersion

TABLE 9.7

SiO2a'b A1203c'd TiO2e Fe203

Acid-Base Properties of Various Oxides

pK 1

pK 2

PZC (pH)

- 2.77 7.7 5.4 6.99

6.77 9.3 6.4 8.4

2.0 8.5 5.9 7.7

a Anderson, M. A., and Rubin, A. J., "Adsorption of Inorganics at Solid-Liquid Interfaces." Ann Arbor Sci. Publ., Ann Arbor, MI, 1981. b Teqari, P. H., ''Adsorption from Aqueous Solution." Plenum, New York, 1981. c Davis, J. A., and Leckie, J. O., ACS Symp. Ser. 93, 299 (1979). d Sposito, G. J., J. Colloid Interface Sci. 91(2), 329 (1983). e James, R. O., and Parks, G. A., Surf. Colloid Sci. 12, 119 (1982).

9.4 Adsorption onto Powder Surfaces

389

of TiO2 particles the normal point of zero charge at pH 5.5 disappears [42] and a new PZC is established above pH 11 as expected for the PZC of BaO. As a result, Ba+2 is a PDI for TiO2. This has consequences for mixed oxides like BaTiO3, PbTiO3, and so forth, SrTiO3, where the PDIs may include Ba, Sr, Ca, Mg, and Pb, as well as, H § and OH-. The adsorption picture for metal salts (e.g., CaCO3 or BaSO4) is always complex. The PDIs for salts are frequently their constituent ions or OH- and H § or both. Any ion that will complex with a metal cation (e.g., EDTA with Ba § NH~ with Cu § will be strongly adsorbed at the solid-liquid interface and add to its charge. Metal cations are easily complexed by ammonium, acetate, oxalate, citrate, peroxide, ethylenediamine-tetra-acetic acid (EDTA), and chloride ions. This leads to a complex, multicomponent adsorption picture for most salts and mixed metal oxides. Each PZC must be evaluated with respect to the PDI that could alter the adsorption and the PZC. The surface after adsorption will be charged with a potential, $s as in Figure 9.14, so that primary adsorption can be treated in terms of a capacitor model called the Stern model [43]. The other type of adsorption that can occur involves an exchange of ions in the diffuse layer with those of the surface. In the case of ion exchange, the primary ions are chemically bound to the structure of the solid and exchanged between ions in the diffuse double layer.

9.4.4.1 Stern Layer Adsorption Adsorption at a charged surface where both electrostatic and specific chemical interactions are involved can be discussed in terms of the Langmuir adsorption isotherm, where the distribution coefficient b is given by the exponential of a sum of the electrochemical a n d electrostatic forces. In this treatment the fractional surface coverage, 0, is given by [43]

[ZeOs + #~]

1-0

- C2 exp /

~

(9.49)

where C2 is the concentration of solute, z is the valence of the ion, e is the electron charge, and $s is the potential at the solid surface (see Figure 9.14). In fact, we have rewritten the adsorption free energy, Q, in the Langmuir equation as a sum of electrostatic and chemical contributions (i.e., zeC~s+ ~). The chemical interactions are contained in the term ~. The size of the electrostatic term, ze%, for an ion of valence 1 adsorbing at a surface with a surface potential of 100 mV is - 0 . 5 ksT. Typical values for the adsorption-free energy are several ksT, thus the chemical interactions contained in ~ are typically the most important term. This

~9~

Chapter 9 Wetting, Deagglomeration, and Adsorption T A B L E 9.8 Adsorption P a r a m e t e r s for Organic Acids on TiO2a at pH 4.0, 0.01 M NaC1, ~0 = +50 mV

Acetate Methyl phosphate Methyl sulfonate a

pK1

pK2

Adsorption energy (kBT)

4.74 2.38 - 1.92

-7.74 --

-5.22 -6.02 -4.36

Morrison, W. H., J. Coatings Technol. 57 (721), 55 (1985).

results in the specific adsorption of one type of ion in preference to another depending upon the chemistry of the surface and the ion. The adsorption of acetate, methyl phosphate, and methyl sulfonate on TiO2 has been studied by Morrison [40]. The adsorption data was fitted by Langmuir isotherms allowing the determination of the adsorption energy, Q = ZeOs + 4~, which is given in Table 9.8. This adsorption data follows the order of increasing PKA, ifpK 2 (and not pK1) dominates the adsorption of phosphate, which is likely. The adsorbed Stern layer is compensated by a compact and essentially fixed layer of hydrated counterions and water molecules which takes the form of a molecular capacitor between the inner and outer Helmholtz planes shown in Figure 9.14. The solid surface adsorbs the Stern layer ions and gives a potential of the inner Helmholtz plane, which is partially compensated by the hydrated counterions and water molecules of the outer Helmholtz plane. The diffuse double layer of Gouy-Chapman starts at the OHP and extends further into the liquid.

9.4.4.2 Diffuse Double Layer of G o u y - C h a p m a n This derivation comes from the book Physical Chemistry of Surfaces by Adamson [2, p. 340] but uses Systeme Internationale des Unites (SI) units, resulting in equations that are the same as those found in Heimentz [44] and Hunter [41]. Once the surface is charged by this type of potential-determining ion adsorption, freely moving counterions associate themselves with the charged surface providing a Gouy [45] diffuse double layer, compensating for the surface charge. The density of ions in the diffuse double layer is given by n_ = n_b ezeo/kT,

n+ = n+b e-zeo/kT

(9.50)

where n-b and n+b are the bulk concentration of ions with a - charge and a + charge, with units of number per M 3, respectively. The whole system remains electrically neutral so that far away from the surface the sum over all N types of ions with their concentration nk and val-

9.4 A d s o r p t i o n onto P o w d e r S u r f a c e s

391

ence zk is zero (i.e., 0 = ~ e z k n k ) . The net charge density (C*m-3) due to freely moving ions at any point within the system is given by [46] N Pf :

N

E ezknk = ~ ezkn~k exp(-ezkO/kT) 1

(9.51)

1

= -2noze sinh [ \lZeOl \kT]

(9.52)

The exponential summation term in this equation is the Boltzmann distribution of ions. The last term in the equation is the Boltzmann distribution of ions for a symmetric salt (i.e., valence z_ :z§ - - 1 : 1 , - 2 : 2 , or - 3 : 3). The surface integral of the dot product of the gradient of the potential, VO, and the unit vector, n, normal to the surface S gives the total excess charge in the solution per unit area (C'm-2): O" :

--ERE 0

V~I"

ridS =

-

pf dx

(9.53)

This is identical to an integration of the charge density from the surface of interest to infinity. Using the Poisson equation and the Boltzmann distribution of ions, the Poisson-Boltzmann equation results: SrS0 V20 = - - P f :

--

E1 ezkn~k exp(-ezkO/kT) = 2noze sinh \ k T ] (9.54)

which is the differential equation responsible for the potential, ~, in the fluid surrounding a charged surface. In this equation S r is the relative dielectric constant of the medium and So is the dielectric permeability of vacuum (8.85 • 10 -12 C2J-lm-1). As a result of solving this Poisson-Boltzmann equation with the boundary condition associated with a constant surface potential or a constant surface charge, we obtain a potential gradient upon moving from the surface into the solution. The exact solution for the Poisson-Boltzmann equation for flat plate with constant surface potential T0 geometry with a symmetrical electrolyte (i.e., valence z_ :z§ = - 1 : 1, - 2 : 2 , or - 3 : 3 ) is given by 1 + exp(-Kx) tanh(~o/4)] ~_ T0 exp(-Kx) = 2 In 1 - exp(-Kx) tanh(T0/4)

(9.55)

where the dimensionless potential 9 = ez$/kBT and K-1 is the scaling distance for the charge distribution called the double layer thickness, defined next. The double layer thickness at 20~ is a function of ion volume and salt concentration as shown in Table 9.9. The approximation just given is valid when T < 1.0 (i.e., 0 < 25 mV), which accounts

392

Chapter 9

Wetting, Deagglomeration, and Adsorption

TABLE 9.9 Double Layer Thickness K-1 (nm) as a Function of Electrolyte Concentration as Calculated from Equation 9.56

Electrolyte Conc.

1 "1

1:2

1 "3

2"2

3 "3

2"3

10 -3 10 -2 10 -1

9.6 3.0 0.96

5.6 1.8 0.56

3.9 1.2 0.39

4.8 1.5 0.48

3.2 1.0 0.32

4.3 1.4 0.43

for the linearization, sinh T ~ ~, in the Poisson-Boltzmann equation. The difference between this linearization approximation and the actual solution is given in Figure 9.15. The double layer thickness [47] is calculated from

\ e2~_,niz2 ]

(9.56)

where n i is the concentration of each ion type of valence z i in the solution, E r is the relative dielectric constant of the solvent, So is the

1

0

10 x(nm)

Shear Plane Potential near the surface of a flat platelet particle using linear and nonlinear Poisson-Boltzmann equation with a surface potential of ~0 = 2.0, (51.4 mV), which is the potential at the outer Helmholtz plane in Figure 9.14. Also showing the shear plane where the zeta potential is measured.

FIGURE 9.15

9.4 Adsorption onto Powder Surfaces

393

permitivity of free space, k B is the Boltzmann constant, and T is the absolute temperature. The surface charge that results from this potential distribution is given by o" 0 -- [ 8 n b ~ r S o k B T ] 1/2

sinh(~o/2)~

(9.57)

F,rF,OKXtt0

Note, this equation is for a symmetric electrolyte. When the dimensionless surface potential, ~o( = e z ~ o / k s T ) , is less t h a n one (i.e., ~o is less t h a n 25 mV with z = 1) corresponding to the linearization approximation where sinh 9 ~- ~, we find the potential distribution [48] reduces to 9 = ~o exp(-Kx) and the charge reduces to (r o = 8rS0K~0 for a symmetric electrolyte. The variation of dimensionless potential, 9 , with position, predicted by this G o u y - C h a p m a n model, is shown in Figure 9.14. For a spherical particle of radius a with the surface potential, ~o, less t h a n 25 mV, we find the following potential distribution given by = ~o K__aaexp[-K(a - r)] Kr

(9.58)

From this equation, we can see that the potential decreases more rapidly for a sphere t h a n a planar surface.

100

4

-100

9

2

i

4

-

Zero (mY)

--

0.0001 M NaCI

:.

0.001 M NaCI

--

0.01 M NaCI

~

9

i

6

6

9

i

8

9

i

10

9

12

pH F I G U R E 9.16 Variation of zeta potential of geothite a-Fe203 9H20 with pH. Data from Aplan and Fuerstenau [49].

394

Chapter 9

Wetting, Deagglomeration, and Adsorption

Increasing the concentration of the counterions decreases the double layer thickness, K-1 (as shown in Table 9.9), compresses the double layer, and increases the potential gradient. Increasing the valence, z, of the counterions very effectively decreases the double layer thickness as seen by equation 9.56. For cation chloride salts with cation valences 1 : 2 : 3 . the decrease in double laver thickness is 1" ~ / k / ~ = 0.63" V 2 / V / ~ = 0.45 times t h a t witl~ a 1" 1 valence salt like KC1. The use of a liquid with a lower dielectric constant or utilizing a lower t e m p e r a t u r e also compresses the double layer. The variation of the zeta potential, ~, m e a s u r e d at the shear plane as a function of pH for geothite, aFe20 3 9H20, is given in Figure 9.16 [49]. For a given pH, the absolute value of ~ potential increases as the salt concentration increases. This is because as salt concentration increases the double layer thickness decreases as shown in Figure 9.15.

Problem 9.4. Surface

Charge

Determine the surface charge in charges per n m 2 t h a t gives surface potentials of 20, 40, 60, 80, and 100 mV for KC1 solutions at concentrations, Ci, of 10 -~, 10 -2, 10 -3 M at room temperature.

Solution

Using the equation

o o - [8nbSr~okBT] 1/2 s i n h ( ~ 0 / 2 ) = [8"C*1000"6.02 • 1023/mole*8.85 •

10-12C2/J/m*78.54*l.38 • 10-23*J/mole/K *298 K] 1/2 sinh(To/2) with ~ o = e z O o / k s T = 1.60 • 10-19C*1"~o/[1.38 • 10-23*J/mole/K *298 K]

we find t h a t (ro/e gives the n u m b e r of charges per unit area. The results follow: Surface charge (charges per nm2) Surface potentials Conc. KC1

20 mV

40 mV

60 mV

80 mV

100 mV

10-1M 10.2 M 10-3 M

2.93 0.92 0.29

6.30 2.00 0.63

10.64 3.36 1.06

16.61 5.25 1.66

25.13 7.95 2.51

Here we see t h a t the surface charge is larger when the surface potential is higher and the salt concentration is higher. It should be noted t h a t this surface charge is an excess surface c h a r g e ~ t h a t is, the n u m b e r of positive charges minus the n u m b e r of negative charges per unit area. Surface densities larger t h a n - 2 0 0 per n m 2 are u n r e a s o n a b l y

9.4 Adsorption onto Powder Surfaces TABLE 9.10

395

Electrokinetic Effects

E Field

Stationary surface

Particle translation

Particle vibration

Applied Induced

Electro-osmosis Streaming potential

Electrophoresis Sedimentationpotential

Electrovibration Vibrationpotential

high, suggesting that this theory breaks down for such conditions. All of the conditions shown in the problem are well within the applicability limits of this G o u y - C h a p m a n theory. 9.4.4.3 M e a s u r e m e n t of t h e Z e t a P o t e n t i a l This material on the m e a s u r e m e n t of zeta potential comes from three excellent books by Adamson [2, p. 340], Heimenz [44], and H u n t e r [41]. The potential at this shear plane, the zeta potential, is measured using one of several electrokinetic phenomena which have in common the relative motion of a charged surface (e.g., a ceramic particle) and the bulk solution as elaborated in Table 9.10. When the electric field is applied, the charged surface experiences a force. When the surface moves, an electric field is induced in the solution. E l e c t r o - o s m o s i s If we have a fixed surface (i.e., a capillary wall or a porous plug) in an electrolyte solution and apply an electric field, E, the mobile part of the diffuse layer will move as shown in Figure 9.17. The velocity, v, at the shear plane using SI units is given by

~,SreOE

V= ~

(9.59)

where ~ is the zeta potential, t~ r is the relative dielectric constant, e 0 is the dielectric permittivity of vacuum, and ~? is the viscosity of the solution. In some cases, the velocity of the fluid is directly measured by a microscope as a function of capillary radius. If the capillary is closed at both ends, there is a parabolic counterflow at the center of the capillary just balancing the flow of the diffuse layer in the opposite direction at the wall. Thus, from the velocity profile and the knowledge that there is no net flow, the velocity at the shear plane can be approximated. If both ends of the capillary are connected, allowing a flow circuit as shown in Figure 9.18, the volumetric flow rate, Q, is dependent on the electric current, I, as follows: Q=

~ereOI v(kb + 2ks~Re)

(9.60)

396

Chapter 9

FIGURE 9.17

Wetting, Deagglomeration, and Adsorption

Movement of the diffuse double layer in a capillary subject to an electric

field, E.

where kb is the conductivity of the bulk electrolyte solution. The conductivity of the surface of the capillary, ks, is often also important and can be isolated by performing experiments with capillaries of different radii, Re.

FIGURE 9.18

Electro-osmosis flow velocity for a porous plug.

9.4 Adsorption onto Powder Surfaces

397

Streaming Potential Streaming potential is the same phenomena operating in r e v e r s e m t h a t is, the flow of electrolyte induces an electric field, E, which is measured. Using transport equations the volumetric flow rate can be related to the pressure drop across the capillary, AP/L, giving ~r~0

AP

E = . ~(k~ + 2ks~Re) L

(9.61)

Noting the similarity between this equation and that for electroosmosis, Onsager [50] developed the following relationship: EL AP

V I

~,r~,O ~(kb + 2ks/Rc)

(9.62)

This equation is the basis of both the electrovibration and vibration potential listed in the electrokinetic phenomena of Table 9.10. In these cases, the root mean square (rms) voltage (= E 9L) is either measured as in the case of the colloid vibration potential or induced by an electrode and the rms pressure fluctuations (= AP) at the same frequency are either induced by an ultrasonic actuator or measured with a pressure transducer, as in the case of electrovibration.

Electrophoresis The most familiar electrokinetic experiment consists of setting up an electric field, E, in a solution containing charged particles and determining their velocity. The particle velocity, v, is measured by direct microscopic observation at the stagnation point (i.e., zero velocity point for electro-osmosis at the radius 0.707Rc) in a capillary as shown in Figure 9.19. The zeta potential is then computed

FIGURE 9.19 Velocity profile in a capillary showing the stagnation planes where the colloidal particle velocity is measured during microelectrophoresis.

398

Chapter 9 Wetting,Deagglomeration, and Adsorption

from the following equation using SI units: v=

2~SrsoEC

37

(9.63)

where C is a constant that accounts for the degree of charge cloud distortion due to the electric field which depends [45] on KR, where K is the reciprocal double layer thickness and R is the particle radius. For small values of KR < 1.0, C is 2 (i.e., the Huckel equation). For large values of KR > 1000, C is 1.0 (i.e., the Helmholtz-Smoluchowski equation). At low zeta potentials, ~ < 25 mV, the values of C as a function of KR are given by the Henry equation [51]. For higher zeta potentials, the values of C are given by as a function of both the zeta potential and KR by Wiersma et al. [52]. Sometimes electrokinetic results are presented as electrophoretic mobility, which is defined as the particle velocity, v, divided by the electric field, E, eliminating the problem of evaluating the constant C. 9.4.4.4 I s o e l e c t r i c P o i n t

For most oxides, as the pH is increased, the adsorption of potential determining ions, H § and OH-, changes in correspondence with the concentration of these species in solution. For each surface, therefore, a point is reached at which the concentration of positive ions and negative ions just balance, the point of zero charge. The pH where the zeta potential, ~, is 0, is called the isoelectric point. The isoelectric point for various ceramic materials is given in Table 9.11. The acidic surfaces of quartz and tungsten oxide are noted in this table, as well as, the basic surfaces of alumina and magnesium oxide. It should also be noted that the method of powder fabrication is important in establishing the structure of the surface and, therefore, the isoelectric point of the powder surface. The IEP for a simple oxide is inversely proportional to the ratio of the valence, z, to the radius, Rcation , of the metal cation making up the oxide. The regression equation for IEP data from Parks [53] is as follows:

[z]

IEP = 1 8 . 6 - 11.5 Rcatio n

(9.64)

Oxidation always reduces the IEP for an oxide, for example ferric (+ 3) versus ferrous (+ 2). For MoO2 through Mo205, the IEP decreases from pH 12 to pH 0.5. The IEP is always higher for hydrated oxides than for freshly calcined oxides.

9.4.5 Adsorption of lonic Surfactants Ionic surfactants are molecules which have hydrophobic and hydrophilic ends. The hydrophobic part is essentially a long chain allophatic

9.4 Adsorption onto Powder Surfaces

TABLE 9.11

399

Nominal Isoelectric Points of Oxides

Material

Nominal composition

IEP (pH) a-c

Antimony pentoxide Molybdenum pentoxide Tungsten oxide Quartz Soda lime silica glass Potassium feldspar Zirconia Apatite Flurapatite Titania (rutile) Titania (anatase) Kaolin Mulite Chromium oxide Hematite Zinc oxide Alumina (Bayer process) Calcium carbonate Litharge Molybdenum oxide Magnesia

Sb205 Mo20~ WO2 SiO2 1.00 Na20" 0.58 CaO. 3.70 SiQ K20"A1203" 6 SiO2 ZrO2 Cas(PO4)3(OH) Cas(PO4)3(F, OH) TiO2 TiO2 A1203" SiO 2 92 H20 3 Al203" 2 SiO2 Cr203 Fe203 ZnO Al203 CaCO3 PbO MoO2 MgO

0.3 0.5 1 2 2-3 3-5 4-6 7 6 4.7 6.2 4.8 6-8 7 8-9 9 7-9 9-10 10.3 12 12

Parks, G. A., Chem. Rev. {}5, 177 (1965). b Patton, T. C., "Paint Flow and Pigment Dispersion," 2nd ed. p. 286. Wiley (Interscience), New York, 1979. c Adamson, A. W., "Physical Chemistry of Surfaces," p. 415. Wiley, New York, 1976. a

group. The longer is the chain, the higher the degree of hydrophobicity. The hydrophilic part is a charged head group like the carboxylic acid ion, - C O O - , the amino ion, - N H ~ , and the sulfate ion, -SO42 . Adsorption of ionic surfactants onto charged surfaces takes place analogous to the counterion adsorption discussed in the previous section. If the solid has a positive charge, the adsorbing species is a cationic surfactant. If the solid has a negative charge the adsorbing species is an anionic surfactant. For the surfactant molecule to be ionized it must undergo either a hydrolysis or dissociation reaction. The hydrolysis reaction for cationic amino group is given by RNH 2 + H20 --~ RNH~ + OH-

(9.65)

The dissociation reaction for alkyl sulfonate is given by RSO2H ~ RSO~ + H §

(9.66)

The dissociation reaction for the carboxylic acid group is given by R CO2H ~ R CO~ + H +

(9.67)

400

Chapter 9 Wetting, Deagglomeration, and Adsorption

W i t h respect to both dissociation a n d hydrolysis, the c o n c e n t r a t i o n s of the ionic species will d e p e n d highly on t h e pH of the solution a n d t h e ionic s t r e n g t h . The degree of ionization is m e a s u r e d by t h e e q u i l i b r i u m c o n s t a n t or more f r e q u e n t l y t h e p K for ionization. Typical p K v a l u e s of several ionizing groups u s e d as ionic s u r f a c t a n t s follow: Ion Product

pK [11]

RCOO- + H § -4.89 RSOO- + H + -0.7 RNH3 § + H § -3.39 F i g u r e 9.20 is a n e x a m p l e of d o d e c y l a m i n e in solution at a concentration of 4 x 10 -5 m o l a r as a function of pH. As the pH increases, we find a decrease in the cationic a m i n e concentration, as a r e s u l t of t h e hydrolysis reaction. The critical micelle concentration (CMC) of t h e dodecyl a m i n e (i.e., the c o n c e n t r a t i o n at which self-aggregation occurs) is also given as a function of pH in F i g u r e 9.20. Because t h e CMC is 1(~ 2 ecylamine 1()3 ,~0.01)CMC -4

10-

\

L o w e r Adsorption ~ Boundary Point (pH8) '~~

-5

10-

\

-6

10

Upper Ads=orption Boundary Point (pH 12.2)

-7

10

6

9

1()

1'1

1'2

9

13

Solution pH FIGURE 9.20 Hydrolysis, critical micelle concentration (CMC), and 0.01 x CMC for dodecylamine as a function of pH. For the hydrolysis curve 4 x 10-5 M dodecylamine was used with the reaction RNH~ + OH- ~ RNH2 + H20. From Novich and Ring [54]. Reprinted with permission from Langmuir [54]. Copyright 1985 American Chemical Society.

9.4 Adsorption onto Powder Surfaces

401

the concentration above which micelle activity in solution is constant, this presents a boundary in the adsorption isotherm. Adsorption of ionic surfactant at a charged surface is covered by the Langmuir adsorption isotherm for ion adsorption at a charged surface. The only difference in this equation is that, for an ionic surfactant, the range over which the concentration of the ionic surfactant is sufficient is drastically limited by the CMC concentration. Novich [54] found that by plotting the adsorption isotherm as a function of the measured concentration divided by the critical micelle concentration (i.e., C/CMC), adsorption isotherms for a homologous series of cationic surfactants on quartz could be reduced to one isotherm over two steps, as shown in Figure 9.21 [55,56]. He also found that the adsorption of dodecyl amine at different pH values could also be reduced to one isotherm when plotted in terms of this reduced concentration, as shown in Figure 9.22. In addition, the first layer of this adsorption isotherm followed the Langmuir adsorption isotherm, as shown in Figure 9.22, with a monolayer coverage that occurred at 0.01*CMC, independent of the length of the hydrocarbon chain. The monolayer coverage, however, was found to be dependent on the length of the hydrocarbon chain. As the chain length decreased, the area per adsorbed molecule approached the area of the charged amine group. It seems that, as the hydrocarbon tail becomes longer, its motion prevents close packed adsorption. At much higher concentrations approaching the CMC, multilayer adsorption is observed with the zeta potential decreasing to zero and then changing sign as this multilayer builds up. Above the CMC, no additional adsorption takes place. Novich also found that this second layer adsorption can take place for a cationic amine on a positively charged surface but only at concntrations above 0.01*CMC. This behavior is also observed with the adsorption of anionic surfactants onto negatively charged surfaces. As a result, Figure 9.20 can now be viewed as a method of determining when sufficient concentration of cationic amine is available for monolayer adsorption from solution. For the case given in Figure 9.20, the upper and lower limits are where a monolayer surface coverage is observed. In between the upper and lower limits are where a monolayer surface coverage is observed. In between the upper and lower limits, greater than a monolayer adsorption is obtained. As a result of understanding the solution chemistry and the aggregation process of micellization, we can predict when the surfactant will adsorb onto a powder surface. Zhu and Gu [57] have developed a relatively simple equation for the multilayer adsorption (i.e., 0 > 1.0) of ionic surfactants which also have the possibility to form complexes in solution in the form of micelles: 0=

ba2[1 + nk2a~-1] 1 + ba2[1 + k2a~-1]

(9.68)

402

Chapter 9 Wetting, Deagglomeration, and Adsorption

FIGURE 9.21 Adsorption isotherms for n-alkylammonium ions onto biotite[K2(Mg, Fe, A1)6(Si, A1)sO20(OH)4] at pH = 5.5, T = 298 K: (a) plotted as a function of the equilibrium concentration of ion adsorbed, Ceq (data from cases [55]; (b) plotted as a function of Ceq/CMC (data from Predali and Cases [56]).

403

9.4 Adsorption onto Powder Surfaces A

o,

'E o

10 -8

I

S_._.eeco.~.nd___Ste.__pM.._..ono___layer Ads___orp___tion

r

m

o

E

1 0_9_

o~ C

m

l d 1~

First S t e p

Monolayer

Adsorption

tO

e~ L_ o

co =

16

<

11

1 3-5

1C}-4

10 -3

1()-2

I(}-I

10

R e d u c e d Equilibrium C o n c e n t r a t i o n , Ceq/CMC F I G U R E 9.22 Adsorption isotherms for Dodecylammonium ion onto quartz, plotted as a function of Ceq/CMC at 25~ over a pH range 6 to 10.5. From Novich and Ring [54]. Reprinted with permission from Langmuir [54]. Copyright 1985 American Chemical Society.

where n is the surface aggregation number (or maximum number of adsorbed layers) for the surfactant and k2 is the equilibrium constant for micellization in solution:

nA+X---~A +nnx Again the

b[= kl/al]

(9.69)

value has the form

b~exp[ zer176 ]ksT

(9.70)

given by a chemical term, ~b, and an electrostatic term, ze%. For many conditions this equation reduces to an equation of the Langmuir form.

9.4.6 Adsorption of Polymers For polymer adsorption the first requirement is that the polymer must be soluble within certain limits in the solvent. The second requirement is that it adsorbs at the ceramic powder surface.

Solubility Polymer solubility can be predicted by the Hildebrand solubility parameter. The Hildebrand solubility parameter, 8, is defined as the square root of the molar energy of vaporization, hE = AH - RgT, 8-

--=--

~ A H - RgT V

(9.71)

404

Chapter 9

Wetting, Deagglomeration, and Adsorption

where AH is the enthalpy of vaporization, V is the molar volume. The concept behind the Hildebrand solubility parameter is that the internal energy/molar volume ratio, hE~V, is the energy that binds together the molecules in a unit volume of material. In solubility theory, a solute will dissolve if it is surrounded by solvent molecules with similar Hildebrand solubility parameters. In practice, solutes and solvents with similar 6 values tend to be soluble; if significantly different, they tend to be immiscible. The heat of mixing a solute and a solvent together is approximated by AH M ~

AE ~)polymer(~solvent[ZpolymerCpolymer + ZsolventCsolvent](~polyme r -- ~solvent )2 (9.72)

where (b is the volume fraction and X is the mole fraction of the polymer and solvent. When the solvent is negligibly small compared to the polymer this equation reduces to AH M ~

AE

=

(~polymerCsolvent(~polymer- ~solvent)2

(9.73)

This equation holds for all but the very high polymer volume fractions. It should be noted that a similarity between two solubility parameters alone is an indication, but by no means an assurance, of solute solubility. That is because the total solubility parameter, 6, is the result of many types of interaction; principally, nonpolar interactions caused by dispersional forces, 8d, polar interactions, 6p, and hydrogen bonding, 8H: 82= 8~ + 8~ + 8~

(9.74)

For true compatability of solute and solvent, matching of all these partial solubility parameters (i.e., 64, 6,, 6H) is necessary. The total solubility parameter can be easily calculated [1, p. 307] from the material's enthalpy of vaporization, vapor pressure as a function of temperature, surface tension, thermal expansion coefficient, critical pressure, and second virial coefficient of its vapor, as well as by calculating its value for the chemical structure of the material. For the calculation of the Hildebrand solubility parameter from chemical structure, we use Small's [58] equation:

~=y1 ~

G~

(9.75)

where Gi are Smalrs molar attractors given in Table 9.12 for different constituent functional groups within the molecule. This equation is reasonably accurate for many solvent and polymer classes except for hydrogen-bonded materials. The unique merit of this method to deter-

405

9.4 Adsorption onto Powder Surfaces TABLE 9.12

Small's Molar Attraction Constants a at 25~

Group Single-bonded carbon: CH3 CH2 CH C Double-bonded carbon: CH2 CH C Triple-bonded carbon: CHC CC Conjugation Cyclic structures: Phenyl Phenylene Naphthyl 5 member ring 6 member ring

Gi

214 133 28 93 190 111 19 285 222 20-30 735 658 1146 110 100

Group

Gi

Cyclic structures: OH H (variable) O (ethers) C1 (mean) C1 (single) C1 in CC12 C1 in CC13 Br (single) I (single) NO2 PO4 CO (keytone) COO (ester) S CN CF2 CF3 SH (thiols) ONO2 nitrates

329 80-100 70 260 270 260 250 340 425 440 500 275 310 225 410 150 274 315 440

a Small, P. S., J. Appl. Chem. 3, 71 (1953).

mine the Hildebrand solubility parameter is that it is applicable to polymers whose their solubility parameters cannot be evaluated by other physical chemical methods. Hildebrand total solubility parameters for many solvents [59] are given in the appendix of this book. The partial solubility parameters, 8d, 8p, 8H, can also be determined by chemical properties of the material like the dipole moment, as well as by using chemical group contributions [1, p. 314]. With these partial solubility parameters more careful matching of the solute to the solvent solubility parameters can be made. In many cases, the simple matching of the Hildebrand total solubility parameter is all that is needed to predict polymer-solvent solubility.

P o l y m e r A d s o r p t i o n Several reviews on the subject of polymer adsorption are presented by Eirich and coauthors [60,61] and Kipling [62]. The adsorption of polymers that have been considered include synthetic rubber, cellulose-type polymers, methacrylate, styrene, vinyl polymers. Most studies have been performed in polar organic solvents, primarily on carbon as a solid, no doubt because of the bias of the rubber industry. Another important point is that the polymers typically used are of a wide molecular weight distribution and their adsorption

4{}6

Chapter 9 Wetting, Deagglomeration, and Adsorption

is more that of a multicomponent system in which fractionation effects can be very important. Another very important aspect of polymer adsorption is that polymers have a very large number of configurations at the interface. As a result of the polymer configuration and its interaction with the surface, it takes several parameters to describe the state of the polymer at the interface. These include the number of points of attachment, the lateral spread as given by the average adsorbed radius, and the thickness of the polymer layer at the surface. These are illustrated in Figure 9.23. A very simple model that can be derived from the mass action approach in which v molecules of solvent are displaced by the adsorption of one polymer molecule at the surface is given by 0 v(1 -

O)~= bC2

(9.76)

This is a simple extension of the Langmuir equation to polymers, however, the Langmuir equation fits most polymer adsorption data within experimental error [2, p. 398]. In Figure 9.24 [63,64] several polymer adsorption isotherms are given, they include (a) the adsorption of polystyrene from benzene onto pyrex glass, (b) the adsorption of hydroxyl-propyl cellulose (HPC) onto SiO2 from aqueous solution, and the adsorption of poly(acrylic acid) onto BaTiO~ from aqueous solution. These experimental data can be fit quite well with the simple Langmuir equation; that is v = 1. The preceding equation will be useful for polymer concentrations below the polymer cloud point, a C~. The complications caused by the cloud point or aggregation of the polymer in solution cannot be accounted for in this simple equation. The preceding equation can also be used for charged polymer adsorption. In these polymers, groups are ionized in the same way that the head group of an ionized surfactant are ionized. This ionization gives the polymer charge and expands the polymer coil in solution as observed by Somasundaran [65] for poly(acrylic acid). This increase in polymer coil volume changes its adsorbed area, FM. This is shown in Figure 9.24(c) [66] for the adsorption of poly(acrylic acid) (PAA) onto BaTiO3 from aqueous solution. At pH 1.5 the poly(acrylic acid) polymer is nonionized and has a tightly coiled conformation as seen in Figure 9.24(d). As a result it has a large monolayer coverage. At pH 10.5, PAA is completely ionized and has a linear conformation because the charges repel each other. This linear conformation gives a lower monolayer coverage. With a charged polymer, the coefficient of adsorption, b, is given by

b=exp[ z'eO~

4)]

(9.77)

F I G U R E 9.23 Structure of polymer adsorbed at a surface showing the number of points of attachment to the surface, the radius of gyration of the polymer, and the span of the polymer as it extends into the solvent. Taken from Rowland et al. [61]. Reprinted with permission from Ind. Eng. Chem. Copyright 1965 American Chemical Society.

408

Chapter 9 Wetting, Deagglomeration, and Adsorption a

L

3

2

~.

950,000Mw

--

110,000Mw

........_..__...

cg } g

<

i 0.0

011

012

01.3

0.4 i

Equilibrium Solution Concentration, wt% PAA

0.5

409

9.4 Adsorption onto Powder Surfaces d

I

1.5

I ....

1 CO0-

I __

-

I

I

I

COO"

coo

COO

r O .m

~

1o

.m O O (D r

........................

o

PAA b

a > 0.75

"10

c 0 I,.. LI.

0.5-

J

PAAa

a<0.22

, "c~ a

0 ..................

- 0 . ,,z ~

1

I

2

I

3

I

4

~

I

5

"~ ...... I

6

I

7

I

8

9

FIGURE 9.24

(a) Adsorption of polystyrene onto pyrex glass from benzene solution at 30~ Data from Rowland and Eirich [63]. (b) Adsorption isotherms for hydroxyl propyl cellulose (HPC) on SiO2 at pH 4.2. From Chang [64]. (c) Adsorption kinetics for polyacrylic acid (Mw 5,000) on BaTiO3 (2.9 m2gm -1) at pH 1.5 and 10.5. From Chen [66]. (d) Conformation ofpoly(acrylic acid) and its degree of ionization, a, as a function ofpH. From Chen [66].

where z' is the number of charges on the segment of the polymer which adsorb at the solid surface, e is the electron charge, and % is the potential at the ceramic surface. Again one term of this equation, ~, contains the chemical interactions and the other term, z'e%, contains the electrical interactions. Because of the many charges on an ionic polymer z' may be large and give an electrostatic term of equal magnitude to the chemical interaction term. The heat of polymer adsorption has been measured microcalorimetrically. Heats of adsorption have been measured by Fowks [19,20] for

410

Chapter 9 Wetting, Deagglomeration, and Adsorption

polymethylmethacrylate and chlorinated poly(vinyl chloride) onto silica and calcium carbonate particles from solutions of various acid, neutral and basic solvents. These heats can be predicted from acid-base interactions using the Drago E and C concept discussed in Section 9.2.4. These heats of adsorption often have little correlation with the heats of adsorption, values of Q, obtained from the best fits of the Langmuir adsorption isotherms due to the very restrictive assumptions used in its derivation. Microcalorimetric studies should always be performed separately to measure the heat of adsorption. The kinetics of polymer adsorption can be described by [67] dO d---t= kin2(1 - O) - k_lO

(9.78)

This equation is a result of considering the adsorption and desorption kinetics described in the equilibrium which establishes the Langmuir equation 9.31. Chen, et al. [68] have studied the adsorption kinetics of poly(acrylic acid) onto BaTiO3 from aqueous solution. Their results are given in Table 9.13. The adsorption kinetics are faster in the case of pH 10.5 because the negative charge on the polymer attracts the positive charge on the BaTiO3 surface. At pH 1.5 there is no charge on the polymer, thus diffusion to the surface is not enhanced.

9.4.7 S e l e c t i o n o f a S u r f a c t a n t An excellent article by Bernhardt [69] tabulates dispersion systems for hundreds of ceramic powders. These dispersion systems consist of a solvent and surfactant with a range of useful concentrations listed. The solvents are both aqueous and nonaqueous and the surfactants are ionic, nonionic, and ionic polymers. This is the most extensive table of established dispersion systems available in the literature today. For organic solvents, a system based on the relative strengths of the hydrophilic and lipophillic (i.e., hydrophobic) portions of the surfactant,

TABLE 9.13 Adsorption Kinetics for Poly (Acrylic Acid) (Mw 5,000) onto BaTiO3 (2.9 m2gm -1) from Aqueous Solution a

pH

kI (%wgt -1 hr -1)

(hr -1)

k_l

(%wgt -1)

Fm (gm PAA/gmBaTi03)

10.5 1.5

0.73 0.31

4 • 10 -3 7.9 z 10 -4

182 _+ 50 40 -+ 24

0.00040 _+ 0.00003 0.0048 + 0.0016

geq

a Data from Chen, Z.-C., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992).

9.4 Adsorption onto Powder Surfaces

411

the HLB system [70], has been considerably successful in selecting surfactants which will adsorb at the solid-liquid interface. The HLB system is based on an arbitrary numerical scale where a value of 0 is assigned to a surfactant that is overwhelmingly hydrophobic and 20 to a surfactant that is overwhelmingly hydrophlic [71]. Each surfactant represents a balance between its hydrophobic and its hydrophilic parts. Extensive tables of surfactant HLB numbers are available in the literature [72]. One method of computing the HLB number for a surfactant is HLB = 7 +

~Hi-~Li

(9.79)

where Hi and Li are the hydrophilic and lipophilic chemical groups listed in Table 9.14. The larger is the number of H or L groups, the greater its contribution to the HLB number for the surfactant. A surfactant HLB number can be estimated roughly by mixing a small portion of the surfactant with water and observing the nature of the mixture. On the basis of Table 9.15 an approximate HLB number can be assigned for these observations. The use of the HLB number to select a surfactant (or mixture of surfactants) is achieved by matching the surfactant HLB number to that of the material being dispersed. Unfortunately, little information is available on the HLB number for ceramic powder surfaces. What data there exists is given in Table 9.16. For ceramic systems, the HLB of the surfactant is usually optimized by experiments with various surfactants.

Aqueous Surfactant Selection The selection of aqueous surfactants follows the general rule that the more ionic is the surfactant (i.e.,

TABLE 9.14 H and L for Hydrophilic and Lipophilic Chemical Groups a

Hydrophilic

Groups

Lipophilic

Groups

Group

Hi

Group

Li

NaSO4 KOOCNaOOCHOOCHOO -(CH2CH20)-

39.0 21.0 19.0 2.1 1.9 1.3 0.36

R3-CH R2-CH2 R-CH3 R2-CH R-(CH2CH(CH3)O)-R

0.47 0.47 0.47 0.47 0.11

a Data from Patton, T. C., "Paint Flow and Pigment Dispersion," 2nd ed. Wiley (Interscience), New York, 1979.

412

Chapter 9

Wetting, Deagglomeration, and Adsorption

TABLE 9.15 Estimation of Surfactant HLB Number by Its Characteristics upon Mixing with Water a

HBL value

Characteristics of mixture

1-4 5-6 7-8 9-10 11-13 >13

Immiscible (no dispersion) Unstable or poor dispersion Milky dispersion after vigorous shaking Stable milky dispersion Translucent or grayish dispersion Clear solution

a Data from Patton, T. C., "Paint Flow and Pigment Dispersion," 2nd ed., p. 288. Wiley (Interscience), New York, 1979.

increase in the Langmuir term ezr and the greater is the affinity of the ion for the powder surface (i.e., increase in the term ~b), the more effective the adsorption. This general rule is applicable to ionic surfactants and polymer surfactants of ionic and nonionic nature. Another important consideration is the aggregation of the surfactant in solution, as in micellization of ionic surfactants or the formation of a cloud point for polymers. The lower is the concentration at which these forms of aggregation occur, the lower the concentration at which adsorption will occur. Therefore, for a particular ceramic powder in aqueous solution, the first thing to know is the IEP of the ceramic powder. If the desired

TABLE 9.16

HLB Values for Ceramic Powders a,b

Ceramic powder

HLB value

Carbon black Fe203 FeO TiO2 Molybdate orange-solid solution of PbMoO4" PbCrO4 9PbSO4 PbCrO4

10-12 13-15 20+ 17-20 16-18 18-20

a Pascal, R. H., and Reig, F. L., Off. Dig., Fed. Soc. Paint Technol. 36, 839-852 (1964). b "Surfactants in Paints," Bull. 764-12. ICI United States, Inc., Specialty Chemicals Division, Wilmington, DE, 1973.

9.4 Adsorption onto Powder Surfaces

413

TABLE 9.17 Chemistry of Polyphosphates Common name

Formula

Orthophosphate Pyrophosphate Tripolyphosphate Metaphosphate

PO 43 P20 74 P30 ~05 Polymeric (PO3)n3n

deagglomeration conditions (i.e., pH and sometimes salt concentration) are above the IEP, then the powder will be negatively charged and a cationic surfactant or a cationic polymer will adsorb at the powder surface. If the desired deagglomeration conditions are below the IEP, then the powder will be positively charged and an anionic surfactant or a anionic polymer will adsorb at the powder surface. For adsorption to occur at a low concentration of the surfactant, the aggregation of the surfactant must be considered--the lower the concentration for surfactant aggregation, the lower the surfactant concentration needed for monolayer coverage.

Polyphosphate Surface Active Agents An important class of adsorbates for ceramic powders in aqueous solution are polyphosphates given in Table 9.17. Phosphates adsorb by three mechanisms, ionic interaction, hydrogen bonding, and chemisorption. Hydrogen bonding of phosphate to an oxide-hydroxide surface is easily effected, because the distance between adjoining oxygen atoms is closely matched to the oxygen-oxygen distance in the polyphosphate. Chemisorption may also occur, because phosphates tend to react with the surface molecules of many ceramic powders to form insoluble phosphates. In general, as the polyphosphate chain gets longer the adsorption is more effective. Adsorption of polyphosphates below the IEP of the ceramic powder leads to charge reversal, due to their multivalent charge. Polyphosphates, with their negative charge, adsorb even above the IEP, where the ceramic powder is negatively charged, due to their strong hydrogen bonding and chemisorption. Just because a surfactant adsorbs on the surface of a ceramic powder does not mean that it will stabilize the dispersion. The adsorbed surfactant will certainly help prevent the reagglomeration of the ceramic powder, like any adsorbed species, but the colloid stability of the ceramic powder-surfactant dispersion must be considered separately. Colloid stability is the subject of Chapter 10 of this book.

414

Chapter 9 Wetting, Deagglomeration, and Adsorption

9.5 CHEMICAL IN A SOLVENT

STABILITY

OF A POWDER

In some solvents, ceramic powders can react with the solvent or decompose chemically or dissolve. Prediction of the chemical stability of a ceramic powder in a solvent is a complex problem which has facets that include thermodynamics and kinetics.

9.5.1 Stability in Water The thermodynamics for the dissolution of an oxide in water are described by chemical equilibria of the form MOx + x H § ~ M §

+ x OH-

(9.80)

which can be written as the difference of two half-cell reactions M O x + xH20 + 2x e ~ M(s) + 2x OH-,

M+2X+2xe~M(s),

so

so

(9.81) (9.82)

or for the reduction of an oxide M O t + 2H + 2e ~ MOx_l + H20,

so

(9.83)

where s ~ is the standard electrochemical potential for the reaction as written. A table of standard reduction potentials is given in the appendix of this book. The standard free energy, AG ~ of these reactions is given by AG O = - n F s ~

(9.84)

where n is the number of electrons on the left-hand side of the reaction equation and F is the Faraday constant (96,490 coulombs per equivalent). For other conditions of temperature and pressure, the equation is given by AG = - n F s = - R ~ T In Keq (9.85) where Rg is the gas constant and Keq is the equilibrium constant. This equation can be rearranged to give the potential, s, for any temperature and pressure as s = s o - RgTlnKeq nF

(9.86)

Note that the spontaneity of a chemical reaction can be judged by the corresponding cell emf as follows, it follows that if AG is negative, is positive"

415

9.5 Chemical Stability of a Powder in a Solvent

BaOz,HzO(s ) 02 (s)

Ti-Ba-HzO S y s t e m ~ ' ~ , ~ T=25~ ['Ba]=rTi]=lM + A

Ba z§ TiOz(s)

> I

BaZ*

I.U

I

~"~

IBaTiO3(s) I 'Ba(OH)2"

-1

-2

0

4

8

Solution

pH

b

0

c

0

Ti-Ba-HzO System T=25oc [Ba]=l M 1()3M

-4

,.-4

3(s)

Ti-Ba-Hz 0

System T=25~

(0

,=,

A 01 0 _J

TiOz(s)

a

12

~

~k k

._1

-8

nn

Ba 2+

-8

k

1" o r

[]

"

-12

Ti(OH)4 8

1()

1'2

14

-1 2

8

1~

1'2

1'.

S o l u t i o n pH S o l u t i o n pH (a) Eh versus pH diagrams for the Ti-Ba-H20 system at 25~ [Ba] = [Ti] = 1.0 M; (b) log[Ti] versus pH diagrams for the Ti-Ba-H20 system at 25~ Eh = 0 volts; (c) log[Ba] versus pH diagrams for the Ti-Ba-H20 system at 25~ Eh = 0 volts. Redrawn with adaptations from Adair et al. [75].

F I G U R E 9.25

416

Chapter 9 Wetting, Deagglomeration, and Adsorption AG

~

Cell r e a c t i o n is

+ 0

+ 0

Spontaneous Nonspontaneous Equilibrium

The dissolution and the reduction of metal oxides as described by these typical reactions are strong function of the pH of the solvent. A traditional method to determine the chemical stability is therefore a plot of the reduction potential, ~ (or - E h oxidation potential) versus pH, which is the classical Pourbaix diagram. Due to the complexity of the many different oxidation states these diagrams can be very complex and must be solved by computer [73]. Many single component ceramic oxide systems have been calculated by Rabenau [74]. Unfortunately, few diagrams exist for multicomponent ceramic oxide systems. Figure 9.25(a) [75] is a Eh versus pH diagram for BaTiO3 with two Eh = 0 cuts of this diagram for the concentrations of Ba (Figure 9.25(b)) and Ti (Figure 9.25(c)) given for the equilibrium with the various solid species at 25~ From this diagram, we can see that BaTiO3 is chemically stable only in a water solution when the pH > 10. TiO2 and the Ba +2 ion are the stable species below pH 10. For other solvents, various dissolution chemical equilibria can be written but very few equilibrium constants are available in the literature. As a result, the prediction of the chemical stability of various ceramic powders in nonaqueous solvents is severely limited. Once the thermodynamics of chemical reaction is determined as spontaneous, the reaction kinetics will establish the importance of this reaction to the degradation of the ceramic powder in the solvent. Reaction kinetics of this type between a solid and a (liquid) fluid were discussed in Chapter 5. Under some conditions the reaction kinetics are very slow, limited by either a slow surface reaction or a slow product layer diffusion. As a result, this reaction can be neglected in its importance to the ceramic powder's chemical stability. Unfortunately little information is found in the literature on the reaction kinetics for ceramic powders reacting with organic solvents. Therefore, trial and error seems to be the only dependable way to determine the chemical stability of ceramic powders in nonaqueous solvents. This is the way that the chemical decomposition of YBa2Cu3Ox in alcohols was determined.

9.6 S U M M A R Y In this chapter, we have discussed the thermodynamic principles of wetting of a ceramic powder by a liquid as a first step for its dispersion in the liquid. Once the ceramic powder is dispersed in the liquid, the

9.6 Summary

417

powder must be deagglomerated by grinding, mixing, or ultrasonics. A simplified model for ultrasonic dispersion was presented. Once the powder is deaggomerated it must be protected from reagglomeration by the adsorption of the solvent, an ion, a surfactant, or a polymer at the newly formed surface. Various solvent and surfactant selection schemes were discussed, as well as, the chemical stability of a ceramic powder in a solvent system.

Problems 1. Bartell and Zuidems [76] found the surface tension of CH212 on talc in air to be 50.8 dyne/cm at 20~ a. Determine the work of adhesion and the spreadir.g coefficient. b. Suppose that the surface of a layer of talc is rough with a value of i = 2.4, calculate the new contact angle. 2. Given the following surface tensions (in dynes/cm): Liquid-Air

")/LV

THg-L TH20-L

H20 Octanol Hexane Mercury

72 28 18 476

375 348 378 0

0 9 50 375

determine which liquids will wet and spread on mercury. 3. Using the data for values of TLV for solutions of sodium dodecylsulfate (SDS); SDS (mol/liter)

2 x 10 -3

10 -3

5 • 1 0 -4

1 0 -4

1 0 -5

0

TLV

33.9

38.26

45.28

60.04

71.44

72.93

a. Calculate the excess surface concentration of SDS in molecules per 1000/~2 using the Gibbs adsorption equation. b. Determine the value ofb for the best fit Langmuir adsorption isotherm. 4. The surface of talc has a charge o f l l . 7 ftC/m 2. Calculate the surface potential, ~0 if the talc is placed in a solution of 10 -3 M in NaC1 at 25~ Also calculate the potential at a distance 50/~ from the surface in the solution. This point is approximately equal to that of the zeta potential. 5. Calculate the HLB number for oleic acid, sodium oleate, and sodium lauryl sulfate. 6. Which of the surfactants in Problem 5 will be good for the dispersion of TiO2 in water? Why? (Answer: sodium oleate)

418

Chapter 9

Wetting, Deagglomeration, and Adsorption

7. For Cr203 in a water solution at pH 10 suggest a dispersant. 8. Using the parachor values listed in this chapter, calculate the liquid surface tension of ethyl acetate (CH3CO2C2H5) Mw = 88.1 gm/mole, p = 0.901 gm/cm 3. (Answer: 23.8 dynes/cm) 9. Using the table of standard reduction potentials, determine if either PbO or PbO2 is stable in a water solution at pH 9. 10. Using the data given in Table 9.8 calculate the chemical term, ~, for the adsorption onto the TiO2 surface of the listed organic acids.

References 1. Patton, T. C., "Paint Flow and Pigment Dispersion," 2nd ed. Wiley (Interscience), New York, 1979. 2. Adamson, A. W., "Physical Chemistry of Surfaces," 3rd. ed. Wiley (Interscience), New York, 1976. 3. Other values are available in Hertzog, E. S., Ind. Eng. Chem. 36, 998 (1944) and Meissner, H. P., Chem. Eng. Prog. 45, 151 (1949) which are summarized in Perry and Chilton [4, pp. 3-240]. 4. Perry, R. H., and Chilton, C. H., "Perry's Chemical Engineering Handbook," 5th ed. McGraw-Hill, New York, 1973. 5. Walden, Z. Elektrochem. 14, 712 (1908). 6. Shafrin, E. G., and Zisman, W. A., Adv. Chem. Se. 43, p. 154 (1964). 7. "Advances in Chemistry Series," No. 43. Am. Chem. Soc., Washington, DC, 1964. 8. Young, T., in "Miscellaneous Works" (G. Peacock, ed.), Vol. 1, p. 418. Murray, London, 1855. 9. Dupre, A., "Theorie Mecanique della Chaleur," p. 368. Paris, 1869. 10. IIler, R. K., "The Chemistry of Silica." Wiley, New York, 1978. 11. "CRC Handbook of Chemistry and Physics," 47th ed. Chem. Rubber Publ. Co., Cleveland, OH, 1966. 12. Parfitt, G. D., "Dispersion of Powders in Liquids," 2nd ed., p. 5. Applied Science, London, 1973. 13. Parfitt, G. D., Powder Technol. 17, 157-162 (1977). 14. Washburn, E. D., Phys. Rev. 17, 374 (1921). 15. Good, R. J.,Aspects Adhes. 8, 107-127 (1971). 16. Drago, R. S., Vogel, G. C., and Needham, T. E., J. Am. Chem. Soc. 93, 6014-6020 (1971). 17. Drago, R. S., Parr, L. B., and Chamberlain, C. S., J. Am. Chem. Soc. 99, 3203 (1977). 18. Bolger, J. C., in "Acid Base Interactions" (K. L. Mittal, ed.), pp. 4-18. Elsevier, New York, 1981. 19. Fowkes, F. M., and Mostafa, M. A., Ind. Eng. Chem. Prod. Res. Dev. 17, 3-7 (1978). 20. Fowkes, F. M., Rubber Chem. Technol. 57, 328-344 (1984). 21. Okuyama, M., Garvey, G., Ring, T. A., and Haggerty, J. S., J.Am. Ceram. Soc. 72(10), 1918-1924 (1989). 22. Gordy, W., J. Chem. Phys. 7, 93 (1939); 9, 204 (1941). 23. Davies, R., and Karuhn, R., Proc. Tech Prog. Int. Powder Bulk Solids Handling Processing, Rosemont IL, 1978, pp. 231-238. 24. Gallager, D., and Aksay, I., private communication. 25. Itoh, H., Pober, R. L., Parish, M. V., and Bowen, H. K., Ceram. Int. 12, 85-93 (1986). 26. Mizuta, S., Cannon, W. R., Bleier, A., and Haggerty, J. S., J. Am. Ceram. Soc. 61(8), 872-875 (1982).

References

419

27. Rehbinder, P. A., Colloid J. USSR 20, 493 (1958). 28. DiBenedetto, A. T., "The Structure and Properties of Materials." McGraw-Hill, New York, 1967. 29. Bartenev, G. M., Iudena, I. V., and Rehbinder, P. A., Colloid J. USSR (Engl. Transl.) 20, 611 (1958). 30. Rozenberg, L. D., "High-Intensity Ultrasonic Fields," pp. 263-340. Plenum, New York, 1971. 31. Plesset, M. S., and Chapman, R. B., J. Fluid Mech., 47(2), 283-290 (1971). 32. Flint, E. B., and Suslick, K. S., Science 253, 1397 (1991). 33. Dooher, J., Lipman, R., Marrone, R., and Poble, H., Ultrason. Symp. Proc., pp. 11-16 (1977). 34. Aoki, M., Ring, T. A., and Haggerty, J. S.,Adv. Ceram. Mater. 2(3A), 209-212 (1987). 35. Fridman, V. M., Ultrasonics 10(7), 162-65 (1972). 36. Gibbs, J. W., "The Collected Works of J. W. Gibbs," Vol. 1, p. 219. Longmans, Green, New York, 1931. 37. von Szyszkowdki, B., Z. Phys. Chem. 64, 385 (1908); Meissner, H. P., and Michaels, A. S., Ind. Eng. Chem. 41, 2782 (1949). 38. Langmuir, I., J. Am. Chem. Soc. 40, 1361 (1918). 39. Kipling, J. J., Q. Rev., Chem. Soc. 5, 60 (1951). 40. Morrison, W. H., J. Coat. Technol. 57, 55 (1985). 41. Hunter, R. J., "Zeta Potential in Colloid Science." Academic Press, New York, 1981. 42. Barringer, E. A., Ph.D. Thesis, MIT, Cambridge, MA. (1983). 43. Stern, O., Z. Elektrochem. 30, 508 (1924). 44. Heimentz, P. C., "Principles of Colloid and Surface Chemistry," 2nd ed., p. 696. Dekker, New York, 1986. 45. Gouy, G., J. Phys. 9(4), 457 (1910); Ann. Phys. (Leipzig) 7(9), 129 (1917). 46. Verwey, E. J. W., and Overbeek, J. T. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, New York, 1948. 47. Debye, P., and Huckel, E., Phys. Z. 24, 185 (1923); Debye, P., ibid., 25, 93 (1924). 48. Chapman, D. L., Philos. Mag. [6] 25, 475 (1913). 49. Aplan, F. F., and Fuerstenau, D. W., "Principles of nonmetallic mineral flotation," p. 371, AIME, New York, 1962. 50. Onsager, L., Nobel Prize (1968). 51. Henry, D. C., Proc. R. Soc. London, Ser. A 133, 106 (1931). 52. Wiersma, P. H., Loeb, A. L., and Overbeek, J. T. G., J. Colloid Interface Sci. 22, 78 (1966). 53. Parks, G. A., Chem. Rev. 65, 177 (1965). 54. Novich, B. E., and Ring, T. A., Langmuir 1, 701 (1985). 55. Cases, J. M., 8th Int. Miner. Process. Congr., Leningrad, 1968, Pap. No. S13 (1968). 56. Predali, J. J., and Cases, J. M., Proc. Int. Miner. Process. Congr. lOth, London, 1973, pp. 473-492 (1974). 57. Zhu, B. Y., and Gu, T., J. Chem. Soc., Faraday Trans. 1 85(11), 3813-3817 (1989). 58. Small, P. S., J. Appl. Chem. 3, 71 (1953). 59. Burnell, H., Off. Dig., Fed. Soc. Paint Technol. 29, 1069-1076 (1957). 60. Ullman, R., Koral, J., and Eirich, F. R., Proc. Int. Congr. Surf. Act., 2nd, London, 1957, Vol. 3, p. 485 (1957). 61. Rowland, F., Bulas, R., Rothstein, E., and Eirich, F. R., Ind. Eng. Chem., 57, 46 (1965). 62. Kipling, J. J., "Adsorption from Solutions of Non-Electrolytes." Academic Press, New York, 1965. 63. Rowland, F. W., and Eirich, F. R., J. Polym. Sci. 4, 2421 (1966). 64. Chang, S.-Y., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Fed~rale de Lausanne, Lausanne, Switzerland (1992). 65. Tjipangandjara, K. F., and Somasundarin, P., Proc. Int. Partic. Technol. Conf., Kyoto, Japan, 1990. pp. III, 206-213.

420

Chapter 9

Wetting, Deagglomeration, and Adsorption

66. Chen, Z.-C., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992). 67. Heller, W., Pure Appl. Chem. 12, 249 (1966). 68. Chen, Z.-C., Ring, T. A., and Lemaitre, J., Proc. "Spec. Ceram. 9," p. 242, British Ceramics Society, London, 1990. 69. Bernhardt, C., Adv. Colloid Interface Sci. 29, 79-139 (1988). 70. Liemerman, E. P., Off. Dig., Fed. Soc. Paint Technol. 34, 30 (1962). 71. "The HLB System," Chemmunique Reprint. ICI United States, Inc., Chemical Specialties Division, Wilmington, DE, 1976. 72. "Atlas Surfactants," Bull. LG-60, ICI Americas, Inc., Specialty Chemicals Division, Wilmington, DE, 1970. 73. Osseo-Asare, K., and Brown, T. H., Hydrometallurgy 4, 217-232 (1979). 74. Rabenau, A., Angew. Chem., Int. Ed. Eng. 24, 1026-1040 (1985). 75. Adair, J. H., Denkewicz, R. P., Arriagada, F. J., and Osseo-Asare, K., in "Ceramic Powder Science II" (G. L. Messing, E. R. Fuller, and H. Hausner, eds.), pp. 135-145. Am. Ceram. Soc., Westerville, OH, 1988. 76. Bartell, F. E., and Zuidems, H. H., J. Am. Chem. Soc. 58, 1449 (1936).

10

Colloid Stability of Ceramic Suspensions

10.1 O B J E C T I V E S This chapter gives the reader an appreciation for the various aspects of colloidal stability as it pertains to ceramic suspensions. First discussed are the van der Waals, electrostatic, and steric interaction energies. These interactions determine the ease with which two particles stick together when they bump into one another by random thermal motion or by shear induced collisions. The kinetics of aggregation is discussed next in two parts: doublet formation and large aggregate formation. The structure of these large aggregates is discussed in terms of the various aggregation mechanisms. Finally, these factors of colloid stability are applied to the processing of ceramic suspensions.

10.2 I N T R O D U C T I O N After a ceramic powder has an adsorbed layer to prevent reagglomeration, its colloid stability must be established. Some types of ceramic 421

422

Chapter 10 Colloid Stability of Ceramic Suspensions

powder processing requires stable colloidal systems and others require flocculated systems. When ions adsorb on the surface of the powder, electrostatic stabilization occurs. When polymers and surfactants adsorb at the surface, steric stabilization dominates. Once colloidally stabilized, a ceramic suspension can be processed. The rheology of this suspension is very important to determine its processing properties. For composites, the constituents of the suspension must be prevented from segregating. This is commonly performed by inducing flocculation in the suspension, thereby locking the well-mixed nature of the suspension into the flocc structure. Flocc structures are then broken by pressing to yield high solid density uniformly packed green bodies. If the ceramic suspension is too low in solid density after these steps then filtration, sedimentation, filter pressing, or centrifugation is performed to dewater the ceramic paste.

10.3 I N T E R A C T I O N E N E R GY COLLOID STABILITY

AND

When two particles collide due to random thermal motion they will stick or rebound, depending on their interaction energy. Therefore, this interaction energy determines the sticking efficiency or the colloid stability. The random thermal motion of a particle in a fluid is called B r o w n i a n motion because Robert Brown was the first to explain it in 1828. Two types of colloid stability will be investigated. One is electrostatic stabilization resulting from the overlap of two diffuse double layers as two colloidal particles collide. The other is steric stabilization resulting from the overlap of polymer or surfactant molecules adsorbed at the particle surface as two colloidal particles collide. Both of these interparticle repulsions are necessary to counteract the attractive forces between particles in suspension due to van der Waals forces. This section discusses the interaction energies present in ceramic systems and their affect on collision kinetics which establish the colloid stability of the suspension.

10.3.1 Van d e r W a a l s A t t r a c t i v e I n t e r a c t i o n Energy H a m a k e r [1] in 1917 showed the potential energy of attraction between two spherical interacting particles of the same material separated by a distance, h, is given by VA(h ) = - A * H(h, geometry)

(10.1)

where A is the Hamaker constant and H(h, geometry) is a geometrical function given in Table 10.1 for several different geometries. Hamaker

TABLE 10.1

Hamaker Interaction Energy a at Separation h

VA(h) = - A * H(h, geometry)

Geometry Two flat plates sizes 6 • 6 • di

A62 [h -2 + (h + d l + d2) -2 - (h + dl) -2 - (h + d2) -2]

Semi-infinite parallel plates 6 • 6

A62 h2

Two spheres of radii a l, a2 Two spheres of equal radius, a Sphere of radius a with an infinite slab a

12~r

2ale2 + In [ h2+2alh+2a2h 1~ h 2 + 2alh + 2a2 h + 4ala2 h 2 + 2alh + 2a2h + 4ala2]J A { 4a 2 2a 2 [ h2+4ah 1], 6 h 2+4ah +h 2 + 4 a h + 4 a 2' + l n h 2 + 4 a h + 4 a 2 ] j

A(

2ala2

+

--6 h 2 + 2alh + 2a2 h

- 1 ~ { ~ -~+h+4a4a + 2 1 n [ h + h 4 a l }

Hiemenz, P. C., "Principles of Colloid and Surface Chemistry" 2nd ed., p. 648. Dekker, New York, 1986.

424

Chapter 10 Colloid Stability of Ceramic Suspensions

made this calculation by considering the dipole-dipole interactions of atom pairs in each particle interacting across the gap. These electromagnetic interactions that travel at the speed of light were then summed over the whole particle volumes resulting in the preceding equation. Phase shifts introduced at large separations by the finite velocity of the wave propagation reduces the degree of correlation and the magnitude of attraction. This pairwise addition is suspect because of the importance of many-body interactions in condensed matter. The continuum theory of Lifshitz [2], accounts for many-body effects by treating the intervening matter as individual macroscopic continuous phases characterized by different dielectric permittivities. These manybody interactions are referred to as retardation effects. The theory breaks down at separations of molecular dimensions predicting an infinite attraction at contact. The theory presented in this section follows the treatment of Russel et al. [3] in their book Colloidal Disper-

sions. 10.3.1.1 t t a m a k e r C o n s t a n t

The H a m a k e r constant for two particles of materials 1 and 2, respectively, interacting across a vacuum is given by [3]" or

A12 = 11/8 ksT

~ N10~l(/~n)N2 c~2(/~n)

(10.2)

n=O

where Ni are the number density of molecules in material 1 or 2, respectively, and aj is the molecular polarizabilities of the two materials. The summation is evaluated at the imaginary values of frequency ~n ---- 2~rnksT/ft, corresponding the poles of coth(fto~/2ksT) in the upper half of the imaginary number plane. For two particles of materials 1 and 2 interacting across a liquid medium (subscript 3) the Hamaker constant is given by oc

A132 = 11/8

ksT ~

[NlC~l(i~ n) - N3a3(i~n)] * [N2~2(i~n) - N3c~3(i~n)]

n=O

(10.3) Alternatively, this result can be expressed in terms of the dielectric constants of the various materials through the relationship [4], ej = 1 + g j a j , as oc

A132 = 11/8

kBT ~ [e:l(/~ n) --

e3(i~n)][e2(i~n)- e:3(/~n)]

(10.4)

n=O

although pairwise additivity remains valid only for (s - 1) ~ 1.0. Explicit calculations of the Eigen frequencies is tedious. In addition, we have effects due to retardation effects. Retardation is the progressive damp-

10.3 Interaction Energy and Colloid Stability

425

ing of higher frequency terms in the summation due to electromagnetic waves becoming out of phase when a liquid medium, with its own multimolecule interactions, is present. This occurs when the time required for the electromagnetic wave to transverse the gap exceeds its temporal period; that is, hsl/2/c > l / ~ n . As a result of these retardation effects, the effective H a m a k e r constant is not a constant but depends on separation h. The effective Hamaker constant at zero separation (i.e., no retardation effects) is given in the preceding equations for various cases. As a result of retardation, the contributions of molecular polarizabilities at high frequencies are less important at large separations. Thus, the effective H a m a k e r constant for polystyrene in water decreases from the value at zero separation of 1.3 • 10 .20 J to 0.3 • 10 .2o J for separations larger than 100 nm. The effective H a m a k e r constant at large separations results from only low-frequency contributions to the summation [5]. At a distance of 100 nm the geometric function also reduces the H a m a k e r interaction significantly. As a result, a simple development for the effective H a m a k e r constant as a function of separation h can be made by considering relaxations in the uv spectra at a frequency of O~uvas the most important in the high frequency limit and the refractive index of the material in the visible spectrum, no, as the most important in the low-frequency limit [6],

~,(i~n)

=

n~- 1 1 + 1 + (~n/(Ouv) 2

(10.5)

Using this substitution and accounting for retardation effects, the effective H a m a k e r constant is given by [7] Aeff(h) = - 127rh2q)(h)

(10.6)

where q)(h) is a very complex function of the geometry of the system and the separation of the surfaces, as well as, the permittivity, Di(o)(=~,o8i((o)E((.o) , where E is the frequency, o~, dependent electric field), of the solid materials and the fluid. For a flat plate geometry, the potential energy per unit area, rPFp(h), is given by [3, p. 145] fo q ~~ l n ~D (i~n) dq ORB(h) = kBT ~ n=O

(10.7)

where the summation is over the spectrum of discrete eigenfrequencies associated with the wave vector, q. For this flat plate geometry, when h = 0 (the nonretarded limit), ORB(h) O~ h -2 cancelling the effect of h 2 in the prefactor and giving a constant value for the effective H a m a k e r constants, Ae~(h = O) = constant. As h increases from zero, riPER(h) decays faster t h a n h -2. This equation has been verified by Isrealachvili and Tabor [8]. For a sphere-sphere geometry, the Derjaquin approxi-

426

Chapter 10 Colloid Stability of Ceramic Suspensions

mation [9] can be used to approximate oc

r

~ 27r rlr2 Ih" ORB(S)ds rl + r2.

(10.8)

Without any electrolyte (i.e., vacuum), r ~ h -1 in both the nonretarded limit as h --~ 0 and in the retarded limit as h ~ ~. In between these limits the retardation varies with position h according to a different relationship. With electrolyte, r decreases to zero as h --~ ~. The Derjaquin approximation clearly should fail for micron-sized particles when the wavelength characterizing the interaction is comparable to the size of the particle. The effective Hamaker constants are calculated for many materials. A list of calculated effective H a m a k e r constants, Aeff(h = 0), for oxides, metals, solvents, and polymers is given in Table 10.2.

10.3.1.2 Pragmatic Theory for the Hamaker Constant For many practical examples in the experimental literature the Hamaker constant is assumed to be a constant and not a function of separation, h, in accordance with Hamaker's original equation [1]: A l l e f f --

~2n2C1

(10.9)

where n is the number of atoms per unit volume and C~ is the London constant taken to be 3hvoa}/4 (aj is the polarizability and hvo is approximately the ionization energy of the material, h is Planck's constant). Because ionization potentials are about 10 to 20 eV and polarizabilities are I to 2 • 10 -24 c m 3, C 1 is in the range 10 .52 to 10 -51 J-cm 6 per atom 2. A comprehensive list of experimentally measured effective H a m a k e r constants is given by Visser [10]. If such a Hamaker constant is already known for one material and the solvent, mixing rules can be applied. For two identical interacting spheres in a liquid, the Hamaker constant is given by the geometric mean mixing rule for the Hamaker constant for the particle 2, interacting with itself in a vacuum and for the liquid 1, interacting with itself in vacuum, given by [11] A121 = (~/A22

-

X/All) 2

(10.10)

The resulting Hamaker constant is always positive regardless of the relative magnitures of All and A22. For two different spherical particles (index i and 2) interacting in a liquid (index 3), the Hamaker constant mixing rule is given by A123 = ( V ~ I 1 - V ~ 3 3 ) ( ~ 2 2 2

- X//A33)

(10.11)

where A~ is the Hamaker constant for each material taken separately and interacting in a vacuum.

10.3 Interaction Energy and Colloid Stability

427

For nonpolar materials the H a m a k e r constant can be calculated from the surface tension or surface free energy of the material, T, and the equilibrium spacing, do, between molecules in the material as follows [11]: All = 247rT d~

(10.12)

This equation was derived by considering the energy of interaction between two blocks of material equivalent to the energy of cohesion given in Chapter 9. The logic used to obtain the Girifalco-Good-Fowkes equation [12] suggests t h a t the dispersion component of the surface tension T d m a y be better to use t h a n T itself when interactions other t h a n London forces the operating between the molecules. The surface free energy of a solid can be determined by noting the solvent where there is no interaction force, t h a t is, A~2~ = 0, and using the preceding equation to give TI~ = T22.

P r o b l e m 10.1. H a m a k e r C o n s t a n t Calculate the H a m a k e r constant from the surface tension of heptane (T = 20.3 m J m -2) and dodecane (T = 25.4 m J m-2). From the density of the solvents and their molecular weights, the molecular spacing can be determined. This allows the calculation of the H a m a k e r constant using equation 10.12, giving the following results: Solvent

Mw

Heptane Dodecane

100.2 170.3

p(gm/cc)

do (nm)

0.684 0.749

0.22 0.18

T(mJ m -2) Al1(• 10 20 J) 20.3 25.4

1.05 0.95

These values are lower by a factor of 4 t h a n those for similar materials given in Table 10.2.

10.3.1.3 Attractive I n t e r a c t i o n E n e r g y for P o l y m e r Coated Particles When the particles are coated with a polymer of thickness 6, the van der Waals attractive interaction energy is calculated by [13-16]

VA-- - ~

1

{ H l l [ V ~ 1 , - %/A22]2 + H22[V~22- V~33] 2 (10.13)

where Hjk is the geometric function given in Table 10.1 for two spheres or two plates. When jk = 11, rl = a + 6 and r2 = a with a separation h + 26, w h e n j k = 22, r 1 = r 2 = a + 6 with a separation h, when jk = 12, r~ = a + ~, and r 2 = a with a separation h + 6.

428 T A B L E 10.2

Chapter 10

Colloid Stability of Ceramic Suspensions

Effective H a m a k e r Constants a-c Aeff(h = O)

Oxides: SiO2 Al203(sapphire) CaCO3(calcite) CaF2(fluorite) Mica Metals: Silver Copper Gold Solvents: Water Pentane Decane Hexadecane Polymers: Poly(methylmethacrylate ) Poly(vinyl chloride) Polystyrene Polyisoprene Poly( tetrafluo roethylene)

Vacuum A11(h=O) 10 -2o J

Water A121(h = O) 10 -20 J

8.8 (quartz) 6.5 (fused quartz) 15.6 10.1 7.2 22

1.7 (quartz) 0.83 (fused quartz) 5.32 2.23 1.04 2.2

50 40 40

40 30 30

3.7c-4.0 b 10.8 4.8 5.2

0.34 0.46 0.54

7.1 7.8 7.9 6.0 10.8

1.05 1.03 1.11 0.74 0.33

Note. It is useful to compare the values of A with kBT which is 0.411 x 10 .2o J at 298 K. a Russel, W. B., Saville, D. A., and Schowalter, W. R., "Colloidal Dispersion," p. 148. Cambridge Univ. Press, Cambridge, UK, 1989. b Parsegian, V. A., and Weiss, G. H., J. Colloid Interface Sci. 81, 285-289 (1981). c Hough, D. B., and White, L. R., Adv. Colloid Interface Sci. 44, 259-272 (1980).

10.3.2 Electrostatic Repulsion The diffuse double layer associated with the surface of a particle in solution is an important force to overcome when the two double layers (one double layer is shown in Figure 9.14) are pushed together during collision of the particles by Brownian motion, for example.

10.3.2.1 Force b e t w e e n Charged Plates To calcualte the repulsive force between two plates, we need the local electrostatic potential, ~, which creates a local stress. The local stress can be calculated from the osmotic pressure caused by the local ionic concentration, which is in turn caused by the local electrostatic potential [17].

429

10.3 Interaction Energy and Colloid Stability

F

F

d -...

SYSTEM n b, 1-Ib

FIGURE 10.1

v

h

, View of t h e r m o d y n a m i c s y s t e m for parallel plate i n t e r a c t i o n e n e r g y cal-

culation.

II + k s T ~ nb_ k k:l

[

1 -- exp

(ez o ] -

ksT]

= IIb

(10.14)

where II is the local osmotic pressure and II~ is the osmotic pressure in solution far away from the two interacting plates, nb-k is the number of ions in the bulk of type k, and zk is the valence of the ions of type k. A force balance is constructed on a system shown in Figure 10.1 bounded by the midplane and a parallel surface far away from the two interacting plates. The forces consist of the force on the midplane and the osmotic pressure on the boundary at infinity, where because of symmetry the electric field is 0. Because the electric stress on the midplane is 0, the force per unit area is as follows [3, p. 111]. F = l[ b - II = k s T ~ nb_ k k=l

[

1 -- exp

ksT ]

(10.15)

where Om is the potential at the midplane. The expression in brackets is simply the excess ionic concentration at the midplane, so that the repulsive force per unit area is equal to the osmotic pressure. If the potential at the midplane is assumed to be just a sum of t h a t from two isolated plates, as shown in Figure 10.2, the force is given by F = 64 k B T nb tanh2(To/4) e x p ( - Kh)

(10.16)

where ~o is the dimensionless potential at the surface of the identical plates. For asymmetric parallel plate cases (i.e., different surface potentials or surface charges), the force has an additional stress term,

430

Chapter 10 Colloid Stability of Ceramic Suspensions

FIGURE 10.2 Electric potential function between two plates for small interaction. The total potential can be approximated by the sum of the two single potentials as shown.

- 1 / 2 s e o E 9E , due to the lack of symmetry at the midplane. This term may change the force from always repulsive to both repulsive and attractive at different separations for different surface potentials on the two plates [3, p. 112]. To determine the potential at the midplane, the Poisson-Boltzmann equation must be solved for the parallel plate geometry. For a z - z electrolyte, the Poisson-Boltzmann equation is given by [18] V2~ = K2 sinh 9

(10.17)

where V9 is the Laplacian operator of the dimensionless potential, xtt = z e $ / k s T , with K the inverse double layer thickness defined in

equation 9.56. For a parallel plate geometry the Laplacian operator is d 2 ~ / d x 2. A double integration of the Poisson-Boltzmann equation gives

[3, p. 112] fcc~

(]~2 __

1)- 1/9(fl _ ~ m ) - 1/2 d f l = ~

Kx

(10.18)

osh~m

This expression can be evaluated numerically with either constant charge or constant potential boundary conditions on the charged plates. For the case where the potential, T, is less than 25 mV, corresponding to a dimensionless potential of 9 < 1.0, the Poisson-Boltzmann equation can be linearized, giving sinh 9 = ~. The solution to the linearized Poisson-Boltzmann equation (i.e., Debye-Huckel linearization) is

10.3 Interaction Energy and Colloid Stability

431

given by [3, p. 114] = A1 cosh Kx + A2 sinh Kx

(10.19)

where A1 and A2 are constants to be evaluated based on the boundary conditions at the two surfaces. For constant potentials, that is, = ~+ = ~_

x = h/2 x = -h/2

(10.20) (10.21)

we have

1 ~++~-

A1 - ~ cosh (Kh/2)'

A2

1 ~+-~_

= 2 sinh (Kh/2)

(10.22)

and for the boundary conditions where the surface charge per unit area, (r = - ~ ~o V$ 9n, are held constant, that is, (r = (r§

x = +h/2

(r = ( r _

x = -h/2

(10.23) (10.24)

we have 1 q++qA1 = ~ sinh (Kh/2)'

A2

1 q+-q= 2 cosh (Kh/2)

(10.25)

where q is a dimensionless surface charge scaled by q

oez eeoksTK

(10.26)

For this low potential case, the force is simply [3, p. 114] F = kBT

nb (A~ - A~)

(10.27)

Figure 10.3 plots the force between two identical charged plates for different boundary conditions and different assumptions for the potential distribution. These force predictions have been experimentally verified by Pasahley and Israelachvili [19] as shown in Figure 10.4 for the nonlinear Poisson-Boltzmann equation with constant potential boundary conditions.

10.3.2.2 Electrostatic Interaction Energy The repulsive electrostatic interaction energy, VE, can be determined directly from the force, F ( h ) , as a function of separation, h, by using V E ( h ) = fh F ( h ) d h

(10.28)

The resulting interaction energy expressions are given in Table 10.3. The energy decreases as the separation increases. As the surface poten-

432

Chapter 10

Colloid Stability of Ceramic Suspensions

F I G U R E 10.3 Electrostatic repulsion force, F/nbk B T versus separation, h, between two identically charged plates with surface potentials of 25 mV in a 1 : 1 electrolyte solution at 0.001 M according to the exact theory with either constant charge or constant potential boundary conditions. Calculated from F = nbks T (A~ - A~) with definitions of Ai given in the text for constant charge and potential boundary conditions.

tial increases, the energy increases. As the double layer thickness, K-1, increases the energy change takes place over larger distances.

10.3.2.3 Force b e t w e e n Two Charged Spheres The repulsive force for two charged spherical double layers is obtained in the same way as that for two plates by considering the force TABLE 10.3 Interaction Energies for Two Flat Plates, VPP(h), Immersed in a z :z Electrolyte, vPP(h) = ksTnbK -1 9f(~o) * g(Kh) Constraints

f (To )

Linear superposition of potentials, constant potential, NLPB using eqs. 10.16 and 10.28 Linear superposition of potentials, constant potential, LPB, using eqs. 9.55, 10.15, and 10.28 Constant potential, LPB, using eqs. 10.22, 10.27, and 10.28

64 tanh2(~0/4)

exp(- Kh)

4~02

exp(-Kh)

2 9 02

g( Kh )

] cosh- 2(x) - sinh- 2(x) dx hK/2

Notes. LPB is linear Poisson-Boltzmann equation V2~ Poisson-Boltzmann equation V2T = K2 sinh(T).

= K2~I~.

NLPB is nonlinear

433

10.3 Interaction Energy and Colloid Stability

5 x 10 -6 M Distilled

103

water ~k= - 150 mV

= - 9 0 mV

,,In"

--, 3 x 10-5M 10 2-

= - 5 0 mV

5x

101

o

10-4M

35 mV 1 O3M

At higher \ conc. f o r c e ~ too weak t O measure

lb

2'o

\? 3'0

40

5'0 610

y

hlnm

FIGURE 10.4 The force measured between two curved mica surfaces in solutions of 2:1 electrolytes (Mg +2, Ca +2, Sr +2, and Ba +2) at pH 5.8. The solid lines are based on the theory for potentials and concentrations shown along with the van der Waals attraction corresponding to a Hamaker constant of 2.2 • 10 -2o J. Redrawn from Pashley and Israelachvili [17]. Reprinted with permission from Academic Press.

at the midplane. This force has two terms" osmotic pressure and the electric stress [3, p. 115]. Integrating these terms over the central plane between two spheres as shown in Figure 10.5 gives

F=

[ ~BT ] 1 ] seo E E - - ~ E . E 8 .ndS

kBTk=IEnb-k 1 - exp +

ndS (10.29)

where Tm is the potential at the midplane and E is the electric field at the midplane, n is the unit vector normal to the midplane and S is the surface of the midplane. Because the nonlinear Poisson-Boltzmann equation has no analytical solution in bispherical coordinates, this formula can be used only with numerical solutions. The linearized Poisson-Boltzmann equation for two spheres has been solven by Oshima [20], using a complicated method of transparent particles and particle reflections.

434

Chapter 10 Colloid Stability of Ceramic Suspensions

-F

oo

FIGURE 10.5 View of thermodynamic system of two spheres for interaction energy calculation, n is the normal vector at the midplane, h is the separation between the two spheres of radius r.

10.3.2.4 N u m e r i c a l Solution to N o n l i n e a r P o i s s o n - B o l t z m a n n E q u a t i o n for B i s p h e r i c a l Coordinates Ring [21] has developed a numerical solution for the nonlinear Poisson-Boltzmann equation in bispherical coordinates, ~ (~, 0, ~). The Cartesian coordinate transformation of such solutions are given in 10.9 and 10.10. In Cartesian coordinates, all contours are not smooth curves as they were in bispherical coordinates because a coarse plotting grid was used. Figure 10.6 shows the potential distribution surrounding two spheres of the same size with different surface potentials as they are moved together with constant potential boundary conditions. At large separations, the particles have potential distributions which are spherical and not affected by the close proximetry of the two particles. At small separations, the potential distribution between the particles is greatly affected by the other particles. For a dimensionless separation of 1.0, there is no minimum potential on the line connecting the two sphere centers, like that at other separations, indicating a large interaction force. Figure 10.7 shows the potential distribution surrounding two spheres of the same size with the same surface potentials, one with constant potential and the other with constant charge boundary conditions as they are moved together. At small separations, the potential distribution between the particles is again greatly effected by the other particles. The surface of the sphere with the constant charge boundary

10.3 Interaction Energy and Colloid Stability

435

Numerical solution for the potential distribution surrounding two spheres with Kal = Ka2 - 10 for constant potential boundary conditions with T1 = 4 and ~2 = 1. The separation, Ks decreases from 5 (figure a) to 4 (figure b) to 3 (figure c) to 2 (figure d) to 1 (figure e).

F I G U R E 10.6

c o n d i t i o n s i n c r e a s e s t h e p o t e n t i a l a t t h e s u r f a c e of t h e s p h e r e to m a i n tain a constant charge. R i n g [21] a t t e m p t e d to u s e t h e s e p o t e n t i a l d i s t r i b u t i o n s s u r r o u n d i n g t h e s e s p h e r e s to c a l c u l a t e t h e i n t e r a c t i o n e n e r g y b u t h i s c a l c u l a t i o n s

436

Chapter 10 Colloid Stability of Ceramic Suspensions

FIGURE 10.6 (continued)

were not sufficiently accurate for this purpose and are in error. This definition of electrostatic energy is different than that just presented. The equation for the force at the midplane just given can, however, be used to accurately calculate the electrostatic interaction energy, VE(h), from integration of F(h) = -dVE(h)/dh, with this numerical solution in bispherical coordinates [22].

10.3 Interaction Energy and Colloid Stability

FIGURE 10.6

437

(continued)

10.3.2.5 Electrostatic Interaction Energy for Two Spheres

Another method to calculate the electrostatic interaction energy, VE, is from the Derjaquin approximation. Derjaguin [9] developed a method to utilize the parallel plate interaction energy, VP(h), as an approximation for the interaction energy between two spheres, VSzS(h). This approximation is given by

VSES(h)= fo 27r h VP(h) dh

(10.30)

and essentially decomposes the curved surfaces into parallel plate rings of various radii interacting across the liquid gap, h, between the two particles shown in Fig. 10.8. This approximation is good when the curvature of the surface is small compared to the double layer thickness (i.e., Ka > 5). Using this result, the interaction energy between two unequally sized spheres is given by [23]

ssoala2(021+O~2)[20~%2 4(al + a2)

[ ( ~ , + ~022) In

(~ + exp(_-Kh)] - exp(-Kh)] (10.31)

+ ln(1 - exp[- 2Kh ]) ]

438

Chapter 10 Colloid Stability of Ceramic Suspensions

FIGURE 10.7 Numerical solution for the potential distribution surrounding two spheres with Kal = Ka2 = 1.0 for constant potential boundary conditions with ~1 = 2 and ~2 = 2. The separation, Ks decreases from 5 (figure a) to 4 (figure b) to 3 (figure c) to 2 (figure d) to 1 (figure e).

where ho is the shortest separation between the sphere surfaces and r r are the surface potentials on the two spheres. The equation can be reduced to 2

VSS(h) = eeoar ln[1 + exp(-Kh)] 2

(10.32)

10.3 Interaction Energy and Colloid Stability

FIGURE 10.7

439

(continued)

when the spheres are the same size (i.e., al = a2 = a), with equal surface potentials (i.e., ~o~ = %2 = %)" For identical spheres, interaction energy is always positive, signifying repulsion. Both of the preceding equations use the linearized Poisson-Boltzmann equation, which is valid only when % / < 2 5 m V . For various boundary conditions and assumptions the electrostatic interaction energy is given in Table 10.4.

440

Chapter 10 Colloid Stability of Ceramic Suspensions

FIGURE 10.7 (continued)

Total I n t e r a c t i o n E n e r g y The total interaction energy is simply the sum of the attractive van der Waals interaction energy and the repulsive electrostatic interaction energy given by [18] Vr(h) = V A ( h ) + V E ( h )

(10.33)

A schematic of the total interaction energy profile for two spheres is shown in Figure 10.9. This figure shows both the repulsive and attractive interactions plus the total, VT. The total interaction energy at large particle separations is 0. Decreasing the particle separation

FIGURE 10.8 Geometrical construction used in the calculation of the interaction be-

tween two dissimilar spherical particles from the interaction of two infinite fiat plates.

441

10.3 Interaction Energy and Colloid Stability T A B L E 10.4

Electrostatic Interaction Energy for Two Identical Spheres Immersed

in a z z Electrolyte, VRE(h) ss = 21rsso L ze J * f(~o) * g(a, Kh) 9

Constraint

f(~o)

Constant potential, LPB, Ka > 5 Constant charge, LPB, Ka > 5 Linear superposition of potentials surrounding spheres, a LPB Nonlinear superposition, NLPB, Ka > 5

g(a, Kh) a ln[1 + e -Kh] - a ln[1 - e -Kh]

~2 (r~ 2xI,~

a 2

h + 2a exp(-Kh) 16 tanh2(~0/4)

a e x p ( - Kh)

Notes. LPB is linear Poisson-Boltzmann equation V2~ = K2xI'. NLPB is nonlinear P o i s s o n - B o l t z m a n n equation V2T = K2sinh(~I') a Bell, G. M., Levine, S., and McCartney, L. N., J. Colloid Interface Sci. 33, 335 (1970).

'§ Repulsion V=ea~o~o/4 In[l+exp(- ~h)]

>,

o1

c uJ ~" .o

E

I

~Energy Barrier, 9max

i

Total Interaction VT--V A+V R

h

Secondary Minimum Attraction V=-A 121a/(24h)

(-)

F I G U R E 10.9 Potential energy of interaction between two particles with electrical double layers. The secondary minimum is not expected when Ka ~ 1.

442

Chapter 10 Colloid Stability of Ceramic Suspensions

a +15

s separation distance

0

.0

200

600

VR/kT=O.O18dZ~ in (l+exp-s/5)l~ ~ ~

"~

800

(A)

b

1000

0

+

d Particle ( d i a m e t e r (p)

o.~

~~" +5-I~ ~

0

400

A

I

I

s Separation distance 200 5

1

/

I

400

~

~

~ 0

rr

~

600 .

(,~)

800

1000

d Particle

- c iam'e' er (p,

5

+5

,--o.1

0

~

0

~ -5-/ '-~ ' d

/

L -10-

Particle

diameter

(p)

i ,~

VA/k T= O.12d/s

=: 0

0.02

0.04

s Separation

0.06

0.08

0.10

-15

0

0.02

0.04

0.06

0.08

0.10

distance (p) s Separation distance (p) FIGURE 10.10 (a) Plot of the attraction and repulsion forces acting on two spherical particles as a function of particle separation for constant dielectric constant, surface potential, and double layer thickness resulting from 10 -6 M 1 : 1 salt. (b) Plot of the sum of attractive and repulsive interaction energies acting on two particles as a function of particle separation given in (a). Copyright 9 1979 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

decreases the energy to a secondary minimum if Kh ~ 1.0. No secondary minimum is observed if Kh > 1.0. Further decreasing the particle separation, a repulsive maximum is reached. At very small particle separations an attractive primary minimum is reached. The repulsive maximum is important because it is essentially an activation energy for the collision of the particles and determines the sticking efficiency or colloid stability as will be seen later in this chapter. Figure 10.10 is a plot of the attractive ( - ) , repulsive (+), and total interaction energy for two spheres, where the size of the spheres is changed. As the particle diameter is increased, the magnitude of the van der Waals interaction energy and the electrostatic interaction energy increase at all separation distances. The resulting total interaction energy shows a positive maximum which increases with particle diameter. Figure 10.11 is a plot of the attractive ( - ) , repulsive (+), and total interaction energy for two spheres where the salt concentration which determines the double layer thickness, K-~, is changed. The attractive van der Waals interaction energy is unchanged by the change in salt

443

10.3 Interaction Energy and Colloid Stability a +15

s Separation distance

o

200

I

I

400

,

I

,

600 I

VR/k T =0.010d~2in (l+exp-s/~) ~ O=4.0;d=O.5Op;~=3OmV

0

,

(/~)

800 I

IO

1000

,

+15-~

5 = (0.34/z)(0/M 0,5) z=l ; 0=4.0

0

(A)

s Separation distance 200

400

600

800

1000

0.08

o. 10

,

L ~-- +10

_M

.2

10_6/

Molarity

(moles

/ liter)

Q. n"

+5

~

+5-

I--

o

O-

.2 L C

i

,~ -lOk-15.

10-

9

0

-5-

.il_

0 L

.

0.02

9

.

0.04

0.06

,

0.08

s Separation distance (p)

9

, o. 10

-15-

0

0.02

0.04

0.06

s Separation distance (p)

F I G U R E 10.11 (a) Plot of the attraction and repulsion forces acting on two spherical particles 0.5 t~m in diameter as a function of particle separation for constant dielectric constant and surface potential. (b) Plot of the sum of attractive and repulsive interaction energies acting on two particles as a function of particle separation given in (a). Copyright 9 1979 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

concentration. However, the electrostatic interaction energy decreases in magnitude for an increase in the salt concentration. Essentially an increase in salt concentration decreases the double layer thickness. The resulting total interaction energy shows a positive (repulsive)maximum for some low salt concentrations, which decreases in magnitude and occurs at smaller separations with increased salt concentration. For higher salt concentrations the maximum decreases to negative (attractive) values at 10 .4 M and disappears completely for 10 .3 M and above. In considering the total interaction energy for a ceramic suspension as a function of pH and salt concentration, we must consider both specific ion adsorption which charges the particles and its effect on the electrostatic interaction energy. To give the reader an understanding of how all these effects play together, Figure 10.12 [24,25] is presented, which is the measured zeta potential for 0.313 t~m TiO2 particles in suspension at various salt concentrations. Added to this curve is the calculated total interaction energy for a two sphere interaction. At the isoelectric point at pH 5.5, there is only an attractive van der Waals interaction due to the disappearance of the electrostatic interaction

444

Chapter 10 Colloid Stability of Ceramic Suspensions

FIGURE 10.12

Total interaction energy for 0.313 t~m TiO2 particles versus separation, h, for various conditions of zeta potential, salt concentration and pH. Hamaker constant = 2.1 • 11-2o J [24]. Zeta potential data from Barringer [25].

energy. At pH values less and more t h a n isoelectrostatic point, there is a positive (repulsive) maximum in the total interaction energy curve that decreases in magnitude and occurs at smaller separations as the salt concentration is increased. Because the positive (repulsive) maximum in the total interaction energy curve is essentially an activation energy for collision we can predict low sticking efficiency or high colloid stability ratio for the cases where the positive (repulsive) maximum is larger than 15 kBT [26, p. 42]. A Boltzmann distribution of collision energies with an average of 1 ksT will have very few (i.e., only exp(-15)) collisions with energies greater t h a n 15 ksT. As a result, high colloid stability will be observed at pH values away from the isoelectric point and at low salt concentra-

10.3 Interaction Energy and Colloid Stability

445

tions for the single component system. Multicomponent ceramic systems consisting of a distribution of particle sizes or different proportions of particles with different chemistries are much more complicated but can be analyzed by the same equations presented here for the different types of particles present in the system.

10.3.3 Steric Repulsion Often in ceramic processing, where the surface potential is small or the double layer thickness is thin, the electrostatic repulsion is not sufficient to stabilize the colloidal suspension against coagulation. As a result another form of stabilization is n e e d e d ~ s t e r i c stabilization. Steric stabilization has been reviewed by two recent books, one by Napper [27] and the other by Sato and Ruch [26]. The following presentation draws heavily from both these books. An absorbed molecule, either a polymer (ionic or nonionic) or surfactant (typically ionic), with its associated solvent molecules can stabilize particle collisions because the adsorbed layer provides a steric hinderance to the close approach of the particles. As we have seen in the preceding chapter, a fraction of the polymer put into a ceramic powder suspension will absorb at the powder surface as an adsorbate and another part will be left in solution. Both the free polymer and the attached polymer play roles in steric stabilization. The attached polymer determines the steric interaction, and the free polymer determines the depletion interaction. In general, the sterical interaction is dependent on the thickness of the adsorbed layer, the chemical nature of the adsorbed molecules, and the chemical nature of the solvent. The depletion interaction is dependent upon only the chemical nature of the adsorbed molecules and the chemical nature of the solvent. It should be noted that the steric and depletion interactions are not dependent on the nature of the underlying ceramic particle but depend only on polymer-polymer and polymer-solvent interactions. This aspect of steric interactions is important to ceramic processing because a single polymer-solvent system can be applicable to several ceramic powders with similar results and a single polymer-solvent system can be used for a mixture of ceramic powders (i.e., a composite ceramic). There are two reasons for steric interactions: (1) osmotic pressure effect due to the high concentration of chain elements in the region of the overlap as shown in Fig. 10.13, and (2) a steric effect due to the fewer possible conformations of the adsorbed molecule in the region of the overlap. These two aspects correspond to the enthalpy and entropy effects of steric stabilization. It has been found for some types of steric stabilization that increasing the temperature destabilizes the system even though for others increasing the temperature stabilizes the sys-

446

Chapter 10 Colloid Stability of Ceramic Suspensions

a

b FIGURE 10.13 The two aspects of steric stabilization of an adsorbed polymer: (a) osmotic effect due to high concentration of chain elements in the region of the overlap, (b) entropy effect due to the restricted conformation of the polymer molecules.

tem. This temperature dependence can be understood in terms of the Gibbs free energy for flocculation, AGE given by

AGE = AHF- TASF

(10.34)

where AGE is the free energy of the close approach of particles with an adsorbed polymer layer corresponding to a state 1, where the particles are far apart, and a state 2, where the particles are close together (see Figure 10.14). Here, AH is the enthalpy and AS the entropy of the changes in the adsorbed layer that take place upon collision and polymer-polymer overlap. Possible combinations of the signs for each of the components of the preceding equation are shown in Figure 10.15. In this figure, we see that there are two types of stabilization, enthalpic and entropic. Enthalpic stabilization can be removed by heating the system. Entropic stabilization can be removed by cooling the system. In addition, when the enthalpy is positive and the entropy is negative, we can have combined stabilization, where no change in temperature will cause flocculation. Many attempts have been made to develop theories to predict the interaction energy between sterically stabilized particles. The details of

10.3 Interaction Energy and Colloid Stability

44 7

a

b

FI G U R E 10.14

Interaction of two sterically stabilized spheres: State 1. Hmixing = Hpolmer/solvent, (2 sphere area 9L m) State 2./-/mixing = Hpolymer/solvent* (2 sphere area 9Lm - interaction volume) + Hpolymer/polmer, (interaction volume) hHm~i~g = Hpo]mer/polmer * (interaction volume) AS = loss of possible polymer conformations due to interaction.

Combined Stabilization

~H ~ Enthalpic Stabilization Stable

Permanently Stable

Heat Unstable

Entropic Stabilization Heat Stable

T,~S

Permanently Unstable

Unstable

F I G U R E 10.15

Schematic representation of the t h e r m o d y n a m i c factors controlling steric stabilization.

448

Chapter 10 ColloidStability of Ceramic Suspensions

b

FIGURE 10.16 Diagramof electrosteric stabilization: (a) negatively charged particles

surrounded by cationic counterions with nonionic polymers adsorbed, (b) positively charged polymers calledpolyelectrolytes intermingled with anionic counterions attached to uncharged particles.

these theories are given in the research monograph entitled Polymeric Stabilization of Colloidal Dispersions by Napper [27] and numerous review papers [28-33]. No one theory, to date, can qualitatively predict the interaction energy between sterically stabilized particles. The reason for this is (1) lack of prediction of the conformation of polymer molecules at an interface, (2) lack of prediction of the inter- and intramolecular excluded volume of adsorbed polymer molecules, and (3) absence of a truly quantitative theory describing polymer solution thermodynamics. A polymer (or surfactant) can effect the colloid stability of a ceramic powder dispersed in a liquid via several means. The colloid stability will be effected differently depending whether the polymer is Physically adsorbed, Chemically attached, or Free (i.e., not adsorbed). In addition, if the polymer is charged as with polyelectrolytes, the charge of the polymer gives rise to electrosteric stabilization-a combi-

10.3 Interaction Energy and Colloid Stability

449

F I G U R E 10.17 Schematic representation of the three domains of close approach for sterically stabilized flat plates: (a) noninteraction domain, h > Ls; (b) interpenetrational domain, Ls <- h <- 2Ls; (c)compressional domain, h < Ls.

nation of electrostatic stabilization and steric stabilization shown in Figure 10.16. Steric stabilization occurs when the polymer is either physically or chemically attached to the surface of the particles. Depletion stabilization occurs when the polymer is free in solution and not adsorbed or chemically bound to the surface. Figure 10.17 is a schematic representation of the three domains [34] of close approach for sterically stabilized flat plates. In this case, the

450

Chapter 10 Colloid Stability of Ceramic Suspensions

polymer is assumed to be attached to the surface and there is no free polymer in solution. These domains establish limits in which different phenomena are clearly recognizable. The first is the noninteractional domain, where the distance between the plates, h, is greater than twice the span, Ls, of the stabilizing moieties (i.e., h -> 2Ls). In this domain no interaction takes place. The second is the interpenetrational domain, where the distance between the plates is intermediate between one and two spans of the stabilizing moieties (i.e., Ls <- h <- 2Ls). The third is the interpenetrational-plus-compressional domain, where the separation between the plates decreases to less than the span of the stabilizing moieties (i.e., h < 2Ls). These domains of the close approach for sterically stabilized flat plates leads to a discussion of the steric interaction interms of the mixing free energy and the elastic free energy. Figure 10.18 is a schematic representation of depletion stabilization in which the polymer is prevented from the zone of close approach between two particles. As a result of this low polymer concentration between the particles due to size exclusion, there is a lower osmotic pressure, which results in (1) an attractive force for greater than theta solvents and (2) a repulsive force for less than theta solvents. Theta solvents will be discussed in the section on the thermodynamics of polymer solutions, but first a discussion of polymer properties.

FIGURE 10.18 Schematicrepresentation of depletion stabilization. Note that drawing of polymers is not to scale.

451

10.3 I n t e r a c t i o n E n e r g y a n d Colloid S t a b i l i t y

10.3.3.1 Types of Polymers, Their Characteristics, and Their Adsorption Conformations Polymers, which received their name from Berzelius in 1833, are macromolecules which have m a n y "mer units" or parts. Each mer unit (A) may have several functional groups but it is repeated n times to form the polymer. T...

-(A)

n -

...

T

Such a polymer has terminal groups T which can be the same or different t h a n the mer unit, A. A polymer composed of several types of mer units, for example, A and B, is called a copolymer. Copolymers may be Statistical: Block: G r a f t (like a comb):

... A A B B A B A B B A B A A B . . . ... A A A A A A B B B B B B B B . . . ...AAAAAAAAAAAAAA... B B B B B B B

Copolymers are interesting steric stablizing polymers because one part of the copolymer can be used to chemically bind or physically adsorb to the surface of the particle and the other part can be used to extend into the solvent to give steric interactions with other particles. Polymers are typically a mixture of molecular weights and not just one. The distribution of molecular weights is often characterized by the n u m b e r average molecular weight,

~, niMi <Mn)_

i

(10.35)

~,ni i

the weight average molecular weight,

~, w~M~ ~, niM~ (Mw) = i =i_2_______ 2 wi ~, niMi i

(10.36)

t

and the ratio of the weight average molecular weight to the number average molecular weight, (Mw)/(Mn}. This ratio ranges from 1 (monodisperse) to very large values (polydisperse) depending on the method of polymer synthesis. Depending on its structure, the polymer, will have different conformations in a mixture with itself t h a n in a mixture with a solvent. The conformation of a polymer is typically coiled. The rms end-to-end distance, (r2) 1/2, is one measure of the diameter of the coil. The rms

452

Chapter 10

Colloid Stability of Ceramic Suspensions

end-to-end distance depends on the molecular weight as seen in Table 10.5 (see also Figure 9.23). Another measure of the size of a polymer coil is the rms radius of g y r a t i o n , R G , defined by

(r2} 1/2 RG -

(10.37)

The radius of gyration of a polymer can be measured directly by light scattering. The span of a polymer chain in a given direction is defined as the maximum extension of the chain in that direction. The conformation averaged span, Ls, is given by

Ls = 0.92 (r2) 1/2

(10.38)

The radius of the smallest sphere which contains all the segments of a polymer coil and is centered at the starting point of the chain is called the Hollingsworth radius. The conformation averaged Hollingsworth radius, (RHoU) , is given by [27] (RHoll } = 1.14 (r2) 1/2

(10.39)

In a solvent the polymer coil tends to swell to some degree depending on the n a t u r e of the solvent. The dimensional expansion is measured by the intramolecular expansion factor, (r2} 1/2

c~- (r2)~/2

(10.40)

where ~/eis the rms end-to-end distance for the reference conformation state (i.e., 0 solvent). In a water solution, the functional groups

TABLE 10.5 Typical Dimensions of Linear Polymer Chains Using the Formula a (r2)l/2/nm ~ 0.06 M 1/2 Mw

l/2/nm

1,000 10,000 100,000 1,000,000

2 6 20 60

a Napper, D. H., "Polymeric Stabilization of Colloidal Dispersions." Academic Press, New York, 1983.

10.3 Interaction Energy and Colloid Stability

453

on the polymer will under some conditions become ionized. As a result, the polymer will uncoil due to charge repulsion. The way a polymer will adsorb at an interface depends on the chemical nature of the polymer molecule, the nature of the ceramic powder surface, and the solvent. Polymers typically adsorb at random points along their backbones. Rarely do they collapse onto the surface, thereby sacrificing entropy in the transition from three to two dimensions. Instead, the adsorbed chain consists of a collection of trains (where each segment contacts the surface), loops (where only the initial and final segments contact the surface), and tails (which begin at the surface and terminate in the solution) (see Figure 10.19). At high surface concentrations, the density distribution function is the only mea~s of characterizing the polymer at the interface as shown in Figure 10.20 [35,36]. The density distribution function can be measured by small angle neutron scattering [36]. The mixing of two of these attached density distributions gives rise to steric stabilization as shown in Figure 10.21 [37]. The thermodynamics of this mixing establishes the force of repulsion (or attraction) between the two surfaces that result.

10.3.3.2 Thermodynamics of Polymer Solutions The earliest and best known theory of polymer solution thermodynamics was set forth by Flory [38] and Huggins [39]. The Flory-Hug-

Conformation of adsorbed polymer molecules with adsorption nomenclature: (a) single point attachment, (b) loop adsorption, (c) fiat multiple site attachment, (d) multilayer adsorption, (d) random coil, (e) nonuniform segment density distribution.

FIGURE 10.19

454

Chapter 10

Colloid Stability of Ceramic Suspensions 1.0

0.5

^

!

P nml

o 0.2--

!

6

12

18

20

30

b

1

0.1-

O.

10

40

Distance/nm FIGURE 10.20 (a) Normalized segment density distribution function determined by neutron scattering for poly(vinyl alcohol) with a hydrodynamic radius of 18 nm, (b) poly(oxyethylene) curve 1 theoretical segment density distributions of the adsorbed polymer, taken from Hesselink [35], and curve 2 experimental results. Both experimental results taken from Cosgrove et al. [36]. Figure redrawn from Napper [27] with permission from Academic Press.

a

>, c.

c

b /~\

5

lO

E

o} G) O0

I

0

40

Plate

9

80

separation

F I G U R E 10.21 Segment density distribution function for two flat plates stabilized by tails for an excluded volume parameter of 5 and 0 (dashed line) according to Edwards [37]. (a) Plate separation 10, (b) Plate separation 80. Figure redrawn from Napper [27] with permission from Academic Press.

455

10.3 Interaction Energy and Colloid Stability

gins theory predicts the Gibbs free energy of mixing, A G M, polymer with a pure solvent: AG M -

ksT{nl In ~bl + n9 In (be + nl (be X1}

a

pure

(10.41)

where ~b~ is the volume fraction of solvent, (be is the volume fraction of polymer, n~ is the number of solvent molecules, ne is the number of polymer molecules (which fill the space of x solvent molecules), and X~ is the dimensionless interaction parameter which is the pivotal parameter in the Flory-Huggins theory. In this expression the first two log terms are due to entropy of mixing, and the third term is due to enthalpy of mixing. The osmotic pressure of a polymer solution can be determined from the Flory-Huggins theory, given by

~r = (RgT/VI)[ln(1 - @e) + (1 - 1/x)~be + X~ ~b~ + . . . ]

(10.42)

or, in terms of polymer concentration, c2 [= 4~2M2/V2],

7r = RgT{c2/M2 + [(V2/M2)2/VI] (1/2 - Xl)C~ + . . . }

(10.43)

where x is the number of solvent molecules displaced by the polymer molecule, which may be approximated by x = V'2/V1. Here, V1 and V2 are the molar volume of the solvent and polymer, respectively, and M 2 is the molecular weight of the polymer. This equation can be simplified for dilute solutions to give

RgT 7r = ~ { 4 ~ 2 / x + ( 1 / 2 - Xl)@~ + . - . }

y~

(10.44)

the R~T4~2/(xV1) [or RgTc2/M2] term in this equation is the classical van't Hoff infinite dilution expression for the osmotic pressure and the second term can be referred to as the second virial term with a second virial coefficient B 2 [= ( 1 / 2 - Xl)@~/V1]. The second virial coefficient becomes 0 when Xl = 1/2, called the Flory point or the theta (0)point. Usually the second virial coefficient becomes 0 at some temperature called the theta temperature. The theta temperature is formally introduced into the Flory-Huggins theory via the equation 1/2 - Xl = 01 - K1 = 01(1 - OIT)

(10.45)

where K~ is the enthalpy of dilution and 0~ is t h e entropy of dilution. Clearly, if T = 0, then is X~ = 1/2, which is the ideal value of X~. Table 10.6 gives a list of 0 temperatures for different polymer-solvent systems. The enthalpic and entropic components of the interaction parameter for poly(methyl methacrylate) in various solvents is given in Table 10.7.

456

Chapter 10 Colloid Stability of Ceramic Suspensions

T A B L E 10.6 Comparison of Critical Flocculation T e m p e r a t u r e s (CFT) with 0 T e m p e r a t u r e s for Some Aqueous and Nonaqueous Dispersions a

Upper or

Stabilizing Moieties Aqueous: Poly(oxyethylene)

Poly(acrylic acid)

Poly(vinyl alcohol) Poly(acrylamide)

Nonaqueous: Poly(isobutylene) Poly(isobutylene)

Poly(methyl styrene) Polystyrene Polystyrene

Poly(dimethylsiloxane)

Mw

1,400 4,700 10,000 13,500 23,000 49,000 96,00O 315,000 800,000 1,000,000 9,800 19,300 51,900 89,700 16,000 26,000 57,OOO 270,000 18,000 60,000 180,000 23,000 150,000 760,000 760,000 760,000 760,000 760,000 9,400 9,400 110,000 110,000 6,000 6,700 22,600 3,200 11,200 23,000 48,000

Dispersion Medium

0.39MMgS04

0.2MHC1

2.0MNaC1

2 . 1 M ( N H 4 ) 2 SO4

2-methylbutane 2-methylbutane 2-ethylpentane 2-methylhexane 3-ethylpentane cyclopentane n-butyl chloride cyclopentane n-butyl formate

n-heptane/ethanol ( 5 1 : 4 9 v/v)

CFT (K)

317 • 320 318 318 314 316 316 315 316 317 287 • 289 283 281 286 • 320 • 301 312 292 • 295 • 280 •

Lower CFT

2

U

2

L

2 3

U

3 5 7

L

325 +_ 1 325 327 381 423 463 455 254 _ 1 403 • 1 280 410 264 • 1 264 _ 1 264 +_ 1 340 340 341 338

U U

L U L U L

L

O(K)

330 327 319 315 315 314 315 314 316 315 287 289 287 287 287 300 300 300

- 10 • 5 +_ 3 • 3 _ 3 +_ 3 _ 3 +_ 3 • 3 • 3 _ 5 • 5 • 5 • 5 _ 5 - 3 - 3 • 3

325 + 2 325 - 2 318 376 426 458 461 263 412 293 427 264 264 264 340 • 2 340 +- 2 340 - 2 340 +_ 2

a D a t a from Napper, D. H., "Polymeric Stabilization of Colloidal Dispersions." Academic Press, New York, 1983.

10.3 Interaction Energy and Colloid Stability

4~

TABLE 10.7 Enthalpic, XH, and Entropic, Xs, Components of the Interaction Parameter for Poly(Methyl Methacrylate) in Various Solvents at 20~ a 0(X)-

XHandC3(Tx-----~)= T dT

dT

Xs, thus X = XH+ XS Solvent

X

XH

XS

Acetone Benzene Chloroform Tetrahydrofuran Toluene

0.481 0.429 0.365 0.446 0.452

0.0228 -0.017 -0.075 0.026 0.028

0.45 0.45 0.44 0.42 0.42

a Data taken from Casassa, E. F., J. Polymer. Sci., A-2 8, 1651 (1970).

The T h e t a point is the t e m p e r a t u r e of a p o l y m e r solution where: 1. The p o l y m e r chains can have a n y conformation w i t h i m p u n i t y insofar as the free e n e r g y is concerned, a n d 2. The p o l y m e r molecules have no i n t e r a c t i o n s w i t h the solvent a n d b e h a v e as if t h e y were volumeless lines in space a d o p t i n g t h e i r u n p e r t u r b e d dimensions. This second definition of t h e t a point is used to describe a solvent. In a solvent w h e r e X~ < 1/2, we have a b e t t e r t h a n t h e t a solvent conditions w i t h a second virial coefficient Be, w h i c h is positive. In a solvent w h e r e Xl > 1/2 we have less t h a n t h e t a solvent conditions w i t h a second virial coefficient B2, which is negative. At the t h e t a point, the p o l y m e r s e g m e n t i n t e r a c t i o n s switch over from being repulsive to being attractive, c a u s i n g p h a s e separation. P h a s e s e p a r a t i o n t a k e s place u p o n h e a t i n g or cooling of a p o l y m e r solution, d e p e n d i n g on the values of K 1 a n d ~1 as s h o w n in Table 10.8.

TABLE 10.8 Thermodynamic Types of Polymer Phase Separation (and Steric Stabilization): limr K1 and lim%_.0Xs = (1/2 - ~1) Stabilization Type

K1

01

K1/~1

Phase Separates (or Flocculates)

Enthalpic Entropic Combined

+ -

+ +

>1 <1 <0

On heating On cooling On heating

458

Chapter 10 Colloid Stability of Ceramic Suspensions

Phase separation and flocculation are similar, suggesting that the Gibbs free energy of mixing of a polymer with the solvent is responsible for both processes, and as a result, phase separation and flocculation have similar effects on heating and cooling. The Flory-Huggins parameter Xl was initially derived as an experimentally determined interaction enthalpy parameter which was supposed to be independent of polymer concentration. However, experiments have shown that it is dependent on polymer concentration. Evans and Napper [40] have shown that the polymer volume fraction dependence of X can be given by X = X~(1 + 2~b2/3 + ~b~/2)

(10.46)

This equation may be considered to be just an extension of the virial expansion and as such is empirical. This equation has been shown to be valid up to a volume fraction of ~b2 - 0 . 2 5 for nonpolar systems. The X parameter can be determined by measuring the activity of the solvent, a~, in a polymer solution using the equation In al ----In r § ~2 + X ~

(10.47)

Barton [41] has assembled a well-referenced source book for the derivation and use of X and cohesion parameters for various polymer solvent pairs. There are many ways to measure solvent activity, the simplest being boiling point elevation, freezing point depression, and osmotic pressure discussed in Section 11.5, "Solution and Suspension Colligative Properties." Because the interpretation of polymers during steric stabilization creates high polymer volume fraction (or concentration) at the point of interpenetration, a good expression for the Gibbs free energy of mixing at high polymer concentration is needed. For this reason, concentrated polymer solutions have been the subject of much theoretical research, which has resulted in the self-consistent mean field theory developed by Edwards [37,42] and the scaling law theory developed by the 1992 Noble prize work of de Gennes [43]. In the self-consistent mean field theory, the polymer chain is characterized by a density of polymer configurations available to its subchains, where each subchain interacts with the others via an interaction energy. The neglect of correlations among segments makes this a mean field approximation. Self-consistency is ensured through the relationship between the local segment density and the probability density. The self-consistent mean field theory is used to develop equations for the steric interparticle interacting energy. The scaling law theory is for the most part beyond the scope of this work, but several results will be used in this chapter. In the scaling law theory, the transition from one regime (i.e., dilute to semidilute

lO.3 Interaction Energy and Colloid Stability

459

or semidilute to concentrated) is expressed as a critical phenomena described by a function f(c2/c~) = 1

if c2/c~ < 1.0

(10.48)

f(c2/c~) = (c2/c~) m

ifc2/c~ > 1.0

(10.49)

where c~ is the critical polymer concentration for the transition and m is the scaling law exponent. For this theory, a thermodynamic property, the osmotic pressure in the semidilute range, for example, is given by 7r = 7rof(C2/C ~ )

(10.50)

where 7to is the dilute solution osmotic pressure defined by equation 10.43. The value of the scaling law exponent, m, is determined by heuristic a r g u m e n t s based on the definition of the critical concentration, c ~ [ - M 2 / N A R~]. As a result [43], m for the osmotic pressure is given by values of 9/4 for a good solvent and 3 for a 0 solvent. Selfconsistent mean field theory gives a value of m = 2 for a good solvent as a comparison. 10.3.3.3 S t e r i c I n t e r a c t i o n

Energy for Two Plates

To determine the steric interaction energy for two parallel plates at a separation, h, the mixing free energy per unit a r e a A , AGM/A, for the approach of two sterically stabilized particles from infinite separation to a separation, V FP ( h ) = A G M ( h ) / A - AGM(h = ~ ) / A

(10.51)

The mixing free energy as a function of the separation, AGM(h), is determined from the volume differential, o A G M / S V . The Flory-Huggins theory for A G M is given by [38,39] A G M --

k s T { n l In 4)1 + n2 In ~b2 + nl 4~2X1 + ..-}

by noting that ~bl = 1 - r On l, we have

(10.52)

and taking the derivative with respect to

OAGM/Onl = ksT{ln(1 - ~b2) + ~b2X1 + ...}

(10.53)

which can be related to 0 V(= A Oh) by the following relationship: c~nl =

(1 - ~2)0V

Yl

(10.54)

giving oAGM/oV

-

k s T V l { l n ( 1 - ~2) + ~2 X1 + ...}/(1 - ~b2)

(10.55)

which may be integrated over the volume between the two plates as shown in Figure 10.17, if the polymer mole fraction distribution, ~2(x), is known as a function of distance from the plate [44]:

46{}

Chapter 10 Colloid Stability of Ceramic Suspensions

T A B L E 10.9

Steric Interaction Energy per Unit Area for Two Parallel Plates:

V FP(H) = 2kBTF2g(H)

Re/. g(H) where H

=

~

is the dimensionless separation

1 Rigid molecule, entropy only at low surface coverage, (i.e., 0 < 0.2) 1

21n[ ] Rigid molecule, entropy only at high surface coverage F2

B2~s(1 -H) 1?2 (1/2 - X) _ (0 - K) 82 =M21? 1 ( 1 / 2 - X) = 171p22 - l?lp--~2

c-f

Flory Huggins theory for constant segment density (2rr/9)3/2 F2(a 2 - 1)LsM(H) + E(H) (8/3rr) '/2

h,i

where M(H) is the mixing function and E(H) is the elastic free energy; see Figure 10.22. The value of the polymer expansion factor, a, can be determined from the Flory X parameter as follows: a5 _ a3 = (9/27r)3/2 Vs2(1/2 -

X)

~71L3/2 where Vs is the segment volume. The equation is known to be too large by a factor of 2 near the theta point, g {E(H) + (u/X/3)M(H)} where u[= (1/2 - X) V2s/V1 F2/(Ls/x/-~] is the excluded volume parameter, M(H) is the distance-dependent mixing function, and E(H) is the distancedependent elastic free energy given in Figure 10.22. Mean field theory for better than-0 solvent. + In - - ~ - j Mean field theory for 0 solvent.

Scaling laws for elastic contributions only.

j,k

10.3 Interaction Energy and Colloid Stability

461

AGM(h)/A = 2kBTV1 ~ ~h/2 {ln[1 - ~b2(y)] + 4~2(Y)X1 +...}/[1 - ~2(y)]0y J0

(10.56) Various theories [27] of polymer mixing are used to determine the polymer mole fraction distribution function, ~2(Y), near the surface of the plate in interaction with one another at a dimensionless separation, H{= [h/2Ls]}, where Ls is the conformational average span of the polymer. The various results for which are given in Table 10.9 as a function of the adsorbed capacity, F2. The mixing and elastic free energy functions given in this table are plotted in Figure 10.22. The equations in this table have not as yet been verified experimentally. Taking the Flory-Huggins expression for the interaction energy, the steric repulsion for two flat plates with a polymer adsorbed on its surface at monolayer coverage is plotted in Figure 10.23 [45] as a function of the interparticle distance for different molecular weights. As the molecular weight increases, the polymer span increases, causing the steric repulsion to occur over a greater distance. 10.3.3.4 S t e r i c I n t e r a c t i o n E n e r g y for T w o S p h e r e s

The Derjaquin approximation [9] can be used to compute the interaction energy for two spheres of radius al and a2, VsSSR(h), if the fiat plate interaction energy, VFR P (h), is known"

V ss (h) ~ 2~r

a la2 aI + a2

s;

FP (s)ds VSR

(10.57)

Conversely the interaction energy can be derived from the integration of with respect to the interaction volume:

VS~ ( h ) ~

SoYO V

dV=

SovkBTV~ {ln(1 -

~b2) + ~b2X~+ ...}/

(10.58)

(1 - q~2) OV

Mackor, E. L., J. Colloid Interface Sci. 6, 492 (1951). b Bagchi, P., and Vold, R. D., J. Colloid Interface Sci. 33, 405 (1970). c Ottewill, R. H., and Walker, T., KoUoid-Z. Z. Polym. 227, 1089 (1968). d Fischer, E. W., KoUoid-Z. 160, 120 (1958). e Dunn, V. K., Ph.D. Thesis, University of Southern California (1974). f Everett, D. H. Faraday Discuss. Chem. Soc. 65, 215 (1978). g Stockmayer, W. H., Makromol. Chem. 35, 54 (1960). h Hesselink, F. T. J. Phys. Chem. Phys. 73, 3488 (1969); 75, 65 (1971). i Maier, D. J., J. Phys. Chem. 71, 1867 (1967). J Dolan, A. K., and Edwards, S. F., Proc. R. Soc. London, Ser. A 343, 427 (1975). k Gerber, P. R., and Moore, M. A., Macromolecules 10, 476 (1977). t Dolan, A. K., and Edwards, S. F., Proc. R. Soc. London, Ser. A 337, 509 (1974). m De Gennes, P. G., Macromolecules 15, 492 (1982). a

462

Chapter 10

Colloid Stability of Ceramic Suspensions

FIGURE 10.22 Mixing, M(H), and elastic, E(H), interactions of polymers attached to two interacting surfaces. Plot of functions used in Table 10.9.

10-

V kBT

0

o

2'o

4'o

6'o

Ho/nm FIGURE 10.23 The distance dependence of the steric interaction energy of latex particles stabilized by poly(vinyl alcohol) in water: (1) 500 nm, (2) 100 nm, and (3) 10 nm. The elastic modulus is 1.4 x 103 nm -2. Data from Sonntag [45].

10.3 Interaction Energy a n d Colloid Stability

463

where the upper limit of the volume integration V[= (1 - H) 2 (3a/Ls + 2 + H)] is the volume of the lens shaped section of polymer overlap between the two identical particles of radius a shown in Figure 10.13(a). Using either calculation method, the interaction energy between two spherical particles of radius, a, is given by [46] g ~ S (h) =

3---~p~

(1)

- X Ls(1 - H ) 2 (3a/Ls + 2 + H )

(10.59)

where F2 is the concentration of the adsorbed layer, k B is the Boltzmann's constant, T is the absolute temperature, P2 is the adsorbate's absolute density, ~?~ is the molar volume of the solvent, Ls is the thickness of the adsorbed layer, and X (= Xs + ~ ) is the dimensionless parameter encompassing both enthalpic (XH) and entropic (Xs) adsorbate-solvent interactions. Reviewing this equation, we find that the value of X > 0.5 leads to attractive interaction energies and colloidal stabilization, therefore, flocculation; and a value of X < 0.5 leads to repulsive interaction energies. A value of X = 0.5 gives an interaction energy of 0 for all separations, h. The two parameters X and L~, together with the strength of the anchoring mechanism, are the three experimental variables that can be manipulated in steric stabilization. Figure 10.24 is a plot of the steric interaction energy, VSSR s (h), for two equal spheres. The interaction

F I G U R E 10.24

V SR ss ~ ~, h ) as a f u n c t i o n of s e p a r a t i o n , h, for t w o e q u a l s p h e r e s of r a d i u s 1 ~ m w i t h v a l u e s of (a) Ls = 2 n m ( M w ~ 1,000), (b) Ls = 6 n m (Mw ~ 10,000), a n d (c) L s = 2 n m (M~ ~ 100,000); F2/P2 = 0.01, X = 0.3, V1 = 18 cc/mole.

464

Chapter 10 Colloid Stability of Ceramic Suspensions

energy increases as the distance decreases for a better than theta solvent and decreases as distance decreases for a less than theta solvent. Other equations given in Table 10.9 for two parallel plates can be changed to an equation for two equal spheres by multiplying g ( H ) by the factor (1 - H ) 2 (3a/Ls + 2 + H ) / L s

(10.60)

10.3.3.5 Interaction Energy Due to Nonadsorbing Polymer Due to the osmotic pressure of a polymer solution, an immersed particle will experience a force acting normal to its surface. For an isolated particle, the integral of the pressure over the entire surface gives a zero force. When the particles are closer together than the radius of the polymer, (r2) 1/2, there is a polymer exclusion zone along the line of centers (as shown in Figure 10.18), which gives an attractive force which results in flocculation. For separations h < (r2) 1/2, the interparticle force is given by the product of the osmotic pressure, rr, and the surface area of the particle in contact with solvent only (i.e., HA). The resulting depletion interaction energy, VD(h), is given by [47] VD(h) = II 8 V

(10.61)

where 6 V is the volume of the solvent only region. For two equal spheres of radius, a, 6 V is given by 6 V = 7r/12{[(r2) 1/2 - h] 2 [2(2a + (r2) 1/2) + 2a + h]}

(10.62)

thus the depletion interaction energy, VD(h), is given by VD(h) = 0 for h >
(10.63)

where c2 is the polymer concentration in solution. By analyzing this equation, we can see that the geometric factor is always positive for all separations and the osmotic pressure term is always negative, because the second virial term is always small compared to the first. The resulting attractive interaction energy rarely exceeds 1 ksT. Asakura and Oosawa [47] also showed that, for flexible macromolecules, the configuration entropy of the chains is decreased in the neighborhood of the particle surface, and this provides a source of repulsion. With their correction for configurational entropy the depletion interaction

10.3 Interaction Energy and Colloid Stability

465

energy is given by VD(h) = 0 for h > 1/2 VD(h) = - R T { c 2 / M 2 + [(V2/M2)2/V1](1/2 - Xl)c~ + . . . } 7r/12{[1/2 - h ]212(2a + 1/2) + 2a + h ]}

[1 - f(h)]

df(h)~

d~nhJ

(10.64)

>~/2

forh <
where f(h) =

(8/7r2) ~

(1/i 2) exp I-]i27r2_(r~/2>2~

/:odd

k

6h2

J

(10.65)

In this equation, the - d f ( h ) / d In h term is the effect of the repulsion due to the decrease in configurational entropy of the polymer molecules at the surface of the particles and the 1 - f(h) term is the augmented osmotic pressure attraction. Asakura and Oosawa have showed that the energy becomes larger when the polymer molecules have an unsymmetric shape. If the molecules are charged (i.e., polyelectrolytes), the energy is much larger. Several theories have been put forward to account for the distribution of polymer segments in the depletion zone. The theories of Feigin and Napper [48] and Scheutjens and Fleer [49] are qualitatively different from the theory of Asakura and Oosawa and de Gennes and coworkers [50,51] in that they predict not only depletion flocculation but also depletion stabilization. Depletion stabilization has not to date been verified experimentally although depletion flocculation has been verified experimentally for several systems [52,53]. The effect of an adsorbed polymer layer [54] and ordered solvent layers [55] on depletion flocculation is also under theoretical attack. The depletion stabilization interaction energy cannot simply be added to the other interaction energy terms to give the total interaction energy.

10.3.3.6 Interaction Energy Due to Electrostatic Effects With polymers that have ionizable groups, adsorption of a polymer will alter the charge of the surface altering the electrostatic interaction energy and also provide steric protection for the colloid, because the ionized groups will give better than theta conditions for the polymer in an aqueous solution. This type of polymer stabilization is called electrosteric stabilization because both the electrostatic and the steric play a role in stabilization. The equations for this total interaction are simply the sum of electrostatic and steric terms as well as the van der Waals attraction.

466

C h a p t e r 10

Colloid Stability of Ceramic Suspensions

10.3.4 Total Interaction Energy If the repulsive energy due to the electrostatic interactions is VER and the steric stabilization is VSR then the total interaction energy VT is given by

VT(h) = VA(h) + VD(h) + VER(h) + VsR(h)

(10.66)

where VD is the depletion energy and VA is the van der Waals attraction. All of these interaction energies are a function of the particle separation, h. This total interaction energy frequently has the form shown in Figure 10.25, where there is zero interaction at large distances, followed by a slightly attractive minimum at smaller separations called the secondary minimum, followed by either a repulsive barrier at short separations when there is a strong repulsion or an attractive region when there is no strong repulsion. At very short distances there is always a repulsive interaction caused by the compression of the last monolayer of ordered solvent molecules. The interaction energy will affect the rate of particle coagulation or flocculation. A weak electrostatic interaction will not be sufficient to overcome a strong van der Waals interaction and result in coagulation.

1 0 . 2 5 Total interaction e n e r g y VT(h) = VA(h) + VER(h) + VsR(h) between two equal spheres with radius 1.0 t~m. F2/P2 = 0.01, X = 0.3, V1 = 18 cc/mole, A12321 = 5 . 7 k s T , T = 298 K Curve A. 0.01 m o l a r KC1, ~ = - 5 0 mV, Ls = 6 n m ( M w ~- 10,000). Curve B. 0.0 m o l a r KC1, ~ = - 2 5 mV, L s = 6 n m (Mw ~- 10,000). Curve C. 0.01 m o l a r KC1, ~ = - 5 0 mV, L s = 2 n m (Mw ~ 1,000). Curve D. 0.1 m o l a r KC1, ~ = - 2 5 mV, Ls = 2 n m (Mw ~- 1,000).

FIGURE

10.4 Kinetics of Coagulation and Flocculation

467

Similarly, a weak repulsive steric interaction in combination with a strong van der Waals interaction will result in flocculation. The kinetics of coagulation and flocculation in the presence of a total interaction energy profile either electrostatic or steric is discussed next.

10.4 K I N E T I C S O F C O A G U L A T I O N AND FLOCCULATION 10.4.1 D o u b l e t F o r m a t i o n Doublet formation is the first step of aggregate or cluster formation. When salt or pH is used to destabilize a colloidal suspension it is referred to as coagulation. When a polymer or surfactant is used to destabilize a colloidal suspension it is referred to as flocculation. The kinetics of doublet formation for both these methods of destabilizing a colloidal suspension is discussed in this section.

10.4.1.1 B r o w n i a n Coagulation The rate at which particles coagulate when the interaction energy between the particle is 0 was first investigated by von Smoluchowski [56]. This condition is defined as rapid coagulation in which the rate of disappearance of primary particles, J0, is equal to the frequency of collision between the particles:

Jo =

dN~ - 4kBTN~ dt 37

(10.67)

where No is the n u m b e r density of primary particles and ~ is the liquid viscosity.

P r o b l e m 10.2. D e t e r m i n e the Half-Life for D o u b l e t F o r m a t i o n for Various I n i t i a l N u m b e r Densities o f P a r t i c l e s in Water Solution The preceding equation can be integrated to give the dependence of N Owith time, assuming an initial condition that at t = 0, No is a constant. The half-life is defined at the time when one-half of the original particles are left. The result of an integration of equation 10.67 gives the half-life for von Smoluchowski coagulation kinetics is tl/2[ = 3~/(4NoksT)]

(10.68)

The half-life is given in the following as a function of initial particle number density, No in water at 25~ [27] that result from tl/2 = 37/ (4NoksT).

468

Chapter 10 Colloid Stability of Ceramic Suspensions tl/2

No

1 ns 1 its 1 ms 1 sec lmin 1 hr 1 day

1.6 1.6 1.6 1.6 2.7 4.5 1

( n u m b e r p e r d m 3) x x x x x x x

10 23 10 2o 1017 1014 1012 101~ 10 3

The half-life of coagulation is strongly dependent on the initial particle number density. When there are energy barriers between the particles, for example, attractive and repulsive interaction energy barriers like those discussed in the previous section, Fuchs [57] showed that the rate of coagulation, J, should be divided by a factor W, the colloid stability ratio, where W is given by [58,59]

W = 2a

f: a

D(h = ~) (VT(h = r - 2a)) dr D(h = r - 2a) exp ksT -~

(10.69)

In this equation, VT is the total interaction energy between the two colliding particles defined in the previous section. The stability ratio, W, for the system gives the ratio of rapid coagulation, Jo, to slow coagulation, J [ = Jo/W]. D(h) is the position-dependent diffusion equation. This diffusion coefficient ratio is a factor that decreases the collision rate because of the difficulty in draining the liquid between the two solid surfaces. This diffusion coefficient ratio is given by [60,61]

D(h = r - 2a) = H

= 2h/a

for h ~ a (10.70)

1

forh >>a

When VT is 0 for all separations between the particles, W is equal to 1. In all practical cases, however, the interaction energy between the particles is not equal to 0, but is either repulsive or attractive or a mixture of both. The rate of collision between two particles is thus increased or decreased by the presence of an attractive or repulsive force between them. If we have only an attractive interaction (VT is negative), the value of W is less than unity. W has been predicted (and measured [62]) to be as low as 0.5 for nonretarded attractive Hamaker interactions [3, p. 276; 63]. If we have only a repulsive interaction (VT is positive), W is larger t h a n unity. Values of W can be obtained from doublet formation kinetic measurements as determined from photon

469

10.4 Kinetics of Coagulation and Flocculation

correlation spectroscopy (PCS) measurements [64] or low-angle lightscattering measurements [3, p. 275]. Experimental colloidal stability ratios for 0.57 t~m SiO2 in aqueous KC1 solutions plotted as a function of pH obtained with KOH and HC1 additions is shown in Figure 10.26. The SiO2 particles are stable (W > 1.0) at pH values above the isoelectric point, that is, pH 2.0. Generally as the salt concentration increases the silica is less colloidally. But there are some complications to this argument near pH 7, where the silica surface is dehydrated at low KC1 concentrations (see the text following equation 10.72.) The W values [65] for a dispersion of A1203 as a function of pH and KNO3 salt concentration are shown in Figure 10.27. The A1203 particles are colloidally stable far away from their isoelectric point (i.e., pH -8.9). As the salt concentration is increased the zeta potential decreases and the colloid stability ratio, W, decreases. Near the isoelectric point there is no electrostatic repulsion, giving a rapid coagulation. The W values [25] for a 0.313 t~m TiO2 as a function of pH and KCL

40

~z~z~,,,, 0000~ 9 :~:~ A.A,A,,,A

30-

~ m ~

3.0M 1.0M 0.5M 0.1M 0.03M

KCl KCl KCl KCl KCl 0.01M KCl

_

O

. w

20 >,~

i

-

. m o E

t

10-

oi 2

4

6

pH velue

8

10

FIGURE 10.26 Colloidal stability ratio for 0.57 t~m SiO2 particles as a function of solution pH for various 1" 1 salt concentrations (i.e., KC1). Taken from Chang [62].

470

Chapter 10 Colloid Stability of Ceramic Suspensions

FIGURE 10.27 (a) Zeta potential as a function of pH for Al203 in an indifferent 1:1 electrolyte solution (i.e., KNO3). (b) Colloid stability ratio for the same A1203 sol as a function of pH. The minimum values correspond to the isoelectric point at pH ~ 9. Data from Wiese and Healy [65].

salt concentration are shown in Figure 10.28. The TiO2 particles are colloidally stable far away from their isoelectric point (i.e., pH 5.5). Near the isoelectric point there is no electrostatic repulsion (i.e., only Van der Waals attraction), giving a rapid coagulation, and the value of the stability factor W is less than 1. When the colloid stability factor, W, is plotted versus salt concentration, as shown in Figure 10.29, we can see that there is a critical salt concentration above which the suspension is unstable (i.e., W - 1.0). This critical coagulation concentration (CCC) is predicted by the Schultz and Hardy rule [66] to depend on the valence of the counter ion to the - 6 power (i.e., the ion of opposite charge to that of the particle):

10.4 Kinetics of Coagulation and Flocculation

471

a 60

~> E

0 0 o

40

v

Ti 02 A

_J
hz i,I p0 s I

1E-1PI K C l IE-2H KCl 1 E - 3 H KC I

0 -20

-40

2

3

4

5

15

7

8

9

+

2.5E-2

10

pH

9

,/~'=

I .5E-2

/-

7.5E-3

H H H

O 0

-1

2

3

4

5

6

7

8

9

pH FIGURE 10.28

(a) Zeta potential as a function of pH for 0.313 tLm TiO2 in indifferent 1 : 1 electrolyte solution (i.e., KC1). (b) Colloid stability ratio for the same TiO2 sol as a function of pH. The m i n i m u m values correspond to the isoelectric point at pH ~ 5.5. D a t a from B a r r i n g e r [25].

CCC a z -6 1-6.2-6.3-6 = CCC(K+I). CCC(Ba+2). CCC(Fe +3)

(10.71)

For the data on the coagulation of TiO2, this prediction holds for Ba § and K § counterions in acid solutions. The S c h u l t z - H a r d y rule can be predicted from the Derjaquin, Landau [67], Verwey and Overbeek [66] (DLVO) theory. The DLVO theory uses a sum of the van der Waals interaction energy and the electrostatic interaction energy to determine the total interaction energy for two spheres. This total interaction energy, VT(h), and its derivative with respect to separation, dVT(h)/dh, defining the m a x i m u m interaction energy, are set equal to

4 72

Chapter 10 Colloid Stability of Ceramic Suspensions

p

KCI

BoCI=

" H

o

:3=

pH

o o .j

-

7.5

KCI -

8.8

1 D =

-1

-5

i

'

-'4

'

-'3

'

-'2

'

!

!

-,

LOG MOLAR CONCENTRATION FIGURE 1{}.29 Colloid stability ratio for different salt concentrations showing the critical coagulation concentration, CCC. Data from Barringer [25].

0. This mathematical condition defines a critical separation in terms of a critical double layer thickness, K-1, which can then be used to relate the critical salt concentration to the valence of the counterion. The result is the relation between the salt concentrations responsible for the CCC's being proportional to the counterion valences to the - 6 power. From experimental values of the stability ratio, the maximum in the interaction energy can be determined according to the calculation [59]

W- W(rapidcoagulation)+ O.25[exp ( VT(max)] ksT

(10.72)

Experimental values of VT(max) are in reasonably good agreement with those predicted with the DLVO theory for many systems, which include the preceding TiO2 particles. However, this is not the case for SiO2. The maximum in the interaction energy profile, VT(max), for 0.57 t~m SiO2 particles is given in Figure 10.30 as a function of pH and KCL salt concentration. It has been calculated from the colloidal stability ratio [68] measured by PCS, which is also shown in Figure 10.26. Comparing the SiO2 results to the TiO2 results, we find (1) a lower pH where W ~ 1.0 (i.e., isoelectric point) and (2) much different pH dependence for each salt concentration. SiO2 has a surface that alters its surface charge by dehydration [69] near pH 7. Other spurious surface chemical effects can be observed with the zeta potential of 0.313 t~m TiO2 particles versus pH for various BaC12 concentrations [25] shown in Figure 10.31. At higher pH values and high concentrations of barium

473

10.4 Kinetics of Coagulation and Flocculation

Activation Energy (~,.=/kT)

5 L

"

e e e e e 3.OM KCI r 1 6 2 1 6 2 O.5M 1 6 2 1 6 2KCI

-

~

4

0.1M

'b.a'a'*''' O . 0 3 M m.,,,m,,,m 0 . 0 1 M

-

"

2'

KCI

.-----0

KCI KCI

:3E-,

-

O'

i

2

t

i

i

r~

i

i

i

i

4.

i

i

i

i

t

i

i

~,l

1.,i

i

6

i

i

i

I

i

i

pH value

i

i

8

!

i

i

i

i

i

I

i

i

i

i

i

i

t

10

F I G U R E 1{}.3{} Coagulation activation energy, Vr(max), for 0.57 ~ m Si02 particles as a function of solution p H for various i" 1 salt concentrations (i.e.,KCI). Calculated from the data in Figure 10.26. Taken from Chang [62].

ion, a BaTiO3 surface is produced which has an isoelectric point near pH 8.5 [70]. The colloid stability ratio as a function of pH and BaC12 concentration reflects this change in surface chemistry, as is also shown in Figure 10.31. Thus Ba § ion plays a role as a potential determining ion for TiO2 in basic solution. 10.4.1.2 B r o w n i a n F l o c e u l a t i o n Studies of the doublet formation rate in polymer stabilized systems are far less numerous t h a n those on electrostatically stabilized systems. Chang [68] has studied the doublet formation rate for 0.57 /~m in diameter SiO2 particles as a function of the amount of hydroxyl propyl cellulose (HPC) adsorbed onto their surface. Figure 10.32 is a plot of the colloid stability ratio as a function of the amount of HPC added to

474

Chapter 10 Colloid Stability of Ceramic Suspensions

a

5

,

,

,

.

,

,

,

,

,

,

TiO 2 A

,

BoCl 2

2

J-

1

o

o

m

,

+ IE-4M 8E-5 M o 5E-5 M

5

>t---

,

0 1E-$ M

"1

-2 - 5

I

5

I

!

4

I

5

I

i

i

6

I

7

i

I

i

8

i

'

9

IO

pH

b 4 + +\ 3

:~ 0 C)

_j

\

IE-4 M 9 8E-5 M

E- 5 H

2 I

0

a~u

pH FIGURE 1{}.31 (a) Electrophoretic mobility (which is proportional to zeta potential) of 0.313 t~m TiO2 as a function ofpH for various concentrations ofBaC12. Ba+2 is a potentially

determining ion for TiO2 sol in basic media. (b) Colloid stability ratio of 0.313 tLm TiO2 as a function of pH for various concentrations of BaC12. The isoelectric point of TiO2 is pH ~ 5.5 while that for BaO is pH ~ 12. Taken from Barringer [25].

the solution. Also plotted is the a d s o r b e d a m o u n t of HPC as a function of t h e a m o u n t of H P C added to the solution. It can be seen t h a t , as the a m o u n t of a d s o r b e d H P C increases from 10 .5 to 10 .3 gm H P C / m 2 SiO2, t h e colloid stability ratio increases from 1, a highly u n s t a b l e suspension, to 50, a very stable suspension. A comparison of these e x p e r i m e n t a l r e s u l t s plotted in t e r m s of a m a x i m u m i n t e r a c t i o n energy, VT(max), w i t h a n i n t e r a c t i o n e n e r g y t h e o r y t h a t includes a t e r m for

475

10.4 Kinetics of Coagulation and Flocculation

10 -2

1000

r

E 0

r

0GO

~:

10 -3"~

100

d

.m ,4~

EL. "1"

~o

v

d

m e,=.,

o

(/)

c: 10 -4 0

10

(..)

++ .o " I-

~.

~

colloid stability adsorption Isotherms

0{ D

,~ 10 -5

0

50

100

150

200

HPC conc. (wt~ of Si02)

250

300

FIGURE 10.32 Colloid stability ratio for 0.5 ftm SiO2 sol at 1 M KC1 at pH 2 with various amounts of hydroxy propyl cellulose (HPC) added. Right axis shows the adsorption isotherm of HPC under these conditions. Taken from Chang [62].

steric repulsion and van der Waals attraction was at best qualitative [68].

10.4.2 Growth a n d Structure of Large Aggregate Clusters In either multicomponent dispersions or aggregated dispersions with populations of doublets, triplets, quadruplets, and so forth, many kinds of interparticle collisions are possible. The collision rate between two different spherical particles of size ai and aj is given by

[ai + aj ] ( a//+ 1 ~) NiNj Jij - 2kBT 3, ~j/j

(10.73)

where Wij, the probability of collision between two particles i and j, is given by [71] oc

Wij = [ai + aj] f[

ai+aj] [ailfl(ui)

1 + ajlfl(uj)]

exp (V~(h = r - [ai + aj])) dr kBT -~

(10.74)

476

Chapter 10

Colloid Stability of Ceramic Suspensions

where }~(Ui) is a result of the position-dependent diffusion equation, which decreases the occurrence of collisions because of the difficulty in draining the liquid between the two solid surfaces. This fl(ui) function is given by [72]

fl(ui) =

6u~ + 13ui + 2 6u~ + 4ui

(10.75)

where ui is twice the distance from the surface of particle i to the plane of symmetry between the two particles given by bispherical coordinates"

ui =

(R - ai) 2 - a~ Rai

(10.76)

where R is the center to center distance. For a two component system (i = 1 and j = 2), the total colloidal stability ratio, WT, is defined by

Y~Ne_ N21 + Y~

+

2Y~Ne

(10.77)

where the values of N1 and Ne are the number density of components 1 and 2, respectively. This approach can also be used for a distribution of particle sizes [71], where the total colloidal stability ratio, WT, is defined as

~V--~j daidaj )-1 WT = ( ~ "~[ai+aj]2P(ai)P(aj)~ JO aiaj

(10.78)

where P(a) is the distribution of particle sizes. The time to coagulate a fraction, f, of initial particles in the distribution, P(a), is given by

tf=27r~WT[kBTd p 1 -ff]

(10.79)

where ~b is the volume fraction of particles in the suspension and is the third moment of the particle size distribution (see equation 3.3). This approach to coagulation can also be used to construct population balances to describe the growth of aggregates containing k spheres and having a radius of ak. The resulting conservation equation [3, p. 276] is k-1

dNk _ l ~ Jij - ~, Jij dt 2 i=l j=k-i i=l

(10.80)

with k = i + j and the boundary conditions N1 = No and Nk = 0 for k > 1. Difficulties in this approach lie in relating ak to k and a~ and in accounting for any size dependence of Wij. Von Smoluchowski [56] obtained an asymptotic solution for the population by assuming that collisions between clusters of approximately equal size dominate~all

477

10.4 Kinetics of Coagulation and Flocculation

the possible collisions during aggregation and Wij is independent of size. This gives

Nk =No

k+l

[1

(10.81)

where t~/2[= 3~W/(4NokBT) = 6~'a~rIW/(~bksT)] is the half-life. The total number density, NT, decreases with time according to NT =

(10.82)

N0 [1 + (t-~/2)] -1

Higashitani and Matsuno [73] have provided a direct test of this theory, which is shown in Figure 10.33. Unfortunately, this approach in describing aggregate growth gives no information about the configuration of the aggregates produced. The configuration of a silica aggregate is shown in Figure 10.34 [74,75]. Properties that depend on aggregate size can be calculated by relating the degree of aggregation to the radius ratio: (10.83)

k = (ak/ao)DR

1t

1

161'

~1 2

$

4

n k

~o

~

15

""

16

k--~l? k

~o-

162

l

( 0 a

) 2O

a

~ 4O

t/min

lOa, 60

0

2'0 b

4'0

6'0

t/min

Aggregationof polystyrene lattices 0.974 t~m in diameter with W = 1.74.Data fromHigashitaniand Matsuno[73].Comparedto the theoryofvonSmoluchowski ~ven by Nk = No (t/tl/2)k-1/(1 + t/tl/2)k+l.

FIGURE 10.33

478

Chapter 10 Colloid Stability of Ceramic Suspensions

10.4 Kinetics of Coagulation and Flocculation

479

F I G U R E 10.35 Computer generated fractal structures resulting from various kinetic growth models [77]. Fractal dimensions are listed for 3-dimensional clusters even though their 2-dimensional analogs containing 1000 primary particles are shown. Simulations by Meakin and Vicsek [78 and 89]. Note for reaction limited cluster cluster aggregation Meakin shows D F = 2.09, whereas Vicsek shows D F = 1.94 corresponding to monodisperse growth and DR - 2.3 for polydisperse growth.

where D F is the fractal dimension of the aggregate. D F has values less than 3, if the dimension for a spherical aggregate with the same volume as k particles; and greater than 1, if the dimension of a linear aggregate. Light, X-ray, and neutron scattering at intermediate wave numbers, q (i.e., 1/ao > q > 1/ak), and photon correlation spectroscopy can be used to measure the fractal dimension [76] of aggregates. The fractal dimension of an aggregate depends on the mode of aggregation, as shown in Figure 10.35 [77,78] and Table 10.10. For a rapid,

F I G U R E 10.34 Electron micrographs of (a) a gold sol with a radius of 7.2--- 0.8 nm [74], (b) silica sols with a radius of 2.7 nm [75] aggregated by Browninan motion showing fractal geometry. Photo taken from Russel et al. [3, p. 282-283].

480

Chapter 10 Colloid Stability of Ceramic Suspensions

TABLE 10.10 Fractal Dimension of Three-Dimensional Aggregate (i.e., d = 3) Growth Model

DE Particle-Cluster Aggregation: Diffusion limited a,b Reaction limitedc (Eden growth) Ballistic Cluster-Cluster Aggregation: Diffusion limited d Reaction limited d Ballistic growthe

d2+ 1 -2.5 d+l 3.0 3.0 DR 1.75 d+2 = 2.3 for polydisperse growth, Dc = 3.4 2(1 + 1/Dc) 1.94 for monodisperse growth 1.95

Meakin, P., Phys. Rev. A 26, 1495 (1983). b Tokuyama, M., and Kawaski, K., Phys. Lett. 100A, 337 (1984). c Schafer, D. W., MRS Bull. 8, 22-27 (1987). d Witten, T. A., in "Physics of Finely Divided Matter" (N. Boccara and M. Daoud, eds.), p. 212. Springer, New York, 1985. Vicsek, T., "Fractal Growth Phenomena," p. 212. World Scientific, London, 1989. a

e

irreversible B r o w n i a n aggregation adding one particle at a time with no s u b s e q u e n t r e a r r a n g e m e n t , c o m p u t e r simulations give a fractal dim e n s i o n of 2.5. If the a g g r e g a t e s grow by c l u s t e r - c l u s t e r aggregation, c o m p u t e r simulations given a fractal dimension of 1.75 to 1.8. These predictions are in good a g r e e m e n t with experiments. Any process t h a t allows the particles to p e n e t r a t e the a g g r e g a t e s t r u c t u r e before sticking will increase the fractal dimension. For example, in slow coagulation, a particle i n t e r a c t i n g with a cluster m a y collide several times before finding a conformation for sticking. This process gives a fractal dimension of ~2. S h e a r forces can reorganize the aggregate by folding a n d bending, f u r t h e r increasing the fractal dimension (see Table 6.4). The average n u m b e r of particles in a typical aggregate, M, is related to the average h y d r o d y n a m i c radius, RG, and the fractal dimension by

M-- ~

k=l ~ kNk = -N-Tk=I~ (ak/ao)DFNk = (RG/ao)DF

(10.84)

w h e r e a0 is the size of the identical p r i m a r y particles. The volume fraction, ~bA, inside an average a g g r e g a t e is given by [3, p. 281]

r

R3 -- Ma~ - (RG/aO)(1-DF)

(10.85)

The volume fraction of particles is larger at the center of the aggregate t h a n at the outside edges. This type of fractal is called a m a s s

10.4 Kinetics of Coagulation and Flocculation

481

fractal. Using von Smoluchowski growth kinetics, discussed earlier, the effective hydrodynamic radius will increase with time according to [3, p. 287]

R v / a ~ = F(2

tl/2/ -

(10.86)

l I D R)

with F the gamma function given in the appendix of this book. For rapid Brownian coagulation, this behavior is observed for many systems [74,79] at long times. For slow coagulation, the fractals produced are more compact (i.e., higher fractal dimension) and the kinetics measured by experiments do not follow this relationship. For slow coagulation, the radius, RG, increased exponentially with time [80]: RG =

ao exp(ket)

(10.87)

where k e is the exponential rate constant. The exponential rate constant can be determined [62] from the doublet coagulation rate constant, k d = 4 k s T / 3 ~ , multiplied by the number density of primary particles, No. As a result the exponential rate constant can be approximated by the reciprocal of the doublet formation half-life, ke ~ 1/t~/2, from equation 10.68. Theories incorporating colloid stability ratios, W~j, which decrease with increasing R v appear to be capable of explaining this kinetic phenomena but not the configurational changes [80]. 10.4.2.1 S o l - G e l T r a n s i t i o n

After a period of growth, these aggregates occupy a large amount of space due to their large volume to mass ratio. The total volume fraction of all aggregates, (PT, is given by

(IE)T-- 4~ra~ 3 ~l= kNk ~k

(10.88)

If the system follows von Smoluchowski growth kinetics, the total volume fraction, q)T, will have the following time behavior [62]: dPT(t) = N T

3

- No 1 + ~

[ao exp(ket)]

for slow RLA (10.89)

*r(t) = NT 3

(

-- No 1 +

r

/ t ~I/DF7

!

Lr(2 - 1/DF)J

,or,ast , (10.90)

The total volume fraction, r a s a function of time for both reaction limited aggregation (RLA) and diffusion limited aggregation (DLA)

482

Chapter 10

Colloid Stability of Ceramic Suspensions

F I G U R E 10.36 Aggregate volume fraction versus time for reaction limited aggregation (RLA) and diffusion limited aggregation (DLA). When ~PT(t) = ~Pc(the percolation limit) gelation occurs.

is given in Figure 10.36. At some point these aggregates will interconnect with one another, creating a continuous network as shown schematically in Figure 10.37, called the percolation limit. There are two types of percolation limit: bond percolation and site percolation. Both of these types of percolation are given in Table 10.11.

F I G U R E 10.37 Schematic diagram of an aggregated colloidal suspension showing a bridging network. The volume fraction of particles at which this bridging network is formed is referred to as the percolation limit. Each sphere in the diagram consists of an aggregate network with a hydrodynamic radius which is shown in the inset.

10.4 Kinetics of Coagulation and Flocculation TABLE 10.11

483

Percolation Limites, a q)c, for Site and Bond Percolation

Structure

~structure

Z

1 Pc ~- ( Z - 1)

Site Pc

Bond Pc

Site ~Pc : Pc~

Face-centered cubic b Body-centered cubic c Simple cubic c Diamond c Random d

0.741 0.680 0.524 0.340 -0.637

12 8 6 4 -8

0.091 0.143 0.2 0.333 -0.143

0.196 0.245 0.3117 0.428 -0.27

0.1185 0.1785 0.2492 0.388

0.147 0.167 0.163 0.146 -0.16

Notes. Z is the coordination number for the structure with a volume fraction (~structure" Pc is the bond or site probability at percolation. (Pc is the volume fraction of aggregates at percolation. a Zallen, R., "The Physics of Amorphous Solids," Chapter 4. Wiley, New York, 1983. b Cox, M. A. A., and Essam, J. W., J. Phys. C. 9, 3985-3991 (1976). c Sykes, M. F., and Essam, J. W., J. Math Phys. 5, 1117-1127 (1964). d Experimentally determined values.

Site percolation is more appropriate for spherical aggregates than bond percolation. For a three-dimensional system, the critical volume fraction for site percolation is (Pc ~ 16% [81]. Therefore, at a total volume fraction, q~T,of 16%, a continuous network is formed. The formation of this continuous network corresponds to the onset of the solto-gel transition. This time is therefore predicted from the preceding equation when (~T reaches a value of (Pc. With further attachment of particles and other aggregates to the continuous percolation network, the structure becomes more rigid. Gelation occurs when the network of particles is rigid or when (~)T : 1.0 [82]. Chang has found that this prediction of the time for gel transition of 7 nm SiO2 suspensions destabilized with 0.5 M NaC1 is similar to experimental observations [83], as is shown in Figure 10.38. The microstructure of solutions after aggregation can be observed by freeze drying the solution. Figure 10.39 [84] shows electron micrographs of A1203 slurries with a solid volume fraction of 0.42 stabilized with various amounts of polyacrylic acid. When the suspension is colloidally stable, the particles are mostly individuals (some degree of aggregation is observed, however, which is due to poor deaggregation at the start of the experiment). The colloidally stable particles are uniformly distributed in the slurry as is observed after freeze drying, Figure 10.39(a). With strong aggregation the particles are organized into aggregates. These aggregates appear as sheets of particles randomly filling the space, Figure 10.39(b). When aggregation is weaker, the aggregate sheets are smaller, also randomly filling the space, Figure 10.39(c).

484

Chapter 10

Colloid Stability of Ceramic Suspensions

12000

10000 0_.9oa - a prediction,~== 1.0 e e e e 9 observotion ,~,~ 8000

-~

6000

0

.=it

r viii

4000

2000

0 ~ IlllllllllllllllWlllllllllIIIIlllJllllJlWlillIIWIIIllIWlWllllJllIllWl

0.00

0.20

0.40

0.60

0.80

1.00

1.20

I

1.40

(SiO,wt~) F I G U R E 10.38 Gelation time versus SiO2 concentration. Two curves are shown (1) experimental observations are for Ludox SM silica particles (7 nm diameter) in 0.5 M NaC1 solution, pH = 8.5, (2) predictions based upon equation 10.90 and (Pc = 1.0 for a rigid gel. Taken from Chang [62]. Concentration

The sheet n a t u r e of the aggregates observed with freeze drying is not expected from the fractal form of aggregates. It is due to either aggregate reorganization during freezing or drying. During freezing, crystals of ice are formed. The ice freezing front excludes particles, forcing t h e m to the edges of the ice crystals, where t h e y collect. Because the ice crystals are hexagonal, the particles will collect on their flat crystal faces, giving the a p p e a r a n c e of a particle sheet. During drying the ice crystal sublimates. W a t e r vapor will flow t h r o u g h the particulate sheet and cause f u r t h e r particle reorientation. These reorganization processes complicate the view of the microstructure of a slurry; however, some useful information can be gleaned from these images. For example, the n u m b e r of particles in each sheet is a m e a s u r e of the n u m b e r of particles in an aggregate and strong aggregation gives large aggregates and weak aggregation gives smaller aggregates.

Electron micrographs of freeze dried A1203 (5.5 m2/gm) slurries with volume fraction 0.42: (a) Colloidally stable due to 1.0% polyacrylic acid, (b) strongly aggregated due to 0.18% polyacrylic acid, (c) weakly aggregated due to 0.30% polyacrylic acid. Taken from Kimura et al. [84].

F I G U R E 10.39

10.4 Kinetics of Coagulation and Flocculation

485

486

Chapter 10 Colloid Stability of Ceramic Suspensions

For anisotropic particles, the percolation limit is a function of the aspect ratio. For ellipsoids of revolution, the percolation limit for a simple cubic lattice was studied by Boissonade et al. [85]. They found as the aspect ratio increases from 1 (a sphere) to 15 (a fiber), the percolation limit decreased from a volume fraction of 0.31 to 0.06 and the correlation length (i.e., aggregate size) did not change (i.e., it was the same as that of the sphere).

10.4.3 ShearAggregation For shear coagulation in laminar flow, the collision rate for Ni particles of size a i with Nj particles of size aj is given by [3, p. 298] Jij = ](ai + aj )3NiNj~/ W~(shear)

(10.91)

where ~ is the shear rate.* This expression is valid for a linear velocity profile or constant shear rate. The colloid stability factor for these shear conditions, W~.(shear), are different than for Brownian motion because hydrodynamic interactions displace the particles from their linear trajectories and the van der Waals attraction converts all close trajectories into spirals that eventually lead to doublet formation [86]. With von Smoluchowski's assumptions (i.e., W~.(shear) = WSH, a constant and ai = aj) then the total number density of particles decreases according to NT--exp(-t) No ~

(10.92)

where tl/2 for shear aggregation is now given by tl/2 -

TfWsH 4~2

(10.93)

where ~ is the particle volume fraction. The exponential decay, characteristic of a first-order process rather than the second-order kinetics seen for Brownian coagulation arises from the invariance of the volume fraction during shear aggregation. Brownian collisions also occur during shear aggregation. The collision rate due to shear is more important than the Brownian collision rate of particles for shear aggregation. This condition occurs for particles larger than a critical size, given by [87] ac

= [ kB Tll/3 [4--~J

(10.94)

* For an aggitated tank, the shear rate should be replaced with the residence time weighted average shear rate. See Kusters [86].

10.4 Kinetics of Coagulation and Flocculation

487

This expression was obtained by setting the shear aggregation rate to the Brownian aggregation rate and solving for the size a to which this corresponds assuming the colloid stability ratio, W is unity.

P r o b l e m 10.3. C r i t i c a l S i z e f o r S h e a r

Aggregation

Determine the particle size above which shear aggregation is domin a n t for a room t e m p e r a t u r e aqueous suspension of particles when the shear rate is varied from 1 to 100 sec -~. S o l u t i o n The radii of particles when the shear aggregation rate is equal to the Brownian aggregation rate are calculated by the preceding equation with ~2 varied between 1 to 100 sec -~, T = 298 K, V = 0.01 poise. The results follow: (sec -1)

Radius (t~m)

1

1.05 0.5 0.4 0.11 0.23

10 20 50 100

Particle sizes larger t h a n those given are those where the shear collision rate is dominant over Brownian collision rate. With 0.05/zm latex particles in a glycerol-salt solution, large-scale shear aggregates have been shown to have the same fractal dimension as that of Brownian aggregates (i.e., D R - 1.8 _ 0.1) when the shear rate is less t h a n 1500/sec [88]. Due to shear aggregation, the aggregation rate for these experiments was much faster t h a n that of comparative Brownian aggregation rates. For turbulent flow, collision rate is given in Table 10.12. Turbulent flow contains eddies which are roughly spherical in which the fluid is T A B L E 10.12

Subrange

Collision Rate for Various Eddy Sizes or Subranges a

Eddy Size = R

Jij

6d < R

Transition

6d -- R < 25d

4 (ai + aj) 3 Ni Nj (spl~) 1/2 3 Wij(shear) (t~)5/12 2.36 (a i + aj)S/3NiNj (p/~Q)l/4

Inertial

25d -< R < L/2

6.87 (ai + aj)V3Ni Nj(s) 1/3

L/2 <- R < L

7.09 (ai + aj)2 N i N j ( s L ) 1/3

Viscous

Wij(shear) Wij(shear) Macro

Wij(shear) a Kusters, K. A., Ph.D. Thesis, Technische Universiteit Eindhoven, 1991.

488

Chapter 10 Colloid Stability of Ceramic Suspensions

swirling. Eddies create a fluctuating velocity profile. In this table the smallest eddy size, R, corresponds to the Kolmogorov micro scale, d, defined as the eddy size when the Reynolds number is 1.0 (i.e., REd = pdhv(rij)/V), where the RMS velocity difference between two points, i and j, separated by a distance rij, hv(rij), is used as the velocity in the Reynolds number definition. The value of hv(rij ) is related to the energy dissipation per unit mass, s. The largest eddy size, R - L, is essentially that of the impeller in a aggitated tank. The various subranges are defined by the size of the eddy. Large eddies the size of the impeller, L, are created by mixing. These large eddies have little viscous dissipation because their Reynolds number, ReL, is large. The large eddies generate smaller eddies for which the Reynolds number decreases. For the eddies of size, d, with Reynolds numbers of 1.0, viscous dissipation takes place. Thus the turbulent energy is continuously withdrawn from the large eddies to the small eddies where energy is removed as heat. The collision rate thus depends on the eddy size. Four different eddy size ranges are given in Table 10.12 as subranges of turbulence.

10.5 C O L L O I D S T A B I L I T Y

IN CERAMIC

SYSTEMS

During the processing of a single component ceramic powder, it is very important to maintain colloid stability until the green body is formed. This colloid stability will give the most uniform packing in the green body without high-pressure pressing, as we will see in the next part of this book. Colloid stability can be achieved by electrostatic or steric stabilization. For composite ceramic suspensions, it is very important to prevent the segregation of the different particles during processing. To prevent segregation of a composite ceramic suspension, the suspension is first stabilized. Then, while being mixed at high intensity, it is flocculated by a destabilizing polymer to lock in the wellmixed nature of the particles in the stabilized suspension. As soon as the polymer is adsorbed onto the surface of the particles, every Brownian collision will result in a flock having been formed. For the best type of flocculation, it is necessary for the polymer to adsorb on all of the constituents of the composite ceramic suspension, thus changing the surface chemistry of each of the particles so that they all look the same and flocculate at the same rate. When equal rates of flocculation occur for the different particles in the composite ceramic suspension, the flocks produced will have a stoichiometry that is the same as that of the suspension. These flocks have a low-density open fractal structure and will need to be broken down with pressing during green body formation. To determine the polymers which will likely destabilize a system, it is necessary to look at the Flory-Huggins X parameter. If the X parame-

10.6 Summary

489

ter is less than 0.5, flocculation will result. Another important aspect is the adsorption of the polymer on the dissimilar surfaces of the composite ceramic suspension. Physical adsorbing polymers are the best for this application because there is no specific chemical interactions of the polymer with the different surfaces in the composite ceramic suspension. Physically adsorbing polymers are forced to adsorb due to their high molecular weight. This high molecular weight causes adsorption at concentrations near the cloud point for the polymer. Thus, it is possible, by adjusting the molecular weight of polymers, to lower the cloud point and increase the adsorption. A higher adsorption density gives a more attractive interaction energy.

10.6 S U M M A R Y In this chapter, we have described the colloid chemistry of ceramic powders in suspension. Colloid stability is manipulated by electrostatic and steric means. The ramifications on processing have been discussed with emphasis on single-phase ceramic suspensions with a distribution of particle sizes and composites and their problems of component segregation due to density and particle size and shape. The next chapter will discuss the rheology of the ceramic suspensions and the mechanical behavior of dry ceramic powders to prepare the ground for ceramic green body formation. The rheology of ceramic suspensions depends on their colloidal properties.

Problems 1. Calculate the Hamaker constant for the collision of two 0.5 t~m spherical A1203 particles dispersed in benzene (A11(h = 0) = 5 • 10 .20 J) with an adsorbed layer of poly(methyl methacrylate) M w = 100,000. Use this H a m a k e r constant to calculate the van der Waals interaction energy as a function of particle separation. 2. Calculate the steric interaction energy between the two spheres given in problem 1. Use Table 10.7 for the value of the relevant Flory-Huggins X parameter. 3. Using the data in Figure 10.27(a) for the zeta potential for A1203 in KNO3 salt solutions with different pH values, determine the electrostatic interaction energy versus separation for 0.3 t~m particles dispersed aqueous solution at pH 5, 10 .3 M KNO3 solution. Compare this electrostatic interaction energy curve to that of the van der Waals interaction energy curve for the same system. Integrate the total interaction energy to calculate the colloid stability ratio, and compare it with that given in Figure 10.27(b). 4. Determine the population of aggregates containing 20 and 50 pri-

490

Chapter 10 Colloid Stability of Ceramic Suspensions

m a r y particles as a function of aggregation time. Assume t h a t the p r i m a r y spherical particles are all 0.1 t~m in d i a m e t e r and the initial particle n u m b e r density is 109 particles per cc. Also a s s u m e t h a t the colloid stability ratio is 0.56. 5. For the d a t a given in problem 4, determine if the system will undergo a sol-to-gel transition if the fractal aggregates have a fractal dimension of 2.15. Determine the time for gelation of this system. 6. An agitated t a n k with a residence time weighted average s h e a r rate of 100 sec -1 is used for the aggregation of 0.313 t~m spherical TiO2 particles at pH 4 in 7.5 • 10 .3 M KC1 solution. See Figure 10.28 for the colloid stability ratio for this system. The initial volume fraction of the TiO2 particles is 0.01. Determine the time for the n u m b e r density of aggregates to reach 1/10,000 of the original number of particles. 9

\

7. An agitated t a n k with a residence time weighted average s h e a r rate of 300 sec -1 is used for the aggregation of 0.57 t~m spherical SiO2 particles at pH 8 in 0.03 M KC1 solution. See Figure 10.26 for the colloid stability ratio for this system. The initial volume fraction of the SiO2 particles is 0.05. D e t e r m i n e if the system will undergo a solto-gel t r a n s i t i o n if the fractal aggregates have a fractal dimension of 2.15. D e t e r m i n e the time for gelation of this system. 8. To m a k e a composite ceramic m a t e r i a l with the stoichiometry of mullite 3A1203.2SIO2, a m i x t u r e of monodisperse spherical 0.7 t~m A1203 particles is mixed with 0.57 t~m SiO2 particles at a volume fraction of 5% in an aqueous 10 .3 M KC1 solution at pH 10. In this solution both types of particles are negatively charged as seen in the table t h a t follows. D e t e r m i n e the total colloid stability ratio for the mixture for these conditions. D e t e r m i n e the best way to destabilize the suspension to m a k e a uniformly mixed aggregate structure. Two ways are possible: (1) increase the salt concentration to 10 -~ M KC1 or (2) decrease the pH to either 9 or 8. Particle pH = 10, 10-3 M KC1 A1203 SiO2 pH = 10, 10-1M KC1 A1203 SiO2 pH = 9, 10-3 M KC1 A1203 SiO2 pH = 8, 10-3 M KC1 A1203 SiO2

K-1 Zeta Potential (nm) ~(mV)

Wll

9.6 9.6

-30 -25

100 50

0.96 0.96

- 15 -5

10 8

9.6 9.6

0 -20

1.0 40

9.6 9.6

+ 25 -20

40 40

References

491

References 1. Hamaker, H. C., Physica (Amsterdam) 4, 1058-1072 (1937). 2. Lifshitz, E. M., Soy. Phys.--JETP (Engl. Transl.) 2, 73-83 (1956). 3. Russel, W. B., Saville, D. A., and Schowalter, W. R., "Colloidal Dispersions," p. 134. Cambridge Univ. Press, Cambridge, UK, 1989. 4. Prieve, D. C., and Russel, W. B., J. Colloid Interface Sci. 125, 1-13 (1988). 5. Parsegian, V. A., and Ninham, B. W., Biophys. J. 10, 646-663 (1970); Nature (London) 224, 1197-1198 (1969). 6. Hough, D. B., and White, L. R., Adv. Colloid Interface Sci. 14, 3-41 (1980). 7. Mahanty, J., and Ninham, B. W., "Dispersion Forces." Academic Press, New York, 1976. 8. Israelachvili, J. N., and Tabor, D., Prog. Membr. Sci. 7 (1971). 9. Derjaquin, B. V., Kolloid-Z 69, 155-164 (1934). 10. Visser, J., Surf. Colloid Sci. 8, (1976). 11. Hiemenz, P. C., "Principles of Colloid and Surface Chemistry, 2nd ed., p. 648. Dekker, New York, 1986. 12. Fowkes, F. M., in "Hydrophobic Surfaces" (F. M. Fowkes, ed.). Academic Press, New York, 1969. 13. Vold, M. J., J. Colloid Interface Sci. 16, 1 (1961). 14. Mathai, K. G., and Ottewill, R. H., Trans. Faraday Soc. 62, 750, 759 (1966). 15. Osmond, D. W. J., Vincent, B., and Waite, F. A., J. Colloid Interface Sci. 42, 262 (1973). 16. Vincent, B., J. Colloid Interface Sci. 42, 270 (1973). 17. Landau, L. D., and Lifshitz, E. M., "Electrodynamics of Continuous Media." Pergamon, O:~ford, 1960. 18. Overbeek, J. T. G., in "Colloid Science" (H. Kruyt, ed.), Vol. 1. Elsevier, Amsterdam, 1952. 19. Pasahley, R. M., and Israelachvili, J. N., J. Colloid Interface Sci. 97, 446-455 (1984). 20. Oshima, H., J. Colloid Interface Sci. 162 (1994) (in press). 21. Ring, T. A., J. Chem. Soc., Faraday Trans. 2 78, 1513-1528 (1982). 22. Ring, T. A., unpublished work. 23. Hogg, R., Healy, T. W., and Fuerstenau, D. W., Trans. Faraday Soc. 62, 1638 (1966). 24. Barringer, E. A., Novich, B. E., and Ring, T. A., J. Colloid Interface Sci. 100, 584 (1984). 25. Barringer, E. A., Ph.D. Thesis, Materials Science Department MIT, Cambridge, MA (1984). 26. Sato, T., and Ruch, R., "Stabilization of Colloidal Dispersions by Polymer Adsorption," Vol. 9. New York, 1980. 27. Napper, D. H., "Polymeric Stabilization of Colloidal Dispersions." Academic Press, New York, 1983. 28. Romo, L. A., J. Phys. Chem. 67, 386 (1963). 29. McGown, D. N. L., and Parfitt, G. D., Faraday Soc. Discuss. 42, 225 (1966). 30. Parfitt, G. D., in "Dispersion of Powders in Liquids," 2nd ed., Chapter 1. Wiley, New York, 1973. 31. Vincent, B., Adv. Colloid Interface Sci. 4, 193 (1974). 32. Ottewill, R. H., in "Nonionic Surfactants" (M. J. Schick, ed.), Chapter 19. Marcel Dekker, New York, 1967. 33. Marumo, H., in "Surface Chemistry of Polymers," Chapter 5. Sangyo Publishing Co., Tokyo, Japan, 1968. 34. Evans, R., and Napper, D. H., Kolloid-Z. Z. Polym. 251, 329 (1973). 35. Hesselink, F. T., J. Phys. Chem. 73, 3488 (1969). 36. Cosgrove, T., Crowley, T. L., Vincent, B., Barnett, K. G., and Tadros, T. F., Faraday Discuss. Chem. Soc. (1982). 37. Edwards, S. F., Proc. Phys. Soc., London 85, 613 (1965); 88, 265 (1966).

492

Chapter 10

Colloid Stability of Ceramic Suspensions

38. Flory, P. J., J. Chem. Phys. 9, 660 (1941); 10, 51 (1942). 39. Huggins, M. L., J. Chem. Phys. 9, 440 (1941); J. Phys. Chem. 46, 151 (1942); J. Am. Chem. Soc. 64, 1712 (1942). 40. Evans, R., and Napper, D. H., J. Colloid Interface Sci. 52, 260 (1977). 41. Barton, A. F. M., "Handbook of Solubility Parameters and Other Cohesion Parameters." CRC Press, Boca Raton, FL, 1983. 42. Dolan, A. K., and Edwards, S. F., Proc. R. Soc. London, Ser. A 337, 509 (1974). 43. de Gennes, P.-G., "Scaling Concepts in Polymer Physics." Cornell Univ. Press, Ithaca, NY, 1979. 44. Bagchi, P., and Vold, R. D., J. Colloid Interface Sci. 38, 652 (1972). 45. Sonntag, H., Abh. AdW. DDR Klasse Chem. 1 2, 517 (1974). 46. Ottewill, R. H., and Walker, T., Kolloid-Z. Z. Polym. 227, 108 (1968). 47. Asakura, S., and Oosawa, F., J. Chem. Phys. 22, 1255 (1954); J. Polym. Sci. 33, 183 (1958). 48. Feigin, R. I., and Napper, D. H., J. Colloid Interface Sci. 71, 117 (1979). 49. Scheutjens, J. M. H. M., and Fleer, G. H.,Adv. Colloid Interface Sci. 16, 361 (1982). 50. de Gennes, P. G., Macromolecules 14, 1637 (1981); 15, 492 (1982). 51. Joanny, J. F., Liebler, L., and de Gennes, P.-G., J. Polym. Sci., Phys. Ed. 17, 1073 (1979). 52. Taube, I., Gummi-Ztg. 39, 434 (1925). 53. Bondy, C., Trans. Faraday Soc. 35, 1093 (1939). 54. Vincent, B., Luckham, P. F., and Waite, F. A., J. Colloid Interface Sci. 73, 508 (1980). 55. Sato, T., and Sieglaff, C. F., J. Appl. Polym. Sci. 25, 1781 (1980). 56. von Smoluchowski, M. Z. Phys. Chem. 92, 129-168 (1917). 57. Fuchs, N. Z. Phys. 89, 736 (1934). 58. Spielman, L. A., J. Colloid Interface Sci. 33, 562 (1970). 59. Prieve, D. C., and Ruckenstein, E., J. Colloid Interface Sci. 73, 539 (1980). 60. Stimson, M., and Jeffery, G. B., Proc. R. Soc. London, Ser. A 111, 110 (1926). 61. Maude, A. D., B. J. Appl. Phys. 12, 293 (1961). 62. Chang, S.-Y., Ph.D. Thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992). 63. Spielman, L. A., J. Colloid Interface Sci. 33, 562 (1970). 64. Barringer, E. A., Novich, B. E., and Ring, T. A.,J. Colloid Interface Sci. 100, 584 (1984). 65. Wiese, G. R., and Healy, T. W., J. Colloid Interface Sci. 51, 427 (1975). 66. Verwey, E. J. W., and Overbgeek, J. T. G. "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948. 67. Derjaquin, B. V., and Landau, Acta Physicochim 14, 633 (1941). 68. Chang, S. Y., Ring, T. A., and Trujillo, E. M., Colloid. Polym. Sci. 269[8], 843-849 (1991). 69. Allen, L. H., and Matijevid, E. J. Colloid Interface Sci., 430 (1970). 70. Gherardi, P., and Matijevid, E., Colloids Surf. 32, 257-274 (1988). 71. Strauss, M., Ring, T. A., Bleier, A., and Bowen, H. K., J. Appl. Phys. 58, 3871 (1985). 72. Honig, E. P., Roebertson, G. J., and Wiersema, P. H., J. Colloid Interface Sci. 36, 97 (1971). 73. Higashitani, K., and Matsuno, Y., J. Chem. Eng. Jpn. 12, 460-465 (1979). 74. Weitz, D. A., and Huang, J. S., in "Kinetics of Aggregation and Gelation" (P. Family and D. P. Landau, eds.), p. 19. Elsevier, Lausanne, Switzerland, 1984. 75. Shaefer, D. W., Martin, J. E., Wiltzius, P., and Cannell, D. S., Phys. Rev. Lett. 52, 2371-2374 (1984). 76. Schaefer, D. W., Polymer 25, 387-394 (1984). 77. Schaefer, D. W., MRS Bull. 8, 22-27 (1988). 78. Meakin, P., in "On Growth and Form" (H. E. Stanley and N. Ostrowsky, eds.), pp. 111-135. Martinus Nijhoff, Boston, 1986.

References

493

79. Sonntag, R., and Russel, W. B., J. Colloid Interface Sci. 113, 399-413 (1986). 80. Weitz, D. A., Lin, M. Y., and Huang, J. S., in "Complex and Supra Molecular Fluids" (S. A. Safron and N. A. Clark, eds. ), pp. 509-549. Wiley (Interscience), New York, 1986. 81. Brinker, C. J., and Scherer, G. W., "Sol-Gel Science," p. 320. Academic Press, San Diego, CA, 1990. 82. Discussions with James Martin, Sandia National Laboratories. 83. Chang, S. Y., and Ring, T. A., unpublished work. 84. Kimura, T., Hirota, M. A. K., and Murata, H.,Kona (Hirakata, Jpn.) 9, 44-53 (1991). 85. Boissondade, J., Barreau, B., and Carmona, F., J. Phys. A 16, 2777-2787 (1983). 86. Kusters, K. A., Ph.D. Thesis, Technische Universiteit Eindhoven, 1991. 87. Ives, K. J., in "The Scientific Basis of Flocculation" (K. J. Ives, ed.), p. 55. Sijthoff & Noordhoff, The Netherlands, 1978. 88. Torres, F. E., Russel, W. B., and Schowalter, W. R., J. Colloid Interface Sci. 142, 554-574 (1991). 89. Vicsek, T., "Fractal Growth Phenomena," World Scientific, London, 1989.

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11

Colloidal Properties of Ceramic Suspensions

11.1 O B J E C T I V E S This chapter discusses the colloidal properties of ceramic suspensions. These properties include sedimentation, Brownian diffusion, colligative properties, and particle ordering. There is an equilibrium between sedimentation and Brownian diffusion which prevents the smallest particles from settling. Colligative properties include boiling point elevation, freezing point depression, and osmotic pressure. The osmotic pressure of electrolyte solutions, polymeric solutions, and colloidal suspensions are discussed in detail. The osmotic pressure of a colloidal suspension is responsible for the structure of the particles in suspension. The structure of particles in suspension plays a role in the suspension rheology, in ceramic casting, and in determining the green body density. 495

496

Chapter 11

Colloidal Properties of Ceramic Suspensions

11.2 I N T R O D U C T I O N Colloid is the term used by G r a h a m [1] to distinguish different types of m a t t e r (i.e., crystalloids and colloids), but after 1861 it became apparent that colloids are not separate types of matter but m a t t e r (i.e., liquids or solids) in a particular state of subdivision in which effects connected with the surface are pre-eminent. Particle sizes in colloidal systems generally range from 1 nm to 10 txm and for this reason have a high surface area to volume ratio. Colloidal systems can be dispersed in either a gas or a liquid. When the dispersing fluid is a gas, the suspension is called an aerosol. When the dispersing fluid is a liquid the suspension is called a sol (for a suspension of solid particles) or an emulsion (for a suspension of liquid droplets). The forces which play an important role in determining the effects connected with the large surface area per unit volume of colloids are given in Table 11.1. In this table, the common denominator used is the Brownian force (= kBT/a), where the thermal energy of random molecular motion is divided by a characteristic length, a, of the particle. Other forces, including electrical, van der Waals attraction, viscous, gravitational, and inertial forces, are given in Table 11.1. All these forces except the inertial force are equally or more important t h a n the Brownian force for a 1/xm TiO2 particle in water at 25~ T A B L E 11.1 M a g n i t u d e s of Colloid Forces on a 1/xm TiO2 Particle in W a t e r at 25~ a Electrical force B r o w n i a n force

ss0g 2 k BT/a

10 a

Attractive force B r o w n i a n force

Aeff/a k BT/a

1.0

Viscous force B r o w n i a n force

qUa ~ kB T/a

G r a v i t a t i o n a l force B r o w n i a n force

hpga 3 k BT/a

10

Inertial force B r o w n i a n force

pU2a 2 k BT/a

10-6

1.0

a For conception of this table, see Russel, W. B., Saville, D. A. and Schowalter, W. R. "Colloidal Dispersions," C a m b r i d g e U n i v e r s i t y Press, Cambridge, U.K., 1989, p. 14. Notes: ~0 = viscosity = 0.01 poise, U = velocity = 1 t~m/sec, Aeff - 10 -20 N" m, ~ - zeta potential = 50 mV, a = radius = 1 t~m.

11.3 Sedimentation

497

These forces will be discussed later in Chapter 11 to establish the important colloidal characteristics of ceramic powder suspensions. These colloidal characteristics include sedimentation, Brownian diffusion, and colligative properties including osmotic pressure. Osmotic pressure of solutions is discussed to set the stage for a discussion of suspension osmotic pressure. Solution osmotic pressure is one of several colligative properties of solutions. Colligative properties of a solution give important information on the solution thermodynamics, including the solution vapor pressure, freezing point, and boiling point. This relation between vapor pressure (as well as boiling point) and solution concentration affects how a green body containing such a solution will evaporate during drying. As we have seen in Chapter 10, the osmotic pressure of a polymer solution allows the determination of an interaction parameter for the interaction between polymer and solvent. This parameter allows the determination of the heat of mixing of polymer and sovlent and the steric interactions for polymers adsorbed on ceramic particles. The osmotic pressure of a suspension is also a useful measure of suspension interactions, including interparticule interactions and the average structure of the particles or the volume fraction of the particles in suspension. Thus, the energy required to create the particle structure can be calculated. Deforming the structure requires energy, which can be used to determine the rheological behavior of the suspension, which is discussed in Chapter 12. The movement of colloidal particles in a solvent is another important characterization of a ceramic powder suspension. There are two competing influences on a particle's movement: thermal agitation and the sedimentation due to gravity (or centrifugal force). Both determine how a ceramic powder will form a consolidated body when it is left to settle. This will also affect a ceramic suspensions "shelf life."

11.3 SEDIMENTATION

The velocity of sedimentation is a result of a force balance shown in Figure 11.1 for a sphere of radius R moving at a terminal settling velocity, Vt. The force of gravity, Fw(= -~R 4 3Psg), is a result of the gravitational field, g, and is directed downward. The buoyant force, FB(= }~R3pfg), is a result of the displaced fluid and is directed upward. The viscous drag force, FK, acts to oppose the movement of the sphere. This force balance is written

F8 +F k =F W

(11.1)

where g is the gravitational constant, py is the fluid density, and Ps is the sphere density. For Reynolds numbers, NR.(= pfVt2R/~) less than

498

Chapter 11 Colloidal Properties of Ceramic Suspensions

FIGURE 11.1 Forces acting on a settling sphere.

1.0 (i.e., laminar flow) the drag force, FK( = 6~vRVt), is known as Stokes's law [2]. For higher Reynolds numbers, the drag force, FK, is given by a more complex equation: FK = (zrR2)( 89 V2t ) f

(11.2)

where f is the friction factor, which is given by Figure 11.2 and is dimensionless. [Note: for laminar flow f = 24/NRe]. Figure 11.2 also shows the friction factor for other particle shapes [3] in particular orientations which give a minimum friction factor for the higher Reynolds numbers. Using Stokes's law, the force balance is given by 41rR3pfg + 6~'~RV t = 47rR3psg

(11.3)

FIGURE 11.2 Frictionfactor, f, versus Reynolds number for a sphere. Adapted from Eisner [3].

11.3 Sedimentation

499

which can be rearranged to give the terminal settling velocity, Vt, for a single sphere"

Vt = 2R2[ps - pf]g 97

(11.4)

which is valid when the Reynolds number, NRe(- pfYt2R/~) less than 1.0 and when the size of the particle is large compared to the size of the solvent molecules. For turbulent flow the terminal settling velocity, Vt, is given by

Vt = / - / 8 R [ p s - pf]g 3fpf

(11.5)

where f must be determined from the Reynolds number using Figure 11.2. Because the friction factor, f, is also a function of the Reynolds number, this requires a trial and error solution. For the case where the Reynolds number is between 1,000 and 200,000, the friction factor, f, has a nearly constant value of 0.44 and is not a strong function of Reynolds number. For ceramic powders, such a high Reynolds number is rarely observed even in centrifugation at high rpm.

Problem 11.1. Terminal Settling Velocity Determine the terminal settling velocity in water at 20~ of two particles both with density 2.0 gm/cc: one particle with a radius of 1 t~m and the other with a radius of 0.1 t~m. (Note, the viscosity of water at 20~ is 0.01 poise (gm/cm/sec) and its density is 1.0 gm/cc). S o l u t i o n Using the preceding equation for the terminal settling velocity for laminar flow,

Vt = 2R2(ps - pf)g 97

(11.6)

we find R = 0.1tLm R = 1.0 t~m

Vt=80t~m/hr Vt = 8 m m / h r

NRe=4.4

x 10 .9

NRe = 4.4 x 10 -s

In both cases the Reynolds number is much less than 1, so that Stokes's law is valid. The difference in settling velocity between 0.1 and 1.0 t~m particles is drastic and is the reason for segregation of particles in a ceramic suspensions. By inspection of this equation, differences in the terminal settling velocity can be due to either density or size differences between the two types of spherical particles. The effects of particle shape asymmetry are considered next.

500

Chapter 11 ColloidalProperties of Ceramic Suspensions

11.3.1 Nonspherical Particle Settling The drag force, Fk, for a nonspherical particles was derived by Perrin [4] and is given by

FK= [TrR2] [ l p f v 2 ]

f~

(11.7)

where fo is the friction factor given in Figure 11.2 for an equivalent sphere (i.e., a sphere of radius Ro with the same volume as the particle) and G(p) is the asymmetry function. For prolate ellipsoid of revolution with semi-axes, a > b = c the axial ratio p is defined as b/a. For prolate ellipsoids where p < 1, the asymmetry function G(p) is

G(p)=p2/3( 1 _ p2)_1/2 in {1 + ( 1 p_ p2)1/2}; p = b/a < 1 a n d R o = ab 2

(11.8)

and for oblate ellipsoids (plates) with semi-axes, c = a > b, the axial ratio p is defined as a/b where p > 1, it is

G(p) = p2/3(p2_ 1)-1/2 tan-l[(p2 _ 1)1/2]; p = a/b > 1 and R o = a2b

(11 9)

Figure 11.3 is a plot of G(p) as a function of p for both prolate and oblate ellipsoids. This expression for the drag force is valid for both laminar and turbulent flow. For laminar flow the drag force becomes

FK=

67r~RoVt G(p)

(11.10)

which gives rise to an average terminal settling velocity of

Vt =

2R2[ps- pf]gG(p) 9~

(11.11)

This average velocity is achieved when all the possible orientations of the particle are equally probable. It represents an integral of the drag force over all possible orientations. When particles are larger than about 1 t~m, their rotational diffusion coefficient is low and preferred orientations occur during settling, corresponding to those close to the orientation with the m i n i m u m drag force. See the book Colloidal Hydronamics by van de Ven [5, p. 270] for further information on the settling velocity vectors of nonspherical particles.

11.3.2 Hindered Settling When the particles are close together, hindered settling occurs. Hindered settling has been studied by m a n y authors [6]. In all cases,

11.3 S e d i m e n t a t i o n

501

F I G U R E 1 1 . 3 G(p) and F ( p ) for (a) prolate (c = a > b, p = b/a < 1.0) and (b) oblate (a > b = c, p = a / b > 1.0) ellipsoids of revolution. G(p) is defined by equations 11.8

(prolate ellipsoids) and 11.9 (oblate ellipsoids) and is used in equations 11.7 to 11.11 and 11.24; F ( p ) is defined in equations 11.42 (prolate ellipsoids) and 11.44 (oblate ellipsoids ) and is used in calculating the rotational diffusion coefficient defined by those equations.

the Stokes's terminal setting velocity is multiplied by a factor h(~); for example, Vt = 2R2[ps - pf]gh(4~) 9~

(11.12)

where (b is the volume fraction of the particles in the fluid. Many expressions for h(r appear in the literature [6]. For simplicity and reasonable accuracy over a large range of the volume fraction from 0

502

Chapter 11 Colloidal Properties of Ceramic Suspensions I

h(r

==

:\

\

= =

o FIGURE11.4

r

o.5

H i n d e r e d s e t t l i n g of spheres, h(~b), w h e r e ~b is the v o l u m e fraction.

to about 40%, the factor h(~b) by Happel and Brenner [7] is given h((b) =

3 + 4.5(~b5/3- ~1/3)_ 34)2 3 + 24)5/3

(11.13)

A plot of h(~) is given in Figure 11.4. Here we see that at low volume fraction the value of h((b) is 1 and at high volume fraction the value of h(4)) is much less than 1.

Problem 11.2. Hindered Settling Velocity At 10 and 20% solids volume fraction, determine the terminal settling velocity in water at 20~ of particles with a density of 2.0 gm/cc: one type of particle with a radius of 1 ftm and the other with a radius of 0.1 t~m. (Note: the viscosity of water at 20~ is 0.01 poise (gm/cm/ sec) and its density is 1.0 gm/cc.) S o l u t i o n Using the preceding equation for the terminal hindered settling velocity for laminar flow, Vt =

2 R 2 [Ps - pf]gh (d~) 9V

(11.14)

we find

R = 0.1 t~m R = 1.0 ftm

Unhindered

H i n d e r e d , ~b = 0.10

H i n d e r e d , 4) = 0.30

Vt = 80 f t m / h r Vt = 8 m m / h r

Vt = 25.7 tLm/hr Vt = 2.57 m m / h r

Vt = 8 . 0 / z m / h r Vt = 0.8 m m / h r

11.3 S e d i m e n t a t i o n

503

Therefore the sedimentation velocity of a ceramic suspension with a high volume fraction is less than that of a single particle. The combined effects of nonspherical shape and hindered settling can be approximated by combining these two effects as follows:

Vt=

2 R 2 (Ps -

pf)gG(p)h(d~) 97

(11.15)

This equation is only a first-order approximation good for the lower volume fractions because the particles will orient themselves, thus lowering their lateral volume fraction.

11.3.3 Centrifugal S e d i m e n t a t i o n In a centrifugal field, the particle velocity, Vt, depends on the radial position, Re, in the centrifuge because the centrifugal force depends on the radial position:

dR~

Vt- dt -

2R2[ps

--

pf]to2Rc

97

(11.16)

where (o (= 27r rpm) is the angular velocity of rotation. In this equation, the gravitational acceleration, g, of the Stokes's law expression has bene replaced by the radial acceleration, (o2Rc, in a centrifugal field. Integration of this equation from two radial positions in the centrifuge, Rcl to Re2, for a duration of time, t, leads to in ~Rc2~ = 2R2(ps - pf )co2t

\Re1/

97

(11.17)

which is the sedimentation equation assuming laminar flow around the spherical particle. Again, the correction, h(~b), for hindered settling at high volume fractions or, G(p), for nonspherical particles can be used.

11.3.4 S e d i m e n t a t i o n P o t e n t i a l When charged particles settle in an aqueous solution, an electric field is induced by the movement of the charged particles relative to the ionic medium, as discussed in Chapter 9. A formula relating the electric field, E (= volts/cm), to the zeta potential, ~, of the particles is given by [8,9] E = C~ss0~g

(11.18)

127rk7 where Cm is the mass of particles per cc, s is the dielectric constant for the liquid, s0 is the dielectric permeativity of the liquid, k is the specific

504

Chapter 11 Colloidal Properties of Ceramic Suspensions

conductance of the solution, and V is the solution viscosity. For centrifugal sedimentation, this equation is written E = Cmss~176 - R2cl) 127rkV(Rc2 - Rcl )

11.4 BROWNIAN

(11.19)

DIFFUSION

The thermal motion of particles in a suspension opposes their sedimentation by creating a counteracting diffusive flux. This thermal motion is called B r o w n i a n motion. Robert Brown in 1828 was the first to note the random movement of pollen particles dispersed in water with a microscope. The average displacement, 2, in three dimensions for a period of time, t, was found [10] to be given by 2 = ~/-2Dt

(11.20)

where D is the Brownian diffusion coefficient. Einstein [11] developed an equation for the diffusion coefficient: D = BkBT

(11.21)

where B is the mobility of the particle defined by B =Vt= FK

l------~6TrR'~

(11.22)

assuming Stokes's law, which leads to the Stokes-Einstein equation for the Brownian diffusion coefficient: D-

kBT

6rrRv

(11.23)

The diffusion coefficient for a suspension of monosized particles can be measured directly by photon correlation spectroscopy [12] (quasielastic light scattering). For distributions of different particle sizes, the average diffusion coefficient is determined by photon correlation spectroscopy.

11.4.1 Nonspherical Particle Diffusion Tanford [13] and Perrin [14] determined the translational diffusion coefficient for ellipsoidal particles: D -

kB----T--TG(p)

67rRo~

(11.24)

11.4 Brownian Diffusion

505

4 3 where Ro is the radius of the equivalent sphere (i.e., ~TrRo = ]7rab 2 for prolate and ~TrR04 3 = 47ra2b for oblate spheroids), G(p) is the asymmetry function of p, the axial ratio. G(p) is defined earlier in this chapter and plotted in Figure 11.3 for both prolate and oblate ellipsoids.

11.4.2 Fick's Laws for Diffusion Brownian diffusion follows Fick's first law for the flux of particles, J (in units of particles per unit area per sec) J = D V. C

(11.25)

and Fick's second law for the change in particle concentration with time dC = V. [D V. C] ~ D V2C dt

(11.26)

where V. C is the gradient of concentration (= d C / d x + d C / d y + d C / d z in Cartesian coordinates) and V2C is the Laplacian of the concentration (= d2C/dx 2 + d2C/dy 2 + d2C/dz 2 in Cartesian coordinates). The gradient and Laplacian for other coordinate systems are given in the appendix of this book. The mean square displacement for a colloidal particle can be obtained from the solution to the diffusion equation as follows: (x 2> = f_ x2C(x, t ) d x = 2Dt

(11.27)

11.4.3 Equilibrium between Sedimentation and Diffusion Consider the flux of particles crossing a horizontal surface A, where the concentration of the particles is C and the concentration gradient is dc/dx. The flux due to gravitational sedimentation on this plane is C d x / d t and the flux due to diffusion is D d C / d x . Because these two fluxes are equal at equilibrium, we have the following equation: C d x / d t = C 2R2(ps 9~

pf)g

=

D dC/dx

(11.28)

which can be integrated from C~ to C2 over a distance x using the Stokes-Einstein relationship for the diffusion coefficient to give C2] 47rR3(ps - pf)g In ~ = 3ksT x

(11.29)

C~ and C2 are the concentrations at two points a distance, x, a part where sedimentation is at equilibrium. Prolonged sedimentation will not change these two concentrations due to this equilibrium.

506

Chapter 11 Colloidal Properties of Ceramic Suspensions

Problem 11.3. S e d i m e n t a t i o n E q u i l i b r i u m For various particle radii between 1 tzm and 0.001 tzm, determine the ratio of equilibrium concentrations, C2/C~, at a distance, x, 1 cm apart. The particles have a density of 2.0 grn/cm a and the liquid is water with a viscosity of 0.01 poise (gm/cm/sec). Solution

Using equation 11.29, we have the following results" C__~2 C1

R (t~m) 0.1 t~m 0.01/~m 0.001/~m

2.17 • 1043 1.105 1

This shows that particles less than ~0.01/~m are subject to sedimentation equilibrium over a distance of 1 cm. Bigger particles sediment and have a clear sedimentation interface. This same equilibrium can be determined for centrifugal sedimentation. The flux due to gravitational sedimentation at this plane is C(dRc/dt), and the flux due to diffusion is D dC/dR c. Because these two fluxes are equal at equilibrium, we have the following equation:

C dRc clt

-

C 2R2(ps 9~?pf )(o 2Rc - D dC/dRc

(11.30)

which can be integrated from C1 to C2 over a distance from Rcl to Rc2 using the Stokes-Einstein equation for the diffusion coefficient: In /--\|C-~] = 27rR3(ps -- Pf)~ \L,1/

(Rc~ - R2cl)

(11 31)

3ksT

C1 and C2 are the concentrations at two points a distance Rcl to Rc2 apart, where sedimentation is at equilibrium. Again prolonged sedimentation will not change these two concentrations due to this equilibrium.

11.4.4 R o t a t i o n a l Diffusion Most traditional ceramic particles are not spherical. As a result they will rotate in solution. This rotation, whether fast or slow, determines if a nonspherical particle can be oriented during ceramic processing. The orientation of particles in a ceramic green body is important in determining its drying, binder burnout, and sintering properties. To examine particle rotation, we will consider a simple shape initially, that of a rod, and then generalize the results to a more general nonspherical form, that of an ellipsoid of revolution. A detailed description of the rotational diffusion of a nonspherical particle is given in the

11.4 Brownian Diffusion

50~

books Colloidal Hydrodynamics by van de Ven [5], Laser Light Scattering by Chu [15], and Dynamic Light Scattering by Berne and Pecora [20]. Consider for a moment a rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, t2 = (0, ~) used to describe its orientation. Because the solvent is expected to frequently collide with the rod, it should exhibit a random walk along the surface of the unit sphere. Debye [16] in 1929 developed a model for the reorientation process based on the assumption that collisions are so frequent that a particle can rotate through only a very small angle before having another reorienting collision (i.e., small step diffusion). Debye began with the diffusion equation

Oc(r, t) = D V2c(r, t) Ot

(11.32)

and constrained the motion to the surface of the unit sphere (i.e., Irl = 1.0). Then, c(r, t) is simply the concentration of rods at the point r = u on the surface of the unit sphere at time t. Because of the spherical symmetry, it is most convenient to solve this equation in spherical polar coordinates (r, 0, r where r = 1.0. The Laplacian of the concentration in spherical polar coordinates for fixed r = 1 is

1[

V2c(r, t) = sin2 0

sin 0

sin 0

c rt )+o2c rt ] O0

O~ 2

"

where all the derivatives with respect to r vanish. Letting c(u, t) sin 0 dO d~p be the fraction of rods with orientation u in the solid angle d2u (= sin 0 dO d,#) at time, t, we obtain the rotational diffusion equation or Debye equation:

ocut=t sin201Esin0(sin0Cut)+2cut]0 where O is the rotational diffusion coefficient replacing D. For an initial orientation of u0 at time t - 0, the solution to the rotational diffusion equation [17] is given by

c(u, t)= ~ exp[-/(/+ 1)Ot]Y~'m(u)Ylm(Uo) lm

(11.35)

= exp[-/~2Ot]c(uo, O) where Ylm(U)[-- Ylm(O, ~p)] is the spherical harmonic Eigen function and its compliment Y~*(u) i 2 (and Iz) are orbital angular m o m e n t u m operators [17], and l(l + 1) and ml are eigenvalues given by i2Ylm(U) = l(l + 1)Ylm(U) iz Ylm (U) = mlYlm(U)

l = 0, 1, 2, 3, . . . . , m~ = - 1 , . . . , 0 , . . . , +1

(11.36) (11.37)

~08

Chapter 11 ColloidalProperties of Ceramic Suspensions

This particular solution of the rotational diffusion equation can be interpreted as the transition probability; that is, the probability density for a rod to have orientation u at time t, given that it had an orientation u0 initially. For one dimension, 4), we have

V2c(r,t)=IO2c(r't) loop2

(11.38)

The solution to this equation for rotational diffusion in one dimension with the boundary condition c(4)=0)=

1.0 a t t =

0forall0

(11.39)

is c(4), t)

1

2 ~

exp ~

(11.40)

and the root mean square (rms) displacement is given by <4)2> = 2 0 t

(11.41)

These results are similar to the results for ordinary diffusion in one dimension (i.e., ~b replaced by x and O by D, the diffusion coefficient). Perrin [4, 18] has put forth a hydrodynamic model of the rotational motion of different shaped particles. For an ellipsoid of revolution with semi-axis a > b = c (i.e., prolate ellipsoid), Perrin has shown that the rotational diffusion coefficient is given by

3kBT F(p)= 3kBT {(2-p2)G(p)/p2/3-1} 16~a a 16rr~a a (1 - p 4) (11.42) where p is the axial ratio, p = b/a < 1.0, V is the solution viscosity, and G(p)defined by equation 11.8 is a function of the axial ratio, which for ellipsoids is given in Figure 11.3 along with G(p)/p2/3and F(p). O=

This equation should be compared to the result O=

3kBT

(11.43)

167r~a 3 for a sphere of radius a. For an ellipsoid of revolution with semi-axis c = a > b (i.e., oblate ellipsoid), Perrin has shown that the rotational diffusion coefficient is given by O=

3kB--TF(p)= 3kBT {(2-p2)G(p)/p2/3-1}

16rr, b a

16~r~b a

~ _--p~

(11.44)

11.5 Solution and Suspension Colligative Properties

509

where p is the axial ratio, p = a/b > 1.0, and G(p) defined by equation 11.9 is a function of the axial ratio which for ellipsoids are given in Figure 11.3 along with G(p)/p 2/3 and F(p). This equation should be compared to the result

O= 3ksT 167r~b 3

(11.45)

for a sphere of radius b. Using the measured values of the rotational and translational diffusion coefficients, the size and shape of ellipsoidal colloidal particles can be determined. Interferometry [19] and quasielastic light scattering [20] can be used to determine the translational and rotational diffusion coefficients of ellipsoidal colloidal particles and thereby determine their form. This rotational diffusion coefficient will be used in Chapter 13 to aid in determining if nonspherical particles will orient in shear flow as they are cast to make a ceramic green body.

11.5 S O L U T I O N A N D S U S P E N S I O N COLLIGATIVE PROPERTIES Solution colligative properties are used to establish the vapor pressure, boiling point elevation, freezing point depression, and osmotic pressure of a solution where the solute is a salt, polymer, or surfactant. Solution colligative properties are used (1) in Chapter 14 to determine the vapor pressure of the solvent over the solution contained in the pores of a ceramic green body during drying and (2) in Chapter 10 to determine the polymer-solvent interaction parameter used in steric stabilization. A suspension's colligative properties of interest in ceramics are essentially its osmotic pressure. The osmotic pressure of a particulate suspension is used (1) in this chapter to determine the particle structure in suspension and (2) in Chapter 12 to determine the rheological properties of a suspension. The information in this section was taken from Physical Chemistry by Castellan [21]. In a solution where the solute is not volatile (e.g., salts, polymers, and surfactants), the vapor pressure of the solvent is limited by the mole fraction of the solvent at the interface. Several other solution properties are also dependent on the mole fraction of the solute, x, only and not on the chemical nature of the solute. These properties are referred to as coUigative properties (from the latin lingare, "to bind," and co, "together") which include vapor pressure lowering, frezing point depression, boiling point elevation, and osmotic pressure. In each case, two phases are in equilibriummone of which is the solution.

510

Chapter 11 ColloidalProperties of Ceramic Suspensions

For equilibrium, the chemical potentials of each phase must be equal. The chemical potential of the solvent, sub-l, in a solution, t~, is given by (11.46)

t~i(T, P) = t ~ ( T , P ) + R~Tln ai

where t ~ ( T , P ) is the chemical potential of the solvent only and a~(= ~x~) is the activity of the solvent in the solution. For an idea! solution (i.e., dilute), the value of the activity coefficient ~/1 is 1.0 and a l = xl. For nonideal solutions ~/~ can be greater than 1 or less than 1, resulting in positive or negative deviations from ideality.

Vapor Pressure Lowering [21] For a pure vapor (considered to be ideal) in equilibrium with a solution, we have (11.47)

l n a i =P_A pO

where p 0 is the vapor pressure of the pure solvent at temperature T o. The vapor pressure of the pure solvent, p o, as a function of temperature, T, is given by P~

= [1 atm] exp [AH~ L Rg

( T~p 1

T)]

(11.48)

where ~-/0ap is the enthalpy of evaporation and TBp is the normal boiling point of the solvent. From this expression, we can see that the vapor pressure of the solvent is lowered in accordance with the activity or concentration of solute in solution.

Freezing Point Depression For a pure solid (i.e., solvent) in equilibrium with a solution, we have In ai =

RgT -

Rg

-

(11.49)

0

where T o is the normal freezing point of the pure solvent and AG~ is the Gibbs free energy of fusion and AH~ is the enthalpy of fusion [21]. From this expresison, we can see t h a t the freezing point of the solution is lowered in accordance with the activity, a ~, or concentration of solute in solution because the enthalpy of fusion, AH~ is positive (i.e., endothermic).

Boiling Point Elevation For a pure vapor (i.e., solvent) in equilibrium with a solution, we have lnal-

Rg

(11.50)

11.5 Solution and Suspension Colligative Properties

511

where T O is now the normal boiling point of the pure solvent and /~/0ap is the e n t h a l p y of vaporization [21]. F r o m this expression, we can see t h a t the boiling point of the solution is raised in accordance with the activity, a 1, or concentration of solute in solution because the e n t h a l p y of vaporation, AH~ is negative (i.e., exothermic).

11.5.1 Osmotic Pressure of Electrolyte Solutions The osmotic pressure of solutions are discussed here as an introduction to the concept of osmotic pressure in suspension [21]. The phenomena of osmotic pressure is illustrated by a semipermeable m e m b r a n e filled with a sugar solution i m m e r s e d in water. The pressure inside the m e m b r a n e , p + 7r, is large t h a n t h a t in the water, p, according to the formula t~~

= t~~

+ 7r) + R g T l n a l

(11.51)

From a f u n d a m e n t a l t h e r m o d y n a m i c equation at constant T, we have dt ~~ = V1 dp, which becomes

Yl dp

+ 7r)- t~~

t~~

(11.52)

where 171 is the molar volume of the pure solvent. If the solvent is incompressible t h e n this equation can be simplified to give 7rV ~ - - R g T In a 1 ~ - R g T In [1 - x 2]

(11.53)

If the solution is ideal a 1 ~ x 1 - 1 - x2, where x2 is the mole fraction of the solute. If the solute is dilute (i.e., x2 ~ 1.0) t h e n we can make the simplification ln(1 - x2) ~ - x 2 = - n 2 / ( n + n2) ~ - n 2 / n , resulting in 7r-

n 2 R g T _ n2RgT nV1 V - c2RgT

(11.54)

where Vis the volume o f t h e ideal solution V -- n1171 + n2V2 ~ nll?l and c2[n2/V] is the molar concentration of electrolyte. The formal analogy between the earlier van't Hoff equation and the ideal gas law should not go unnoticed. The solute molecules of n u m b e r s n2 are dispersed in the solvent analogous to the gas molecules dispersed in an empty space. The solvent is analogous to the empty space. A 0.001 M solution will have an osmotic pressure, rr, of 0.0244 atm, which corresponds to a column of w a t e r 24 cm high. The m e a s u r e m e n t of the osmotic pressure is useful to determine the molecular weight, M2, of dilute ideal solutions; for example, if w2 g r a m s of solute is dissolved in a liter of solution t h e n w2RgT rr - ~ o r M

M2

2=

w2RgT

(11.55)

512

Chapter 11 Colloidal Properties of Ceramic Suspensions

This equation is good for ideal solutions. For an ionic surfactant solution, the solution is nonideal even at very low surfactant concentration and gives a highly nonlinear dependence of osmotic pressure on concentration. This is expected because ionic surfactants have a high affinity for the interfaces of solution-vapor, solution-solid, and solut i o n - m e m b r a n e as well as for themselves (i.e., micellization).

11.5.2 Osmotic Pressure of Polymer Solutions Osmotic pressure of polymer solutions was discussed in detail in Section 10.3.3 because it is important to steric stabilization. The final result of that development presented here for completeness. The osmotic pressure of a polymer solution determined from the FloryHuggins theory [22] and is given by

7r = (RgT/V~)[ln(1 - 4~2) + (1 - 1/x)d)2 + X~(b~ + . . . ]

(11.56)

where ~bl is the volume fraction of solvent, ~2 is the volume fraction of polymer, n l is the number of solvent molecules, n2 is the number of polymer molecules (which fill the space of x solvent molecules), and X~ is the dimensionless interaction parameter which is the pivotal parameter in the Flory-Huggins theory. In the preceding expression, the first two terms are due to entropy of mixing and the third term is due to enthalpy of mixing. In terms of polymer concentration, c2(= d~2Mw2/V2), the osmotic pressure is given by

7r = RgT[c2/Mw2 + [(V2/Mw2)2/V~](1/2- X~)c~ + . . . ]

(11.57)

where the number of solvent molecules needed to displace a polymer molecule may be approximated by x = l?2/l?1. l?1 and l?2 are the molar volume of the solvent and polymer, respectively, and Mw2 is the molecular weight of the polymer. Comparing this equation with that for the osmotic pressure of an electrolyte solution, we see that the mole fraction for electrolyte solutions has been replaced by the volume fraction for polymer solutions plus a term for the interaction parameter. The preceding equation can be simplified for dilute solutions to give

R~T 7r = ~ { d ~ 2 / x

+ ( 1 / 2 - Xl)~b~ + . . . } .

(11.58)

The RgT&2/(xV1) (or RgTc2/Mw2) term in this equation is the classical van't Hoff infinite dilution expression for the osmotic pressure, and the second term can be referred to as the second virial term with a second virial coefficient B2[ = (1/2 - Xl)d~/V,]. The second virial coefficient becomes 0 when X~ = 1/2. This point is called the Flory point or the

11.5 Solution and Suspension Colligative Properties

513

theta (0)point.

Usually the second virial coefficient becomes 0 at the theta temperature.

11.5.3 O s m o t i c P r e s s u r e o f the D o u b l e L a y e r in a Colloidal Suspension An aqueous colloidal suspension also has an osmotic pressure associated with both the double layer of the particles in solution and the structure of the particles. The osmotic pressure term for the structure is given in Section 11.6 for both ordered and random close packing. The osmotic pressure associated with the double layer surrounding the ceramic particles in aqueous solution is discussed here. If we consider a spherical cell containing at its center a spherical particle of radius, a, and a shell of fluid of radius, fl, as a one-particle example of the suspension with a volume fraction ~b [= (a/fl)3], then the osmotic pressure of the suspension is given by [23]

[ezO(fl )] IIDL = 4ekBT sinh2[ 2ksT

(11.59)

where c is the concentration of a binary symmetric electrolyte of valence z, e is the fundamental charge, and O(fl) is the potential at the outer boundary of the fluid. (Note: this equation is the same for the platelet particle case [24], where a is the half thickness of the particle and fl is the half thickness of the solution between the platelet particles.) Because the particle lattice is assumed to be static, the osmotic pressure term due to particle conformation for the disordered suspension derived by C a r n a h a n and Starling [25] and used by Dickinson [26] and Evans and Napper [27] must be added to the previous equation:

3ksTrb( 1 +

Hc = 47ra 3

(b + (b2 - 4)3) (1 - ~b)3

(11.60)

where 6 is the volume fraction of solids. This equation is derived in the Section 11.6. The II c term is small compared to the 1-IDL term when the electrical double layer is thick and highly charged. The value of O(fl) can be determined by solution to the PoissonBoltzmann equation:

d2O-~ 2dO

V2O = ~

r dr

2ezc sinh [ ezO l F,rF~0 [kBTJ

- ~

(11.61)

for the spherical shell with the boundary conditions $(a) = %

d--~-OI

dr r=~

= 0

(11.62)

514

Chapter 11 Colloidal Properties of Ceramic Suspensions

For potentials ~ less t h a n 25 mV (z = 1), the value of O(fl) is given by [28] O(fi)

=

A exp [K(fl - a)] + B exp[-K(fi - a)] Kfl

(11.63)

where K-~ is the scaling distance for the charge distribution called the double layer thickness. The double layer thickness [29] is calculated from

_~=( -s~-~ .~~/2 (11.64)

and A and B have the following definitions for the constant surface potential boundary condition:

A = [ezOo~ (_~ Ka exp[-2K(fl - a)] ) \ksT] K f i - 1 + e x p [ - 2 K ( f i a)]

(11.65)

+1

B

=

ezOo

Ks k - ~ - A

(11.66)

Strauss et al. [28] has developed a numerical method for the nonlinear Poisson-Boltzmann equation T > 25 mV for this spherical particle in a spherical cell geometry. Figure 11.5 is a plot of the osmotic pressure for a suspension of identical particles with 100 mV surface potential and Ka - 3.3. In this figure, the configurational osmotic pressure is also given and is much smaller t h a n that of the osmotic pressure due to the double layer. The osmotic pressure increases with increased volume fraction due to the further overlap of the double layers surrounding each particle. In addition, Strauss's work developed the means to determine the osmotic pressure for a polydisperse suspension corresponding to a lognormal size distribution. With this system a particular osmotic pressure, H(= 4CkBT sinh 2 [ez~(fi)/2ksT]), is assumed specifying a value of O(fl). The volume fraction is then calculated. For a particular particle size, a, the volume of fluid associated with the particle is determined by the specified value of O(fl) and the boundary conditions. With the outer boundary condition, dO/drlr=~ - 0 and O(fl) = constant for a specific value of the osmotic pressure, the total volume fraction, can be determined by summation of the volume fraction associated with

515

11.5 Solution and Suspension Colligative Properties 0.1

0.08

0.06

ui ft. oo 0.04 oo LU tr n 0.02

0

0

0.2

0.4

0.6

VOLUME FRACTION

FIGURE 11.5

Osmotic pressure versus volume fraction for r = 100 mV and Ka = 3.3" (a) body-centered cubic DLVO, (b) face-centered cubic DLVO, (c) linear PB equation, (d) nonlinear PB equation, (e) configurational osmotic pressure. Reprinted from [28] with permission by Academic Press.

0.12

0.1

0.08

u.[ n" c/) LU

n"

13.

0.06

0.04

0.02

0

0

0.2

0.4

0.6

0.8

VOLUME FRACTION

FIGURE 11.6

Osmotic pressure versus volume fraction for various widths of log-normal particle size distribution, qJ0 = 100 mV, 0.001 M KCI: (a) monodisperse with geometric m e a n diameter 0.1/~m, (b) (r z = 0.25, (c) (r z = 0.5, (d) (rz = 0.75. Note: ~g = exp((rz). Reprinted from [28] with permission by Academic Press.

516

Chapter 11 Colloidal Properties of Ceramic Suspensions

each size, a, in the normalized size distribution, P(a): or

d~ = f o (~

P ( o~) d o~

(11.67)

Figure 11.6 is a plot of the osmotic pressure versus volume fraction for a suspension with different widths, (rz = In crg, of the log-normal particle size distributions where ag is the geometric mean size and crg is the geometric standard deviation. As the width of the size distribution increases, the osmotic pressure decreases for a particular volume fraction. With this osmotic pressure, we can evaluate the order-disorder transition for an electrostatically stabilized suspension, which is discussed next.

11.6 O R D E R E D S U S P E N S I O N S "Monodisperse" particulate systems have been known to form ordered particulate arrays under slow sedimentation conditions, as shown in Figure 11.7. An important question is, How does polydispersity affect the ordering of a suspension? To explain this, Lindemann's rule for melting was adapted by Strauss [28] to particulate systems. Lindemann's rule states that, when the root mean square displacement of an atom from its lattice position due to thermal fluctuations reaches 10% of the lattice spacing, the ordered lattice melts into a disordered liquid. For colloidal suspensions (using the spherical cell approximation), Lindemann's rule will be interpreted slightly differently. The average radius, fl, of the spherical cell is equated with the lattice parameter, and the deviation of rms fl value from the mean fl value, , is given by firms = V(fl 2)

-

(fl)2 > 0.1 (fl)

(11.68)

when disorder begins. For a suspension of particles with hard sphere interactions, fl can be considered to be the radius of the size distribution. As a result, a particle size distribution with a relative standard deviation greater than 0.1 will not form an ordered array. The volume fraction at which ordering takes place for a hard sphere interaction is 0.5, as will be discussed later. Strauss used this equation as a condition for ordering a polydisperse suspension with an electrostatic interaction energy. Figure 11.8 is a plot of the flrm~/(fl) for different values of the suspension volume fraction and widths of the particle size distribution. When flr~s/(fl) is less than 0.1, ordering can take place. In Figure 11.8, this occurs for (rz = 0.15 at volume fractions, ~, less than 0.3, when c = 0.001 M KC1, 00 = 100 mV, and ag = 0.1 /zm and much lower volume fractions (i.e., 4) < 0.1 by extrapolation) for broader size distribu-

517

11.6 Ordered Suspensions

F I G U R E 11.7 Ordered arrays formed from filtered latex suspensions (Photo from Interfacial Dynamics Corporation, Portland, Oregon. Reprinted with permission.)

tions given. This ordering volume fraction (~ < 0.3) is drastically different than that for a hard sphere interaction (i.e., 0.5) and is a result of the softness and the long-range nature of the electrostatic interaction. Ordering at very low volume fractions has been experimentally observed with very low salt concentrations.

11.6.1 Osmotic Pressure (and Other Thermodynamic Properties) of a Ceramic Suspension This derivation has been taken from the book C o l l o i d a l D i s p e r s i o n s by Russel, Saville, and Schowalter [30]. The thermodynamic properties of a system can be evaluated from the Helmholtz free energy, A, as a function of the volume and temperature: A = E - TS

(11.69)

518

Chapter 11 Colloidal Properties of Ceramic Suspensions 1 0.90.8-

0.7uJ N_ u) _J

_l ~ m U.I

0.6" 0.5" 0.4-

0.3-

~

0.2-

o.10

C

~

,,,

,,

'

"

~

d

,~.+-..-~ ~

0.2

0.4

0.6

0.8

VOLUME FRACTION

FIGURE 11.8 Root m e a n s q u a r e (rms) cell size v e r s u s volume fraction for various w i d t h s of a log-normal particle size distribution with geometric m e a n d i a m e t e r 0.1/~m, ~0 = 100 mV, 0.001 M KCI: (a) ~rz = 0.75, (b) crz = 0.5, (c) ~rz = 0.25, (d) r z = 0.15. Note: (r e = exp((rz). R e p r i n t e d from [28] w i t h p e r m i s s i o n by Academic Press.

where E is the internal energy and S is the entropy. Using equilibrium statistical mechanics [31,32] the Helmholtz free energy can be given by A = -ksTlnQ

(11.70)

where Q is the partition function for the particle assembly given by Q = ~1. j~. . . j~ exp( - Ek~ ) dr1 ...drg

(11.71)

where EN is the internal energy of a volume V containing N particles in the configuration (r~ ... rN). From the Helmholtz free energy the osmotic pressure, II, of the particle assembly can be determined by l] = (~--~)T

(11.72)

and the Gibbs free energy can be determined from G = A + II V

(11.73)

These thermodynamic functions can be related directly to the interaction energy between particles, (P(r~j), and the equilibrium position of

11.6 Ordered Suspensions

519

the particles in the assembly characterized by the equilibrium radial distribution function, g(rij). By assuming pairwise additive interaction energies the internal energy, EN, of the configuration is given by (11.74)

EN = ~ r

with rij2 = (r i _ 1)) 9(ri -- 1)). Substitution of this expression into that for the Helmholtz free energy and taking the derivative with respect to volume at constant temperature gives the following expression for the osmotic pressure: II = N k B T -

N e

r3 g(rij)-~riidrij

(11.75)

where g(rij) is the equilibrium radial distribution function of particles in the suspension defined as 1

N 2 g ( r ) = ( N - 2)--------~ f'" "f P y d r l

. . . dry

(11.76)

where PN is the probability density corresponding to a particular configuration of particles. The equilibrium radial distribution function depends on the volume fraction, except in the dilute solution limit due to interactions among three or more particles. To determine the most stable phase of two possible phases, both thermodynamic and mechanical equilibrium between phases must be established. To establish thermodynamic and mechanical equilibrium, the Gibbs free energy and the pressure of the two phases must be equal. Hence, the calculation requires evaluating G and II as a function of the volume fraction of particles, ~b, for a particular interparticle interaction energy, r Then the intersection of a plot of G versus H identifies the point where the phase is in thermodynamic and mechanical equilibrium. This coexistence point corresponds to a particular interparticle interaction energy, q)(r) and temperature. A complete phase diagram is constructed by varying r and can be represented by plotting the volume fractions, ~b, of the coexisting equilibrium phases as functions of a dimensionless temperature, such a s - k s T/(I)mi n. This procedure will be discussed in detail for a hard sphere interaction potential and in general for a soft sphere (electrostatic interaction potential. 11.6.1.1 Hard Sphere Interaction The hard sphere interaction energy is an accurate approximation for short-range interactions between particles. This occurs when we have steric stabilization [33,34] due to polymer adsorption and electrostatic stabilization with a thin double layer [35,36] (i.e., high ionic

~2~

Chapter 11 ColloidalProperties of Ceramic Suspensions

strength). At dilute concentrations, the probability of a three-body interaction is small. Therefore, the equilibrium radial distribution functiong(r) is independent of concentration to a first approximation. Under these circumstances, the hard sphere potential is given by {0

(P(r) =

r < 2a} 2a > r

(11.77)

which produces a radial distribution function:

g(r) = H(r - 2a)

(11.78)

where H is the Heaviside step function. Interaction energies with long ranges alter the pair distribution function for r > 2a. When the interaction is attractive, g(r) is greater t h a n 1. When the interaction is repulsive, g(r) is less than 1. In either case, when q~ --* 0 as r --* ~, g --~ 1. This hard sphere dilute solution radial distribution function gives rise to a virial coefficient series in the volume fraction, ~, for the compressibility of the suspension II - (1 + A2~b + . . . ) NkB T

(11.79)

where the second virial coefficient is defined as A2 = ~

{1 - g(r/a)}(r/a)2d(r/a)

(11.80)

For concentrated suspensions of hard spheres, the radial distribution function for the fluid phase is generated from the solution to the P e r c u s Yevick [37] equation using a Heaviside step function multiplied by a nearest neighbor geometric function for a disordered fluid. The result is a function for the compressibility derived by Carnahan and Starling [25]: II NkBT

1 + (b + (b 2 -

(1 - r

~b3

(11.81)

Figure 11.9(a) is a representation of the radial distribution function for the disordered fluid for various volume fractions. For all volume fractions, the radial distribution function is highest at the surface of the central sphere and decreases with increasing radial distance. At high volume fractions there are also peaks for g(r) at r/(2a) equal to 2 and 3, corresponding to short-range order. The preceding equation is valid for volume fraction less than 0.5. For higher volume fractions, an ordered phase has the lowest Gibbs free energy. Various simulations [38-41] for hard spheres of the same size predict a transition from a disordered fluid with ~b < 0.50 to an ordered face-centered cubic structure for 0.55 < ~ < 0.74. Figure 11.9(b) is a representation of the

521

11.6 Ordered Suspensions

4.0

~=0.4 7 3.0

~

gHS

2.0

~=0.31

1.0

0

!

1.0

i

2.0

!

3.0

4.0

r/2a

b 15

~o

~=0.68

gHS

0 1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

9 1.9

r 2.0

r-2a FIGURE 11.9 Radial distribution function for suspensions of hard spheres: (a) in the disordered state and (b) in the ordered state. Taken from Russel [30, pp. 339-340]. Copyright 9 1989 by Cambridge University Press. Reprinted with the permission of Cambridge University Press.

radial distribution function for this ordered packing for various volume fractions. Here we see peaks at regular radial intervals. Within the ordered structure the compressibility is given by [42]

II NkBT

-

2.558 + 0.125fl + 0.176/32 - 1.053fl 3 + 2.819fl 4 - 2.922fl 5 + 1.118fl

6 + 3(4-

fl)/~

(11.82)

522

Chapter 11

where fl = 4 (1 verges near

Colloidal Properties of Ceramic Suspensions

~b/0.74) indicating that the compressibility diII

NkBT

2.22 0 . 7 4 - ~b

(11.83)

where ~ = 0.74 is the volume fraction of the face-centered cubic structure when spheres are touching. Figure 11.10 is a plot of the Gibbs free energy versus osmotic pressure for both the disordered and the ordered face-centered cubic structure. The phase with the lowest Gibbs free energy is the most stable. Figure 11.11 is a plot of the compressibility for the disordered fluid and the face-centered cubic ordered structure as a function of volume fraction. At ~b = 0.5, a coexistence between the disordered fluid and the ordered structure is shown. Other thermodynamic functions can be generated from these compressibility relationships. The small differences in the volume fractions between the two phases makes the disorder-order transitions a subtle one to observe experimentally except by iridescence of the ordered system as shown in Figure 11.12. Iridescence is a result of the diffraction of light by the ordered phase. It will be discussed in the next section. Although a face-centered cubic-ordered phase appears at equilibrium for r > 0.50, disordered dispersions of hard spheres can persist for

FIGURE 11.10

Gibbs free energy, G, versus osmotic pressure, II, for a suspension of hard spheres showing the intersection of the disordered and the ordered curves corresponding to the disorder-order transition with 47ra3no/3 = 0.74. Adapted from Gast et al. [63]. Reprinted with permission by Academic Press.

11.6 Ordered Suspensions

523

extended times. Simulations [43,44] reveal an osmotic pressure that increases smoothly from the values predicted from the disordered fluid to ~ -< 0.50 but diverges according to II NkBT

1.85 0 . 6 4 - 4)

(11.84)

This divergence occurs at random close packing of ~ ~ 0.63 - 0.64, consistent with the idea that higher volume fractions require some degree of order to be thermodynamically stable. 11.6.1.2 Soft S p h e r e I n t e r a c t i o n For electrostatic stabilization with low ionic strength, the hard sphere interaction is not applicable. In this case, we have a soft sphere interaction to consider. The softness of the interaction alters the crystallike structure with the lowest Gibbs free energy (i.e., most stable) from that of a face-centered cubic (FCC) structure to that of a body-centered cubic (BCC) structure [45,46]. Prediction of this structural transition has also been made by Hone et al. [47]. Shih et al. [48] have given phase diagrams of the most stable structure with a given number density, D[m-3], as a function of salt concentration, Psalt, and charge on the particles, Z. Figure 11.13 is an example of such a calculation. Here we see that, at low density, the particles take on a random, liquidlike structure for all salt concentrations. When the particle density increases, the BCC structure is formed at low salt concentrations,

41

l i m m w l i i i B i

z 21

immm

ri

imiw'.,d V . a l m IePL2;,di mm 0

0.2

0.4

0.6

FIGURE 11.11 Compressibility factor, H/kBT, for suspensions of hard spheres inc]ud-

ing the fluid and solid curves and the coexistence region for 0.50 < ~ < 0.55. Using the equation [25] H/NkBT - (1 + ~ + ~2 _ ~3)/(1 _ ~)3 for the disordered region and II/NkB T = 2.558 + 0.125fl + 0.176fl 2 - 1.053fl 3 + 2.819fi 4 - 2.922fl 5 + 1.118fl 6 + 3(4 - fl)/fl, w h e r e fl = 4(1 - 4)/0.74) for t h e o r d e r e d r e g i o n [42] w i t h a f a c e - c e n t e r e d cubic s t r u c t u r e .

524

Chapter 11 Colloidal Properties of Ceramic Suspensions

FIGURE 11.12 Settled dispersion of colloidal particles showing a disordered region on top and an ordered region below. Photo by K. E. Davis. Copyright 9 1989 by Cambridge University Press. Reprinted with permission of Cambridge University Press.

11.6 Ordered Suspensions

~2~

0.75x1018 Z-400

;\

'E

A (0 V

..... ---------.

0.50 .

2 a = O . 1 0 9 x 166m 2 a : O . 2 3 4 x 166m 2a=O.400xllSSm

FCC .'~,'

a

9

..;;,>.

"-'3" o25|',.,-p 0

,

.....

2.5

Psalt

5.0

7 . 5 x 1 0 -8

(M)

FIGURF, 11.13 Phase diagram plotted as particle density, D, versus salt concentration for particle charge Z = 400: (a) particle diameter of 0.109 t~m, (b) particle diameter of 0.234 tLm, (c) particle diameter of 0.400 t~m. The BCC-FCC phase boundaries are the same for the three cases, but the liquid-solid phases boundaries are pushed to lower densities as the particle size is increased. Taken from Shih et al. [48]. Reprinted by permission by Elsevier Science Publishing.

and at higher salt concentrations, the FCC structure is formed. The BCC structure is a delicate structure, which is not stable as the density is increased at a particular salt concentration. In Figure 11.13, the FCC structure is more stable at higher particle density. Nevertheless, the BCC structure is the most stable structure even for different particle sizes at low particle density. At higher particle charges, the BCC structure is even more stable. Experiments by Hachisu et al. [36] for the order-disorder transition of polystyrene particles in electrolyte solution are in agreement with this theory, as verified by Russel [49]. Experiments [45,46] confirm the FCC structure for the ordered phase at high particle density. The softness of the interparticle interaction alters the crystal structure to the BCC structure [45,46], in agreement with the theoretical findings displayed in Figure 11.13. Order formation in binary mixtures ofmonodisperse latex of different particle sizes was studied by Yoshimura and Hachisu [50]. When the volume fraction of particles exceeds a certain limit, a order~disorder

526

Chapter 11

Colloidal Properties of Ceramic Suspensions

phase transition takes place to deposit an ordered phase or phases with alloy structures. Various structures were observed with light microscopy, which include A1B2, NaZn~3, CaCus, MgCu2, and another type which has a composition ofAB 4 with P63/mmc symmetry. The structure is determined by the ratio of effective diameter of the constituent particles and is a realization of a state of good packing if not the closest packing.

11.6.2 Measurement of Ordered Array Structure Using light diffraction from an ordered suspension (see Figure 11.14 [51]), Krieger and coworkers [52,53] have shown that the interparticle spacing in an ordered packing of particles could be obtained by using the Bragg equation (commonly used for X-ray analysis of crystalline materials): n~ = 2d sin 0

(11.85)

where n is the order of diffraction, k is the wavelength of the incident

FIGURE 11.14

Photomicrograph of ordered structure in deionized polystyrene latex with particle diameter 0.33 ~m and ~b = 0.01. Taken from Kose et al. [51].

11.6 OrderedSuspensions 5.0

527

k o= 652.8 nm ng = 1.474

e,'

1,0 8

8,

:~ 0.5 e,

m o o

o

Si 02

0.05

,o

2'0

3'0

I

.o

5'0

Angle, 8 (degrees)

1

60

FIGURE 11.15 Light intensity (detector output voltage) versus scattering angle for ordered monodisperse TiO2and SiO2dispersions, showing first- and second-order diffraction peaks and the critical angle for total internal reflection, X = 632.8 nm. Taken from Barringer [55].

light, d is the distance between the diffracting lanes, and 0 is the angle between the incident light and the diffracting planes. In the usual experiment X is known and 0 is measured allowing the calculation of d when n is assumed to be 1. Depending on the sequence of different d values, the structure can be determined in addition to the unit cell dimension, as shown in Figure 11.15, where corrections have been made to account for the diffraction of the cell [54]. Barringer [55] has measured the iridescence or light diffraction of ordered arrays of monodisperse SiO2 and TiO2 particles. The results are shown in Table 11.2. The structure determined is t h a t of a face-centered cubic structure in both cases, with a unit cell dimension of 0.583 ftm for the 0.35 ftm SiO2 particles and 0.717 ftm for the 0.5 ft TiO2 particles.

11.6.3 D e f e c t s i n O r d e r e d A r r a y s Ordered arrays are not completely uniform but have defects, as do crystals, as shown in Figure 11.16. These defects fall into four main categories: (1) point defects or vacancies (i.e., places where a particle is missing); (2) line defects or dislocations, (3) planar defects (i.e., grain boundaries), and (4) volume defects like cracks. The point defects are

528

Chapter 11

Colloidal Properties of Ceramic Suspensions

TABLE 11.2 Line Broadening Results of Figure 11.15, Showing Ordered Domain Size, Using Equation 11.86 a

0 Degree 0

O'

Detector output (volts) Baseline

Results for 0.35/xm SiO2 33.8 28.5 0.05 61.2 60.8 0.05 Results for 0.5/xm TiO2 27.6 23.2 0.17 56.4 55.1 0.17

Peak height

fl(20) (radian)

Ordered domain size (txm)

3.8 0.267

0.01745 0.03316

43.6 39.6

0.61 0.28

0.0262 0.0436

27.3 26.2

a Reprinted by permission of Chang and Ring [56]. Copyright 1988 American Chemical Society.

inherent to the equilibrium state of the ordered array and therefore are uniquely determined by the thermodynamic variables for the system such as temperature, pressure, and volume fraction. The presence and concentration of the other defects, however, depends on the way

FIGURE 11.16 SEM micrograph of the packing on the top surface of the SiO2 ordered structure analyzed diffraction peaks in Figure 11.15 (bar = 5 ttm). Reprinted by permission of Chang and Ring [56]. Copyright 1988 American Chemical Society.

11.6 Ordered Suspensions

FIGURE 11.17 ticles.

529

Energy per point defect in an ordered structure of monodisperse par-

the ordered array was formed. The introduction of point defects in an ordered array of particles increases its enthalpy AH because each particle is an energy well. Although the AH term increases, the Gibbs free energy of point defect formation, A G, decreases due to the effect of the TAS term. For any increase in disorder of the array brought about by the introduction of point defects, AS, the entropy associated with the addition of a point defect is positive. This situation is schematically shown in Figure 11.17. The increase in the concentration of defects, Np, increases the AH term linearly and decreases the - T A S term exponentially. As a result AG corresponding to the sum AG = AH TAS, first decreases then increases. The minimum of this curve gives the equilibrium concentration of defects. From Figure 11.16 the equilibrium point defect concentration can be estimated to be ~3 per 380 particles or 0.8%.

11.6.4 Processing Effects on O r d e r D o m a i n Size The growth of an ordered colloidal structure is similar to that of the growth of a crystal. In this case, the rate at which particles arrive at

530

Chapter 11 ColloidalProperties of Ceramic Suspensions

a growth site depends on the rate at which the particles arrive at the surface of the ordered structure and the rate at which the particles diffuse across the surface of the ordered structure to the growth site. If the particle arrival rate is small compared to the rate of surface diffusion, then the ordered domain size will be large. If the particle arrival rate is large compared to the rate of surface diffusion, then the ordered domain size will be small. Because the surface diffusion rate of a particular particle size is dependent on only the volume fraction at the surface (which is relatively constant at 50 to 55%) then it is relatively constant. The particle arrival rate, however, can be controlled by the sedimentation rate, centrifugation rate, filtration rate, or electrophoretic rate. This gives rise to different ordered domain sizes, depending on the way in which the suspension was processed. Slower cake build-up processing conditions lead to larger ordered domain sizes.

11.6.5 Measurement of Ordered Domain Size by Light Diffraction Diffraction line broadening has been adapted to light diffraction to measure ordered domain size by Chang and Ring [56]. The first treatment of diffraction line broadening was by Scherrer [57], who derived the equation fl(20)

kX

=L cos 0

(11.86)

where fl is the peak breadth at half-height in radians, L is the ordered domain size (the volume average size perpendicular to the diffracting planes), and k is a constant about which there has been considerable disagreement. Values given include [58] 0.94 (Scherrer [57]), 0.89 (Bragg [64]), 0.92 (Seljalkow [65]), and 1.42 (Laue [66], Jones [67]). The value of k depends on the shape of the ordered domain and the particular diffraction peak used for breadth determination. For ordered domains made of spherical particles, Stokes and Wilson [59] gave k = 1.0747. Because all these values of k are very near 1.0, most users of this theory simply neglect k in the preceding equation and refer to L as the "crystallite" size. In addition, diffraction line breadth contains information on lattice strain, lattice defects, and thermal vibrations of the crystal structure. The chief problem to determine crystallite size from line breadth is the determination of fi(20) from the diffraction profile, because broadening can also be caused by the instrument. To correct for the instrumental broadening on the pattern of the sample, it is convenient to run a standard peak from a sample in which the crystallite size is large enough to eliminate all crystallite size broadening. By use of a convolu-

11.6 Ordered Suspensions

531

tion integral method with these two patterns, the crystallite size broadening can be isolated. Another way to correct for the instrumental broadening on the pattern is to determine the broadening due to the instrument optics. Barringer [55] has measured the light diffraction of ordered arrays of SiO2 and TiO2 particles. The results are shown in Figure 11.15. The structure determined in both cases is that of a facecentered cubic structure. The peak broadening in this figure is also tabulated in Table 11.2 and gives rise to a crystallite size of 40 ftm for the SiO2 system and 27/xm for the TiO2 system. For the SiO2 system, the ordered domain size is clearly seen in Figure 11.16.

11.6.6 Effect of Ordering a n d D o m a i n Size on Ceramic Processing Ceramic suspension rheology is effected by ordering of monodisperse particles or random close packing of broad size distribution particles or network formation of gels. At the point of particle-particle contact, the viscosity at low shear rate increases drastically, approaching infinity. This divergence in viscosity gives an effective yield stress and other visco-elastic effects, including thixotropy and memory effects. These memory effects are a direct result of ordered domain size for monodisperse particles, the size of the random close packed regions for broad size distributions and the network structure for gels. Ceramic suspension rheology will be discussed in Chapter 12. Because ordering and domain size play such an important role in rheological properties of high volume fraction ceramic suspensions, we will have to return to this subject in Chapter 12. 11.6.6.1 E f f e c t o f V a c a n c i e s a n d D o m a i n S i z e on Sintering

When an ordered array of monosized particles is sintered, the vacancies and grain boundaries create pores, cracks, and other flaws which are difficult to remove during sintering. Pores are flaws about the size of the original ceramic particles in the final sintered ceramic. Grain boundaries are flaws that lead to cracks of a size equivalent to the size of the ordered domain. Both these flaws lead to the failure of the final ceramic piece according to Griffiths's analysis: (r = ~

(11.87)

where (r is the yield stress, c is the size of the flaw, and Kic is the toughness of the ceramic. When the flaw size increases the ceramic breaks with less stress. The bigger is the ordered domain size, the larger the flaw size.

532

Chapter 11 Colloidal Properties of Ceramic Suspensions

As a result of these vacancies and, more important, the ordered domains of large size in the packing, the most uniform packing of ceramic powders which sinters without flaws is obtained with random, close packed particles. To prevent ordered packing that gives random, close packed particles, the particle size distribution must not be monodisperse but broad enough so that order is prevented. This can be predicted by using Lindemann melting rule and the analysis of Section 11.6 and shown in Figure 11.8 and by Dickinson et al. [60]. In a simplified form this analysis suggests that the size distribution of particles needed for random close packing must have a relative standard deviation greater than 0.1. If the relative standard deviation is less, ordered domains are always possible at high particle volume fractions typical of ceramic green bodies. Suzuki et al. [61] have shown both theoretically and experimentally that a particle size distribution with a relative standard deviation between 0.1 and 0.3 can be optimally packed to relatively high green densities of approximately 60% without ordering, Broad size distributions are not desirable for many ceramic applications because they can cause cannibalistic or exaggerated grain growth, as will be discussed in the chapter on sintering. Exaggerated grain growth also leads to large flaws of a size of these large grains. As a result of these considerations, the optimum relative standard deviation of a particle size distribution desirable for a ceramic powder is between 0.1 and 0.3 [62].

11.7 S U M M A R Y In this chapter, we have described the colloid properties of ceramic powders in suspension. The ramifications of colloidal properties on ceramic processing have been discussed with emphasis on single phase ceramic suspensions with a distribution of particle sizes and composites with their problems of component segregation due to density and particle size and shape. The next chapter will discuss the rheology of the ceramic suspensions and the mechanical behavior of dry ceramic powders to prepare the ground for ceramic green body formation. The rheology of ceramic suspensions depends on their colloidal properties.

Problems la. Derive the equation for the terminal settling velocity for a sphere when NReis larger than 1.0. lb. Derive the equation for the position in a centrifuge as a function of time for a sphere when NReis larger than 1.0.

References

533

2. Determine the equilibrium between Brownian diffusion and sedimentation for a centrifuge operating at 10,000 rpm, using various particle radii between 1 t~m and 0.001 t~m. What is the ratio of equilibrium concentrations, (CJC1), at distances, Re2 = 10 cm and Re1 = 11 cm. The particles have a density of 2.0 gm/cm 3 and the liquid is water with a viscosity of 0.01 poise (gm/cm/sec). 3. Determine the osmotic pressure of an aqueous 1 : 1, 2 : 1, 2 : 2 and 2 : 3 salt solution at 0.1 molar. Assume that the solutions are ideal. 4. Determine the osmotic pressure of 0.1 molar polymer solution in water with molecular weights of 1,000, 10,000, and 100,000. The Xl value of all polymers is 0.6. 5. Determine the osmotic pressure for a suspension of 0.001 t~m hard spheres at a volume fraction of 0.74. 6. Using a linearized flat plate solution to the Poisson-Boltzmann equation for the potential distribution of a sphere, what is the osmotic pressure of a 0.01 volume fraction suspension of 0.1 t~m spheres immersed in a 1:1 salt solution at 0.1 M. The surface potential of the particle is 25 mV. Compare this value with that for the salt solution only. 7. Determine the boiling point of 0.1 molar polymer solution in water with molecular weights of 1,000, 10,000, and 100,000. The Xl value of all polymers is 0.6. 8. Determine the freezing point of an aqueous 1:1, 2:1, 2 : 2 and 2 : 3 salt solution at 0.1 molar. 9. Determine the settling velocity in water of a kaolin plate particle 0.01 t~m thick and 3 t~m in diameter. Compare this value with that of a sphere of the same diameter. Assume that the particle rotates fast so that all orientations are possible during sedimentation. 10. Calculate the equilibrium between sedimentation and diffusion for the particle given in problem 9 assuming hindered settling at =

50%.

References 1. Graham, T., Phil. Trans. Roy. Soc. 151, Part I, 183-224 (1861). 2. Geiger, G. H., and Poirier, D. R., "Transport Phenomena in Metallurgy." AddisonWesley, Reading, MA, 1972. 3. Eisner, F., Proc. Int. Congr. Appl. Mech. 3rd, 1930, p. 32 (1932). 4. Perrin, J., "Brownian Motion and Molecular Reality." Taylor & Francis, London, 1910. 5. van de Ven, T. G. M., "Colloidal Hydrodynamics," Colloid Sci. Monogr. Ser. Academic Press, London, p. 270, 1989. 6. Concha, F., and Almendra, E., Int. J. Miner. Proc. 6, 31 (1979).

534

Chapter 11

Colloidal Properties of Ceramic Suspensions

7. Happel, J., and Brenner, H., "Low Reynolds Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, NJ, 1965. 8. Davies, J. T., and Rideal, E. K., "Interfacial Phenomena." Academic Press, New York, 1963. 9. Rutgers, A. J., and Nagels, P., Nature (London) 171, 568 (1953). 10. Perrin, J., Nobel Prize (1926). 11. Einstein, A., "Investigations in the Theory of the Brownian Movement." Dover, New York, 1956. 12. Weiner, B., in "Modern Methods of Particle Size Analysis," (H. G. Barth, ed.), Chapter 3. Wiley, New York (this book is part of a series entitled "Chemical Analysis," Vol. 73). 13. Tanford, C., "Physical Chemistry ofMacromolecules," p. 327. Wiley, New York, 1961. 14. Perrin, F., J. Chem. Phys. 10, 1036 (1942). 15. Chu, B., "Laser Light Scattering," p. 219. Academic Press, New York, 1974. 16. Debye, P., "Polar Molecules." Dover, New York, 1929. 17. Dicke, R. H., and Witke, J. P., "Introduction to Quantum Mechanics." Addison-Wesley, Reading, MA, 1960. 18. Perrin, F., J. Phys. Radium 5, 497 (1934); 7, 1 (1936). 19. Dublin, S. B., Clark, N. A., and Benekek, G. B., J. Chem. Phys. 54, 5158 (1971). 20. Berne, B. J., and Pecora, R., "Dynamic Light Scattering," p. 143. Wiley (Interscience), New York, 1976. 21. Castellan, G. W., "Physical Chemistry." Addison-Wesley, Reading, MA, 1964. 22. Napper, D. H., "Polymeric Stabilization of Colloidal Dispersions." Academic Press, New York, 1983. 23. Bell, G. M., Levin, S., and McCartney, L. N., J. Colloid Sci. 33, 335 (1970). 24. Verwey, E. J. W., and Overbeck, J. T. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, New York, 1948. 25. Carnahan, N., and Starling, K., J. Chem. Phys. 51, 635 (1969). 26. Dickinson, E., J. Chem. Soc., Faraday Trans. 2 75, 466 (1979). 27. Evans, R., and Napper, D., J. Colloid Interface Sci. 63, 43 (1978). 28. Strauss, M., Ring, T. A., and Bowen, H. K., J. Colloid Interface Sci. 118(2), 326334 (1987). 29. Debye, P., and Huckel, E., Phys. Z. 24, 185 (1923); Debye, P., ibid. 25, 93 (1924). 30. Russel, W. B., Saville, D. A., and Schowalter, W. R., "Colloidal Dispersions," p. 332. Cambridge, Univ. Press, Cambridge, UK, 1989. 31. Castillo, C. A., Rajagoplan, R., and Hirtzel, C. S., Rev. Chem. Eng. 2, 237-348 (1984). 32. Van Megen, W., and Snook, I., Adv. Colloid Interface Sci. 82, 62-76 (1984). 33. Vrij, A., Jansen, J. W., Dhont, J. K. G., Pathmananoharan, C., Kops-Werkhoven, M. M., and Fijnaut, H. M., Faraday Discuss. 76, 19-36 (1983). 34. de Kruif, C. G., Jansen, J. W., and Vrij, A., in "Complex and Supramolecular Fluids" (S. A. Safron and N. A. Clark, eds.), pp. 315-343. Wiley (Interscience), New York, 1987. 35. Hachisu, S., and Kobayashi, Y., J. Colloid Interface Sci. 46, 470-476 (1974). 36. Hachisu, S., Kobayashi, Y., and Kose, A., J. Colloid Interface Sci. 42, 342-348 (1973). 37. Smith, W. R., and Henderson, D., Mol. Phys. 19, 411-415 (1970). 38. Alder, B. J., and Wainwright, T. E., J. Chem. Phys. 27, 1208 (1957). 39. Wood, W. W., and Jacobson, J. D., J. Chem. Phys. 27, 1207-1208 (1957). 40. Alder, B. J., Hoover, W. G., and Young, D. A., J. Chem. Phys. 31, 459-466 (1968). 41. Haymet, A. D. J., and Oxtoby, D. W., J. Chem. Phys. 84, 1769-1777 (1986). 42. Hall, K. R., J. Chem. Phys. 57, 2252-2254 (1972). 43. Woodcock, L. V., Ann. N. Y. Acad. Sci. 371, 274-298 (1981). 44. Pusey, P. N., and van Megan, W., in "Complex and Supramolecular Fluids" (S. A. Safron and N. A. Clark, eds.), pp. 673-698. Wiley (Interscience), New York, 1987. 45. Williams, R., and Crandall, R. S., Phys. Lett. 48A, 255-256 (1974). 46. Ackerson, B. J., and Clark, N. A., Phys. Rev. Lett. 46, 123-126 (1981).

References

535

47. Hone, D., Alexander, S., Chaikin, P. M., and Pincus, P., J. Chem. Phys. 79, 14741479 (1983). 48. Shih, W. Y., Aksay, I. A., and Kikuchi, R., J. Chem. Phys. 86, 5127-5132 (1987). 49. Russel, W. B., "Dynamics of Colloidal Systems." Univ. of Wisconsin Press, Madison, 1987. 50. Yoshimura, S., and Hachisu, S., J. Phys. Colloq. (Orsay, Fr.) C3, Suppl. 3, Tobe 46, p. C3-115-C3-126 (1985). 51. Kose, A., Ozaka, M., Takano, K., Kobayashi, Y., and Hachisu, S., J. Colloid Interface Sci. 44, 330-338 (1973). 52. Hiltner, P. A., and Krieger, I. M., J. Phys. Chem. 73, 2386-2389 (1969). 53. Krieger, I. M., and O'Neill, F. M., J. Am. Chem. Soc. 90(12), 3114 (1968). 54. Tomita, M., Takano, K., and van de Ven, T. G. M., J. Colloid Interface Sci. 92, 367 (1983). 55. Barringer, E. A., PhD. Thesis, Massachusetts Institute of Technology, Cambridge, MA (1984). 56. Chang, S. Y., and Ring, T. A., Langmuir 4, 1128 (1988). 57. Scherrer, P., "Kolloidchemie by Zsigmondy," 3rd ed., p. 394. Leipzig, Berlin, 1920. 58. Lipson, H., and Steeple, H., "Interpretation of X-Ray Powder Diffraction Patterns," p. 261. Macmillan, London, 1970. 59. Stokes, A. R., and Wilson, A. J. C., Proc. Camb. Phil. Soc. 38, 313 (1942). 60. Dickinson, E., Miln, S. J., and Patel, M., Powder Technol. 59, 11-24 (1989). 61. Suzuki, M., Oshima, T., Ichiba, H., and Nasegawa, I., Kona (Hirakata, Jpn.) 4, 4-10 (1986). 62. Dirksen, J., and Ring, T. A., in "High-Tech Ceramics: View Points and Perspectives" (G. Kostorz, ed.), p. 29. Academic Press, London, 1989. 63. Gast, A. P., Hall, C. K., and Russel, W. B., J. Colloid Interface Sci. 96, 251-267 (1983). 64. Bragg, W. L., "The Crystalline State, A General Survey." Bell, London, 1933. 65. Seljalkow, N., Z. Phys. 31, 439 (1925). 66. von Laue, M., Z. Krist. 64, 115 (1926). 67. Jones, F. W., Proc. R. Soc. (A) 166, 16 (1938).

This Page Intentionally Left Blank

PART

IV GREEN BODY FORMATION MISE-EN FORME

With the information given in Part III just completed. We are ready to discuss the methods of ceramic green body formation. A ceramic green body is a molded shape made from a multitude of ceramic particles. The adjective green comes from the slightly green color that a form made from a porcelain paste often has when wet. A green body can be made from a ceramic suspension, a ceramic paste (i.e., a high-volume fraction suspension), or from a dry (or, more likely, spray dried) ceramic powder. It is molded into a shape with forces applied to either the individual particles, as in drag and gravitational forces, or to the mass of particles as, in ramming a dry powder onto a die or extruding a paste. After drying, the ceramic green body contains (1) the ceramic components; (2) a polymeric binder system including dispersant, binder, and plasticizer, and (3) residual solvent used to make the initial ceramic suspension or paste or used to spray dry the ceramic powder before dry pressing. In Chapter 12 of this book, the mechanical properties of ceramic suspensions, pastes, and dry ceramic powders are discussed. Ceramic suspension rheology is dependent on the viscosity of the solvent with polymeric additives, particle volume fraction, particle size distribution, particle morphology, and interparticle interaction energy. The interparticle forces play a very important role in determining the colloidal stability of the suspension. If a suspension

538

Part I v

Green Body Formation--Mise-En Forme

of monosized spheres is colloidally stable, it has rheological properties when dilute and when concentrated that are, to a reasonably high degree, predictable by presently available theories. Extensions of these theories to broad particle size distribution suspensions often gives reasonable results. If the same suspension is colloidally unstable, its rheological properties are not predictable even when dilute. Suspensions of monosized platelets or fibers when colloidally stable have rheological properties w h e n d i l u t e that are also predictable by presently available theories. The mechanical properties of dry ceramic powders are determined primarily by interparticle forces (i.e., friction and cohesion). These mechanical properties are important in designing the equipment necessary to process the ceramic suspensions, pastes, and dry powders into ceramic green bodies. Each of the different raw materials

Woodblockprint of the stamping of a pattern on the surface of a bowl and two piece forming of a jar. Taken from T'ien Kun K'ai Wu 1637 print from Bushnell, S. W. "Description of Chinese Pottery and Porcelain" [translation of J'ao Shuo], Oxford University Press, 1977.

FIGURE IV.1

Part IV Green Body [~ormation--Mise-En Forme

539

for green body formation is used with different methods of ceramic mise en forme in Chapter 13. For example, (1) ceramic suspensions are used with slip casting, drain casting, tape casting, dip coating, filter pressing, and electrodeposition; (2) ceramic pastes are used with extrusion and injection molding; and (3) dry ceramic powders are used in die pressing, dry bag isostatic pressing, and wet bag isostatic pressing. In addition, ceramicists are interested in manipulating the green body microstructure and its uniformity during green body fabrication by controlling particle segregation and anisotropic particle orientation. Both prediction of the forces necessary for fabrication and prediction of microstructure control can be done with a detailed knowledge of interparticle flow in green body manufacturing processes. Chapter 13 ends with a discussion of experimental methods used in the characterization of ceramic green bodies and their uniformity.

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12,

Mechanical Properties of Dry Ceramic Powders and Wet Ceramic Suspensions

12.1 O B J E C T I V E S The mechanical properties of dry ceramic powders and wet ceramic suspensions must be controlled to have (1) dry powders that flow into molds for dry pressing and deform during pressing, (2) suspensions that flow into slip cast molds, and (3) pastes that can be stamped into objects, extruded into shapes, or injection molded. The mechanical properties of a dry powder are discussed in terms of bulk solids flow. The rheology of suspensions is discussed in terms of constitutive equations that are Newtonian (i.e., linear) and non-Newtonian (i.e., nonlinear). The rheology of thick pastes is discussed in terms of visco-elasticity. With an understanding of the mechanical properties of dry powders, ceramic suspensions and pastes, the stage is set for Chapter 13, "Ceramic Green Body Formation." 541

542

Chapter 12

Mechanical Properties of Powders and Suspensions

12.2 I N T R O D U C T I O N

Considering a mass of ceramic powder about to be molded or pressed into shape, the forces necessary and the speeds possible are determined by mechanical properties of the dry powder, paste, or suspension. For any material, the elastic moduli for tension (Young's modulus), shear, and bulk compression are the mechanical properties of interest. These mechanical properties are schematically shown in Figure 12.1 with their defining equations. These moduli are mechanical characteristics of elastic materials in general and are applicable at relatively low applied forces for ceramic powders. At higher applied forces, nonlinear behavior results, comprising the flow of the ceramic powder particles over one another, plastic deformation of the particles, and rupture of

b

L1

L2

r

F

A

rjlf

(a) Young's modulus Y = OF/A/OL/L, (b) shear modulus G = aF/A/OLJ L2, (c) bulk modulus K = OF/A/OV/V.

FIGURE12.1

12.3 Equations of Motion

543

the particles, especially aggregates. These nonlinear mechanical characteristics are also dependent on the rule at which the force, F (or strain), is applied. The shear and Young's moduli are related to the shear stress, AF/A = T i j , and normal stress, A F/A = ~ii, respectively. For dry ceramic powders, there is little predictive capability for the nonlinear mechanical characteristics from first principles. As a result, the stress strain relationship must be measured for each ceramic powder formulation. This chapter deals with the basics of the mold filling process. First the equation of motion is given which is a simple force balance on a differential volume element. This equation shows the interrelationship between the forces that act on the differential volume element and the rate of m o m e n t u m transfer into and out of the differential volume element. This m o m e n t u m is simply the product of the mass (density z volume) and the velocity. The forces acting on the differential volume are pressure, gravity, and stress. The stress can be related to the strain acting on the ceramic powder or suspension through one of several rheological equations. The rheology of ceramic suspensions is measured in terms ofvicosity, which is a function of the shear stress. This function can be linear (i.e., Newtonian) or nonlinear (i.e., non-Newtonian). For a dry ceramic powder, the mechanical behavior is measured by the principal stress required to cause flow, which is called the Coulomb yield criterion. With the equation of motion and the mechanical behavior of the material, the velocity profile or strain rate profile in the material can be predicted, thus allowing prediction of the speed of powder casting and pressure distribution inside and at the walls of the mold.

12.3 E Q U A T I O N S

OF M O T I O N

The flow of materials is accounted for with two balances: conservation of mass and conservation of momentum transfer. The most important is a m o m e n t u m balance, which is also called the equation of motion. The mass balance (also called the continuity equation) makes sure that mass is conserved.

12.3.1 Continuity Equation The continuity equation in vector notation is given as [1] apat ~ v ' ( p v ) = 0

or

Dp_ Dt -pV.(v)

(12.1)

where p is the material density, v is the velocity vector, and Dp/Dt is the substantial derivative of the density. The substantial derivative is

544

Chapter 12 Mechanical Properties of Powders and Suspensions

the Op/Ot of a volume element which moves with the material at a velocity, v. In Cartesian coordinates the continuity equation is given by op + OpVx + OpVy + Opv z = 0 Ot Ox Oy Oz or

op

op

op

[OVx OVy OVz~

DP=O--P-P + v x +v + Vz =03, --~zJ Dt Ot -~x -~y -~z P ~ Ox + +

(12.2)

12.3.2 M o m e n t u m B a l a n c e The momentum balance is essentially a force balance using Newton's law, F = ma. Where both the mass, m, and the acceleration, a = d v / d t , is grouped into the derivative as follows: F_d[mv] dt

(12.3)

giving (my), the momentum. The term d ( m v ) / d t is the rate of momentum transfer. Performing this force balance (or equivalently a balance on the rate of momentum transfer) in a differential volume element, the equation of motion is obtained: D(v) 0(v) P Dt = p ~ +pv'Vv=-vP-V'r+pg

(12.4)

where p[D(v)/Dt] is the density (i.e., mass per unit of volume) times acceleration, VP is the pressure force on the differential element per unit volume, V 9r is the viscous force acting on the differential volume element and p g is the gravitational force acting on the differential element. Both forces are on a per unit of volume basis. This equation is the famous Cauchy equation of motion. When the Newtonian relations for the stress tensor are substituted for r in the Cauchy equation of motion, we have the famous N a v i e r - S t o k e s equation. For various coordinate systems, the equation of motion is given in the appendix of this book. The equation of motion and the continuity equation are used to determine the velocity profile in the material caused by the pressure and the gravitational forces. In principle these two equations can be solved if the stress tensor, r, can be written as a function of the velocity vector v (or the time derivative of the displacement, ~ -- d x / d t ) . The equation that relates the stress tensor, r, to the velocity vector v or to the time derivative of the displacement, i , is called the constitutive equation o f the material.

12.3 Equations of Motion

545

12.3.3 Constitutive Equations for Dry Powders The constitutive equation for a dry powder is a governing equation for the stress tensor, r, in terms of the time derivative of the displacement in the material, ~ (= v = d x / d t ) . This displacement often changes the density of the material, as can be followed by the continuity equation. The constitutive equation is different for each packing density of the dry ceramic powder. As a result this complex relation between the stress tensor and density complicates substantially the equation of motion. In addition, little is known in detail about the nature Of the constitutive equation for the three-dimensional case for dry powders. The normal s t r e s s - s t r a i n relationship and the shear s t r e s s - s t r a i n relationship are often experimentally measured for dry ceramic powders because there are no known equations for their prediction. All this does not mean that the area is without fundamentals. In this chapter, we will not use the approach which solves the equation of motion but we will use the friction between particles to determine the force acting on a mass of dry powder. With this analysis, we can determine the force required to keep the powder in motion.

12.3.4 Constitutive Equations for Fluids There are two general types of constitutive equations for fluids: Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, r, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior: (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. For Newtonian fluids the relationship between stress tensor and the velocity vector is given in Table 12.1 for various coordinate systems [1]. These relationships can be generalized to be r = -~h

(12.5)

where V is a constant, the viscosity and h is the symmetrical "rate of deformation tensor," which for Cartesian coordinates is Ovi Ov~ h 0 = -~- ~ ~

(12.6)

T A B L E 12.1

C o m p o n e n t s of the Stress Tensor for Newtonian Fluids

Rectangular Coordinates (x, y, z):

Oex 2 ] T==--V 2-~X -- g (V'v)

(A)

Tyy=--~ 2-~-y- g(V-v)

(B)

%z

-~z

[O.x o~,]

(C)

Txy= Tyx =----T~ Tyy -- C]X]

(D)

[o., <]

(E)

-r;

9, z = % = - n k O 9z x = , x z = - ,

[o.o.] z +-~y

(F)

7x+W

(V v) = ~OVx + _ _ o% + _ _ OVz Ox 03, Oz Cylindrical Coordinates (r, O, z): 9

(G)

]

.=_.[2(lO.

(A)

_2

(B)

9zz = - ~

]

(C)

OlJr

2

Trr----T~ 2 7 r - ~(V-v)

2~-z

- g(v.v)

TFO=TOr= --~/ [ r ~ r (-~)-1 -!O1)r] r OOJ [Ovo + l OVz] roz = rzo = - n k az

7-~

[ O'UzOl~r ] Trz=Tzr=--T~ ~r-~-~Z 10 1 OVo Or, (V " 1]) = 7 ~rr (rVr) nt- -- ~

r 00 -t- -0z

(D) (E) (F) (G)

Spherical Coordinates (r, 0, ~b): _

Trr

_

r~r__~.~]

(A)

9QL Or

(B) (C)

r** = - ~ [2kr sin 0 ~ + r + r - 5(V .v) r o [1)0\ 1 01)rl TrO = TOr : -~q [r -~r ~ r ) + r - ~ ] [sinOO__(v~) 1 ovo1 r o ~ = r o ~ = - T L r 00 ~ +rsin00~bJ [ 1 01)r rO__(~)] Td~r -- Trrb -~ --gQ r sin 0 a'~ + Or 10 (V.1)) = -~-~-.~.-r(r21)r) + ~ 1 -~o(1)osinO)+ l Ov--2

(D) (E) (F) (G)

r sin 0 04)

The Function -(r:V1)) = ~q)~ for Newtonian Fluids Rectangular:

r,,~vx.~. _,_,,ovy.~. _,_,,o,-,z.~.l _,_r~,,, _,_O,,xl~ + [o,,~ + o,,yl~ _,_[O,,x + _~]~ _ ~ [,~,,x ..,__ +,~,,y o,.,r

(I)'=2L\ax/

\0y/

\ az / j

Lax

Oy j

7y

az ]

-~z

3 7x

Oy

(A)

az j

Cylindrical:

[ (~1)r~2 (~_~1)0 ~)2 + (~1)z~21+ [r ~ (~)1+ -- 01)r12-t- r!0~1)z -k 01)012-k [01) r -{-(~1)z12-- 2 [ 1

rb. = 2 \ Or / + \r aO +

\ az ] J

-~r

r O0]

[ r O0

Oz J

-~z

Or J

-3

(~ 101)0 01)z12 - -k c]z J -~r (FUr)-~---r -O0

(B)

Spherical: r(~

2 -t- ( 101)0 -k ~ ) 2 -k

O'=2L\Or ]

-~

(

10v~ +--+VrvocotO)2] -k \Or rsin0~-

r

r

~rr

(~)

+--lOVr12+ r o0J

01)r+ rO(~r)]~ 0 1 ~0 (1)0sin 0) + a1)619 + [ rl sin 0"~ - 32 1 1~r(reVr)+rsinO Or Odp]

\sin0 (__ 0 v,) + 1 o-0 s-~n0 rsin0

+ 01)o12 0#)J

(C)

548

Chapter 12 Mechanical Properties of Powders and Suspensions Pseudo-plastic

with Yield stress

Dilatant

with Yield stress

L_ 4~ U~ L__

Pseudo-plastic

r r U~

Dilatant

Strain Rate, dVy/d• FIGURE 12.2 Constitutive equations for ceramic suspension rheology.

where the component OVi/C~ j is the ij component of the strain, 4/ij. The Newtonian coefficient of viscosity, 7, depends on the local pressure and t e m p e r a t u r e but not on z or A. A one-dimensional version of the Newtonian model is %

= -~

dvx

m

dy

-

-v

~/xy

(12.7)

This type of flow is illustrated in Figure 12.3. A fluid in shear consists of infinitely thin layers sliding one on top of the other. The velocity gradient, dvx/dy, is called the shear rate, (4/) and is m e a s u r e d in units of sec :~. The viscosity, V, is the ratio of the force in the y direction per unit of area (Fy/A) to the shear rate, rxy. The units of viscosity are Pascal 9sec (SI) or poise [gm cm -1 sec -1] (cgs). The stress for pseudo-plastic and dilatent fluids is not a linear function of shear rate. For non-Newtonian fluids, the relation between z and A is not a simple proportionality because the viscosity is a function of h. For a Bingham plastic fluid, the following relationship holds:

z

=

-

A= 0

~o

Zo

+

IX/(A'A)/2I

for (z"2 z) < z2

}A

for (Z :2z) > z2

(12.8) (12.9)

12.3 Equations of Motion

~4~

FIGURE 12.3

Newtonian concept of viscosity: F is the applied shear force, A is the area of the plates, r is the shear stress (r = F/A); the shear rate, ~, is dVx/dy; the viscosity, 7, is defined as r/~.

where (T " T) = E E "$ijTji i j

(12.10)

( A " A ) - E E Aij Aji i j

(12.11)

A one-dimensional version of the Bingham plastic model is

where To is the yield stress and Vo is the Bingham plastic viscosity. For a power law fluid valid for dilatant and pseudo-plastic fluids, the following relationship holds: ~" =

--{~p]~v/(A"A)/2]n-1}A

(12.13)

where n is the power in the power law and Vp is the power law viscosity which is equal to the Newtonian viscosity when n = 1. Note that, for the power law fluid, the viscosity, Vp, is not the same units as the viscosity for a Newtonian fluid. A one-dimensional version of the power law fluid model* is

* O s t w a l d - d e Waele model.

550

Chapter 12 Mechanical Properties of Powders and Suspensions

Here we see that the viscosity is a function of the shear rate:

For the Reiner-Philippoff model [2] for pseudo-plastic fluids, T

= -

~+

7o 1+

) A 2~c J

(12.16)

where % is an adjustable parameter, ~/o is the viscosity at the low shear limit, and ~/~ is the viscosity at the high shear limit. A one-dimensional version of the pseudo-plastic fluid model is --

"

--

_.=

d v

x

1 + rxy | ~yy

(12.17)

To J

This equation is similar to the Cross equation [3] for polymers to be discussed later. The Cross equation is also one of the key constitutive equations for ceramic suspensions at high solids loading. All the non-Newtonian constitutive equations just given are simplifications of the most general time-independent constitutive equation for isotropic, incompressible non-Newtonian fluids that do not exhibit elasticity [4,5], r = -~?h - 1/2~c{A. A}

(12.18)

where ~?c is the "cross-viscosity" and both Vc and ~7 are functions of the velocity profile. More general expressions that include elasticity effects can be written [6-9] but they are not presented or discussed here. For more details as to the formulation of the relationship between stress, r, and the velocity vector, the book Transport Phenomena by Bird, Stewart, and Lightfoot [1] should be consulted. At this point, we will examine the shear and normal stress in a ceramic suspension. We will use simple one-dimensional flow examples to elucidate various points. For two- and three-dimensional examples, the full formalism given previously must be used. Due to the complex mathematics, numerical methods are typically the only route to a solution for the velocity profile, wall stress, and pressure distribution in a fluid.

12.4 C E R A M I C S U S P E N S I O N R H E O L O G Y All of the different rheological behaviors just discussed are observed in ceramic suspensions and so is time-dependent behavior. We need to know the rheological behavior of ceramic suspensions to predict how

12.4 Ceramic Suspension Rheology

551

a suspension will flow into a mold in slip casting or in an extruder. The rheological properties are used with the Cauchy equation of motion to predict the velocity profiles, wall shear stress, and the pressure distribution in the mold. In this section, we will discuss the rheology of ceramic suspensions for dilute and concentrated suspensions, as well as suspensions that are colloidally unstable.

12.4.1 Dilute Suspension Viscosity A dilute ceramic suspension has Newtonian rheology. Thus an important characteristic of a dilute suspension is its viscosity. The viscosity of a dilute suspension, ~, is always higher t h a n that of the pure solvent, V~. Using pure hydrodynamics Einstein [10,11] derived an expression relating the viscosity to the volume fraction, ~, of the dispersed phase: =

T~s

-- ~ - ~ ] s p h e r e - -

(1 + ~/2)(1 + ~ + ~2 + . . . ) 2 ~ 1 + 5/2 ~b + . . . (12.19)

This equation was derived for spherical particles and consists of a Taylor series in their volume fraction ~b. This equation is valid for a mixture of different sized spheres at dilute concentrations. If a polymer is the disperse phase, it is convenient to convert the volume fraction to concentration, c2, using the relationship ~ = c2(V2/Mw2), where V2 is the hydrodynamic molar volume and Mw~ is the molecular weight of the polymer. If the polymer sphere swells in the solvent, then the swollen volume fraction must be used for ~. Einstein's equation can be generalized to nonspherical shapes using the formula ~?s

= 1 + a~b + . . .

(12.20)

where a is a constant whose value depends on particle shape (e.g., = 2.5 for spheres and a = 4 for cubes*). At volume fractions above 0.02, deviations from Einstein's simple equation are observed for all shapes.

12.4.1.1 Suspensions of Anisotropic Particles It has been shown experimentally [13] that the viscosity of anisotropic particle suspensions increases proportionally to the square of the ratio between the large and small axes for ellipsoids of revolution when the particles are prolate and increase directly proportional to the

* Simha and Kuhn have developed these relations but Fig. 12.4 comes from Hiemenz [12, p. 596] where there is no reference.

552

Chapter 12 Mechanical Properties of Powders and Suspensions

A l p h a , a, from t h e e q u a t i o n *l/~ls = 1 + a (b + . . . v e r s u s axial r a t i o for ellipsoids of r e v o l u t i o n a c c o r d i n g to S i m h a t h e o r y [12].

FIGURE 12.4

ratio between the large and small axes for ellipsoids of revolution when the particles are oblate [14], as shown in Figure 12.4. As we will see, the reason for this effect is that anisotropic particles rotate in the shear field and have an effective volume fraction larger t h a n that of the particles themselves. R o t a t i o n a l D i f f u s i o n i n a S h e a r F i e l d Consider for a moment rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, t2 = (0, ~) used to describe its orientation. Because the solvent molecules are expected to frequently collide with the rod, it should exhibit a random walk on the surface of the unit sphere (i.e., Irl = 1.0). Debye [15] in 1929 developed a model for the reorientation process based on the assumption that collisions are so frequent that a particle can rotate through only a very small angle before having another reorienting collision (i.e., small step diffusion). The rotational motion of such a rod by diffusional reorientation without shear was solved in Section 11.4.4. With shear, we have the differential equation

Oc(r, t) t 3) sin0 0c(r, t) 0t

2

d6

=

DrV2C(r, t)

(12.21)

where ~ is the shear rate. Thus, c(r, t) is simply the concentration of rods at the point r = u on the surface of the unit sphere at time t. Because of the spherical symmetry, it is most convenient to solve this equation in spherical polar coordinates (r, 0, ~), where r = 1.0. The

12.4 Ceramic Suspension Rheology

l

553

O
Pe~

FIGURE 12.5 Steady state solution of the rotary convective diffusion equation for ellipsoids of revolution (schematic). The orientation probability of the particles is random for Per = 0 and becomes more aligned as Per --* ~. But even at Per - ~ the particles rotate continuously, giving a finite probability for all angles.

steady state solution [0c(r, t)/Ot = 0] to this equation is a rod which rotates with a period of (12.22)

T = 2__~(re + r e 1)

where r e is the axial ratio of the rod (or for ellipsoids of revolution r e = b / a < 1 for prolate spheroids and re = a / b > 1 for oblate spheroids). The temporal solution of the differential equation Oc(r, t) t ~/ sinO Oc(r, t) = DrV2C(r, t) Ot 2 d6

(12.23)

with the boundary conditions c((b = O) = 1.0 a t t = 0

for all 0

(12.24)

is not available except numerically; but this differential equation shows that two dimensionless time scales are of interest: tl = D r t and t2 = ~/t/2. The relative importance of these two time scales can be shown by the rotational Peclet number, Per, Per = ~ Dr

(12.25)

When is Per --~ ~, diffusion is negligible and the particle rotates but has preferential orientations as shown in Figure 12.5, and when

554

Chapter 12 Mechanical Properties of Powders and Suspensions

Per ~ 0, Brownian motion allows all orientations the same probability, also shown in this figure. E f f e c t o f P a r t i c l e R o t a t i o n o n t h e E f f e c t i v e V o l u m e Fract i o n The rotation of particles cause the effective volume fraction of an anisotropic particle to be much larger than that of the volume fraction of the particle itself. For example, the volume of the effective sphere of revolution divided by the particle volume is given by Ysr

_

Yparticle Ysr

_

Yparticle Ysr

gparticle

4/37ra 3

4/37ra 3 4/37ra2b

_

4/37ra 3

for a rod

(12.26)

for oblate spheroids

(12.27)

= 16/3(a/b)2

7r / 41r b 2a = (a/b)

= (a/b)2

for prolate spheroids

(12.28)

4~37tab 2

which explains the intrinsic viscosity in Figure 12.4 because a = 2.5 gsr/gparticle.

12.4.1.2 E l e c t r o - V i s c o u s E f f e c t

Another reason Einstein's equation is inaccurate is because of the electrical double layer surrounding the particles in aqueous solution. The presence of a double layer gives an electro-viscous effect which causes (1) an extra force to be needed to move two similarly charged double layers past one another and (2) a larger effective volume of the particle, due to its double layer of immobilized solvent molecules. Von Smoluchowski [16] derived an equation for the primary electro-viscous effect

where s is the permitivity (F,rF,O), ~ is the zeta potential, h is the specific conductance of the solution, and a is the particle radius. This equation is valid for large values of Ka, where this effect is the smallest and is simply a correction of the first-order term in the Taylor series expansion for the viscosity derived by Einstein. Booth [17] derived another equation for the electro-viscous term which is smaller than that of von Smoluchowski and becomes negligible regardless of the value of the potential when the double layer thickness, K-~, is very small in compari-

12.4 Ceramic Suspension Rheology

555

son to the particle radius. This equation is given by

~---= 1 + ~b ~s

I

1+

t

1

~(Ka)(Ka)2(1 + (Ka)) 2

\4~0ka ~

(12.30)

The limiting form of ~(Ka) for large Ka was given by Booth as _=(Ka) = 3/2 7r(Ka) 4

(12.31)

which reduces to von Smoluchowski's expression except that the second term in parenthesis should be multiplied by 3/2. See Hunter's book [18] for more details. 12.4.1.3 Effect of Surfactants

on Viscosity

Surfactants below their critical micelle concentration (CMC) add to the solvent viscosity according to the Einstein equation

~s

= 1 + 5/2 4~ss

(12.32)

where 4~ss is the soluble surfactant volume fraction. In addition, the surfactant will adsorb at the surface of the ceramic powder increasing the effective volume fraction of the ceramic. The increase in the effective volume fraction of the ceramic particles due to surfactant adsorption is ~bc[1 + (Ls/a)3], where Ls is the span of the surfactant layer adsorbed on the particle of radius a. As a result of the two effects, the Einstein equation can be rewritten as ~? = [1 + ac[1 +

7~ss

(Ls/a)3]4~c]

(12.33)

where ac is the value of a for the ceramic particles and "~ssis the viscosity of the surfactant solution. Upon substitution, this equation becomes ~Ts

= [1 + 5/24~ss][1 + ac[1 + (Ls/a)3]4~c]

(12.34)

where Vs is the pure solvent viscosity, ~c is the volume fraction of ceramic powder, 4~ssis the volume fraction of spherical surfactant molecules in the solvent at equilibrium with those adsorbed at the surface, ac is the geometric constant for the shape of the ceramic particles, and [1 + (Ls/a) 3] is the volume fraction correction for the adsorbed layer on the ceramic particles. Above the CMC, the surfactant molecules aggregate and produce another phase, micelles, with their own volume fraction, ~M. Accounting for the volume fraction ofmicelles, the viscosity

556

Chapter 12 Mechanical Properties of Powders and Suspensions

of this complex ceramic suspension is given by ~s

= [1 + 5/2r

+

aM~M][1 + ac[1 + (Ls/a)3]d~c]

(12.35)

where a M is the geometric constant for the shape of the micelles. Micelles are often, but not always, spherical giving a value of 5/2 for aM. At very high surfactant concentrations, micelles can have other shapes including rods and lamella. The preceding equation is good for only colloidally stable suspensions where ~tot -- ~c + (~M + (~ss <~ 0.02. 12.4.1.4 Effect of P o l y m e r s on V i s c o s i t y When polymers are used as steric stabilizers or binders in a ceramic suspensions, they have two effects: (1) to increase the viscosity of solvent and (2) to increase the effective volume of the particle due to specific adsorption. The increase in solvent viscosity is accounted for by the Einstein equation Vps= 1 + 5/2~bp ~?s

(12.36)

where the volume fraction of the swollen polymer left in solution, ~b~,is =

Np

(12.37)

and Np is the number of polymer molecules dissolved in a volume V of the solvent. The radius of the polymer molecule in solution, (rZ) 1/2, is larger than the effective molecular radius in the solid state for a greater than theta solvent, due to the swelling and unfolding of the polymer molecule. The increase in the effective volume fraction of the ceramic particles due to polymer adsorption is ~bc[1 + (Ls/a)3], where Ls is the span of the polymer layer adsorbed on the particle of radius a. (Note that L s = 0.92(rZ) l/z, the "polymer brush" assumption.) As a result of these two effects, the Einstein equation can be rewritten as Vps

= {1 + ac[1 + (Ls/a)3]d~c}

(12.38)

where Vps is the viscosity of the polymer solution. Upon substitution this equation becomes ~s

= [1 + 5/2~bp]{1 + a c [ 1 + (Ls/a)3]~bc}

(12.39)

where ~?s is the pure solvent viscosity, ~bc is the volume fraction of ceramic powder, ~bp is the volume fraction of spherical polymer molecules in the solvent at equilibrium with those adsorbed at the surface,

12.4 Ceramic Suspension Rheology

557

ac is the geometric constant for the shape of the ceramic particles and [1 + (Ls/a)3] is the volume fraction correction for the adsorbed layer on the ceramic particles. Again, this equation is only good for colloidally stable suspensions. Fleer et al. [19] verified this equation for cubic AgI particles with poly(vinyl alcohol) adsorbed at the surface. For polymer solution concentrations (i.e., ~bp) that give essentially monolayer coverage of the particle surface, the value of [1 + (Ls/a)3] is nearly constant for a wide range of ceramic powder concentrations (i.e., ~bc). These equations are good for only low-volume fractions of all the added phases.

12.4.1.5 Dilute but Slightly A g g r e g a t e d S u s p e n s i o n s When a suspension is colloidally unstable, coagulation (in the presence of salts) and flocculation (in the presence of certain polymers) results. The aggregates formed give rise to a bridging network at very low volume fractions, schematically shown in Figure 12.6. The primary difficulty in dealing with colloidally unstable suspensions, both experimentally and theoretically, is the nonequilibrium nature of the structure. Consequently, the rheology is often subject to memory effects over very long time scales. This memory effect is referred to as thixotropy. For weakly aggregating suspensions, that is, V(rij)max/ksT < 15, the structure recovers to a reproducible rest state after shear. Such dispersions would separate out an ordered particulate phase, if the particles were monosized and spherical, given sufficient time, but rheological studies performed relatively soon after mixing these components show a metastable structure is formed which changes negligibly during rheological experiments.

O

w

I

I

o

~

O

I

I

FIGURE 12.6 Schematic diagram of an aggregated colloidal suspension showing a bridging network at gelation. The volume fraction of particles at which this bridging network is formed is referred to as the percolation limit.

558

Chapter 12 Mechanical Properties of Powders and Suspensions

Using the Cross equation [3] for a dynamic equilibrium for flocc entanglements gives the suspension viscosity ~(~) = ~ + ~?o- ~/~

(12.40)

where Yc and m are fit parameters, V0 is the low shear limit viscosity and ~ is the high shear limit viscosity. Due to strong aggregate networks the low shear limit viscosity may become large and ramify itself in experiments performed at finite but low ~ as a yield stress, Zo, as seen in Bingham plastic rheology. The Cross equation is plotted in Figure 12.7 [20]. For dilute suspensions that are colloidally unstable, there is a particle volume fraction where a bridging network is formed. This network is characterized by a probability of filled sites,p. The critical probability,

Tl0

TI~176

=,

,t=, L_ m

u~ .C

l

RateofStrain,7

= oo Network

Aggregates

0000 0 0

Individual Particles

FIGURE 12.7 Schematic plot of the viscosity given by the Cross equation showing the low shear viscosity, ~0, and the high shear viscosity, ~ . Also included is the Michaels and Bolger [20] concept of network formation and destruction upon shear. The application of a sufficiently high shear rate disrupts the network, forming aggregates which break up further at yet higher shear rates. Upon cessation of the shear, Brownian motion brings the flocks together into a network of first aggregates which enlarge until a percolation limit is reached, giving another bridging network.

559

12.4 Ceramic Suspension Rheology

Pc, at which a spanning cluster occurs is called the percolation threshold. There are two versions of percolation, site and bond. With bond percolation, the sites are initially filled and the bonds are added to connect the sites. With site percolation, a grid placed over a region is gradually filled with spheres. The percolation threshold is lower for bond percolation than for site percolation because a bond is attached to two sites while a site is connected to a maximum of z bonds. Taking the coordination number, z, around the sites, the threshold for bond percolation is seen to be close to that of the classical theory [21] Pc

1 z-1

(12.41)

The percolation probability has different values based on the classical theory site or bond percolation for different structures, as shown in Table 12.2. This critical percolation volume fraction, ~*, is calculated from the percolation threshold and the space filling factor. The volume fraction for site percolation for various structures is essentially the same as follows. In three dimensions, the site percolation threshold occurs at - 1 6 % volume. Near the percolation threshold the average cluster size diverges as does the spanning length of clusters. At the percolation limit, the rheological behavior of the suspension changes from Newtonian to either the Cross equation with a low shear limit viscosity or the Bingham plastic equation with an apparent yield

TABLE

12.2

Structure Facecentered cubic

Percolation V o l u m e Fractions, (b, for Different Geometries a

Coordination z

Probability of percolation 1 z - 1

Probability of site percolation p~i~

Probability of bond percolation p c b~

Space filling factor V

Percolation volume fraction dp* --

v p site

12

0.091

0.196 c

0.1185 a

0.741

0.147

8

0.143

0.245 b

0.1785 d

0.680

0.167

6 4

0.2 0.333

0.3117 ~ 0.428 ~

0.2492 ~ 0.388 ~

0.524 0.340

0.163 0.146

Bodycentered

cubic Simple cubic Diamond Random, close packed

-8

-0.143

--

-0.27

-0.637

-0.16

a Zallen, R., "The Physics of Amorphous Solids," Chapt. 4, Wiley, New York, 1983 with additional information from other sources. b Brinker, C. J., and Scherer, G. W., "Sol-Gel Science," Academic Press, New York, p. 321, 1990. c Cox, M. A. A., and Essam, J. W., J. Phys. C: Solid State Phys. 9, 3985-3991 (1976). d Sykes, M. F., and Essam, J. W., J. Math Phys. 5, 1117-1127 (1964).

560

Chapter 12 Mechanical Properties of Powders and Suspensions

stress. This yield stress to be overcome in the Bingham plastic rheology is dependent on the strength of the bridging network. The yield stress corresponds to the maximum force per unit area that the network can withstand before rupturing. This is proportional to the maximum force acting between each pair of particles multiplied by the number of particle interactions per unit area. By assuming that yielding requires the rupture of only one particle-particle interaction, the Bingham plastic yield stress [22] is given by

"ro~ ~

[V(rij )]max

(12.42)

This simplistic argument does not account for the variation of the pair distribution function with volume fraction. Beyond the percolation limit, the bridging network is more concentrated. Below the critical volume fraction, no continuously bridging networks are formed and the viscosity is low. As shown in Figure 12.7, this bridging network breaks up as the shear rate increases, giving different viscosities at different shear rates. As a result, this gives low and high shear limit viscosities observed at steady state for concentrated polymer solutions and concentrated particulate suspensions (discussed later). For slightly unstable suspensions, the value of the viscosity at high shear depends also on the dynamic structure of the networks formed and broken at high shear rate as seen in Figure 12.7. But for the most part, the flocc structure is destroyed at large shear rate. At intermediate shear rate the rheological behavior is pseudo-plastic as predicted by the Cross equation. At low shear rates the viscosity diverges as shown in Figure 12.7, giving a yield stress. The point of divergence depends on the volume fraction of particles as predicted by the percolation limit. 12.4.1.6 E l e c t r o s t a t i c a l l y S t a b i l i z e d S u s p e n s i o n s as a F u n c t i o n of pH

At the isoelectric point, IEP, a suspension of particles undergoes coagulation which gives rise to aggregates of ever-increasing size. These aggregates lead to a network structure which has a rheology as decribed by either the Cross equation or the Bingham plastic equation discussed previously. Away from the IEP, the suspension is colloidally stable and behaves as a Newtonian suspension with a viscosity described by the Einstein equation. The general view of viscosity as a function of pH for TiO2 particle is shown in Figure 12.8. As the pH increases or decreases away from the IEP, the zeta potential, ~, increases in magnitude, colloid stability increases and the viscosity, V, decreases. As the pH is further increased

12.4 Ceramic Suspension Rheology

561

FIGURE 12.8 Schematic of the low shear viscosity of TiO2 as a function of pH. Near the zero point of charge (ZPC) the rheology is non-Newtonian for dilute suspensions, conforming to the Cross equation, which suggests that aggregation is responsible for this increase in viscosity. Away from the ZPC, the rheology is Newtonian for dilute suspensions.

or decreased away from the IEP, the zeta potential becomes a constant value then the ionic concentration increases, giving rise to a decrease in the double layer thickness, K-1, a decrease in colloid stability, and an increase in the solvent viscosity. A specific example of this behavior is shown in Figure 12.9 [23] where the viscosity of A1203 suspensions is plotted as a function of pH. Near the isoelectric point, the viscosity is high due to the colloid instability and the formation of floccs, and away from the IEP the viscosity is low due to colloid stability.

562

Chapter 12 Mechanical Properties of Powders and Suspensions 10000

1000

Reed

100

9 Healy+Liddell

10

I

6

9

9

7

I

8

9

9

9

I

9

10

11

pH

FIGURE 12.9 Viscosity (at low shear rate) ofA1203 suspensions at different pH values adjusted by the addition of various amounts of 0.2 M A1C13. Near the isoelectric point the suspension is not colloidally stable, giving a high viscosity. Data taken from Reed [23].

12.4.2 Rheology of Concentrated Ceramic Systems This section on concentrated suspensions discusses the rheological behavior of systems which are colloidally stable and colloidally unstable suspensions. For stable systems, the rheology of sterically stabilized and electrostatically stabilized systems will be considered. For sterically stabilized suspensions, a hard sphere (or hard particle) model has been successful. Concentrated suspensions in some cases behave rheologically like concentrated polymer solutions. For this reason, a discussion of the viscosity of concentrated polymer solutions is discussed next before a discussion of concentrated ceramic suspensions.

12.4.2.1 Concentrated Polymer Solutions At high polymer concentrations, polymer molecules entangle, producing pseudo-plastic rheological behavior. This occurs at a polymer concentration, c~ = 3Mw/47rNAR~, comparable to that in the polymer coil according to scaling law theory [24]. With a distribution of polymer molecular weights, a more complicated expression is necessary because the low molecular weight chains will not entangle at any concentration. Hence, only the fractions of polymers with molecular weights above a critical molecular weight will contribute to the entanglement. The state

12.4 Ceramic Suspension Rheology

~6~

of entanglement is a function of shear rate giving rise to a shear rate dependent viscosity. Increasing the shear rate leads to structural breakdown with a concomitant decrease in viscosity. Cross [25-27] derived an equation for the viscosity of non-Newtonian suspensions based upon the assumption that pseudo-plastic flow is associated with the formation and rupture of structural linkages" V(~) = ~ +

~o+~

(12.43)

where ~0 is the viscosity with subscripts 0 corresponding to the low shear limit and ~ corresponding to the high shear limit, ~ is the shear rate, and "Yc and m are constants. Soong and Shen [28] assumed that this transient entangled network is constantly undergoing formation and disengagement as a result of random thermal motion and shear. When the system reaches a steady state, the entire network is in a dynamic equilibrium. The probability for a given entanglement configuration at a given shear rate can be determined by equations for the rates of entanglement, d N / d t (driven by thermal diffusion and independent of shear), dN

~

dt

=

ke(go - N ) rsm

(12.44)

and disengagement, d N / d t (driven by shear rate only) at steady state, dN dt - kd~/mN

(12.45)

The disengagement rate was assumed to be proportional to ~ m, where the parameter m is related to the elasticity of the medium and depends on the polymer molecular weight distribution. An empirical equation for m is

m =/.~_\~~ ,~/5

(12.46)

\l~w/

where Mn is the number average molecular weight and M w is the weight average molecular weight of the polymer. The ratio m is thus a measure of the breadth of the molecular weight distribution. For a monodisperse polymer m = 1.0. From this dynamic equilibrium analysis, Soong and Shen [29] show that ~c from the viscosity expression is given by (ke~ 1/m -1 ~c = \kdd] ~s

(12.47)

/564

Chapter 12 Mechanical Properties of Powders and Suspensions

where kd is the disentanglement rate constant, k e is the entanglement rate constant, and ~s is the characteristic time constant for the rate of segment diffusion. In general, this model [30] gives a time dependent viscosity as would be expected due to the kinetic nature of the entanglement and disengagement. For the dynamic equilibrium discussed previously, the critical shear rate is related to a single time constant. This approach can also be used for the entanglement of aggregate networks of particles, as will be seen later.

12.4.2.2 Colloidally Stable Concentrated Suspensions This section draws heavily from two good books: Colloidal Dispersions by Russel, Saville, and Schowalter [31] and Colloidal Hydrodynamics by Van de Ven [32] and a review paper by Jeffrey and Acrivos [33]. Concentrated suspensions exhibit rheological behavior which are time dependent. Time dependent rheological behavior is called thixotropy. This is because a particular shear rate creates a dynamic structure that is different than the structure of a suspension at rest. If a particular shear rate is imposed for a long period of time, a steady state stress can be measured, as shown in Figure 12.10 [34]. The time constant for structure reorganization is several times the shear rate, ), in flow reversal experiments [34] and depends on the volume fraction of solids. The viscosities discussed in Sections 12.4.2.2 to 12.4.2.9 are always the steady shear viscosity and not the transient ones.

- ~ =o.55

0.,50

3"II

1.0 9

~

0,30

0.8

" 06 ~ X

9

0.4

~t 0

l

05

,

1.0

~

1.5

i

l

l

2.0 P_5 30

.... 1

3.5

l

I

4.0 4.5

I

l

50

5.5

lilt FIGURE 12.10 Reduced torque versus strain following flow reversal for hard sphere suspension of various concentrations. Polystyrene spheres 45/~m in diameter in silicone oil. Taken from Gadala-Maria and Acrivos [34].

12.4 Ceramic Suspension Rheology

~6~

Monodisperse Spheres The rheology of concentrated ceramic suspensions is very important for good mold filling. For concentrated suspensions that are colloidally stable (by steric means, giving a hard sphere model), there is a particle volume fraction (i.e., ~b = 0.63 for monosized spheres) where the particles come into contact, giving a random close packed network, as shown in Figure 12.11. This random, close packed network gives a viscosity due to interparticle forces measured by the osmotic compressibility. Section 11.5.3 discussed these interparticle forces and their contribution to osmotic compressibility. The osmotic compressibility, Z, is defined by Z -

[IV NkB T

(12.48)

where II is the osmotic pressure of the ceramic suspension and V is a volume containing N particles. The osmotic pressure of the particle assembly can be determined from the derivative of the Helmholtz free energy, A, with respect to volume, V:

n=

5-y

(12.49)

Using equilibrium statistical mechanics [35, 36] the Helmholtz free energy can be given by A = - k B T In PN + ~,

(12.50)

where PN is the probability density corresponding to a particular configuration (r~ ... RN) of particles and s is the sum of all interparticle potentials, V(rij). Batchelor and coworkers [37-39] have used the

FIGURE 12.11 Packing structures of cubic close packing, hexagonal close packing, and random close packing of 0.31 t~m diameter TiO2 spheres.

566

Chapter 12 Mechanical Properties of Powders and Suspensions

change in the Helmholtz free energy induced by an arbitrary homogeneous deformation, s, to determine the thermodynamic stress, " dA = Vs: <~>

(12.51)

This then gives a direct link between the thermodynamic stress and the osmotic pressure (or the compressibility) of the suspension. As a result of this stress, the viscosity will depend directly upon the structure, PN, and the interparticle potential, V(rij ). Using this interrelationship Batchelor has been able to evaluate the ensemble averages of both the mechanical and thermodynamic stresses by renormalizing the integrals. As a result, he has developed truncated series expressions for the low shear limit viscosity, Vo, and the high shear limit viscosity, V~, corresponding to 7_20= 1 + 2.5~ + 6.2~ 2 + O(4~3) ~s

(12.52)

= 1 + 2.5~ + 7.6~b2 + O{~b3)

(12.53)

~s

for hard sphere interaction potentials where ~?sis the solvent viscosity and identical results for V~ + =V1 + s2"5~b

( 2 . 5 + 3 s ~ ) ~ 2 + O(83)

~--~ ~s = 1 + 2"5r + (2"5 + 4-~s~) ~b2 + rO (

(1~.54)

(12.55)

for electrostatic interaction potentials, where the radius~ so,~ of the excluded shell (i.e., double layer) is included in the ca}~ation. These equations are good for So > 2. For charged spheres at high Peclet number (Pe is defined later), the radius, So, of the excluded sheII corresponds to the separation at which the electrostatic repulsian~ balances the maximum viscous force due to the shear flow [38~40]: 3Irma 2~so - 47rSSo~2s(aK)2 1(aKso) + aKso 2 exp(2aK - aKso)

(12.56)

The value of s0 can be determined by trial and error from this expression. For a finite repulsion that decreases with increasing separation, the characteristic separation, So, will vary inversely with shear rate and the viscosity will be shear thinning leading to a Newtonian high shear viscosity. Ample experimental data exhibit exactly this behavior [41], as will be discussed in later part of this section.

Hard Spheres at High Concentration Because viscous forces perturb the microstructure against the restoring effect of Brownian

567

12.4 Ceramic Suspension Rheology

motion, the translational Peclet number, Pet, is defined in terms of the relative velocity, a2~ and the Brownian diffusion coefficient for the particle, Dr(- kBT/67ra~s) as follows: pe t

-

a2~

-

Dt

67r'~sa3~/

(12.57)

kBT

The Peclet number gauges the magnitude of the departure from equilibrium configuration of the particles. (Note that the rotational Peclet number, Per, for a sphere has nearly the same definition only 6 is replaced by 8.) As such the Peclet number can be used in the Cross equation to determine the value of ~c[- 8], giving ~?()) = V~ + To - V.._.~ Pet 1+ 8

(12.58)

which is plotted in Figure 12.12. This relationship is identical to that of the Cross equation for polymers with m = 1 (monodisperse system). Colloidal dispersions depart from equilibrium at low shear rates (i.e., 1 sec -~ < ~ < 1,000 sec-~). In fact, for colloidal particles, it is difficult to attain the equilibrium state because the translational Peclet number is always small, even for high shear rates. These theoretical expressions have been tested by experiments on monosized polymer lattices [42] and monosized silica [43] suspensions dispersed in water and other liquids. Both the low and the high shearlimiting viscosities are shown in Figure 12.13 to increase monotonically

26

24

,,~

C"

22

2o te e6 14 t2 I0

'

O.Ol

,

.

!

O.lO

,

I

LO

.

,,

'

IOLO

9

Pet/6n= Per/8n FIGURE 12.12

Steady shear viscosity as a function of Peclet number for polystyrene lattices of radii of 54, 70, 90, 37, and 55 nm at 50% by volume in different solvents (--, H20; O, benzyl alcohol; and 0, meta-cresol), where T0 = 24.7Vs, V~ = 13.9~s. Data from Kreiger [42].

568

Chapter 12 Mechanical Properties of Powders and Suspensions

100

q/rls

o

10

1.0

-/qs

"

0

I

i

~1

0.2

, ~3

l 0.4

, 0.5

[

t

0.6

0.7

FIGURE 12.13 Low and high shear viscosity as a function of volume fraction, showing the divergence at different volume fractions. Data from Kreiger [42].

with volume fraction with shear thinning (i.e., nonlinear behavior) detected for ~b > 0.25-0.3 for hard spheres [43]. This data has been best fit to the following expressions for hard sphere interactions: 7 o _ 1 + 2 . 5 ~ b + 4 4 ) 2+424) 3 + . . . ~ ~s V--~-~= 1 + 2.54~ + 44)2 + 254~3 + . . . ~ '~s

( (

1-0

1 -,,

3

1

(12.59)

(12.60)

These volume fraction expansions point out that shear thinning arises from three-body and higher order interactions. This reflects the lack of long-range order but significant short-range order in dilute suspensions of hard spheres. Short-range order generates thermodynamic stresses at low shear. At high shear, viscous forces dominate. The shear stress, ~, characterizing this change over in shear thinning is characterized by Pet(= 67r~sa3~//ksT) = 8. Using the approximations at the right in the preceding equations, we can see that the viscosity diverges at a lower volume fraction in the low shear limit than in the high shear limit. The lower shear limit occurs at ~b = 0.63, which corresponds to random close packing where the osmotic compressibility diverges for a disordered fluid because the particles are in contact. Note that the divergence of the low shear viscosity does not take place at an order-disorder transition, which occurs for monosize spheres at

12.4 Ceramic Suspension Rheology

569

cubic close packing, where the volume fraction is 0.52. The fact that the dispersions flow at high shear rates for ~b > 0.63 means that the shear must orient and enlarge the hexagonal close packed ordered domains (with volume fraction 0.74) that exist in the microstructure. The viscosity at the high shear limit diverges at a volume fraction slightly less than that for hexagonal close packing (i.e., ~b = 0.71). These packing structures are shown in Figure 12.11 and the volume fractions are noted here. Packing

~b

Cubic close packing Random close packing Body-centered cubic packing Face-centered cubic packing

0.5236 0.61-0.637 0.6802 0.7405

Problem 12.1. H a r d Sphere S t r e s s - S t r a i n Curve Determine the shear thinning stress-strain curve (i.e., r vs. ~) for ethanol suspensions of monodisperse spherical SiC powders at 20~ if the spheres are 0.3/zm and 1.5 ftm in radius. Make these calculations at two volume fractions of 50% and 55% solids. S o l u t i o n We will use the Cross equation for the viscosity as a function of shear, p, rate: V(~) = V~ + 7 0 - V _._._____2~ 1+ k

(12.61)

9c This gives the shear stress as r = V()))

(12.62)

For ethanol at 20~ the viscosity is 1.2 mPa- sec (= 1.2 cP). For the suspensions, we can calculate the low and the high shear viscosities for the two volume fractions with the following equations: VO Vs

1-

--t

1-

~

0 3

(12.64)

ns

giving

T0 ~

(12.63)

~b = 0.40

~b = 0.60

9 m P a . sec 6 mPa.sec

529 m P a . s e c 50 m P a . s e c

570

Chapter 12 Mechanical Properties of Powders and Suspensions 60

100

(Pa)

T(Pa)

~=0.6 J f

50

30

....--

J

f

,"-

-

.....-- /

I 500

0

j./-

~

,=0.4

/___ _ . . . . . . . . . . . . - ~ ~

0

1000

500

0

y(l/sec)

FIGURE 12.14

--

,.,--

r

0

J

~=0.6 , t f

S t r e s s - s t r a i n curves for Problem 12.1.

1000

y(I/sec)

To determine the critical shear stress, ~)c, we note t h a t the t r a n s l a t i o n a l Peclet number: Pet[ = 61r~qsa34/c/ksT] = 8, allowing the calculation of the critical shear stress:

Tc

a =0.3/~m

a=

1.5/~m

53/sec

0.4/sec

With this data we can plot the s t r e s s - s t r a i n curves given in Figure 12.14. For the 1.5 t~m particles, the critical shear rate is small, so t h a t only the high shear rate is shown on the plot of shear rate from 0 to 1000 sec -~. But for the 0.3/~m particles, the critical shear rate is 53 sec -1 and we can see the shear t h i n n i n g taking place over the shear rates 0 to 1000 sec -1 in the plot. 478.

9

/

i

rj

x ''x t~

2

,j' ,/,..,,.*

11 , . p X "

.,C

m 4.78

++'

.-. . ~ |

0.478

Legend

E)12 VOL ~'. TIE)= + 27.2 VOL ~. "r'io= n 3 ~ 0 V O L % TiO= X 42 V O L % TiO= 9 47 VOL ~'. TiO=

I 10

1 O0

1000 Shear Stress,

sec.

-1

FIGURE 12.15 Shear thickening found by Metzner and Whitlock [44] for aqueous TiO2 dispersions 1 t~m diameter. Data taken from a review paper by Jeffrey and Acrivos [45].

12.4 Ceramic Suspension Rheology .'-I

~. . . . . . i "

"

' ' ~ .... I

'

" ......

I:",~'

'''""I

"-''

/571 ....

VOL. FRACT. PVC. -

9

102

).. I-==. (n 0 0 U~

N

1o I

~_

/ 0.57 o 0.55 0.53 v 0.51 x 0.49 ci 0.47

~"

A

(~45

-

! a

s

,I 16 m

I

I

. . . . . . i

i~ io

_I

I~

i Io z

io s

SHEAR RATE (sec"I)

FIGURE 12.16 Shear thickening found by Hoffman [47] for 1.25 t~m diameter polyvinyl chloride (PVC) latex particles in di-2-ethylhexyl phthalate with the discontinuous dependence of viscosity on shear rate. Redrawn from Hoffman [49].

Not all experiments on concentrated suspensions have found shear thinning behavior characterized by the Cross equation. For example, Metzner and Whitlock [44] have demonstrated shear thickening or dilatant behavior for TiO2 spheres of 1 t~m diameter at very large shear rates as shown in Figure 12.15 [45]. Sacks [46] has shown similar dilatant behavior for silicon powders. Reed [23, p. 243] suggests that dilatant behavior is caused by hindered rotation and mutual interference of particle motion above a particular shear rate. In addition, Hoffman [47,48] has demonstrated dilatant behavior for 1 t~m monodisperse PVC spheres at volume fractions above 0.5 and high shear rates between 10 and 500 sec -1 as shown in Figure 12.16 [49]. These data show that dilatant behavior occurs at a lower shear rate if higher volume fractions are used. Hoffman further carried out light diffraction studies on the suspensions. For the low shear rates, he found that the suspensions were ordered. At high shear rates in the dilatant region, the structure was more disordered. Indeed Gadala-Maria and Acrivos [34] have noted that a steady state shear induced particle structure develops which is very reproducible for volume fractions of hard spheres greater than 0.3. Before steady state is established, the viscosity is erratic. This shear induced structure is put into place by shear induced migration of particles. Leighton and Acrivos [50,51] have measured shear induced self-diffusion in concentrated suspensions under shear and found it to be proportional to a2~ and to have the asymptotic form of 0.5a2~b 2 in the dilute limit as ~b--~ 0.

572

Chapter 12 Mechanical Properties of Powders and Suspensions

Soft Spheres at High Concentration At the maximum volume fraction, ~b~, of the electrostatically stabilized suspension low and high shear viscosities also diverge. The maximum volume fraction is determined from the equation ~m~

Nzr r m 3

6

(12.65)

where r~ is the particle radius plus its charge cloud a n d N is the number per unit volume of particles. This maximum volume fraction can also be calculated [52,53] from the initial volume fraction and the interaction energy, V ( s ) , between the particles separated by a distance s [31]. This maximum volume fraction can be used in the general equation for the steady state low and high shear viscosities of the form ~

1-

(12.66)

where n is the order of the divergence, which typically has values between 2 and 3 for electrostatically stabilized suspensions. For monosized hard sphere interactions, a value ofn = 2 has been experimentally observed [43]. For electrostatically stabilized suspensions, this maximum volume fraction will depend primarily on the dimensionless double layer thickness, Ka, and the electrostatic interaction energy through the dimensionless surface potential, Ts( = eOs/kBT). Because the dimensionless double layer thickness can be large (i.e., 10-100) when the salt concentration is low, the effective volume fraction at which the maximum volume fraction is reached can be very small. For charged particles with electrostatic interactions, the interparticle potential is long ranged. At moderate ionic strengths (i.e., Ka >> 1), where the suspensions are still colloidally stable, the samples lie in the ordered regime of the phase diagram, see Figure 11.13. Their rheology resembles that for hard spheres [42] with an apparent yield stress, as shown in Figure 12.17. Reducing the ionic strength to K ~ a, the ordered structure disappears and produces dramatic changes in the rheology, giving Newtonian behavior. The new feature in Figure 12.17 is the transition evident at intermediate shear rate and electrolyte levels. For the system at 4 x 10 .5 M HC1 and ~ ~ 50 sec -~, the viscosity abruptly increases and the particle order disappears. Lindsay and Chaikin [54] suggest that at rest the particles are in a three-dimensional symmetry, either FCC or BCC. Steady shear flow rearranges the structure to an ordered flow of crystal planes. At a critical stress, i.e., ~ 50 sec -~, this structure becomes unstable, producing an ordered state with a higher viscosity.

12.4 Ceramic Suspension Rheology

1.0

00

+

0130 n ~.

O~

+ +

+,+

+ ++

0.5

9

~00

00

oo

~

A

+ 000 000

ooooo

o~176176+ , + + + ,** oo

~ AA

5~3

~~ ~ ~ ~

~

~

a a ~aa A

@ hO

T o l A' 0

0 0

00 nix~ AAA

"

,, 40

J 80

i, 120

, I 200

160

240

280

V, sec FIGURE 12.17 S h e a r stress a s a function of s h e a r r a t e for polystyrene lattices (a = 45 nm) at ~ = 0.04; s q u a r e s in deionized water, + in 1 x 10 .5 M HC1, hexagons in 4 x 10 .5 M HC1, h in 5 x 10 .5 M HC1. D a t a from Lindsay and C h a i k i n [54].

These data dramatically illustrate that long-range electrostatic repulsions alter the rheological behavior of dispersions. The primary effect is to produce an ordered colloidal crystal at low volume fractions at rest. These colloidal crystals have low shear viscosities which are exceedingly large or diverge giving a yield stress. At finite stress, the solids flow by crystal plane sliding and ultimately become disordered at a critical stress. This disordering drastically increases the suspension viscosity. Broad Size Distributions Suspensions of broad size distributions of ceramic particles show in general that the low shear viscosity, 7o, and the high shear viscosity, ~ , diverge according to -n

77

1 --

(12.67)

at a maximum volume fraction, ~m. The order of the divergence, n, is often found to be near 2, which is that for a monodisperse suspensions. For broad size distributions the value of ~b~ increases with the width of the log-normal size distribution as shown in Figure 12.18. In this plot the experimental data for alumina filter cakes gives lower packing density because the particles are aggregates and are not completely dense inside.

/574

Chapter 12 Mechanical Properties of Powders and Suspensions

F I G U R E 12.18 Maximum packing fraction, 4~m,as a function of the geometric standard deviation, (r~, of log-normal particle size distributions. Data taken from Reed [23, p. 191]. 1000

I

0

-I-

olil r~

]~ ~k

@

r,~ om

100

'

!

I

I

I

I

I

I

tetra-modal tri-modal bi-modal mono-modal infinite modal 1+512 f

I

,

I

i

9

,

i.

-t-

9 8ZX

Einstein(eq. 12.19)

om

r

m

=r k

r

10

,

r/] @

~

II

A

"

0.0

'

I

0.2

'

'

I

A &

A '

'

0.4

I

0.6

"

'

0'.8

1.0

§ F I G U R E 12.19 Effect of a mixture of particles of different sizes on low shear relative viscosity, ~O/~s -- (1 - ~b/~m)-2, for multimodal systems. Data for r in multimodal systems taken from Table 12.3. This figure is similar to one given in Farris [55].

12.4 Ceramic Suspension Rheology

0.5

dp/Dp

575

0.5

=

O.4 ~

0.3

0.6

0.7

o =

-

zO

,

40

,i,

| _

,

6~

_|_,,

-

so

9_

c

io5

,

Volume % of L a r g e r c o m p o n e n t

FIGURE 12.20 Experimental void fraction of a two-component particle mixtures both having initial void fractions of 0.5. The numbers on the curves refer to the ration of the two particle sizes. Reprinted with permission from Furnas [57] copyright 1931 American Chemical Society.

F i g u r e 12.19 [55] is a plot of t h e viscosity as a function of volume fraction for e v e r - b r o a d e n i n g particle size d i s t r i b u t i o n s [56]. F u r n a s [57] h a s described t h e m a t h e m a t i c a l r e l a t i o n s h i p for t h e particle size d i s t r i b u t i o n w i t h t h e m a x i m u m volume fraction s h o w n in F i g u r e 12.20 for two sized particles a n d F i g u r e 12.21 for up to four components. This size d i s t r i b u t i o n is also t h e one t h a t gives t h e m i n i m u m viscosity. The c u m u l a t i v e particle size distribution, F(a), composed of particles

0.4

Initial voidage

t-

.o

tl,q) E >0 E " E X

/

0.6

0.8

~

1.Ore

10 -5

~

i

. . . . . .

7

_

10 -4

.

.

.

0.001

.

.

.

.

.

0.01

.

.

.

.

.

0.1

.

.

.

1.0

Smallest d i a m e t e r / l a r g e s t diameter

FIGURE 12.21 Calculated minimum void fraction for two, three, and four component particle mixtures. Reprinted with permission from Furnas [57] copyright 1931 American Chemical Society.

~ ~6

Chapter 12 Mechanical Properties of Powders and Suspensions

from a n u m b e r of discrete size intervals between a S and an, is given by F(a)

a j - aJs a jL _ ajs

(12.68)

.

Table 12.3 shows the m a x i m u m volume fraction for size distributions composed of different n u m b e r s of discrete sized particles, which is graphically shown in Figure 12.21. F a r r i s [55] calculated the value of exponent j of the size distribution as In ~bl_ ln(1 - ~b) J = Ink In k

(12.69)

where r is the volume fraction of the solid and ~bl is the volume fraction of the liquid. The value of k (= a s / a n) is the ratio of the small to the large particle diameters. The n u m b e r of discrete size intervals, m2, into which the distribution is broken is described by the following equation:

(a;) where b is the size ratio between d i a m e t e r s of adjacent size. A packing of these types of particles will give the highest packing fraction and, as a result, the lowest viscosity for a given volume fraction. If we are to use a direct analogy of suspension rheology to the Cross equation derived for polymer solutions, we should consider t h a t the

TABLE 12.3 Effect of Particle Size Distribution on the Maximum Packing Density a Diameters in the mixture

Weight % of each size

Max. solid volume fraction

d

100

74

d 0.143d

84 15

86

d 0.143d 0.020d

75 14 11

95

d 0.143d 0.020d 0.0029d

72 14 10 3

98

McColm, I. J., and Clark, N. J., "High-Performance Ceramics," p. 184. Blackie and Sons, Glasgow, 1988. a

12.4 Ceramic Suspension Rheology

577

viscosity follows the relationship (12.71) 1 9

m

When ) is given in dimensionless form as the translational Peclet number, the dimensionless critical shear rate Peclet number, ~c, is 8, as given in the following equation: v ( ~ ) = v~ + n o - n~

(12.72)

I + Pet 8 where m = 1 for monodisperse spheres. To apply this analogy completely, the value of m should not be 1.0 but a measure of the breadth of the particles size distribution as follows" m =

(12.73)

where D n is the number average size and Dw is the weight average size of the particles size distribution. Unfortunately, no authors to date have noticed this inconsistency in the analogy with the Cross equation with the additions of Soong and Shen [28] and the findings of de Kruif et al. [43] for the low and high shear viscosities as a function of the ratio of the volume fraction and a maximum volume fraction for polydisperse particulate systems. There appears to be no experimental work on this approach to fit experimental data in the literature.

Anisotropic Particles Anisotropic particles that are sterically stable have been shown to exhibit thixotropy where the increasing shear rate curve is different than the decreasing shear rate curve. For a 3.534% wgt suspension of hectorite platelets, such a plot is shown in Figure 12.22 [58]. The decreasing shear rate curve has steady rheological properties like that predicted by the Cross equation with its low and high shear viscosities. The low and high shear viscosity diverge as a function of the particle volume fraction, as is shown in Figure 12.23. This plot shows the steady state low and high shear viscosities as a function of the volume fraction, r for monosized platelet particles of hectorite, a synthetic mineral, with a diameter of 33 nm and axial ratio of 45. The volume fractions where the low and high shear viscosities diverge can be calculated from the radius of the sphere of revolution for the particles. For these platelet particles, the volume fraction of

578

Chapter 12

Mechanical Properties of Powders and Suspensions

F I G U R E 12.22

S h e a r stress v e r s u s s h e a r r a t e for a 3.534% wgt suspension o f h e c t o r i t e platelets w i t h a = 33.175 nm, b/a = 45 _ 2 showing thixotropy. The i n c r e a s i n g s h e a r r a t e curve is different t h a n the d e c r e a s i n g s h e a r r a t e curve. The decreasing s h e a r r a t e curve h a s rheological behavior corresponding to t h a t of the Cross equation. D a t a from C h a n g [58].

1000

120

,<.-

100

800

80 600

60 400

40 -r

200

(

20

0 ~Z~-_~--:::~.--_---=-~.-~, 0.00 0.01

9

0 0.02

Volume Fraction

F I G U R E 12.23

Plot of low a n d high s h e a r viscosity v e r s u s volume fraction for hectorite platelets with fit e q u a t i o n s t h a t follow for a = 33.175 nm, b/a = 45 _+ 2. ~--Q= r~s

1-

w h e r e 4~mo= 0.0147 = o.63 9b/a

~ -

1-

where

D a t a from C h a n g [58].

(~m~ =

o.o153 = o.71 9b/a

12.4 Ceramic Suspension Rheology

~9

the sphere of revolution, (~sr, is given by

(12.74)

(~sr = (~ (b/a)

where b/a is the axial ratio of the platelets assumed to be oblate ellipsoids of revolution. The critical sphere of revolution volume fractions for divergence of the low and high shear viscosities was found to be 0.63 and 0.71, in accordance with equations 12.59 and 12.60, respectively, as was the case for hard spheres discussed previously. The maximum volume fraction of anisotropic particles where their spheres of revolution reach the maximum volume fraction, Ohm ~ 0.71(b/a), gives an actual volume fraction, which is much less by the factor b/a. Because b/a is 45 in this case, the actual volume fraction is only 0.0153. This is still in the dilute range of volume fractions, however, the suspension does not have the Newtonian rheology typical of a dilute suspension but follows a concentrated suspension rheology given by

v(~)= v~ +

v0-

v~

(12.75)

1+~ )c where 3)c is given by the critical translational Peclet number for the sphere of revolution of radius a, Pet,c = 6~'~lsa3#/c/ks T = 8. At much higher actual volume fractions the particle rotation is restricted by the presence of other particles and the suspension first becomes dilatant, increasing its viscosity with shear rate. These properties depend on the type of particle packing that can result as well as interparticle forces. The maximum packing of platelet particles can be drastically changed if the packing is like a house of cards or a deck of cards. The only theoretical approaches available to attack the steady state rheology of concentrated suspensions of anisotropic particles are those developed for liquid crystalline polymers, which we will not discuss here. For the most part only experimental results are available for the rheology of concentrated suspensions. A plot of the s t r e s s - s t r a i n relationship for an H § ion-exchange kaolinite also follows the Crossian rheology with very high low shear viscosities. In fact, the data show a yield stress at the lowest shear rate measured of ~) ~ 1 sec -1 and a high shear viscosity, which are both a function of the volume fraction of the clay, as shown in Figure 12.24 [59]. A plot of the stress-strain relationship for an Na § ionexchanged kaolinite also shows a Crossian rheology. Figure 12.25 is a plot of the high shear viscosity and apparent yield stress at 3) ~- 1 sec -~ for an aqueous suspensions of Na § ion-exchanged kaolin at various weight fractions. Comparing Figures 12.24 and 12.25, the onset yield stress, r0 = 1 dyne/cm e, was at 2% solids by weight for H-clay and 1.1%

580

Chapter 12 Mechanical Properties of Powders and Suspensions

F I G U R E 12.24 Viscosity (at ~ ~ 50 sec -1) and apparent yield stress (at ~ = 1 sec -1) versus suspension solids fraction for H § ion-exchanged kaolinite. Data taken from Langston and Pask [59].

10

10000 Viscosity(poise) Yield stress(dynes/cm2)

1000

~E

1

I O0

.1 10

.01

1

10

100

~D

"ID

I

Wgt%

F I G U R E 12.25 Viscosity (at ~ ~ 50 sec -1) and apparent yield stress (at ~ = 1 sec -1) versus suspension solids fraction for Na § ion-exchanged kaolinite. Data taken from Langston and Pask [59].

12.4 Ceramic Suspension Rheology

581

The charge on the edge and face of a kaolinite platelet as a function of pH. These charges give rise to different aggregate structures also shown.

FIGURE 12.26

solids by weight for Na-clay. These solid contents correspond to the percolation limit for these different clays. With anisotropic crystalline particles, the different crystal faces have different chemical compositions and therefore different IEP values. As shown in Figure 12.26 [60], kaoline has a face which is predominantly SiO2 with an IEP at pH 2.0 and an edge which is predominantly A1203 with an IEP at pH 8.0. At different solution pH values, different aggregate structures develop as a result of this anisotropy. For example, at pH 4, a "house of cards" aggregate structure is produced by the (+) edges attaching to the ( - ) faces. The percolation limit for this structure is at a very low volume fraction. At pH 8, a "deck of cards" aggregate structure develops because the uncharged edges can aggregate but the ( - ) faces cannot. At pH > 12, the suspension is colloidally stable. These types of floccs give rise to a plastic rheology with a yield stress.

582

Chapter 12 Mechanical Properties of Powders and Suspensions

Deck of Cards >,, ,,m

m 0

II

(n >

% Aggregates

House of Cards

% Aggregates FIGURE 12.27 Effect of aggregate structure on the viscosity of a sernidilute suspension of anisotropic platelet particles [61].

When the suspension is unstable, the degree of aggregation alters the viscosity of the suspension depending on the structure of the aggregates, as shown in Figure 12.27 [61]. When the aggregates are of the "deck of cards" type, the viscosity decreases as the volume fraction of aggregates increases. When the aggregates are of the "house of cards" type, the viscosity increases as the volume fraction of aggregates increases. The addition of a small amount of this type of clay to another suspension will render it more plastic and easier to mold. In a multicomponent slip like that shown schematically in Figure 12.28, large silt (10-60 t~m), clay (0.2-4 t~m), and small and large flocculated colloids form a traditional ceramic suspension. The viscosity for such a multicomponent suspension is far too complicated to allow theoretical prediction of its rheological behavior due to (1) the complex ion and polymer adsorption, (2) different electrostatic charges of the different exposed crystal faces, giving a complex colloid stability, resulting in complicated aggregate structures. In this picture, all the fine particulate additives flocculate, occupying more than their initial volume fraction as aggregates have a dramatic effect on the suspension viscosity. For this reason, small amounts of fines are added to ceramic

12.4 Ceramic Suspension Rheology

583

FIGURE 12.28 Diagram of a soil magnified 10,000 times. The resemblance of loam to ceramic bodies is readily apparent.

suspensions to adjust the rheological properties to those with more plasticity desired for casting; that is, either a high value of the low shear viscosity or a large Bingham yield stress.

12.4.2.3 Unstable Concentrated Suspensions Aggregation, coagulation, and flocculation in viscosity analysis is a dynamic process that entangles and disengages the network. Aggregates are well known to be mass fractals with fractal dimensions between 1 and 3. Two classic theories of aggregation [62] give fractal dimensions, DR, of 1.6-2.0 for aggregation cluster by cluster, either reaction limited (RLCCA) or diffusion limited (DLCCA), and D R - 2.5 for aggregation particle by particle. The number of individual particles, N, contained in a mass fractal of aggregate of size R is

N a R DF

(12.76)

The volume fraction, ~bA,inside an aggregate of radius R is (~A Ol (R/ro)3-DF

(12.77)

where r0 is the radius of that individual particle and ~0 is the initial volume fraction of individual particles. As aggregation proceeds, the aggregate volume fraction eventually reaches a percolation limit at a

584

Chapter 12 Mechanical Properties of Powders and Suspensions

o

Gel Time Time

FIGURE 12.29 As aggregation proceeds, the low shear viscosity increases drastically as the percolation limit is reached at the gel time, tg.

critical overlap radius [63], Rc,

Rc = ro4)o1/m-DF)

(12.78)

where a drastic increase in viscosity results.* This drastic increase in viscosity is schematically shown in Figure 12.29 with the low shear viscosity of Vo of the aggregating suspension increasing to infinity at the gel time, tg, according to [64] 70 = (tg- t) -k

(12.79)

where k is the scaling law [65] exponent for viscosity. The gel time can be predicted if the initial particle number density, No, the colloid stability ratio, W, and the fractal dimension, DF, are known [66]: 1.0= NT43 Ra = N O( 1 + tg l-1 ~ (ao)aexp[(-DF- 3)(tg/tl/2)] t:/2] (12.80) for slow RLCCA

1.0=NT 47rR33 - N o

(

F /' tg ~1/'~13 1+tl/2/ tg ~-1 ?

It(2 1/D )J/ -

(12.81)

for fast DLCCA where the half-life, t~/2 = ~r~?~a3W/(d~kBT) for Brownian diffusion, or tl/2 = 7rW/(4~/) for shear induced aggregation. * Note: this is true for only mutually opaque aggregates with D F > 1.5, which is most often the case.

12.4 Ceramic Suspension Rheology

585

105 -

104 I

n o m nUnoman

103

9mm UUmm

mmmm n

102 QQ 9

mn

II I

n

9 9 ago 0 9

101

o

100 _. 10-2

o

o

o

o

Oo 9 o

o

9 o

I

l

!

10-1

100

101

j

102

.,.

L

103

Txy,Pa F I G U R E 12.30

S t e a d y s h e a r v i s c o s i t y v e r s u s s h e a r r a t e for p o l y s t y r e n e l a t t i c e s (a = 220 n m ) in w a t e r a t 0.06 M NaC1 a n d 1.5% w g t T r i t o n X-405 w i t h soluble d e x t r a n (Mw = 600, rg = 33 n m ) a d d e d ; 9 ~b = 0.2q)min = -1.5kBT, O, ~b = 0.2 (~min = - 2 0 k s T, i , & = 0.3q)min = - 2 0 k s T. D a t a from P a t e l a n d R u s s e l [80].

At higher shear rates, this network is broken up further and further, giving rise to a well-dispersed colloid at higher shear rates with a viscosity of V~ as seen in Figure 12.30. This well-dispersed colloid at the high shear limit is, however, not without small temporal aggregates. Cessation of the shear rate again brings together the floccs into a new network which may have a different structure and therefore a different initial viscosity Vo, because the shear has been shown to increase the fractal dimension of floccs [66]. This shows a memory of the high shear state but not of the initial structure before the shear was begun. A "giant flocc" model for the viscosity of a flocculated suspension has been developed by Schreuder and Stein [67]. This model shows that, at low shear rates, the shear is not distributed homogeneously but only on certain shear planes. Then, with an increasing shear rate, the distance between successive shear planes diminishes to the size of the individual particles. This decrease in the distance between successive shear planes is related to the size of the aggregates present in the suspension under shear.

12.4.3 Ceramic Paste Rheology Pastes are ceramic suspensions with particle volume fractions near the maximum packing value for the particular particle size distribution.

586

Chapter 12 Mechanical Properties of Powders and Suspensions

At this volume fraction, the viscosity diverges because the shear stress is now given by the particle-particle contact in the tightly packed structure. As a result, we obtain a fluid with visco-elastic properties similar to polymeric solids. In ceramic processing, we extrude and press these pastes into green shapes. As a result, the rheology of ceramic pastes is of importance. The rheology of very concentrated suspensions is not particularly well developed, with the exception of model systems of monodisperse spheres. This section first discusses visco-elastic fluids and second the visco-elastic properties of ceramic pastes of monodisperse spheres. The material on visco-elastic fluids draws heavily from the book C o l l o i d a l D i s p e r s i o n s by Russel, Saville, and Schowalter [31]. Visco-elastic models have been developed for the nonlinear mechanical properties of fluids and solids. For a viscous f l u i d in simple shear flow, the shear stress, Zxy(~/), is a function of the effective viscosity, ~?()) and the shear rate, ~, as follows: Zxy(~)) = ~(~)~)

(12.82)

The shear rate, ~), can be used to determine the local velocity, Vx(y)(=y~/) in simple shear flow. For the simple shear flow, the following conditions on the shear stresses apply: Zxy(~) = -Zxy(-~) Zyz(~/ ) = Zxz(~/ ) = 0

(12.83) (12.84)

and the normal stresses must be even functions of ~),

Zxx(~/ ) = Zxx(-~/ ) Zyy(~/ ) = Zyy(-~/ ) Zzz(#/) = Zzz(-~/)

(12.85) (12.86) (12.87)

Two other material functions are related to the normal stresses. They are the first and second normal stress differences defined by NI(~)) = Zxx(~/ ) - Zyy(~/ ) N2(~/ ) = Zyy(#/ ) - Zzz(~/ )

(12.88) (12.89)

For Newtonian fluids, the effective viscosity is a constant V(~)) = ~? and N~(~)) = N2(~) = 0, suggesting that the normal stress differences arise from elasticity or memory of the material. For monosized ceramic suspensions, experimentally it has been found that N1 > 0 and N2 < 0. For a viscous f l u i d in sinusoidal oscillatory flow,

Vx(y) = y #/sin cot

(12.90)

the shear stress is given by a viscous and an elastic contribution: G'(co) . Zxy(~) = ~?'(oJ) ~ sin (ot - ~ 7 cos o~t oJ

(12.91)

12.4 Ceramic Suspension Rheology

~8~

where V'(co) is the dynamic viscosity which is in phase with the shear rate, ~ sin o~t, while the elastic contribution, G'((o), is 90 ~ out of phase with the shear rate and in phase with the strain. This equation is valid for only small frequencies (i.e., co < V/py2). For Newtonian fluids, the dynamic viscosity is equal to the viscosity (i.e., V'(oJ) = 7) and the elastic contributions are equal to 0 (i.e., G'(o~) = 0). For a viscous fluid undergoing creep, the shear stress is suddenly applied at time t - 0, rxy(~) = 0 rxy(~) = (r

t= 0 t> 0

(12.92) (12.93)

and the shear strain, Sxy(t), is monitored. The creep compliance, J(t), is then defined as J ( t ) = 2 Sxy(t)

(12.94)

For Newtonian fluids, the creep compliance is simply t/~. For a simple elastic solid, the same methodology can be used as that for a fluid but the viscosity is infinity (i.e., V(~) = ~), the dynamic viscosity is 0 (i.e., ~'(o~) = 0), the elastic contribution is a constant (i.e., G'(oJ) = G), and the creep compliance is equal to the reciprocal of the elastic contribution (i.e., J ( t ) = 1/G). For viscoelastic fluids, the formalism of a viscous fluid and an elastic solid are mixed [31]. The equations for the effective viscosity, dynamic viscosity, and the creep compliance are given in Table 12.4 for a viscous fluid, an elastic solid, and a visco-elastic solid and fluid. For the viscoelastic fluid model the dynamic viscosity, V'((o), and the elastic contribution, G'(oJ), are plotted as a function of (~o) in Figure 12.31. With one relaxation time, X, the breaks in the two curves occur at ~,o~. For ceramic suspensions little is predictable beyond the viscosity of the suspension making measurements of ~'(oJ), G'(o~), and J ( t ) necessary for almost all practical ceramic pastes. Experiments on the viscoelastic properties of monodisperse silica spheres at high concentration with hard sphere interactions show that a visco-elastic model with a single relaxation time, X = (Vs - V ' ) / G ' , correlate the data well, as seen in Figure 12.32 [68]. In this figure, the shear modulus, G~, is scaled by k s T / a 3 and the x coordinate is 61r~sa3(o/ksT , which is the translational Peclet number. For electrostatically stabilized suspensions, the shear moduli increase with increasing volume fraction and decreasing particle size or ionic strength. This can be accounted for by considering the energy required to displace the particle pairs from their equilibrium separation, rm. This energy is the elastic energy caused by the deformation [31], giving G~ ~ N2 r 2 d2V(rm) m dr 2

(12.95)

T A B L E 12.4

Mechanical Behavior of Viscous Fluid, Elastic Solid, Visco-elastic Fluid, and Visco-elastic Solid

Viscous fluid

Elastic solid

Visco-elastic fluid

Steady shear

~ = ~

~ = ~

V = ~o

Small-amplitude oscillations

~': ,

,' = o

,-:

Creep

G' = G

j=t

j t~

1 =G

G' = G"

~?=~

1 + ~-- (hoJ)2 ~0 1 + ()~0~)2

~0

G' = 0

Visco-elastic solid

V--= 1 V0 l+(hoJ)2

()~o~)._______~ 2

G'

1 + ()~(o)2

j=l

1-~-~) G"

)~- v o - V " G"

G"

i-e

v'~

+t

Go~G"

-}- (~O)) 2

1 + ()~(o)2

J=Gool-(~oo-~')exp(-~

-~h/GOtl

V0

k-

v~ V t _

G0

Taken from Russel, W. B., Saville, D. A., and Schowalter, W. R., "Colloidal Dispersions." Cambridge Univ. Press, Cambridge UK, 1989. Reprinted with the permission of Cambridge University Press.

12.4 Ceramic Suspension Rheology

589

F I G U R E 12.31

Shear modulus and dynamic viscosity as a function of frequency. Mechanical behavior of a visco-elastic fluid and visco-elastic solid from models in Table 12.4.

qollls .-----. J--*-+~.... r1'Irls

101

' "+~+<4.~+ n

Q

r1-1qs 0

0

1oo

lO--1 n

I

10"-1~ /

/ -

n

,/

I

10~

I

101

I

102

l

103

6nrlsa 3 BT

I0-'

F I G U R E 12.32

Shear moduli and dynamic viscosities m e a s u r e d for silica spheres at (b = 0.46, 9 9 a = 28 _+ 2 nm, 9 + a = 76 _ 2 nm (Mellema et al. [68]). The broken lines correspond to the infinite shear viscosities (de Kruif et al. [43]) and the solid curves to the frequency dependence predicted by the visco-elastic fluid model of Table 12.4 with the measured values of V0, V~, and G ' Redrawn from Russel et al. [31]. Reprinted with the permission of Cambridge University Press.

590

Chapter 12

Mechanical Properties of Powders and Suspensions

where N2 is the number of particle pairs per unit volume and V(r) is the interaction energy. Using simple approximation for the electrostatic repulsion between particles in an aqueous suspension gives correlations for the dependence on ionic strength and volume fraction. The empirical formula of Goodwin [69] r 2 V 0 rm

4Ir s 8o Ss

: (~Krm

(12.96)

is very useful. Using this approach to shear modulus scaling for polymers adsorbed at the surface of ceramic particles, we find [53]

k BT

- f

r m --

2a

(12.97)

where L~ is the thickness of the adsorbed layer. This functionality has been successfully used to collapse data taken from polymethyl methacrylate (PMMA) spheres in decalin [70]. To use the mechanical properties of visco-elastic materials to predict flow, we need to have the values of G~, G', ~?o, V~, and h. Once these values are determined experimentally for the ceramic paste of interest, the equations for creep given in Table 12.4 can be used to determine the defromation versus time for a particular applied force. Such predictions can be used for paste extrusion, stamping, and for die rebound because ceramic pastes at low solid factions behave like visco-elastic liquids and at high solids fractions behave like visco-elastic solids. All these subjects will be discussed in detail in Chapter 13.

12.5 M E C H A N I C A L P R O P E R T I E S OF D R Y CERAMIC POWDERS The pressing of dry ceramic powders follows the sequence shown schematically in Figure 12.33" 1. At low force, the powder particles remain fixed in position and the particles deform elastically. This takes place over a very small range of deformation for dry ceramic powders. This behavior is referred to as a compact body deformation. This deformation can be estimated from the elastic properties of the particles, the void fraction of the powder packing, and the nature of the liquid or binder occupying the voids [71]. 2. With higher forces the friction between particles is overcome and the particles slide with respect to one another. This behavior is referred to as a plastic body deformation. The start of this particle movement

12.5 Mechanical Properties of Dry Ceramic Powders

FIGURE 12.33

591

Behavior of a spray dried powder during die pressing.

is predicted by a first yield criteria for the powder. The particle movement is obtained by a decrease in the packing density for overconsolidated packings and an increase in the packing density for under consolidated packings. For this type of deformation, the stresses do not depend on the rate of deformation. (This type of particle flow also takes place during mold filling.) 3. After the precedings movement is essentially complete, the particles themselves deform elastically and plastically. If the particles are

592

Chapter 12 Mechanical Properties of Powders and Suspensions

aggregates of smaller particles, there may be a second yield criterion for particle rearrangement inside of the aggregates. If high rates of flow are achieved, on the order of 1 m/sec, the stresses are dependent on the deformation rate. This behavior is referred to as a flowing medium. The packing then behaves like a fluid, where frictional and inertial forces participate in stress transmission. 4. If there is little deformation, much higher forces can result. At the very highest forces, the particles rupture, creating smaller particles which fill the interparticle voids between other particles and further increase the density of the powder. The next sections of this chapter discuss these various deformation regimes. In all these cases, the packing is considered as a mechanical continuum, and the stresses, deformations, and rates of deformation that arise are represented by the methods of elasticity theory, plasticity theory, and fluid mechanics. This continuum approach has long been common practice in soil mechanics. Jenike [72] was the first to apply this approach to the study of bulk solids and thereby to deal theoretically with the flow of such solids in bunkers. This topic has been reviewed at length by Schwedes [73] and in a condensed form by Rumpf [74]. The approach has also found application to the flow of dry ceramic powders into a die and the flow resulting from the pressing of a ceramic powder [75, 76]. The problems posed in this field are of a quite different nature going from (1) the flow of granular material in mixers, chutes, dies, and bunkers, where the stresses of interest are low, to (2) the compaction of a ceramic powder in a press to (3) the stable static behavior occurring in soil mechanics, where the stresses of interest are very high. Moreover, the powders that have to be investigated are different. In soil mechanics, the soils of interest are coarse, sand, and fine-grained moist clay. In process technology, we are concerned with very different types of bulk solids such as plastics, flour, cement, paint pigments, ceramic powders, and the like, as well as moist materials such as filter cakes and agglomerates. Here a distinction must be made between dry bulk solids and wet powders. Ceramic powders fall in both of these categories due to the easy flow of spray dried powders and the difficult flow of the particles inside these aggregates held together by the polymeric binder.

12.5.1 Coefficient o f Pressure a t Rest Using continuum mechanics, ceramic powder packing can be analyzed for its mold filling capability. Powder packings are different from liquids in one important way, which is described by the coefficient of pressure at rest.

12.5 Mechanical Properties of Dry Ceramic Powders

593

Force

F I G U R E 12.34 Schematic of snap-through buckle of a particle arch during pressing of a ceramic powder. The force exerted by the particles on the wall in the vertical direction is a result of the coefficient of pressure at rest.

The coefficient of pressure at rest is a descriptive index used in soil mechanics. It shows t h a t the flow of powder packings cannot be regarded as analogous to the behavior of liquids. Let us consider a die with a large square cross-section. The y axis points vertically downward. Let rxx be the horizontal and Zyy the vertical compressive stress at any given depth acting along the x and y axes, respectively.* For reasons of symmetry, these stresses are also p r i n c i p a l stresses. The coefficient of pressure at rest is ~o = Zxx/Zyy. When the material is a liquid, the same hydrostatic pressure prevails on all sides, giving ~o = 1. In a liquid, the molecules are completely free. With a purely elastic solid, the load in the vertical y direction can be applied without a load being applied in the x direction; therefore, ~,o = 0 for elastic solids. A powder packing behaves neither like a liquid nor an elastic solid. Under a load in the verticaly direction, the particles are displaced in the horizontal direction, so that an additional horizontal stress Zxx arises, as if shown in Figure 12.34. However, the particles lack complete freedom of movement, so it follows t h a t Zxx < Zyy and 0 < ~'0 < 1, with a typical value being ~o = 0.5. Limiting values hold for the extreme conditions of the packing, such as for a fluidized bed ~o = 1 and for a highly compacted pressings ~o = 0. The effect of particle

* Note Zxx = Zzz for this geometry.

594

Chapter 12 Mechanical Properties of Powders and Suspensions T A B L E 12.5 The Effect of Particle Size, Particle Morphology, and Binder Content on the Coefficient of Pressure at Rest, ~0, for Zirconia Powder Mean size ~,0

0.33 tLm 0.406

Powder morphology h0 Binder content ~,0

0.53 tim 0.429

Irregular 0.406 4% PVA 0.488

0.92 t~m 0.392

Spherical 0.500 4% PVA + 2% stearic acid 0.498

Data from D. Bortzmeyer, "Dry Pressing of Ceramic Powders," NKV Summer School, September 1991, Petten, The Netherlands.

size, particle morphology, and binder content on the coefficient of pressure at rest for various zirconia powders is given in Table 12.5. All values were shown not to be functions of axial pressure and fall near to 0.5. Particle size and morphology have the greatest effects on the coefficient of pressure at rest.

12.5.2 Compact Body The moduli for a dry ceramic powder in this regime of very small deformation are given by the moduli for the particles and the volume fraction, ~c, filled by the ceramic powder as follows: Young's modulus: (12.98) t~Tyy * * Ol/l = Ypowder = Yparticle (1 - s ) : Yparticle ~)

Shear modulus: (12.99) O'$xy

--

0ll/12

OTi___.._..L__

Bulk modulus" OV/V

Gpowder

=

*

Gparticle(1 ,

__

s)

=

*

Gparticle(~ *

gpowder - Kparticle(1 - e) = gparticlet~

(12.100)

The equations for the deformation of an elastic solid given in Table 12.4 can be used for a determination of the creep and small amplitude oscillations. When a polymer is adsorbed onto the surface of the ceramic powder, the particle moduli are replaced by those of the polymer until there is sufficient force to deform the ceramic particles.

12.5 Mechanical Properties of Dry Ceramic Powders

595

12.5.3 Plastic Body For a plastic body, we are interested in the point where the forces acting on the powder packing will start to cause the particles to move with respect to one another. On a discreet particle basis, Kuhn et al. [77], have analyzed the critical force required to buckle an arch of particles within a powder compact as shown in Figure 12.34. This force is formulated by the Hertz contact theory [78], assuming frictionless spheres, and depends on the number of particles making up the arch and the principle stress (or pressure) acting on the arch from the powder surrounding the arch. The more particles there are in the arch, the lower the force; the higher the pressure that is applied, the lower the force required to buckle the arch giving powder compaction. Using the continuum approach to the onset of particle movements with respect to one another, the yield criteria for packings is obtained. The approach that is followed here is identical to that used in Particle Technology by Rumpf [74]. For a particular volume element, the forces acting on the surfaces of the element consists of normal and tangential (i.e., shear) force, forming a stress sensor. At rest the tangential stress and normal stress can be summed into three principle stresses, (rl, (r2, and (r3, which act in the principal directions perpendicular to the surfaces of the volume element.* If the forces occurring in a packing are to be described by the methods of continuum mechanics, then the volume element must be large compared with the particle size but small compared with the size of the processing equipment. To completely specify the state of stress at a point in a three-dimensional field (the stress produced by any force), six quantities are necessary: the three principle types of stress and three angles specifying the principal directions.** The manipulation of three-dimensional states of stress will not be considered here because it is very difficult. We shall restrict ourselves to two-dimensional states of stress, which is identical to the case discussed e a r l i e r - - a die with large cross-section in the x and z directions with the vertical direction corresponding to the y direction. To describe the generalized forces in this geometry, we shall choose a volume element with a right-angled triangular cross-section whose shorter sides are perpendicular to the directions of the principal stress ((r 1 and (r2) and whose hypotenuse is perpendicular to the x direction

* In the theory of elasticity and plasticity, as well as in fluid mechanics, tensile stress is chosen to be positive and compressive stress is chosen to be negative. In soil and powder mechanics, compressive stress predominates, therefore, it is customary to define compressive stress as positive. ** Or equivalently, the three normal and three tangential forms of stress.

596

Chapter 12 Mechanical Properties of Powders and Suspensions

in one instance and to the y direction in the other. We shall adopt the convention that 0.1 is always greater t h a n 0.2. The normal types of stress in the x and y directions are Zxx and Zyy, and both tangential forms of stress Zxv have the same value. The following equations are obtained by considering the equilibrium of the forces: Txx -- 0.1 COS20L -~- 0.2

sin2a

=

7"yy ---- 0.1 s i n 2 a + 0.2 cOS20L --

ZxY =

[(0.1 -~- 0"2) -~- (0.1 -- 0"2)COS

2a]

[((rl + ~r2) - ((rl - ~2)cos 2a]

[(0.1 + 0.2) sin 2a] 2

(12.101) (12.102) (12.103)

These equations show the relationship between the normal and shear types of stress on the principal types of stress and the angle a t h a t represents a force balance for the volume element. The equations can be represented by the Mohr stress circle (Figure 12.35) with a radius of (0.1 - 0.2)/2 and its center at (0.1 + 0.2)/2 on the abscissa. The shear stress rxy is plotted on the ordinate and the normal forms of stress rxx and ryy are plotted on the abscissa. The points of the circle intersection with the abscissa give the principal types of stress 0.1 and 0.2- The normal stress Zxx and the associated shear stress, Zxy, are fixed by the radius arm which is rotated through an angle 2a from the abscissa. The normal stress Zxx now appears as the projection of the radius arm onto the abscissa, and the shear stress Zxy appears as the projection onto the ordinate. It can be seen from the Mohr circle that the shear stress has its maximum values for a = 45 ~ and 135 ~

12.5.4 Yield Criteria for Packings If a solid body is compressed by a normal force Zyy against a flat support and subjected to a tangential force rxy, it will start to move if rxy >- t~Zyy (t~ = tan 8 is the coefficient of friction and 8 is the angle of solid friction). In this case, the frictional force cannot become greater t h a n t~Zyy. This the Coulomb friction criterion for solid bodies experienced over a wide range of sliding velocities. A similar approach is made for powder compacts. The yield point is established by the friction between the particles and gives the shear stresses that lead to irreversible, plastic, deformation. This shear stress at the yield point is proportional to the normal stress but only in special cases, when the velocities are small and each shear stress is independent of velocity. These special cases and the fact that the coefficient of pressure at rest ho is less t h a n

12.5 Mechanical Properties of Dry Ceramic Powders

l

597

Txy

\1 /

~'XX, .ram.ram

Ox]G,

Tyy

b G1

v

TxYl Oy

FIGURE 12.35 The Mohr stress circle (a) is a representation of a two-dimensional state of stress in a powder compact (b). The Coulomb yield criterion is also plotted as a straight line in the Mohr stress circle.

unity are characteristics that distinguish the flow of particulate solids from that of the flow of liquids.' A yield criterion similar to Coulomb's criterion for the friction between solid bodies has long been used in soil mechanics. It will be discussed in the next section. Later, an account will be given of Jenikes modification to this criterion for bulk solids.

12.5.5 The C o u l o m b Y i e l d C r i t e r i o n The general form of the expression for the Coulomb yield criterion used in soil mechanics is rxy = rxx

tan 6 + c

(12.104)

598

Chapter 12 Mechanical Properties of Powders and Suspensions

where ~xyis the shear stress at which the soil begins to deform plastically under a normal stress rx~, 8 is the angle of friction, and c is the cohesion stress, which is a measure of the stickiness of the particles. The designation cohesion is misleading since it does not represent a tensile stress but a shear stress when the normal stress is 0. The cohesion is a concept that is useful in soil mechanics and in the flow of powders. A graph of the Coulomb yield criterion on the Mohr diagram is the straight line in Figure 12.35(a). Its intercept on the ordinate is the cohesion. The Coulomb line therefore denotes those related shear and normal stresses at which yield, or flow, occurs. Above the Coulomb line, no states of stress are possible because at a given normal stress, the material begins to flow as soon as the matching shear stress of the Coulomb line is reached. States of stress with a Mohr circle below the Coulomb line are stable and do not flow. States of stress with a Mohr circle tangential to the Coulomb line cause flow of the powder. States of stress with Mohr circles that cut the Coulomb line are physically impossible because such Mohr circles would have normal and shear stresses above the yield point. The following relation for the principal stresses that lead to plastic flow (yielding) are obtained by simple geometrical analysis: 1 + sin 8 cos 8 + 2c (rl = (r21 - sin 8 1 - sin 8

(12.105)

For cohesionless solids, we obtain the special case of a Coulomb line going through the origin and having only one characteristic parameter, the angle of friction 8. This angle of friction can be demonstrated experimentally when a powder is poured into a pile. The angle of the resulting cone shown in Figure 12.36 is called the angle of repose of the powder.

FIGURE 12.36 Angle of repose of a heap of powder. An acute angle of repose (left) is observed with a powder that does not flow well. An obtuse angle of repose (right) is observed with a powder that flows well.

12.5 Mechanical Properties of Dry Ceramic Powders

599

The angle of repose is equal to the angle of friction for cohesionless solids. For cohesive solids, the angle of repose is less reproducible and is not a measure of the angle of friction. In fact, the cohesive material is retained by adhesive forces and piles up erratically. The influence of particle morphology on the angle of repose has been widely studied for cohesionless powders. The general findings are that broad size distribution powders give lower angles of repose. In addition, powders consisting of angular and platelet particles give lower angles of repose compared to powders consisting of spherical particles. For the special case of cohesionless powders with c = 0, the previous yield relation is reduced to 1 + sin 6 (rl = (r21 - sin 6

(12.106)

In soil mechanics, the pressures lie in a range up to and over 2 MPa, whereas in powder mechanics they are usually below 0.1 MPa. For this range of pressure, the Coulomb criterion is generally not applicable. Only cohesionless solids exhibit Coulomb yield behavior at low pressures.

12.5.6 Y i e l d B e h a v i o r o f P o w d e r s a t L o w P r e s s u r e Jenike developed the idea that no single line represents the yield but rather a curve called the yield locus. The yield behavior depends on the packing density of the powder when it is caused to flow under the action of normal and shear stress. Figure 12.36 shows a yield locus for a given porosity, s. A Mohr circle for the stage when yielding starts is characterized by the principal stresses ~1 and ~2. The point E at the end of the yield locus lies on the Mohr circle ((rl, (r2) and pertains to steady state flow. The point c gives the cohesion of the material at the given porosity, and the point t (the x intercept) gives the tensile strength. There is also a Mohr circle that starts at the origin and touches the yield locus. Its second point of intersection with the abscissa is designated fc. The corresponding state of stress has obviously only one principal stress. The other principal stress is 0. This circle represents a uniaxial compression test. The compressive strength of the material is, therefore, fc. The effective yield locus for zirconia powder [79] at 3.06 gm/cc density is given by the following points (Fig. 12.37). Normal stress rzz (MPa) Shear stress Txz (MPa)

-0.4 0.4

0 1.8

10 10

120 50

210 0

For each porosity, there is a particular yield locus, a family of three ~eld loci is shown in Figure 12.38. Many experiments [72] have established that the envelope of the Mohr circles through the points E~ that lead to steady state flow for different porosities is, to a very close

600

Chapter 12 Mechanical Properties of Powders and Suspensions

TE

Txx,

0

0"2

FIGURE 12.37

~]~I

fc 0"2

0"I

F/

Jenike's yield locus for a particular porosity, s.

approximation, a straight line through the origin. This line thereby represents an extremely simple and convenient physical law for steady state flow. It is called the effective yield locus. Its angle of inclination 8e is known as the effective angle of friction. This line gives the relation for the principal states of stress as 1 + sin (~e o-~ - o21 - sin 8e

(12.107)

which can now be regarded as the physical law for steady state flow. In addition to this relation, the factors of interest concerning the problems of process technology are the porosity, s, prevailing during steady state flow and the proper compressive strength, fc, as functions of the consolidating pressure. For this it is useful to choose the larger

~y E3 E3

E1

E2

TXX,

0 FIGURE 12.38 the family.

f:

"yy

Yield loci for different porosities, s, with the effective yield locus for

12.5 Mechanical Properties of Dry Ceramic Powders

601

0"1

fc

O'1

FIGURE 12.39 The dependence of the porosity and the compressive strength, fc, on the largest principal stress, r occuring during steady state flow.

of the principal states of stress Or 1 of steady flow for the yield law. Each yield locus is specified by two pair of values (8, ~rI ) and (f~, r 1). From the family of yield loci, curves can be constructed for a given material, as shown in Figure 12.39. These two curves describe the principal stress required to give a particular density (or void fraction, e) in the mold and the compressive strength of the compact t h a t results. These two curves are then exactly what the ceramic powder processor needs to know. However, the yield locus for every void fraction m u s t be known for the powder packing to make these plots. The yield locus for every void fraction cannot be predicted, in general, for ceramic powders and must be measured. This can be done during uniaxial compression. Using the effective yield locus, which can be obtained from one stress required to cause a ceramic powder packing to yield at one void fraction, the whole pressing curve can be predicted if the interparticle friction is the same for all volume fractions. This is not often the case for agglomerated or spray dried ceramic powders because two types of interparticle friction are involved, that between agglomerates and that between the particles that make up the aggregates. In general, a material flows poorly if its strength fc is large relative to the consolidating stress ~ . Jenike called this ratio of (rl to fc, the flow function (F = ~l/fc ). It is generally not constant because, as a rule, the compressive strength fc is not proportional to the principal stress cry. This ratio can be used to make a classification of the flow according

602

Chapter 12 Mechanical Properties of Powders and Suspensions

to the following different ranges: F < 2 for cohesive nonflowing materials, 2 < F < 4 for the flow of cohesive materials, 4 < F < 10 for easily flowing materials, 10 < F < ~ for freely flowing materials. The angle for wall friction 8~ for powders can also be measured in a shear cell. Experimental results have shown that the shear wall stress ~xy~ is proportional to the normal stress at the wall ~xx~ giving a relation rx~w = tan 8w

(12.108)

~xxw which shows that Coulomb's friction relation is also valid at the wall. In general, this Coulomb yield criterion can be used to determine what stress will be required to cause a ceramic powder to flow or deform. All that is needed are the two characteristics of the ceramic powder: the angle of friction, 8, and the cohesion stress, c, for each particular void fraction. With these data, the effective yield locus can be determined, from which the force required to deform the powder to a particular void fraction (or density) can be determined. This Coulomb yield criterion, however, gives no information on how fast the deformation will take place. To determine the velocity that occurs during flow or deformation of a dry ceramic powder, we need to solve the equation of motion. The equation of motion requires a constitutive equation for the powder. The constitutive equation gives the shear and normal states of stress in terms of the time derivative of the displacement of the material. This information is unavailable for ceramic powders, and the measurements are particularly difficult [76, p. 93].

12.6 SUMMARY

In this chapter, we described the fundamentals of suspension rheology from dilute suspensions to concentrated suspensions. Attention has been paid to interparticle forces and the structure of the suspension because these things drastically influence suspension rheology. In addition, visco-elastic properties of concentrated suspensions including ceramic pastes have been discussed. Finally, the mechanical properties of dry ceramic powders have been discussed in terms of the Coulomb yield criterion, which gives the stress necessary for flow (or deformation) of the powder. These mechanical properties will be used in the next chapter to predict the ease with which dry powders, pastes, and suspensions can be made into green bodies by various techniques.

12.6 Summary

603

Problems 1. Determine the shear stress, ~xy, versus strain rate, ~, relationship for a monodisperse SiC powder at a volume fraction of 45% dispersed in chloroform. Assume that a hard sphere model can be used for SiC in chloroform because this solvent has excellent wetting properties for the oxidized SiC surface. Data: the viscosity of chloroform is 0.58 cP. 2. An aqueous solution of dodecylamine (DDA) at pH 12 has a CMC at 10 .5 M. If a solution initially at 10 .2 M DDA pH 6 is titrated to pH 12, what is the volume fraction micelles which are produced. Assuming these micelles are spherical, determine the viscosity of the solution. Neglect the effect of the dissolved DDA molecules because they are in a dilute concentration at pH 12 and have a low molecular weight. In this DDA solution, 0.5 ftm SiO2 particles corresponding to 0.01% volume were added. After titration to pH 12, what is the viscosity of the solution. Data: the viscosity of water is 1 cP, the density of SiO2 is 2.2 gm/cc. 3. For a dilute aqueous suspension of spherical 0.05 ftm SiO2 particles (10 s particles/cc) coagulating by RLCCA under solution conditions that give a colloidal stability ratio of 10, determine the time required for gelation. Schematically, draw a curve that represents the low shear viscosity as a function of time from the onset of mixing to gelation. Data: the viscosity of water is 1 cP, the density of SiO2 is 2.2 gm/cc. 4. A monodisperse 0.5 t~m BaTiO3 powder is placed in aqueous suspension at pH 10.5 and ~b = 0.01. Polyacrylic acid (PAA) ( M w = 5,000) is added at 0.05 wgt% and absorbs onto the surface of the BaTiO3 powder surface by the Langmuir adsorption isotherm with K = 182(wgt%) -1 and F m = 0.00014 gm PAA/(m 2 BaTiO3). Determine the Newtonian viscosity of this multicomponent suspension. Data: the density of BaTiO3 = 5.8 gm/cc. 5. Determine the shear stress, ~xy, versus strain rate, ~, relationship for a monodisperse A1203 hexagonal platelet powder at a volume fraction of 0.01% dispersed in isopropanol. Assume that a sphere of revolution model in the semiconcentrated regime can be used for A1203 platelets because this solvent has excellent wetting properties for the A1203 surface. Data: the viscosity of isopropanol is 2.3 cP, the A1203 platelets are 7 ftm in diameter and 0.2 t~m thick. 6. Determine the exponential time constant that will be observed during stamping of a monodisperse 0.2 ~Lm Si3N 4 powder at a volume fraction of 58% dispersed in chloroform. This suspension can be

604

Chapter 12 Mechanical Properties of Powders and Suspensions 0.6

Aggregate Density

0.5

A

o~ U)

r-

O

0.4

"O m m

o I._

o.

0.3

ir-TapDensity 0.2

-

9 .1

. . . . . . .

.

9

m

m

9

m

www|

40

1

m

m

9

mm

ram|

loo

Die Pressure (MPa)

FIGURE 12.40

Compaction of spray dried alumina at 92% relative humidity. Data from Reed [23, p. 337].

considered a visco-elastic fluid with hard sphere interparticle interaction energy because this solvent has excellent wetting properties for the oxidized Si3N4 surface. Use of the data given in Figure 12.32 for the calculation of the quantities needed for the creep equation for a visco-elastic fluid given in Table 12.4. Data: the viscosity of chloroform is 0.58 cP, the density of Si3N4 = 3.44 gm/cc. 7. A spray dried yttria stabilized ZrO2 powder at 48% solids by volume has a cohesion of 0.15 MPa due to the polyvinyl alcohol (PVA) binder and an angle of internal friction of 60 ~. (a) Determine the compressive stress required to deform the powder to a higher volume fraction. (b) Determine the tensile strength of the same ZrO2 powder slip cast into a ceramic green body at 48% solids with PVA as a dispersant. 8. For spray dried Alcoa A1203, the green body density versus die pressure is given in the graph in Figure 12.40. The die pressure can be considered the compressive strength of the green body at a

References

605

particular density or void fraction. Determine the effective yield locus for this data on a Mohr circle. From the Mohr circle, calculate the cohesion of the A1203 particles with their binder polyvinyl alcohol.

References 1. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena." Wiley, New York, 1960. 2. Philippoff, W., KoUoid. Z. 71, 1-16 (1935). 3. Cross, M. M., J. Colloid Interface Sci. 33, 30 (1970). 4. Rivlin, R. S., Proc. R. Soc. London, Ser. A 193, 260-281 (1948); Proc. Cambridge Philos. Soc. 45, 88-91 (1949). 5. Reiner, M., Am. J. Math. 67, 350-362 (1945). 6. Oldroyd, J. G., in "Rheology" (F. R. Eirich, ed.), Chapter 16. Academic Press, New York, 1956. 7. Lodge, A. S., "Elastic Liquids." Academic Press, New York, 1964. 8. Fredrickson, A. G., "Principles and Applications of Rheology." Prentice-Hall, New York, 1963. 9. Bird, R. B., Chem. Eng. Prog., Symp. Ser. 58, 61 (1965). 10. Einstein, A., Ann. Phys. (Leipzig) 19, 289 (1906). 11. Einstein, A., Ann. Phys. (Leipzig) 34, 591 (1911). 12. Hiemenz, P. C., "Polymer Chemistry." New York, 1984. 13. Simha, R., J. Appl. Phys. 23, 1020 (1952). 14. Voyutsky, S. (translated by N. Bobrov), "Colloid Chemistry," p. 368. Mir Publishers, Moscow, 1978. 15. Debye, P., "Polar Molecules." Dover, New York, 1929. 16. von Smoluchowski, M. KoUoid-Z. 18, 190 (1916). 17. Booth, F., Proc. R. Soc. London, Ser. A 203, 533 (1950). 18. Hunter, R. J., ed., "Zeta Potential in Colloid Science: Principles and Applications," p. 192. Academic Press, New York, 1981. 19. Fleer, G. J., Koopal, L. K., and Lyklema, J., KoUoid-Z. Z. Polym. 250, 689 (1972). 20. Michaels, A. S., and Bolger, D., Ind. Eng. Chem. Fund. 3(1), 14-20 (1964). 21. Brinker, C. J., and Scherer, G. W., "Sol-Gel Science," p. 319. Academic Press, San Diego, CA, 1990. 22. Russell, W. B., MRS Bull. 16(8), 27-31 (1991). 23. Reed, J. S., "Introduction to the Principles of Ceramic Processing," p. 244. Wiley (Interscience), New York, 1988. 24. Napper, D. H., "Polymeric Stabilizatio of Colloidal Dispersions," p. 80. Academic Press, New York, 1983. 25. Cross, M. M., J. Colloid Interface Sci. 33, 30 (1970). 26. Cross, M. M., J. Colloid Sci. 20, 417 (1965). 27. Cross, M. M., J. Appl. Polym. Sci. 13, 765 (1969). 28. Soong, D., and Shen, M., J. Polym. Sci., Polym. Lett. 17, 595 (1979). 29. Soong, D., and Shen, M., Polym. Eng. Sci. 20, 1177 (1980). 30. Liu, T. Y., Soong, D. D., and Dekee, D., Chem. Eng. Commun. 22, 273 (1983). 31. Russel, W. B., Saville, D. A., and Schowalter, W. R., "Colloidal Dispersions." Cambridge Univ. Press, Cambridge, UK, 1989. 32. Van de Ven, T: G. M., in "Colloidal Hydrodynamics" (R. H. Ottewill and R. L. Rowell, eds.), Colloid Sci. Monogr. Ser. Academic Press, Boston, 1989.

606

Chapter 12 Mechanical Properties of Powders and Suspensions

33. Jeffrey, D. J., and Acrivos, A., AIChE J. 22(3), 417-432 (1976). 34. Gadala-Maria, F., and Acrivos, A., J. Rheol. 24(6), 799-814 (1980). 35. Castillo, C. A., Rajagoplan, R., and Hirtzel, C. S.,Rev. Chem. Eng. 2, 237-348 (1984). 36. Van Megen, W., and Snook, I., Adv. Colloid Interface Sci. 82, 62-76 (1984). 37. Batchelor, G. K., J. Fluid Mech. 41, 545-570 (1970); 83, 97-117 (1977). 38. Russel, W. B., J. Fluid Mech. 85, 209-232 (1978). 39. Batchelor, G. K., and Green, J. T., J. Fluid Mech. 56, 401-472 (1972). 40. Blachford, J., Chan, F. S., and Goring, D. A. I., J. Phys. Chem. 73, 1062-1065 (1969). 41. Kreiger, I. M., and Eguiluz, M., Trans. Soc. Rheol. 20, 29-45 (1976). 42. Kreiger, I. M., Adv. Colloid Interface Sci. 3, 111-136 (1972). 43. de Kruif, C. G., van Iersel, E. M. F., Vrij, A., and Russel, W. B., J. Chem. Phys. 83, 4717-4725 (1986). 44. Metzner, A. B., and Whitlock, M., Trans. Soc. Rheol. 2, 239-254 (1958). 45. Jeffrey, D. J., and Acrivos, A.,AIChE J. 22(3), 417-432 (1976). 46. Sacks, M. D., Ceram. Bull. 63, 1510-1515 (1984). 47. Hoffman, R. L., Trans. Soc. Rheol. 16, 155-173 (1972). 48. Hoffman, R. L., J. Colloid Interface Sci. 46, 491-506 (1974). 49. Hoffman, R. L., MRS Bull. 16(8), 32-37 (1991). 50. Leighton, D., and Acrivos, A., J. Fluid Mech. 181, 415-439 (1987). 51. Leighton, D., and Acrivos, A., J. Fluid Mech. 177, 109-131 (1987). 52. Mewis, J., Frith, W. J., Strivens, T. A., and Russel, W. B.,AIChE J. 35, 415-422 (1989). 53. Buscall, R., J. Chem. Soc., Faraday Trans. 87, 1365-1370 (1991). 54. Lindsay, H. M., and Chaikin, P. M., J. Phys. (Paris) C3 46, 269-280 (1985). 55. Farris, R. J., Trans. Soc. Rheol. 2, 281-301 (1968). 56. Sadler, L. Y., and Sim, K. G., Chem. Eng. Prog. March, p. 68 (1991). 57. Furnas, C. J., Ind. Eng. Chem. 23, 1053 (1931). 58. Chang, S.-Y., Internal report, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992). 59. Langston, R. B., and Pask, J. A. Natl. Acad. Sci. Natl. Research Council Publ. 566, 23-28 (1958). 60. Scholfield and Samson, (1954). 61. Michaels, A. S., in "Slip Casting," Chapter 2, American Ceramic Society, 1963. 62. Witten, T. A., and Cates, M. E., Science 232, 1577 (1986). 63. Martin, J. E., and Wilcox, J. P., Phys. Rev. A 39, 252 (1952). 64. Kozuka, H., Kuroki, H., and Sakka, S., J. Non-Crys. Solids 95/96, 1181-1188 (1987). 65. de Gennes, P.-G., "Scaling Concepts in Polymer Physics." Cornell Univ. Press, Ithica, NY, 1979. 66. Chang, S.-Y., Ph.D. Thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992). 67. Schreuder, F. W. A. M., and Stein, H. N., Rheol. Acta 26, 45-54 (1987). 68. Mellema, J., de Kruif, C. G., Blom, C., and Vrij, A. Rheol. Acta 26, 40-44 (1987). 69. Goodwin, J. W., and Hughes, R. W.,Mater.Res. Soc. Symp. Proc. 177,187-198 (1990). 70. Russel, W. B., Ceram. Powder Sci. 3, 361-373 (1990). 71. Schubert, H., Chem.-Ing.-Tech. 45, 396 (1973). 72. Jenike, A. W., Bull. Univ. Utah 53(26), 1 (1964). 73. Schwedes, J., "Fleissverhalter von Scht~ttgt~ttern in Bunkern." Verlag Chemie, Weinheim/Bergstrasse, 1968. 74. Rumpf, H., "Particle Technology" (translated by F. A. Bull). Chapman & Hall, London, 1990. 75. Bortzmeyer, D., Powder Technol. 70, 131-139 (1992). 76. Takahashi, M., and Suzuki, S., in "Handbook of Ceramics and Composites" (N. P. Cheremisinoff, ed.), Vol. 1, pp. 65-96. Dekker, New York, 1990.

References

607

77. Kuhn, L. T., McMeeking, R. M., and Lang, F. F., J. Am. Ceram. Soc. 74(3), 682685 (1991). 78. Timoshenko, S., and Goodier, J. M., "Theory of Elasticity," pp. 409-414. McGrawHill, New York, 1951. 79. Bortzmeyer, D., Abouaf, M., Chane-Ching, J. Y., and Paraud, N., in "Ceramic Powder Processing Science-Proceedings of the Second International Conference" (H. Hausner, G. L. Messing, and S. Hirano, eds.), pp. 561-568. Deutsche Keramische Gesellschaft, Koln, 1988. 80. Patel, P. D., and Russel, W. B., J. Rheology 31, 599-618 (1987).

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13

Ceramic Green Body Formation

13.1 O B J E C T I V E S This chapter is devoted to the formation of a ceramic green body from (1) a dry ceramic powder, (2) a suspension of ceramic powders in a solvent solutions, or (3) a paste of a ceramic powder in a limited amount of solvent solution. In Chapter 12, we saw the various rheological properties of dry powders, suspensions, and pastes. These rheological properties will now be used to predict the flow behavior of dry powders, suspensions, and pastes in molds in order to make a particular ceramic shape called the ceramic green body. Several shaping methods can be used to make a ceramic green body. These include slip casting, filter pressing, dip coating, and tape casting for ceramic suspensions; extrusion and injection molding for ceramic pastes; and dry pressing, isostatic pressing, ramming, and stamping for dry ceramic powders. Each of these methods will be discussed in this chapter. In addition, a section on the characterization of the packing (and other component) uniformity in the ceramic green body is discussed. 609

610

Chapter 13 Ceramic Green Body Formation

FIGURE 13.1 Detail of a combustion chamber (a) with the slip casting mold (c) used for its fabrication. Taken from Garret Turbine Engine Co. Fabricated by Norton Co. (b) Detail of a ceramic turbocharger.

13.2 INTR OD UC TION T h e process of m a k i n g a c o m p l e x s h a p e (such as a t u r b o c h a r g e r b l a d e or a c o m b u s t i o n c h a m b e r , b o t h s h o w n in F i g u r e 13.1) f r o m a d r y c e r a m i c powder, a s u s p e n s i o n , or a p a s t e is a p r o b l e m w h e n one consid-

13.2 Introduction

611

F I G U R E 13.1 (Continued)

ers that the ceramic powder should be uniformly packed into every nook and cranny of the ceramic shape. This uniformity is essential for the uniform sintering of the green body. Any differences in the packing density of the ceramic powder will cause different shrinkage during sintering, which leads to warping or cracking of the shape during sintering. This chapter is broken down into different methods of shape manufacture, depending on the initial state of the ceramic material, which can be either a dry powder, a suspension, or a paste. We will start with dilute ceramic suspensions and the methods of slip casting and filter pressing and proceed to ceramic pastes with extrusion and injection molding and finish with dry powders with pressing and stamping. These methods are illustrated in Figure 13.2. Each of these methods will be discussed in detail later in this chapter. After the ceramic shape is formed, methods of green body characterization are discussed. Particular attention is paid to the uniformity of powder packing, binder distribution, and solvent distribution because these play an important role in the next phases of processing: drying, binder burnout and sintering. Inhomogeneities in these properties will lead to warping and cracking of the ceramic piece during further processing. This chapter draws from many sources, including books (both proceedings [1-4] and textbooks [5-10]), review papers [11-13], and recent journal articles. Much good work has been done in the last 10 years to

612

Chapter 13

FIGURE 13.2

Ceramic Green Body Formation

Molding methods used for ceramic objects.

put this field on a more quantitative basis. At present, with high-speed computers, the solution of these complex fluid flow equations can be performed, and these processes can be designed more precisely than ever before. However, computer calculations are limited by the constitutive equations for the ceramic suspensions which describe how the applied forces are redistributed by the fluid. For ceramic suspensions with high particle loadings, the constitutive equations are not available. Only a few simple fluids have well-defined constitutive equations, these include dilute ceramic suspensions which are colloidally stable.

13.3 G R E E N B O D Y F O R M A T I O N W I T H CERAMIC SUSPENSIONS A ceramic suspension consists of ceramic powder, a solvent, often a dispersant to stabilize the ceramic powder against agglomeration, a polymeric binder to provide green strength after the green body has been dried, and often a plasticizer to lower the glass transition of the polymeric binder. All these additives must be compatible so the ceramic suspension has the desirable properties needed for green body fabrication. Many of these formulations used in industry are very secretive,

13.3 Green Body Formation with Ceramic Suspensions

613

however, several are well known, having been published in the literature. The ceramic powder often is a mixture of powders, as is the case in porcelain, earthenware, and other common domestic ceramics. In some high tech ceramics, like ceramic-ceramic and metal-ceramic composites, the ceramic powder is a mixture of different materials which easily segregate due to particle density, particle size, or particle morphology. In these cases, the dispersant is often replaced by a flocculant. The flocculant causes agglomeration of the different particles in the composite formulation, keeping the well-mixed nature of the suspension when it was colloidally stable in the resulting floccs. These floccs give a low packing density to the green body, which can be increased by pressing. These multicomponent suspensions have unique rheological properties which can be Newtonian if very dilute, non-Newtonian if more concentrated, and visco-elastic if very concentrated. The suspension rheology plays a very important role is some green body fabrication methods. In other fabrication methods, the ceramic powder volume fraction plays a more important role. This section deals with the various types of suspension-based green body fabrication methods. These include slip casting, filter pressing, tape casting, sedimentation and centrifugal casting, electrodeposition, and dip coating. Generally, these can be broken down into two categories: those methods using dilute suspensions (e.g., <5% solids) where slip casting, sedimentation, and filter pressing are examples and those using thick suspensions (e.g., >5% solids) where tape casting, centrifugal casting, and dip coating are examples.

13.3.1 Slip Casting With slip casting, a porous mold is used to give shape to the green body. Slip casting can be performed with two options, drain casting or solid casting, as shown in Figure 13.3. In drain casting, the mold is filled with a dilute slip, typically less than 5% solids, and a portion is dewatered by the porous mold giving a cast layer on the mold wall. The excess suspension is drained from the mold, and the cast layer is allowed to dry and shrink away from the mold. In solid casting, a thick slip is poured into the mold, where it completely dewaters. In both drain and solid casting, the mold is filled with a ceramic suspension, and a combination of hydrostatic pressure and capillary suction dewaters the ceramic suspension adjacent to the mold as shown in Figure 13.4. This is analogous to filtration, as noted by Adcock and McDowall [14]. With slip casting there are two resistances to flow: the cake and the mold [15]. Considering only one porous medium, the flow rate, Q,

614

Chapter 13

Ceramic Green Body Formation

(a) Schematic of slip casting fabrication process, consisting of mold assembly slip pouring and casting, mold drainage, trimming the top, mold disassembly, and removing the finished piece. (b) Schematic of solid casting, consisting of mold assembly, mold filling, slip absorption, and finished piece removal from the mold and trimming.

F I G U R E 13.3

t h r o u g h porous media of area, A, and thickness, L, caused by a pressure drop, AP, is given by Darcy's law [5, p. 385]" AP L

- ~?acV~

(13.1)

where ac is the specific cake resistance equal to the reciprocal of the cake permeability, Kc( = 1/ac), ~ is the viscosity of the filtrate, and Vo

13.3 Green Body Formation with Ceramic Suspensions

FIGURE 13.4

615

Schematic diagram of slip casting with a mold.

(= Q / A ) is the superficial velocity of the filtrate. Using a bed of spheres with diameter Dp as a model of a porous media, the pressure drop over a bed of depth L for flow with a superficial velocity, Vo, is given by [16]

AP L-

150~?v0(1- s) e 1.75pv02(1- s) D~ s 3 + Dp s3

(13.2)

where p is the density of the filtrate and s is the void fraction of the porous medium. The first term in this equation is dominant for Reynolds number {NRe = PVoDp/[V(1 - s)]} < 10 or laminar flow, which gives Darcy's law with 150 (1 - s) z ac - Dp2

s3

(13.3)

This equation can be used to determine the specific cake resistance from the average particle size of the cake. Using a bundle of tubes to model the porous medium, the sphere diameter, Dp, in the preceding equation is replaced by Dp -

6

(13.4)

av

where av is the total pore surface area per unit volume of the porous medium. With this substitution, the specific resistance of the porous mold can be determined from the total pore surface area per unit volume. Using the filtration concept with two resistances to flow, Darcy's law becomes ~PT L = V(ac + OLm)VO

(13.5)

where ac is the specific resistance of the consolidated layer or cake and am is the specific resistance of the mold. The specific resistance of the mold is generally larger t h a n that of the consolidated layer of ceramic particles because the mold is thicker and has sufficiently small pores

616

Chapter 13

Ceramic Green Body Formation

that the ceramic powder is filtered by the mold. If the ceramic particles infiltrate the pores of the mold, the mold will become clogged. The total hydrostatic pressure, APT, is given as a sum of the applied pressure, AP, and the suction pressure of the mold as follows: ~ 9 T -- ( ~

(13.6)

-Jr- 2~/sJr m)

where ~/s~is the interfacial surface tension between the pore wall and the solvent with all of its additives, including dispersants, wetting agents, and binders, and rm is the mean pore radius of the mold. An estimate of the mean pore radius for packings of spherical particles of diameter d is given by [17] rm = d/K, where K is a factor depending on the packing a r r a n g e m e n t of the spherical particles (K - 12.9 for hexagonal close packing, and K = 4.8 for simple cubic packing). By performing a differential mass balance at the suspension-cake interface, we find the build-up of cake of mass, me, with the volume of filtrate, V: (13.7)

dmc = p w dV

where w

(c)

= 1 - cm

is the mass of dry cake per unit mass of filtrate,

p is the density of the filtrate, m is the mass ratio of wet cake to dry, and c is the mass fraction of solids in the slurry. This cake build-up is related to the increase in thickness, x, by the following relation: (13.8)

d m c = (1 - S ) p s A d x

where Ps is the density of the solid. Equating the two preceding expressions, we have (1-

s ) dx - pw dV k dV p~ A -A=

k

Q

dt - k vodt

(13.9)

where k ( - ~/~2) is the ratio of the solid volume fraction of the cake to the solid volume fraction of the filtrate. Integrating from 0 to time t and from 0 to thickness L, we have

Vo =

L(1 - ~) kt

(13.10)

Substituting this equation for v0 into Darcy's law for the combined resistance of cake and mold (equation 13.5), we have ~9 T

L

- V(ac + a~)

L(1 - s) k~

(13.11)

13.3 Green Body Formation with Ceramic Suspensions

617

which upon rearrangement gives

L2=[

kAPT ] t = F t

(13.12)

~(~c + ~ m ) ( l - s )

from which the well-known rule of cast layer thickness a~/time is obtained. Typical results showing this result for a 3-mole%-yittria stabilized tetragonal zirconia powder [18] being slip cast is shown in Figure 13.5. A more advanced view of slip casting with two types of cake, one dense and the other dilute, has been presented [19, 20]. But the results are of the same form as the preceding equation with F, the proportionality constant, having a more complicated definition. The packing obtained during slip casting is essentially that of the collapsed slurry. When the dewatering rate is slow, the particles have ample time to diffuse laterally to find a packing location which minimizes the defects in the structure before they are locked into location by the arrival of the next layer of particles. Conversely, when the dewatering rate is fast, the particles have too little time to rearrange themselves before they are fixed into a packing location by the arrival of the next layer. For this reason, the rate of dewatering controls the packing uniformity of the particles in a slip cast piece. This analysis assumes a uniform cake. Nonuniform cakes can be caused by cake clogging and cake compression. Cake clogging can occur

~'~ E E .,,._..

6

r r r

",= I,,--

4

,.be r

o

2

a

ca r

0 0

9

I

2

9

• i

4

9 80%wt

9

I

6

9

i

8

9

i

10

9

12

Time(min)

FIGURE 13.5 Cake thickness versus casting time for yittria stabilized tetragonal zirconia suspension. Data taken from Moreno et al. [18].

618

Chapter 13

Ceramic Green Body Formation

during filtration due to the migration of small particles through the cake. These small particles can be caught in the mold if the pores of the mold are smaller than such particles. Tiller and Chow [21] have given some empirical equations to describe the average cake resistance of a clogged cake. For cake compression, a simple proportionality for the cake resistance, ac, is used [22]: ~c~P s

where the power S is the cake compressibility. The value of S varies from 0 for incompressible cakes to 1 for highly compressible cakes. The proportionality constant is determined largely by the size of the particles forming the cake. As a result of this equation, we can see that the factor F in equation 13.12 becomes a nonlinear function of pressure, which will accentuate the flow inhomogeneities and lead to a nonuniform cake build-up and nonuniform particle packing in the cake. Suspensions of anisotropic particles (either platelets of fibers) behave in different ways during slip casting. Weymouth and Williamson [23] studied kaolin platelet orientation in slip cast earthenware. They used a special peeling technique and a polarized light microscope to investigate particle orientation. They found that kaolin platelets tend to align their basal planes to that of the surface through which the filtration proceeds. When the mold had two mutually perpendicular faces, each face was associated with its own aligned deposit, with a sharp boundary between the two orientations. When viewed under crossed polarizers, the sharp boundary between the two orientations can be easily noticed. This boundary is a plane of weakness in the green body because it leads to anisotropic drying shrinkage stress. According to Williamson [24], the difference in the drying shrinkage is due to the difference in the thickness of water between the face-to-face stacked particles and the edgeto-edge stacked particles at the boundary. This phenomena has the following explanation. Anisotropic particles (both platelets and fibers) arrive at the cast layer with all possible orientations. For any angle which is not perfectly perpendicular to the cast layer, hydrodynamic drag forces on the particle force it to align either with its basal plane, if it is a platelet, or its length with the cast layer, if it is a fiber. The resulting fiber mat will therefore be randomly oriented in two dimensions but not in the third. Williamson has shown that the orientation of mullite needles in slip cast green bodies is the cause for differential shrinkage during firing.

13.3.2 Filter Pressing In pressure filtration, a piston is used with a frit bottom to allow the excess liquid to leave the chamber, catching the ceramic powder

13.3 Green Body Formation with Ceramic Suspensions

FIGURE 13.6

619

Filter press apparatus.

on the frit, as shown in Figure 13.6. The consolidated layer of ceramic powder grows according to a parabolic rate law, derived from Darcy's law. The consolidated layer thickness, L, is given by [25] ~l(C~c + C~m)(1

_

~)

t = Ft

(13.13)

This equation identical to equation 13.12 for slip casting, where am is now the resistance of the frit. Using this equation, a plot of the consolidated layer thickness, L, versus t ~/2 gives a straight line where F is the slope, from which the permeability of the cake of consolidated ceramic powder, ac, and frit, am, can be determined. The permeability decreases as the applied pressure increases, due to an increase in the packing density [26]. Flocculation of the ceramic suspension before filtration gives much higher permeabilities, often by a factor of 10. To account for flocculation during filtration, a two-layer

620

Chapter 13 Ceramic Green Body Formation

cake model is used [27], where ac is replaced by acl + ac2 (where one value is for the consolidated layer and the other is for the flocculated layer adjacent to the suspension). Increasing the applied pressure decreases the permeability and increases the packing density to a much larger degree with flocced suspensions than with stable suspension. To increase the low packing density inherent in the floccs and aggregates and approach the packing density associated with individual particles, pressures in excess of 80 MPa are commonly used. Like slip casting, the packing of filter cakes obtained is essentially that of the collapsed slurry. When the dewatering rate is slow, the particles have ample time to diffuse laterally to find a packing location which minimizes the defects in the structure before they are locked into location by the next layer of arriving particles. Conversely, when the dewatering rate is fast, the particles have too little time to rearrange themselves before they are fixed into a packing location by the arrival of the next layer. For this reason, the rate of filtering controls the packing of the particles in the cast layer. But, unlike slip casting, the high pressures applied in filtration can cause further consolidation of the cast layer, increasing the packing density, especially when the suspension has been flocculated before filter pressing. To the extent possible, the orientation of particles in the cast should be avoided if the final product is to be homogeneous in microstructure and physical properties. This is difficult if the particles have a large aspect ratio and are cast for a well-dispersed dilute slip. This phenomena may, however, be used to advantage to (1) design molds so that the inner flat surfaces do not meet at sharp angles, which gives a weak junction of the two particle orientations at the apex of the angle, and (2) enhance mechanical properties in one direction of the final cast piece, assuming that the piece survives the differential stresses occurring during drying and sintering.

13.3.3 Tape Casting Tape casting is used to produce a green body which consists of a thin layer of a dried ceramic suspension. These green layers can be cut into a near-net shape and sintered to give a useful ceramic object. In addition, these thin ceramic layers can be layered to produce a multilayer structure like that of the multilayer capacitor shown in Figure 13.7 or that of a multilayer integrated circuit package. Electrodes of a particular design are often screen printed on a layer with a metal powder paste. This process is called metalization. If a connection is necessary between layers, the green tape can be punched before metalization. After layering the multilayer structure is cofired to sinter together the metal powder and the ceramic layers into a single piece.

13.3 Green Body Formation with Ceramic Suspensions

FIGURE 13.7

621

Schematic of a multilayer ceramic capacitor.

The fundamentals of tape casting have been studied by several authors for Newtonian [28-30] and non-Newtonian fluids [31-33]. This anlaysis of tape casting comes from a paper by Ring [32]. Tape casting is a common way to produce thick film ceramic green bodies. A tape casting machine (see Figure 13.8) consists of a moving belt onto which a ceramic powder slurry is spread by a stationary doctor blade, generally with a rectangular form. Control of the thickness of the cast is of critical importance in tape casting. This is done by first precisely controlling the solids concentration in the thick suspension. Typical solids loading are between 10 and 50% by volume. To determine the cast thickness, it is necessary to know the velocity profile in the gap between the doctor blade and the moving belt. In an attempt to simplify the geometry of the tape casting machine, the zone between the bottom of the flat tape casting blade and the moving belt surface is isolated for analysis. In this zone, shown in Figure 13.8(b), a momentum balance is performed, giving dzxy

- dp/dx

(13.14)

where rxy is the shear stress and the d p / d x is the pressure difference across the doctor blade from x = 0 to x - W. This momentum balance is done at a steady state far from the entrance and exit zones. (Some doctor blades may not be wide enough for these conditions to apply.) The pressure difference across the doctor blade can be established from the height of fluid in the doctor box dp/dx = -(pgL

+ Papplied)/W

(13.15)

where p is the fluid density, g is the gravitational acceleration constant, L is the height of the fluid behind the doctor blade, and W is the width

622

Chapter 13

Ceramic Green Body Formation

FIGURE 13.8 (a) Tape casting apparatus with (b) detail of the flow of fluid under the doctor blade. Note opposite flow directions in (a) and (b).

of the doctor blade. As given in this equation, an applied pressure, Papplied, is also included in the pressure gradient. Applied pressure in this case is the applied gas pressure in the doctor blade box. For Newtonian fluids the shear stress, rxy, is given by [34] Txy = - ~ dVx/dy

(13.16)

where ~ is the viscosity of the fluid and Vx is the velocity in the x direction. For Bingham plastic fluids, the shear stress, Zxy, is given

13.3 Green Body Formation with Ceramic Suspensions

623

Zxy = Zo - ~ dVx/dy

(13.17)

by [34]

where Zo is the yield stress. For the Cross equation, which is valid for monodisperse suspensions of hard spheres, the shear stress, Zxy, is given by

txy = -~(~/) dVx/dy

(13.18)

V(~)) = ~ § 7o - ~

(13.19)

where ~) = dVx/dy and

19 where 7c is critical shear rate for shear thinning given by the Peclet number, Pet (= 67rrJsaa4/c/ks T) = 8 for hard spheres and ~ and r~0 are the high and low shear viscosities, respectively. Both the low and high shear-limiting viscosities have been shown to increase monotonically with volume fraction, with shear thinning (i.e., nonlinear behavior) detected for ~ > 0.25-0.3 for hard spheres [35]. This data has been best fit to the following expressions for hard sphere interations: ~o = 1 + 2.54~ + 44~2 + 424~a + . . . . ~s

rl____~= 1 + 2.54~ + 44~2 + 254~a + . . . ~

( 1 -~-~ 1 - 0--.-.-~

(13.20)

(13.21)

~s

where ~ is the viscosity of the solvent solution. All of these rheological expressions (equations 13.16, 13.17, and 13.18) can be used to analyze the flow under the doctor blade in tape casting. Using the m o m e n t u m balance equation 13.14 and one of the preceding equations for the shear stress, the differential equation which governs the velocity Vx, can be determined. For Newtonian fluids, the solution is given by U = Y + p(y2 _ y)

(13.22)

where U (= Vx/'(,r) is the dimensionless velocity, Y (= y/h) is the dimensionless y coordinate and P [= h2(-dp/dx)/(2~oV)] is the dimensionless pressure. This equation is identical to that developed and experimentally verified (for the zero applied pressure case) by Chou et al. [29]. This equation is valid for doctor blades with W > h2Vo/rl as developed by Longwell [36]. In the preceding equation, two major terms are responsible for the velocity. One is a result of the velocity of the moving belt, 12",and the other is a result of the pressure gradient dp/dx. A plot

624

Chapter 13

Ceramic Green Body Formation

F I G U R E 13.9

Velocity profile in the tape casting gap for a Newtonian fluid at different dimensionless pressures. For cast thickness versus pressure, see curve A in Figure 13.10(b).

of this velocity distribution is shown in Figure 13.9. When no pressure gradient exists, the velocity profile is linear between the moving belt and the stationary doctor blade. When a pressure gradient exists, the velocity distribution takes on a parabolic shape offset from the center of the gap. Increasing the pressure sharpens the parabola of the velocity distribution. With a Bingham plastic, a similar analysis must be performed, with the added constraint that the velocity gradient is large enough to overcome the Bingham yield stress, ~o, so that flow indeed occurs. For the Bingham plastic, it is useful to isolate two cases for analysis. The first is the case of zero pressure gradient. Under these conditions, we have a simple velocity profile equation to deal with. When there is sufficient velocity to overcome the Bingham yield stress, such as V > roh/V, the solution is identical to the Newtonian flow case; namely, a linear velocity profile between the moving belt and the stationary doctor blade: U = Y

(13.23)

When there is insufficient velocity to overcome the yield stress, such as V < ~oh/V, the velocity distribution consists of two parts: a stagnant zone near the stationary doctor blade and a moving zone near the

13.3 Green Body Formation with Ceramic Suspensions

62~

moving belt, described by* U = O, Y < Yc = yc/h U = Y ' = ( Y - Yc)/(1 - Yc) = (Y - Yc)/(h - Yc),

Y > Yc

(13.24a) (13.24b)

where Yc is the height of the stagnate fluid zone, defined by 0 <-Yc <- h

(13.25a)

0 <- Yc <- 1.0

(13.25b)

Yc = V ~ l z o ,

or in dimensionless form, Yc = 1/r~,

where r~ [= roh/(~?V)] is the dimensionless Bingham yield stress. When a pressure gradient is present, either by hydrostatic or applied pressure, a more complex velocity profile emerges. When there is sufficient velocity to overcome the Bingham yield stress, such as V > roh/~?, the solution is identical to Newtonian flow case shown in Figure 13.9. When there is insufficient velocity to overcome the yield stress, such as I7"< roh/V, the velocity distribution consists of two parts: a stagnant zone near the stationary doctor blade and a moving zone near the moving belt, shown in Figure 13.10(a) and described by, U=0,

Y
Y > Yc

(13.26a) (13.26b)

where Y ' = (y - y c ) / ( h

-Yc)=

(Y-

Yc)/(1 - Yc)

and Yc is given by Yc = [r~ - 1 + P]/(2P),

0 <- Yc <- 1.0

(13.27)

The shape of the velocity distribution is similar to t h a t for Newtonian flow but is truncated to a zone of fluid near the moving belt. Using the Cross equation for suspension rheology, the solution must be obtained by numerical integration of the m o m e n t u m balance equation after substitution for the Crossian rheology of the system. This integration has been done by the author [33] with the results shown in Figure 13.11(a). At zero applied pressure, the velocity profile is identical to t h a t for Newtonian rheology case, using the high shear viscosity from the Crossian rheology. But at higher applied pressure, the Crossian velocity profile is much larger t h a n that of the Newtonian fluid counterpart. This is because the Crossian rheology has a very high viscosity for low shear rates near the m a x i m u m in the velocity * Note: this assumes that the belt is set into motion after the doctor blade gap is filled with fluid. Other solutions will exist with other initial conditions.

626

Chapter 13 Ceramic Green Body Formation

FIGURE 13.1{} (a) Velocity profile in the t a p e casting gap for a B i n g h a m plastic fluid w i t h insufficient belt velocity to overcome the B i n g h a m yield stress: (A) P = 0, (B) P = 1, (C) P = 2, (D) P = 3. Note: a a n d h are not the s a m e for all curves! (b) C a s t t h i c k n e s s as a function of total p r e s s u r e g r a d i e n t for different dimensionless yield stresses: (A) r~ = 0 ( N e w t o n i a n fluid), (B) r~ = 1, (C) r~' = 2, (D) r~ = 3, (E) r~ = 4.

profile. This maximum velocity tends to give a plug of near zero shear rate near the center of the gap. This plug is similar to that obtained with the Bingham plastic but it occurs in the center of the gap and not near the moving belt. Because the object of this work is to describe the thickness of the cast tape far from the doctor blade, it is necessary to calculate the

13.3 Green B o d y F o r m a t i o n w i t h C e r a m i c S u s p e n s i o n s

627

P=4

P=2 U (y)

P=2 P=O

0

b

s

t"

3

y

1

n

._o I0

2' Newt0nian

f. -

0

l

-

e

1

-

2

i

3

,,,

"

,

I

4

"

9

5

Pressure

(a) Velocity profile for flow between moving belt and doctor blade for various dimensionless pressures for Newtonian, '~/'~s = 11.4 and Crossian rheology, ~b = 0.5, ~c = 0.5, m -- 1 for the Cross equation with monosized hard spheres, giving "~o/'~s = 23.4, V:~/~s = 11.4, where Vs is the solvent viscosity. (b) Cast thickness as a function of the dimensionless pressure for Newtonian and Crossian rheology, ~ = 0.5, ~c = 0.5, m = 1 for the Cross equation with monosized hard spheres giving Vo/'~s = 23.4, ~ / ~ s = 11.4, where ~s is the solvent viscosity.

FIGURE

13.11

average velocity of the fluid in the gap. The thickness of the cast tape, 8, is related to this average velocity by f.

1

8 = h I U(Y) d Y Jo

(13.28)

628

Chapter 13 Ceramic Green Body Formation

The results of this analysis are shown in Figure 13.10(b) for Newtonian (curve A) and Bingham plastic rheologies with different yield stresses. The thickness of the cast tape depends on the rheology of the casting fluid, the pressure in the doctor blade box, and the velocity of the moving belt. With Newtonian fluids without external pressure (curve A, r~ - 0), the cast thickness is always one-half the height of the doctor blade from the moving tape. This observation is confirmed by both tape cast operators and researchers [29].* For Newtonian fluids with a pressure graident, the thickness of the cast tape is given by 6/h =

1/2 - P / 6

(13.29)

which is shown in Figure 13.10(b). With Bingham plastic fluid flow, the following relationship is used when the pressure is not equal to 0, 6/h =

(1/2 + P/6)[1 - ( ~ - 1 + P ) / ( 2 P ) ]

(13.30)

and when the pressure is 0, 6/h =

(1 - 1/r~)/2

(13.31)

As the pressure gradient is increased, the cast thickness increases for all Bingham yield stresses. At a constant pressure, the cast thickness decreases as the yield stress increases. The thickness results for Crossian rheology are shown in Figure 13.11(b) and are compared to those for Newtonian rheology with the viscosity given by the high shear viscosity in the Crossian rheology. In this case, the Crossian result gives much thicker tapes t h a n the Newtonian case, due to this low shear rate plug at the center of the velocity profile. When the pressure is small and fluctuating, an unstable condition results because two equations are valid, equations 13.30 and 13.31 for Bingham plastic fluids with much different dependencies of 6 / h on P. As pressure decreases to 0, we jump from a tape cast thickness defined by equation 13.30 to one defined by equation 13.31. The results of equation 13.31 are shown as letters on the y axis at zero pressure on Figure 13.10(b). These values are discontinuous from the curves of 6 / h versus P for all cases except when r~ is 0. This problem haunts the double doctor blade design, with one doctor blade following the other [37], sometimes used in industry (see Figure 13.8(a)). In addition, variation in r~ can result from subtle changes in the colloid chemistry of the ceramic suspension (e.g., loss of solvent) or variation in the gap height. For the Crossian rheology the steep cast thickness versus pressure function shown in Figure 13.11(b) will lead to thickness variations for small pressure variations or for variations in the volume fraction because the powder volume fraction dictates the Crossian viscosity. * Mistler, R. E., Keramos Industries, Inc., Private communication.

13.3 Green Body Formation with Ceramic Suspensions

629

For these reasons, operation of a nearly dry second doctor blade may lead to tape cast thickness fluctuations. Much more stable operation can be obtained from operating with only one doctor blade with a full doctor box and control of tape speed and applied pressure. For Bingham fluids, these thickness fluctuations will self-level to a height, AS', equivalent to the height of fluid supported by the yield stress or h 6 ' ~ ro/(pg)

(13.32)

where h8 is the thickness of the ripple, p is the density of the fluid, and g is the acceleration due to gravity. Newtonian and Crossian fluids will self-level to a flat surface given sufficient time without evaporation. The packing and orientation of the particles in tape casting is highly influenced by shear. Consider for a moment the orientation of particles in the gap. With a linear velocity profile, nonspherical particles will orient in the direction of the shear gradient, OVx/Oy. This orientation is responsible for differential x - y shrinkage upon sintering,* which is a problem when the cast layers are stacked together as in multilayer capacitors and multilayer silicon chip substrates. Particle orientation, the quantitative aspects of which are discussed in Section 13.3.6.1, can be minimized with a parabolic velocity profile because near the maximum velocity, the velocity gradient is small, leading to a small orientation force on the particles in this zone. Parabolic velocity profiles, caused by the pressure head in the doctor box, minimize differential x - y shrinkage during sintering.

13.3.4 Sedimentation Casting and Centrifugal Casting In sediment casting, a suspension of ceramic powder is placed in a vessel and allowed to settle until a cake is formed. The supernatant is decanted and the wet green body allowed to dry. Depending on the colloid stability, different types of settling behavior will be observed, as shown in Figure 13.12. When the particles are well dispersed, the settling is slow and persistent, giving a high density cake that resists redispersion. The supernatant is cloudy for dilute free settling suspensions but will be clear for high volume fractions when hindered settling takes place. For flocculating suspensions, the particles flocculate and settle faster into a loosely packed cake, which is easily redispersed. For flocculated suspensions, the supernatant is clear. * Sanchez has shown no differential x - y shrinkage during sintering for tape casts of monodisperse spherical silica powders [38].

630

Chapter 13 Ceramic Green Body Formation

FIGURE 13.12 Schematicofthe settling ofcolloidal stable and flocculated suspensions.

The terminal settling velocity, Vt, for a single sphere with radius, R, was discussed in Section 11.3 and is given by

Vt = 2R2[ps - pf]g 9~?

(13.33a)

which is valid when the Reynolds number, NRe (= pfYt2R/Ti) less than 1.0 and when the size of the particle is large compared to the size of the solvent molecules. In the preceding equation, ~ is the solution viscosity, Ps is the solid and pf is the fluid densities, respectively, and g is the gravitational constant. For turbulent flow, the terminal settling velocity, Vt, is given by

Yt

_/8R(pspf )g 3fpf

(13.33b)

where the friction factor, f, must be determined from the Reynolds number using Figure 11.2.

13.3 Green Body Formation with Ceramic Suspensions

631

The combined effects of nonspherical shape and hindered settling can be approximated by Vt = 2R~[ps - pf]gG(p)h(4)) 97

(13.34)

where h(~) is the hindered settling factor, which is a function of ~, the volume fraction of the particles in the fluid, and G(p) is the particle asymmetry function. For an ellipsoid of revolution with a major semiaxis, a, and a minor semi-axis, b, the axial ratio p is b/a. For prolate ellipsoids where p < 1, the asymmetry function G(p) is G(p)=p2/3( 1 _ p2)_1/2 in [1 + ( 1 P_ p2)1/2] ;

p < 1,Ro = ab 2

(13.35)

and for oblate ellipsoids (plates), where p > 1, it is G(p) = (p2 _ 1)1/2p2/3 arc tan[(p 2 - 1)1/2];

p> 1,R0=a2b

(13.36)

Figure 11.3 is a plot of G(p) as a function of p for both prolate and oblate ellipsoids. For simplicity and reasonable accuracy over all ranges of the volume fraction, the factor h(~b) is given by Happel and Brenner [39] as h(~) = 3 + 4.5(~ 5/3 - ~1/3) _ 3~2 3 + 2~b5/3

(13.37)

At low volume fraction the value of h((b) is 1, and at high volume fraction the value of h(~) is much less than 1. Equation 13.34 is only a first-order approximation to the terminal settling velocity and is good for the lower volume fractions because the particles will orient themselves, allowing higher volume fractions. For centrifugal casting, the same process occurs in a centrifugal field (as shown Figure 13.13) to speed up the cake formation. The gravitational acceleration, g, of the Stokes's law expression has been replaced by the radial acceleration, co2Rc, in a centrifugal field. In a centrifugal field, the terminal settling velocity, Vt, depends on the radial position, Re, in the centrifuge because the centrifugal force depends on radial position Vt-

dR c 2R2[ps -- pf](.o2Rc dt 9n

where oJ (= 2~r rpm) is the angular velocity of rotation.

(13.38)

632

Chapter 13

F I G U R E 13.13

Ceramic Green Body Formation

Centrifugal casting of an annular green body shape.

13.3.4.1 Effect of P a r t i c l e Size D i s t r i b u t i o n To illustrate the effect of multiple particles sizes on settling, consider the mixture of particles (p = 3 gm/cc) shown in the particle size histogram in Figure 13.14. In this figure the percentage mass of each particle size is given in addition to the time needed to settle 10 cm, assuming unhindered settling in a Newtonian fluid of 1.0 cm poise viscosity with a density of 1 gm/cc. If a well-mixed suspension is added to a 10 cm cylinder, the situation after 1.2 days of settling is as shown in Figure 13.14. Here we see that the 5 ftm particles have completely settled to the bottom. Most of the 4 ftm particles have settled to the bottom and less of the 3 ftm and the 2 ftm particles have settled to the bottom. Nearly none of the 1 ftm particles have settled to the bottom. As a

13.3 Green Body Formation with Ceramic Suspensions

633

FIGURE 13.14 (a) Graph of a ;mixture of five fractions of spherical particles having radii of 1, 2, 3, and 5 t~m.~Particle density 3.0 gm/cc and liquid is water. The time for complete settling a distance~flO cm is also given. (b) Theoretical settling conditions at the end of 1.2 days for theseparticles. Data for graphs obtained from Section 13.3.4.

result there is a large amount of segregation of the particles in the sediment cake. For a given particles size distribution by weight, f(d), the total weight, W, settled to the bottom at time t is made up of two fractions,

634

Chapter 13

Ceramic Green Body Formation

one part greater than and the other less than a given diameter dl"

W=

f~ dl

1 t x2(ps - Pf)gl f(x)dx+f:f(x)-~[ i8-~ j

dx

(13.39)

The term in the square brackets in this integral is the terminal settling velocity for a dilute suspension of spheres. The size, d~, is the size which would just fall the full distance h in time t and is given by 18h~

d~ = (Ps- pf)gt

(13.40)

Therefore, at any time, the cake is composed of particles to some degree segregated by size, as shown in Figure 13.15. The largest particles fall first and are mixed with some smaller particles, and the smaller particles fall last. The principle shown in Figure 13.15 is also used for the determination of particles size distributions. This same analysis can be used for centrifugal casting where the weight, W, of cast at time t is made up of two fractions, one of particles greater than and the other less than a given size d~:

ix2 s, 21 18 In

\Rc~]

The term in the square brackets in this integral is the reciprocal of the time for a particle of size x to settle from two radial positions, Rcl to Re2, in a centrifuge. The size, dl, is the size which would just fall the full distance from Re1 to Re2 in time t and is given by

dl

=

~/18 In ~Rc2~

\Re:! v

~] ~p:-_ pf)o~2 t

(13.42)

Therefore, at any time, the cake is again composed of particles to some degree segregated by size. The largest particles fall first and are mixed with some smaller particles; the smaller particles fall last. Segregation of particles in these types of casting has disastrous effects during sintering. The fine particle zone of the ceramic green body will sinter at a lower temperature than the large particle zone. When the green body dries and sinters, this segregation will lead to differential shrinkage which warps or cracks the piece. For this reason, segregation must be prevented during sedimentation and centrifugation casting by utilizing nearly monodisperse powders or flocculating the suspension before casting. Chang et al. [40] have shown that colloi-

13.3 Green Body Formation with Ceramic Suspensions cz r,,4 A

100%

100 "~ +10% ll~m

6~

at 29 day~ v

'o +20% ,i.. 4)

9 21~m 8O

'

(n o< c-

so ~ +40%

.0

3gin

:

i

o m

L_ 0

9-, .C O't

'~

; : ,

40 " /

: :

i

9

2pro

3p.m

~' 4p.m r 9 +20H "~ 20

t

4p.m 51~m

E 0

10%

:

i,

::

'

:

i

5~m

0

:

o

. ~

i

~.

~~///

i

i

~

S

--

ettli

n

.

g

.

~

Time

.-J"

.

days)

,

....

lO

Time to completely settle a part cu ar s ze b

s

~=a o~ S S

sTangent

S

o o a=a

l:

W(particles(< r)

W(particles(> r)

== ~9 r

"-= 1

-I

Settling Time, t tr, Time for all particles of size r to settle out ~3

(a) Percent of sediment deposited versus sett|ing time. Settling curve (solid line) and tangent projections (clotted lines) give the percentages of the panicle mixture settled. (b) Graphical derivation os the equation relating weight fractions to cumulative weight settled.

FIGURE 13.15

636

Chapter 13

Ceramic Green Body Formation

dal suspensions of A1203 and A1203/ZrO2 composite suspension can be centrifugally cast without segregation at high salt concentration which coagulates the suspension. The body forces on the particles in the centrifugal field cause particle rearrangement, giving a high green density. Flocculation with polymers usually gives rise to low green densities because the floccs are low density packing of the particles. Higher density can then be achieved by pressing the cast piece and breaking the floccs or aggregates down to individual particles that pack to a higher density.

13.3.4.2 Effect of Anisotropic Particles As the sediment cake builds up, anisotropic particles arrive, the largest one first, with all possible orientations. If the particles have sufficient time to fall to their lowest energy position (i.e., platelets parallel to cake surface) before the next layer arrives, then an oriented layer will be deposited. Thus, the orientation of the layer can be controlled by the concentration of particles in the colloidally stable slip and the sedimentation or centrifugation rate. Orientation and size segregation led to inhomogeneities in the green body, which lead to differential shrinkage in both drying and sintering.

13.3.5 Electrodeposition Electrodeposition was introduced into ceramics processing to make green bodies more quickly. During sedimentation casting and slip casting of very fine particles, long time periods are required to build up a deposit. An electric field is used to enhance the deposition rate of ceramic particles in the cake. More recently, this technique has been used to produce unusual green body shapes or ceramic coatings for irregularly shaped electrodes. Electrode coatings are particularly useful in the manufacture of electroceramics because the electrode is an integral part of the device, as shown in Figure 13.16. The electrode can be used to form one end of the electric field during deposition to direct the particles to only its surface. This particle movement is similar to the electrophoretic velocity or streaming potential described in Section 9.4. The particle velocity, v, due to an electric field, E, is given by [41] v=

2~Srs0EC 37

(13.43)

where ~ is the zeta potential, V is the solution viscosity, and C is a constant that depends [42] on KR, where K is the reciprocal double layer thickness (see equation 9.56 for the definition), and R is the particle radius. For small values of KR < 1.0, C is ~ (i.e., the Huckel

13.3 Green Body Formation with Ceramic Suspensions

637

FIGURE 13.16 Schematic diagram of the electrodeposition of a negative charged ceramic powder onto the cathode of an electrolyte deposition cell.

equation). For large values of KR > 1000, C is 1.0 (i.e., the HelmholtzSmoluchowski equation). At low zeta potentials, ~ < 25 mV, the values of C as a function of KR are given by the Henry equation [42]. For higher zeta potentials, the values of C are given as a function of both the zeta potential and KR by Wiersma et al. [43]. Early work on this deposition process are discussed by Overbeek [44] and Hill et al. [45]. More recent work is given by Brown and Salt [46], Nass et al. [47], Clasen [48], and Furono et al. [49]. The deposition of aqueous dispersions of polymers onto dielectrics is discussed by Tikhonov et al. [50]. Deposition from nonaqueous solvents, frequently used for nitride and carbide ceramics, is discussed by Kolesov et al. [51] and Petrov [52]. Malov et al. [53] applied a correction to the deposition equations to take account of the nonuniformity of the electric field in the neighborhood of the deposit. Hein et al. [54] have developed an improved electrophoretic technique for the deposition of polycrystalline YBa2Cu3OT_x (and Bi-Sr-Ca-Cu-O) layers 10-20 t~m thick on arbitrarily

638

Chapter 13

Ceramic Green Body Formation

shaped silver substrates. Applying a magnetic field of 8 Tesla during the electrodeposition process, a high degree of c-axis orientation parallel to the magnetic field was observed with X-ray diffraction, giving low values of the surface resistance at 77 K at 251.5 GHz, which makes these deposits good for microwave cavities and hydrogen maser applications and presumably improved superconducting values of T c and Jc for the orientated electrophoretic deposits.

13.3.6 Dip Coating In dip coating, a nonporous substrate is dipped into a dilute ceramic suspension and removed at a constant velocity. Neglecting surface tension, Deryagin and Levi [55] derived the momentum balance for a section of liquid near the moving substrate, as shown in Figure 13.17. This momentum balance is given by Oz xy _ p g Oy

sin a

(13.44)

where Z xy is the shear stress acting on the fluid which is flowing with a velocity u ( y ) at location y along the thickness of the film, p is the solution density, and a is the dipping angle. Using the Newtonian

h-

FIGURE 13.17

Schematic of dip coating with nomenclature.

13.3 Green Body Formation with Ceramic Suspensions

rheology this momentum balance can be rewritten as ~2u(y) pg sin a = - ~ 02y

639

(13.45)

where ~ is the suspension viscosity. Using the following boundary conditions,

du u = U aty = 0 and ~yy = 0 aty = h

(13.46)

the equation can be integrated to give the velocity profile:

u(y, h) = U + pg sin______~(h a - y)2 _ Pg sin______~(ah ) 2 2~ 2~

(13.47)

This equation is valid for any lateral section of the film. The uptake of fluid is given by /-

h

Q(h) = lo u(y, h) dy

(13.48)

and is a constant at steady state. If the force of gravity were 0, the velocity would be U for the complete thickness of the film. But, with gravity, the fluid away from the substrate is slowed down. This results in an uptake which is dependent on the thickness, h, of the film. As the thickness increases, the uptake increases from 0, goes through a maximum, decreases back to 0, and then becomes negative as the film thickness increases. The film thickness, hmax, corresponding to the maximum uptake, Q m a x - f:max u(y, hmax) dy, is given by ~ hmax= ~ p g U~no~

(13.49)

and the coated thickness far from the suspension-air interface is given by h u ( y , h)

h~ = f0 ~

dy

_

Qmax

h~axU

U~7

pg ~m o~

-- ~

pg U~ sin

-_2~hma x

(13.50)

This h~ ~ V U l a w is commonly observed in the dip coating of Newtonian fluids at all dipping speeds and angles [56]. For other rheological behavior, the same methodology can be used. Deryagin and Levi [55] provide solutions for the coating thickness for Bingham plastic and power law fluids. In addition, their book solves coating problems for cylindrical substrates (i.e., wires and pipe walls). Also their book provides solutions for coating problems in which the surface tension plays its role as another force acting on the a i r -

640

Chapter 13 Ceramic Green Body Formation

suspension interface. Surface tension forces are important only when the layer thickness is very thin. For Crossian rheology, the momentum balance can be integrated to determine the velocity profile [33], which is given by

~qO4/c pg cos (y2/2 - hy) u (y , h ) = U + - ~ y + 2~q~ 1 - -

2~

[F(y)

-

F(y

=

0)]

(13.51)

where

F(y)

(2cy + b ) V X + 1 sinh_ 1 [ (2cy + b) ] 4c 2k kffcc [~/--~ac-- b 2

X = a + by + cy 2,

4c k = 4ac - b 2

(13.52a) (13.52b)

with c = (pg cos a) 2, b = pg cos a [2~O~c + 4W~c], and a = (pgh cos a)2 - pgh cos a (2~?O~c + 4W~c) + (W)c) 2. This solution gives a velocity profile that is affected less by gravity than the Newtonian velocity profile because the low shear viscosity is very high. To determine the thickness, we must first determine the uptake, Q(h) = foh u(y, h) dy, which is a constant at steady state. This results in an uptake which is dependent on the thickness, h, of the film. The uptake again increases as the film thickness increases from 0, then goes through a maximum, and decreases through 0 to negative values as the film thickness increases. The film thickness, hmax, corresponding to the maximum uptake, Qmax[- fohmaxu(y, hmax) dy], is obtained by finding the maximum of Q(h) using the derivative dQ(h)/dh and setting it equal to 0. This gives the value of hmax. The coated thickness far from the suspension-air interface, h~, is given by

h~ =

f] u(y, h) Qmax _/ U~qs ~ dy - hmax U ~] pg sin a

(13.53)

This new factor, Qmax/hmaxU, is highly dependent on the values of ~o, V~, and ~c for the Crossian rheology. This new factor is plotted as a function of volume fraction for monodisperse hard spheres in Figure 13.18. At zero volume fraction, a Newtonian solution is observed with the viscosity of the solvent giving the value for the factor ofw Increasing the volume fraction, increases the viscosity and the shear thinning behavior, which results in a thicker coating. Because dip coating is a rather low shear process, we can approximate the increase in coating thickness by the Newtonian relation for the thickness but using the low shear viscosity. This approximation has been used to obtain Figure

13.3 Green Body Formation with Ceramic Suspensions

641

100

/

3

Facto

r (~

)---

/

Z

1

m

l

m

1

J

O

J

J

J ~

O.

6

Factor [ = Qmax/(hmax/U)] versus volume fraction ~ for dip coating a shear thinning suspension of monodisperse hard spheres. This plot is good for steady state dip coating and not at start-up or shut-down and for ~/max = X/Psg(sin a)U/Vs < ~c, defined by the critical translational Peclet number, Pet = 61r~sa3~c/KsT = 8. Under these conditions, factor QmaxlhmaxU= 213 ~/'~ol'~s = 213 (1 + ~/0.63)-1.

F I G U R E 13.18

13.18. The Crossian rheology, when compared to the Newtonian rheology using the low shear viscosity, has a larger thickness and the same dependence on substrate velocity (i.e., h~ ~ ~/-U. To date, this theory has not been verified experimentally. 13.3.{}.1 P a r t i c l e O r i e n t a t i o n i n S h e a r Dip coating experiments with anisotropic platelet particles were performed by Albers-Werk and Ring [57]. The results of one of these dip coated layers is shown in Figure 13.19. Here we see that all of the platelet particles are aligned in the plane of the substrate surface. The orientation of anisotropic particles during dip coating can be analyzed by considering the rotational diffusion of these particles in shear. Rotational diffusion in shear flow has been reviewed by Van de Ven [58]. The ratio of the shear rate, ~, to the rotational diffusion coefficient, O, defines the rotational Peclet n u m b e r ( P e r = ) / O ) . When the rotational Peclet number is small (i.e., near zero), the anisotropic particles are randomly oriented by diffusion. When the rotational Peclet number is large, the particles rotate but have a preferential orientation aligned with the shear. The period of rotation is given by

where a / b is the axial ratio of the particles. Perrin [59,60] has put forth a hydrodynamic model of the rotational motion of different shaped particles. For an ellipsoid of revolution with

642

Chapter 13 Ceramic Green Body Formation

FIGURE 13.19 SEM micrograph showing a horizontal view of the dip coated layer of mica particles. Photo courtesy of M. Albers and T. A. Ring, LTP-DMX-EPFL, Lausanne, Switzerland. semi-axis a, b, b (i.e., oblate ellipsoid plate), Perrin has shown that the rotational diffusion coefficient is given by

3kBT [(2 - p2)G(p)p2/3 - 1 ] (1 - p4) 0 = 167rVsb 3

(13.55)

where p is the axial ratio, p = a/b, Vs is the solution viscosity, kB is the Boltzmann constant, and T is the absolute temperature. G(p) is an a s y m m e t r y function (see Figure 11.3). For oblate ellipsoids (plates) with semi-axes, a, b, b, the axial ratio p is defined as a/b where p > 1, it is

G(p) = ( p 2 _ 1)-1/2 tan-l[(p2 _ 1)1/2];

p > 1

(13.56)

For our case the average (3 t~m • 3 t~m • 0,1 t~m) platelet particle has a rotational diffusion coefficient of 5.3 x 10 .7 sec -1. The shear rate for dip coating has been analyzed by Deryagin [61]. The shear rate is greatest at the level of the liquid bath and varies from a maximum value to 0. The m a x i m u m value of the shear rate is given by

/psg(sin a)U

(13.57)

where Ps is the solution density, Vs is the solution viscosity, g the acceleration due to gravity, a is the dipping angle, and U is the dip velocity. For the dip velocity of 0.18 cm/sec and the aqueous suspension of mica

13.4 Extrusion and Injection Molding of Ceramic Pastes

643

used in these experiments (i.e., Ps - 1.5 gm/cc and Vs = 0.01 poise), the shear rate is 163 sec -~. Calculating the rotational Peclet number, we obtain 3 x l0 s, which is clearly in the oriented particle regime (i.e., greater than 1.0), in agreement with these experiments. It appears that particle consolidation by dewatering and drying does not disturb the particle orientation induced during dip coating. This rotational Peclet number analysis can also be used to determine if particles will orient during other casting methods, such as tape casting. This analysis is valid for only dilute suspensions where the rotation and orientation of a particle does not affect its neighbors.

13.4 E X T R U S I O N AND OF CERAMIC PASTES

INJECTION

MOLDING

Extrusion and injection molding are typically used in polymer processing. They are also used in ceramic green body manufacture. In extrusion, a ceramic paste is continuously pushed through a die in the shape of a cylindrical drainage tile or a honey comb catalyst shape. Some extruded shapes are shown in Figure 13.20. An infinitely long extrudate can be produced, as shown in Figure 13.21. Because of the high shear rates imposed upon this viscous mass of ceramic paste, temperatures high enough to soften or melt the polymeric binder are

FIGURE 13.20

Extruded ceramic shapes.

644

Chapter 13

FIGURE 13.21

Ceramic Green Body Formation

Schematic of extrusion of ceramic drainage tiles.

often obtained. Upon leaving the die, the paste is no longer under shear and it solidifies into the desired shape. Before it completely solidifies, the extrudate is cut periodically to give green bodies of the same length. With injection molding, the extrudate flows into a mold. When the mold is filled, the mold is either removed from the extruder and another empty mold is put in its place or the extruder is stopped. In some cases, an extrudable and injectable paste may consist of 65% vol. ceramic powder and 35% vol. polymeric binder. In others, an extrudable paste may consist of a highly loaded aqueous suspension of clay particles such that its rheology is plastic. The low shear (i.e., <100 sec -1) viscosity of such a paste is between 2000 and 5000 poise at ambient temperature. Highly nonlinear stress strain curves are typical of ceramic pastes, as well as time dependent thixotropy. In many cases, pastes behave like visco-elastic fluids. This complex rheological behavior of ceramic pastes has made theoretical approaches to these problems difficult. For this reason, the discussion in this chapter is limited to Newtonian fluids where analytical solutions are possible, with obvious consequences as to accuracy of these equations for nonNewtonian ceramic pastes. This section discusses the flow of paste through an extruder, through a die, and in a mold in various detail allowing the reader to comprehend the basics of these green body fabrication methods.

13.4.1 F l o w in the E x t r u d e r An extruder can be of the piston type or screw type. The flow in the piston type can be modeled by the flow in a die, discussed in Section 13.4.2, with an applied pressure. A screw extruder (Figure 13.22(a)) consists of an Archimedian screw that revolves inside a close fitting

13.4 Extrusion and Injection Molding of Ceramic Pastes

FIGURE 13.22

645

(a) Schematic of an extruder for ceramics. (b) Detail of the screw action on the ceramic mix. (c) Velocity due to drag and pressure with the net flow. Ideas for this composite figure were taken from Mutsuddy [62].

646

Chapter 13

Ceramic Green Body Formation

heated cylinder. The paste is fed through a hopper at one end and is transported to the die at the other. In a single screw extruder, the flow of a viscous material (Figure 13.22(c)) can be seen as being composed of (1) drag flow induced by the moving surface, (2) the pressure flow induced by the axial pressure gradient, (AP/AL) in the extruder, and (3) the leakage flow induced by the clearance, H C, between the outside diameter of the screw and the barrel. For the isothermal flow of a Newtonian fluid, the total flow (Figure 13.22(c)) can be presented as [62] Q=

(TrD~)2NH sin 0 cos 0 7rDbH3 sin 2 0 (AP/AL) 2 12vL

(13.58)

(TrDb)2EH3 tan 0 (AP/AL) 12vL where Db is the diameter of the barrel, N is the rotational speed of the screw, 0 is the pitch of the screw (typically 17.66~ ~ is the viscosity of the paste, E is a factor to allow for eccentricity of the screw, and H is the difference in the diameter of the barrel and the screw. It is worth noting that this equation is an oversimplification of the flow of a nonNewtonian ceramic paste. More complicated mathematical models are available [63] for non-Newtonian fluids. These solutions are the result of numerical solutions to the momentum balance equation for this complex geometry. For each particular, suspension, and operating condition another velocity profile is obtained. We do not discuss these results in this chapter. 13.4.2 Flow in the Extrusion

Die

In the extrusion die, the flow is driven simply by the pressure drop, as shown in Figure 13.23 and Figure 13.24. This type of flow can be analyzed relatively easily, even for complicated rheological behavior. We will look at two geometries in detail: flow between two flat plates and flow in the annular space between two cylinders. Both these geometries are used commonly to make floor tiles, cylindrical pipe pieces, and by cutting in half, semicylindrical roof tiles.

FIGURE 13.23 Flow of ceramic paste between two flat plates under the influence of a pressure gradient. Both the shear stress, Txz, and velocity, Vz, profiles are given.

13.4 Extrusion and Injection Molding of Ceramic Pastes

647

Flow of ceramic paste in the annular space between two cylinders under the influence of a pressure gradient. Both the shear stress, rxz, and velocity, Vz, profiles are given.

F I G U R E 13.24

13.4.2.1

Flow

between

Two Flat Plates

The m o m e n t u m balance for the flow between two fiat plates shown in Figure 13.23 is given by rxz

dx

-

AP

AL

(13.59)

Upon integration, the s h e a r stress is given by AP rxz = --A-LX + c

(13.60)

where c is a constant which m u s t be evaluated from the b o u n d a r y conditions rxz -

0

Vz = 0

at the midplane at x = 0 at the wall at x = ___8/2

which allows the d e t e r m i n a t i o n of the constant, c, which results in the following expression for the s h e a r stress: AP rxz = -A-LX

(13.61)

Using the rheology of the ceramic system, we can write rxz in t e r m s of the velocity g r a d i e n t d v z / d x . Then, by integration, the velocity profile can be obtained. These velocity profiles for Newtonian, B i n g h a m plastic, and Crossian rheology follow. hP dvz Newtonian rxz = - 7 dx

AL

Vz = - ~

[(8/2)2 - x 2]

(13.62)

648

C h a p t e r 13

B i n g h a m plastic rxz = ro - ~

Ceramic Green Body Formation

de z

(13.63)

Vz = 0 i f A ~ 8 / 2 < ro

ifh~- 8/2 > %. There is a plug in the center with a 2T 0

AP constant velocity at x > Xo = h-~ AP AL This velocity is Vz = W-[(8/2) 2 - X2o] for all other x values there is a parabolic flow n e a r walls:

the

AP AL 2] ro Vz = W'- [(8/2) 2 - x + - - [ x - 8/2] z~

Crossian rx~ = -~?

-~-

AP ~?OYc AL vz = ,_-z--_+ W-" [x2/2 - y(8/2)] ,-al~

-

(13.64)

- -

2~?~

[F(x)

-

F(x

=

0)]

where F ( x ) = (2cx + b) V X

+

4c

Z - - a + bx + cx 2

(AP~ 2 with c = \ ~ - ~ ] ,

(2cx + b) 1 2 k V~c sinh-' L Y r e -

k =

4c ~ 4ac - b 2

AP b = ~--~ [2V0yc + 4 ~ y c ] ,

AP )2 AP ~-~8/2 -~-~8/212~O~c + 4~?~c] + ( ~ c ) 2

anda =

These velocity profiles are used to determine two key factors: the average flow rate and the shear stress at the wall. The average velocity, /)~, is obtained using the following integral: 812

i)z = 2/8 (

Vz(X) dx

(13.65)

ao

The shear stress at the wall is obtained from AP rxz - ~-~ 8/2

(13.66)

13.4 Extrusion and Injection Molding of Ceramic Pastes

649

The shear stress at the wall is used to determine the thickness of the h a r d e n e d metal die necessary to w i t h s t a n d the forces acting on it.

13.4.2.2 Flow in the Annular Space b e t w e e n Two C o n c e n t r i c Cylinders For the other die geometry, t h a t of a a n n u l a r space between two cylinders shown in Figure 13.24, we have the following m o m e n t u m balance:

d (r'rrz) _

AP

--r

(13.67)

r/2 + c/r

(13.68)

dr

AL

This m a y be integrated to give AP

Trz = - ~

where c is a constant. The constant cannot be determined because we have no information on ~'rz at either of the fixed surfaces r = ~ or r = R. The most we can say is t h a t there is a m a x i m u m velocity corresponding to rrz -- 0 at r -- kR which allows c to be evaluated as follows" APR

(13.69)

[ ( r / R ) - ~2 ( R / r ) ]

~, is now our constant of integration. Using the rheology of the ceramic system, we can write rrz in terms of the velocity gradient d v z / d r . Then, by integration, the velocity profile can be obtained with the boundary conditions vz= Oatr vz = 0

= ~R

at r = R

These velocity profiles for Newtonian and B i n g h a m plastic rheology follow. The velocity profile for Crossian rheology has not yet been developed for this geometry. AP R2 dvz Newtonian rrz = - ~ dr

Bingham plastic rrz = ro - ~

Vz = dvz

Vz -

AL

4~

[

1 - (r/R

.~AP R (

01~-~

)2

l-K2 + 2 ln(1/K)

1 - K2

ln(r/R)] (13.70)

1 - 21n-(1~i! < 70

(13.71)

For both geometries, the velocity profiles are schematically plotted in Figures 13.23 and 13.24 along with the shear stress profiles. With these velocity profiles, we see t h a t large gradients of the velocity will play a significant role in orienting aniostropic particles, as was dis-

650

Chapter 13

Ceramic Green Body Formation

u.I v

_z 3: (n .J

1

<_ IZ

ntr u_ M.

outside radius

(L

l

1

w

2

l

3

i

4

l

8

I

6

EXTRUSION BLANK RADIUS

I

7

1

I

9

10

(inches)

FIGURE 13.25 Differential shrinkage during sintering of porcelain hollow pipes of various sizes made by extrusion. In the ceramic powder mixture used for porcelain are kaolin platelet particles which orient in the shear of extrusion. From Funk [64]. Reprinted by permission of the American Ceramic Society.

cussed in the section on dip coating. The effects of platelet particle orientation during extrusion of porcelain on the differential shrinkage during sintering of a hollow cylinder has been measured by Funk [64] (see Figure 13.25). The differential shrinkage is lowest at the outside radius where the highest velocity gradient is predicted by the preceding equations, both Newtonian and Bingham plastic. This gives the largest rotational Peclet number which predicts the largest particle orientation. When platelet particles are oriented, they can be packed to a higher density than when they are randomly oriented, resulting in lower differential shrinkage upon sintering. At smaller radii, where the maximum velocity is predicted, the differential shrinkage is also a maximum. This is consistent with the idea that this region has a velocity gradient near 0, giving a rotational Peclet number near 0 and a random orientation of platelet particles. At the inside radius, we also have a high velocity gradient which orients the particles and lowers the differential shrinkage. For thin walled pipes, the region of near-zero velocity gradient is very thin and as a result little random orientation of

13.4 Extrusion and Injection Molding of Ceramic Pastes

651

particles is predicted with a resulting lower maximum in the differential shrinkage. Once the paste flow has left the extrusion die, there is a relaxation of stress. The yield stress of the paste maintains the final extruded form of the green body. Depending on the rheology of the paste and the pressure used for extrusion, the relaxation of the extrudate may increase or decrease the thickness of the green body in comparison to that of the gap in the die. At this point the green body is cut to length. Cutting usually takes place with a wire or knife (as shown in Figure 13.21) passing quickly through the extrudate at the surface of the extruder as extrusion takes place.

13.4.3 Flow into the Injection Molding Die Newtonian (and non-Newtonian) flow into a die is a complicated process due to the free surface of the fluid where the boundary condition of the shear stress, being 0, is defined and this free surface position changes with time. This boundary condition is used with the momentum balance equation to determine the velocity profile in the mold at any

FIGURE 13.26 Numerical solution to the flow of a Newtonian fluid in a mold with an obstacle, viscosity is 10 poise. Taken from Funk [64, p. 188]. Reprinted by permission of the American Ceramic Society.

652

Chapter 13 Ceramic Green Body Formation

FIGURE 13.27 Microstructure of an A1203 whisker (NiA1) metal matrix composite fabricated by polymer injection molding. This fiber orientation was established by proper gate placement and design. This micrograph was provided by David Alman of RPI. Figure taken from German, and Hens [67]. Reprinted by permission of the American Ceramic Society.

given time. This situation is a classic moving boundary problem for differential equations, which is notoriously difficult to solve. As a result, this process can be modeled only by numerical methods [65]. An example of a Newtonian fluid filling a sample mold is given [66] in Figure 13.26. Here we see that the fluid enters from the left from a wide inlet. This fluid has a free surface as seen in b. As the fluid encounters the obstacle in the die it bulges around it as seen in d and e. The zones where there is a high concentration of points is the point where a high pressure exists. High-pressure regions are potential high-stress areas for the mold and potential high powder packing density for the green body. This information can be used to better design molds and eliminate packing inhomogeneities in the green pieces produced by injection molding. Once the die is filled, the flow is stopped and the die is removed from the extruder. Molds for injection molding are segmented and as a result can be disassembled so that the green body can be removed. Anisotropic particles will tend to orient in the shear flow as the die is filled. A1203 whiskers have been oriented in the injection molding of

13.5 Green Body Formation with Dry Powders--Dry Pressing

653

the metal (NiA1) matrix alumina whisker composite shown in Figure 13.27 [67]. This orientation gives this composite enhanced mechanical properties in the direction of the whisker alignment, which is an important advantage. This advantage was obtained by careful design of the mold and its points of injection.

13.5 G R E E N B O D Y F O R M A T I O N WITH D R Y POWDERS--DR Y PRESSING There are two types of dry pressing: isostatic and die pressing. In both types of pressing, a spray dried powder is usually but not solely used. Both types of dry pressing are shown in Figure 13.2. In isostatic pressing, the spray dried powder is allowed to flow into a rubber mold. Then the material is vibrated or tapped to give uniform particle packing density distribution. After tapping, the mold is sealed and then placed into an fluid bath. The pressing action takes place when the fluid is placed under hydrostatic pressure. With isostatic pressing, the pressure is evenly distributed over all the surface of the green body inside the rubber mold. After a relatively short period of time at pressure, the pressure is released and the mold removed from the oil and the green body, now much smaller than that of the mold cavity, is released from the mold cavity. Isostatic pressed green bodies can be of any geometry because the only limitation is that of shaping the rubber mold which is quite easy. In die pressing, the spray dried powder is allowed to flow into a hardened steel mold. Then the materials are pressed by a die. This pressing action can be either uniaxial or biaxial, depending on the type of press. After a short period of time under pressure, the pressing action is decreased and the press is used to eject the green body from the press. With novel die and press designs, die pressed shapes can be rather complex. Large production runs are almost always made with die pressing and not isostatic pressing. In addition, there is a hybrid form of pressing called ramming. In this case, a ceramic powder (often spray dried) is placed in a thick walled rubber mold and a steel mandrel is rammed into the mold, forcing the powder into a gap between the steel mandrel and the now deformed rubber mold, as shown in Figure 13.28. The pressure acting on the powder particles is a result of the elasticity of the rubber mold. To have a uniform pressure acting on the ceramic powder, the wall thickness of the mold must be designed appropriately. Ramming is often used for the fabrication of crucible shaped ceramic objects, as shown in Figure 13.28. Ramming, die pressing, and isostatic pressing have several aspects

654

Chapter 13

FIGURE 13.28

Ceramic Green Body Formation

Schematic diagram of ramming of ceramic powders.

which are similar, and for that reason, they are treated together in this chapter. Two aspects are different. One is the tapping or vibration step used in isostatic pressing, and the other is the type of force used in pressing. The isostatic force is a uniform pressure on the surface of the green body in the mold but die pressing is either an uniaxial or biaxial force on green body. Ramming is a combination of both types of force, as you might expect. As a result, we will treat all types of dry pressing in the same way and point out these differences as we go along. We will start with a discussion of tapping or vibration. Powder compaction has been the subject of many review articles [68,69]. But since the field is still developing, the author has found that the journal literature was most useful in writing this section.

13.5.1 TappedDensity Tapping provides a method to (1) increase the packing density and (2) eliminate inhomogeneities in the packing, which is by far the more important consideration. If the mold is not filled homogeneously, the final pressed shape will be very different from that of the mold. In addition, the tap density of a powder is one of the most frequent methods of powder characterization. The tap density reveals differences in powders that are not predicted by particle size measurements. In addition to the tap density, the rate at which the density changes with tapping is another characteristic of a powder.

13.5 Green Body Formation with Dry Powders--Dry Pressing

655

A paper by Takahashi and Suzuki [70] gives an empirical mathematical model for tapping kinetics. The density increases as the number of taps, N, increases because unfilled holes in the powder compact are gradually filled. The number of taps is releated to the time by the frequency of taps, oJ, as follows" N = (ot. The rate of density increase is given by d(p - Po) = _ ( p _ po)k dlnN

(13.72)

where k is the rate constant and Po is the density at N = 0. After integration from Po to p at N and noting that at N = ~, p = p~, we find that (p - p0) = e x p ( - k / N )

(p~- po)

(13.73)

This equation can also be written in terms of a volume compaction V* = (Vo - V)/(Vo - V ~ ) w i t h the kinetics dV* = kV* d(1/N)

(13.74)

V* = e x p ( - k / N )

(13.75)

giving

In any real powder, there will be a range of void sizes filled at different rates. This means that the overall volume compaction will be a sum of all the rates for the different sizes of holes. Converting to a continuous distribution of hole sizes with a continuous distribution of rate constants, f ( k ) for these holes we can write ~c

V* = fo f ( k ) e x p ( - k / N )

(13.76)

where f o f ( k ) = 1. Takahashi and Suzuki assumed that a Weibull distribution could be used for f ( k ) as follows: f ( k ) = b m k m-~ e x p ( - b k m)

(13.77)

With experimental data for V* as a function of 1/N, they were able to determine by inverse Laplace transform, the function f ( k ) and the values of m for the Weibul distribution for Kibushi clay (m = 1), TiO2 (m = 1), and SiO2 (m = 0.5). Such studies require that each powder have a reproducible initial packing, which can be achieved by fluidization with air. If the tap frequency is too high, consolidation will not take place so frequencies of i Hz are often used with an amplitude of tap of i cm. For these conditions, it is often found that p~ is achieved in 1000 to 2000 sec.

656

Chapter 13 Ceramic Green Body Formation

13.5.2 D i e P r e s s i n g With die pressing, a powder after flowing into a mold is subjected to high pressure as is shown in Figure 13.2. The uniaxial die compaction of a sprayed dried A1203 is shown in Figure 13.29(a). When this powder is compacted, it first reorganizes into a close packed structure. At higher pressure, it undergoes plastic deformation of the agglomerates to further increase the packing density. At higher pressure still, the remaining pores are filled by comminution products. This series of structural reorganizations is schematically shown in Figure 13.30, which can be followed by the change in the pore volume distribution of the A1203 powder compact [71], as shown in Figure 13.29(b). During the reorganization period, the flow of agglomerated particles over one another is controlled by the Coulomb yield criterion described for the agglomerates. The equations that are often used for this Coulomb yield criterion are those for cohesionless dry solids (i.e., Tij ---- Tii tan 6 + c, where c = 0). During the deformation of the agglomerates, the flow inside an agglomerate is controlled by the Coulomb yield criterion for the individual grains that make up the agglomerate. The equations often used for this Coulomb yield criterion are those for powders with cohesion because agglomerates are held together by the polymeric binder used in spray drying. These two types of deformation are associated with two different sizes of pores, as is shown in Figure 13.31 which decrease at different pressures. When the localized forces on the particles are sufficiently large, particle comminution starts as in Stage III of Figure 13.29(a). The pressures required for stage II compaction depend on the choice of polymeric binder system used. The hardness of a binder system is related to its mechanical properties. These mechanical properties of a polymer depend on the molecular weight of the polymer, its molecular weight distribution, and its chemistry. One important property is the glass transition temperature of the binder system. Above the glass transition temperature for the binder system, the polymer flows easily. Below the glass transition temperature, the polymer is brittle. The glass transition temperature of a binder system can be lowered by lower molecular weight additives, which are called plasticizers. In the case of polyvinyl alcohol (PVA) used as a binder for A1203, DiMilia and Reed [71] have shown that water adsorbed from the atmosphere acts as a plasticizer and lowers the glass transition temperature to below room temperature. This lowering of the glass transition temperature gives a lower apparent yield point for compaction as is shown in Figure 13.32. Polyethylene glycol also lowers the glass transition temperature of PVA and acts as a plasticizer. The density of a powder after pressing has been the subject of much empirical work. Some 15 different empirical equations [72] have been

0.6 Stage I

Stage II

> <(

Stage III

::><

92%RH

0.5 A

r 0.4 0 r Q.

E

0 0

0.3

(

0.2

=,.,=,~_.=..

IP y

Vibrated Density

~f 9

9

.. . . . . .

19

01

I

.

Punch b

.

.

.

.

.

1

.

.

I

.

.

.

.

.

.

10

.

.

I

.

.

.

.

.

.

.

.

100

1000

Pressure(MPa)

0.8 Filled Die 10 MPa

--

70 MPa

0.6 Intragr

0.4

I

P 0.2

0.0

9

01

9

.

.

.

.

.

.

I

.1

9

9

9

9

. . . .

!

1

"

9

9

I ' ' ' = F

10

9

,

,

w , , , . ~ .

100

9

9

9

,

, , , ,

1000

Pore Size (pm) FIGURE 13.29 (a) Compaction of spray-dried alumina at 92% relative humidity. (b) Cumulative pore size distributions by Hg porosimetry for spray dried A1203 (Alcoa A-17-1) for different pressing pressures. Data taken from Reed [6, p. 337].

658

Chapter 13 Ceramic Green Body Formation

FIGURE 13.3{} Schematic behavior of a spray dried powder during die pressing.

developed. The pressed density, p, usually follows a simple function of the applied pressure, P, over a limited range of pressure [6, p. 337]: P p = p~ = m I n m

By

(13.78)

where m is the compaction constant, Py is the apparent yield pressure of the granules, and p~ is the ultimate density. For glass spheres, there is only one ultimate density, that of the densest packing of the particles. For a spray dried powder with its agglomerates, there are two ultimate densities, one for the granules and the other for the primary particles

13.5 Green Body Formation with Dry Powders--Dry Pressing W

"

w

'

,

"

,

I

i

i

-

659

!

o 9 2 % R H - V 11 915% RH-V 11 o

o 0.3 E

"

-V 1

9 "

-V 1

3

0 0.2

8 7

0.0

'

' 10 6

'

10 7

10 8

A P P L I E D P R E S S U R E (Pa)

FIGURE 13.31 Pore volume for compaction of Alcoa A-17 Al203 with 2.1% PVA binder at different humidities: VI, intergranular and, Vll, intragranular. Data taken from DiMilia and Reed [71]. Reprinted by permission of the American Ceramic Society.

9 PVA 2.7%wt 9 2.1%wt --'g- 1.0%wt A

40 n_ :E

I= o n_ "o m 4) ).. .i.

I= 0s... t o. o. ,4

0

20

40 Relative

60 Humidity

80

100

(%)

FIGURE 13.32 Apparent yield point, Py, versus relative humidity used to equilibrate the PVA binder used in the die pressing of Al203 powder. Data taken from Reed [6, p. 337].

66{}

Chapter 13 Ceramic Green Body Formation

well packed together. For these two u l t i m a t e flow densities, there are two types of compaction flow: 1. I n t r a g r a n u l a r flow 2. I n t e r g r a n u l a r flow. For two s p r a y dried ceramic powders, the u l t i m a t e densities of the a g g l o m e r a t e s and the aggregates, as well as the a p p a r e n t yield p r e s s u r e are given in Table 13.1. The K a w a k i t a compaction e q u a t i o n is a n o t h e r e q u a t i o n which is often used for ceramic powder pressing. It can be derived by considering t h a t compaction is similar to packing by tapping, where the compaction pressure, P, is directly s u b s t i t u t e d for the n u m b e r of taps, N, in the a n a l y s i s in Section 13.5.1. The K a w a k i t a equation is a special case, where the value of m in the Weibul distribution function for t a p p i n g is 1. In the K a w a k i t a equation, the compaction, C, defined as the relative reduction in volume is given by [72] c=

V~

V= I

Po _

V0

p

abP

(13.79)

(1 + bP)

where a a n d b are constants. R e a r r a n g i n g , this equation can be linearized to d e t e r m i n e a a n d b from e x p e r i m e n t a l d a t a as follows: P

1

P

C

ab

a

(13.80)

The c o n s t a n t s a and b have significance which can be d e t e r m i n e d by comparison with the C o o p e r - E a t o n equation [73]. b = 1/Py is the a p p a r e n t yield p r e s s u r e a = p~ is the u l t i m a t e density

TABLE 13.1 Apparent Yield Pressures and Associated Densities for Compaction of Different Ceramic Powders Ceramic powder

aggregates

Py

Py primary

particles

p~

A1203

21.4 MPa

344.8MPa

0.5

MgO SiO2 CaCO3 Si

16.5 MPa 16.3 MPa 9.8 MPa -MPa

337.9MPa 367.3MPa 285.7MPa 20 MPa

0.65 0.6 0.68 0.52

p~ primary

aggregates particles 0.85 0.90 0.85 ~1.0 ~1.0

m in equation 13.78

4.2-6.7 depends on R.H.a 1,b 1,~ 1,b 0.21,c no lubricant

DiMilia, R. A., Reed, J. S., Adv. Ceram. 9 (1984) b Cooper, A. R., and Eaton, L. E., J. Am. Ceram. Soc. 45(3), 97-101 (1962). c Haggerty, J. S., Flint, J. H., Garvey, G. J., Lihrmann, J.-M., and Ritter, J. E., Proc. Int. Symp. Ceram. Mater. Compon. Eng., Lubeck-Travenmunde, FRG, April 1986. a

13.5 Green Body Formation with Dry Powders--Dry Pressing

661

These equations show the steady state density achieved for a particular pressure. The relations suggest that inhomogeneities in the stress distribution within a die will lead to inhomogeneities in the packing density of the ceramic green body.

13.5.3 Stress D i s t r i b u t i o n in the Ceramic C o m p a c t Let us assume that we are considering a cylindrical green body being die pressed, as is shown in Figure 13.33. Under load the particles in a volume element will densify if the Coulomb yield criterion Tij -- 7"ii

tan 6 + c

(13.81)

has been exceeded. In this equation, rij is the shear stress at which the powder begins to deform plastically under a normal stress rii, 6 is the angle of friction, and c is the cohesion stress, which is a measure of the stickiness of the particles. In a ceramic green body being pressed, the powder will continue to deform until the force is sufficient for movement (i.e., the Coulomb yield criterion is exceeded.) To determine the normal stress distribution in the green body, a force balance is performed on the generalized volume element shown in Figure 13.33. This z component of the equilibrium force balance with

FIGURE 13.33

Geometry, normal stress, and shear stress in uniaxial die pressing.

662

Chapter 13 Ceramic Green Body Formation

negligible body force is given by O~zz_ 1 O(r ~rz) Oz r Or

(13.82)

where compressive stress like ~zz is taken to be positive. (For other geometries, this force balance equation is given by h . ~ = 0, see the appendix. As Schwartz and Weinstein [74] point out, "rrz(r

=

0 , z ) -- 0

(13.83)

for axial symmetry. Thus O%z/OZ = 0 at r = 0, which upon integration gives %z(r = 0, z) = constant. If equation 13.82 is integrated from 0 to R, we obtain Rrrz(r = R, z) = fo~ r O%z Oz dr

( 13.84 )

where rrz(r -- R, z) is the wall shear stress. The wall shear stress is given by rrz(r=R,z) = tan 6w rzz(r = R , z)

(13.85)

where 6w is the angle for wall friction. This shows that Coulombs friction relation is also valid at the wall. Using the coefficient of pressure at rest, Xo, we can relate r r r ( r = R, z) _ %z(r = R , z ) - Xo

(13.86)

at the wall, r = R. (See the definition of ~o in Section 12.5.1.) Using these definitions, we find ~,0tan 6w

R r

= R , z) = fo~ r O~zz Oz dr

(13.87)

Therefore the normal stress distribution, r z), can be obtained from the solution to this integral equation if the axial load for the surfaces of the die is known. Thompson [75] has assumed that a parabolic radial distribution of axial pressure on the top and bottom punch faces of the cylindrical die, in accord with Unkel's experimental measurements [76]. In addition, Thompson has assumed that this parabolic distribution is valid for the length of the cylindrical die as well, giving Tzz(r, z) - r2f(z) + c(z) = r2f(z) + C

(13.88)

where c(z) is the compaction pressure on the central axis of the compact, which we have shown to be constant, C. Using this distribution and

13.5 Green B o d y F o r m a t i o n with Dry P o w d e r s - - D r y Pressing

663

equation 13.87, we find

R2f(z) + C =

R3

df(z)

(13.89)

4hotan 6w dz

which has the solution f(z) = B exp(4hotan 6w

z/R) + C/R 2

(13.90)

giving for the normal stress

z/R) + C[1 - (r/R) 2] (13.91)

9zz(r, z) = B r 2 exp(4hotan 6w and for the shear stress rrz(r,z)

=

Bhor3tan 6w exp(4h ~ tan 6w R

z/R)

(13.92)

Both the radial and the normal stress distributions are given in terms of two constants, B and C. Boundary conditions are used to evaluate these constants. These are as follows: 1. The integral of the compaction pressure at the punch face is equal to the punch force, measured for the experiment: R

Fo = 7rR2Po = fo 21rr~zz(r'z = zo) dr

(13.93)

where Fo is the punch force, Po is the average compaction pressure at the punch face, and Zo is the axial position of the punch face when the powder is fully compacted. This boundary condition gives 1 B = ~-~ (2P o - C)exp(-4 ~,o tan 6w

zo/R)

(13.94)

This result allows the evaluation of rzz(r, z) and Trz(r , Z) in terms of C only. 2. The density distribution in a compact is such that it has a constant mass regardless of the extent to which it is pressed. This amounts to the continuity equation for pressing. Prior to compaction, the die was filled to a level, zi, with a powder of uniform density, Pi, giving the mass, m ~'R2ziPi . After compaction is complete, the same mass can be calculated from the density distribution p(r, z) as follows: =

R

m = fo f[~

(13.95)

Using a compaction equation valid for the material of interest, t~ -- tOi =

g(r

(13.96)

664

Chapter 13

Ceramic Green Body Formation

the value of C may be determined because rzz is written in terms of only this one unknown parameter. (The equation is related to the effective yield locus of the powder at different densities.) This value of C is dependent on the length and diameter of the die and the initial density of the powder and its compaction properties. The value of C differs depending on g ( r z z ) . Thompson used a simple function for g ( r z z ) = T(~-zz) l/n, giving C as C = [pi(zi/z o -

(13.97)

1)/T] n at the centerline

Using equation 13.95, the value of C which maintains a constant value for the powder mass, m, is obtained. This series of equations allows the determination of the shear and normal stress distribution in the die and also, due to the relationship, P - Pi = g ( r z z ) , the green density distribution in the mold after compaction. Figure 13.34 shows Thompson's results for a cylindrical die with

a

z

b z

t_.., L _ _

I

0

_

I

0.2

J

I

0.4

I

0.6

I

0.8

I

1.0

I

I

0

0.2

I

I

0.4 0.6 dR

I

I

0.8

1.0

dR F I G U R E 13.34 Density distribution calculated for a cylindrical die pressing Cu powder: (a) single action press, (b) double action press. Conditions: ~0 = 0.5 and tan 8 - 0.3, L/D = 1.0. Contour lines correspond to a constant density (gm/cc) as indicated on line. From Thompson [75]. Reprinted by permission of the American Ceramic Society.

13.5 Green Body Formation with Dry Powders--Dry Pressing

665

its length to diameter ratio of 1.0. These results are similar to those of Schwartz and Weinstein [74]. For uniaxial pressing with a single plunger, these results show that the highest pressures (and therefore the highest densities) occur at the top periphery of the cylinder forming a ring. The lowest densities occur at the bottom of the cylinder, also forming a ring. These calculated density distributions are close to those measured experimentally [77,78], as shown in Figure 13.35 [79,80]. Errors are a result of the elastic and plastic nature of the ceramic particles themselves. Several other authors [81,82] have calculated the pressure distribution in cylindrical dies and other forms using finite element numerical methods. Bortzmeyer [81] has incorporated cohesion, elastic, and plastic deformation of the particles into finite element calculations for more complicated geometries, as shown in

m m m m m

m m m m ~

m m m m m

m m m m m

m m m m m

m m m m m

m m m m m

E E E E E

m m m m m m m E

.- '<' hX

"6 -1-

,ot' \ \

/N

j,-

2<.2

Radial d i s t a n c e in m m FIGURE 13.35 (A) Packing density measured by a quantimet on rectangular sections from a transfer section of a powder pressed in a die from both sides. Data from Afanasjev et al. [79]. (b) Density distribution in a sample pressed from one side. Density has been inferred by microhardness measurements. Data taken from Torre et al. [80].

666

Chapter 13

Ceramic Green Body Formation

(a) Density distribution as measured by a quantimet in the region of an undercut in a die pressing Cu powder. Data taken from Perelman and Roman [78, p. 43]. (b) Density distribution predicted by finite element methods. Taken from Bortzmeyer [82].

FIGURE 13.36

Figure 13.36. These numerical methods also give results t h a t are similar to experiments. Other approaches use the continuity equation and equation of.motion developed for fluid flow (see C h a p t e r 12) with a constitutive equation for the powder mass. The constitutive equation for powder flow is a problem t h a t has no solution at this time. Several simple formulas in t e r m s of tensor invariants and deviation tensors [83]

13.5 Green Body Formation with Dry Powders--Dry Pressing

667

have been proposed in soil mechanics. Nagao [84-86] has attempted to derive the constitutive equation in general terms by considering the (1) geometrical relation between porosity and coordination number in random packing and (2) deformation in terms of the particles sliding against themselves and cohesive forces between particles. With a constitutive equation for a dry powder, the time dependence of the deformation that results from the pressing action can be predicted in the same way that the velocity profile can be predicted in the flow of a fluid. For a schematic view of the transient deformation with applied pressure, we have to know about the visco-elastic properties of the compact.

13.5.4 Deformation of Visco-elastic Solids a n d Fluids For visco-elastic solids, the equations for the effective viscosity, dynamic viscosity, and the creep compliance are given in Table 12.4. For a die undergoing compressive loading and unloading, the deformation given by these equations is schematically shown in Figure 13.37. Here we see elastic deformation gives complete rebound and visco-elastic deformation gives partial rebound. Both types of behavior are observed with ceramics. With ceramic pastes the visco-elastic fluid behavior is typically observed. Dry ceramic powders being pressed under high pressure behave more like visco-elastic solids. One of the most important parameters in industrial die pressing is the punch speed. Increasing the punch speed usually decreases the green density because the binder behaves viscously and cannot deform fast enough to allow particle rearrangement when the punch speed is too high. In addition, if the punch is too fast, the air trapped in the powder will lack sufficient time to escape, producing a large pore pressure which prevents compaction. Another parameter of interest is the loading sequence. Industrially monotonic increasing load followed by monotonic decreasing load is used. It has been shown that higher green density can be achieved by cyclic loading and unloading. In fact, a cyclic 40 MPa load can be as effective as a monotonic load of 600 MPa. This idea is sometimes used when industrial equipment applies a second pressure on the die or ultrasonic waves are used during pressing.

13.5.5 Die Ejection a n d Breakage After compaction, the green body must be ejected from the die. During die ejection the green body is released from compression at high pressure to very low pressure and then pushed out of the mold. The primary force to be overcome during die ejection is that of wall friction.

668

Chapter 13

Ceramic Green Body Formation

Compressive Load

r .O m

-

r

E k.

o

~

14,q)

a

~

Rebound

Removed Time

Visco-elastic Deformation

Compressive Load

to r

Rebound

E I,..

qO. , G)

r

Removed Time

Elastic Deformation

FIGURE 13.37

Pressing cycle for visco-elastic solids.

Wall friction depends on the roughness of the die wall and its lubrication. For walls with a scale of roughness larger than the diameter of the powder, the wall friction is essentially the same as the internal friction of the powder. This is because the rough wall traps small amounts of the powder in its reguosity and the sliding of the powder mass against this trapped powder produces the effective wall friction. If the wall has a scale of roughness smaller than the diameter of the powder, the wall friction is controlled by particle-wall friction. In both cases, the wall friction can be decreased by lubricants applied to the wall prior to pressing. The thickness of the lubrication layer plays an important role in reducing the wall friction. If the wall roughness is smaller than the diameter of the particles, the thickness of the lubrication layer need only exceed the diameter of the particles to form a lubricated powder layer against the wall and thus decrease the wall

13.5 Green Body Formation with Dry Powders--Dry Pressing

669

friction. When the wall roughness is larger than the diameter of the particles, the thickness of the lubrication layer needs to exceed the wall roughness to decrease the wall friction. Under these conditions there is a powder-powder slip plane which is well lubricated giving the angle for wall friction 6w -< 6, the angle on internal friction for the powder. This effect has been verified experimentally by Strijbos [87] with the die pressing of 0.04 t~m ferric oxide powders. Stearic acid lubricant applied to the hardened steel die wall with a 0.22 t~m roughness decreased the wall friction angle from 32 ~to 14.7 ~ Other lubricants include oleic acid, paraffin, wax, metal stearates, naphthenic acid, graphite, and talc. The wall friction is most troublesome when the part starts to clear the die. Here radial springback puts the ejected section of the green body into tension as seen in Figure 13.38, where it is at its weakest. This springback can be several percent in the axial direction. As a consequence, there is a great difference in the pressure density curve inside and outside the mold. These movements are not homogeneous among the particles. All macroscopic isotropic deformation occurs through complex particle rearrangement. Particle morphology has a great influence on springback. Particles with a high aspect ratio or highly agglomerated particles are likely to store a large amount of bending energy during pressing, which is released during springback. Common defects observed in die pressing are shown in Figure 13.39. These include lamination, end cap delamination, and ring cap delamination. The reason for these problems is the nonuniform density distribution caused by pressing. Releasing the stress causes tension on the

FIGURE 13.38 Die ejection of a green body.

670

Chapter 13

Ceramic Green Body Formation

V

Lamination FIGURE 13.39

End Cap

Ring Cap

Commondefects in die pressed compacts.

green body, and the zones of lowest density are prone to cracking. Bortzmeyer [82] has noted that this problem can be overcome if, during ejection, the green body is maintained in compression by pressure applied by the upper punch as shown in Figure 13.40. Other reasons for breakage during ejection include internal pressure due to air included in the sample during pressing and friction stress on the mold during unloading and ejection caused by powder stickiness. The internal air pressure can be eliminated by deairing the powder before compaction. The use of a die lubricant is necessary to decrease mold friction, as was discussed earlier. Powder stickiness can be reduced by drying the powder before compaction.

FIGURE 13.40

Maintainingthe green body in compression during die ejection.

13.5 Green Body Formation with Dry Powders--Dry Pressing

671

13.5.6 Isostatic Pressing There are two major types of isostatic pressing. One is called wet bag and the other is called dry bag. With wet bag isostatic pressing, a rubber mold is filled with a ceramic powder, tapping is used to homogenize the packing density, and the mold is vacuum sealed. The sealed mold is placed in an oil (or water) reservoir. The reservoir is sealed and the pressure is increased. Depending upon the equipment used, the maximum pressure ranges from 35 to 1400 MPa with common production units operating at 400 MPa or less [88]. With isostatic pressing, the rubber mold transfers the pressure from the oil to the surface of the green body uniformly. The pressure is released gradually and the rubber mold expands away from the surface of the now compacted green body making removal from the mold easy. With dry bag isostatic pressing, the mold has two types of surfaces, metal surfaces onto which the powder is compressed and rubber surfaces which separate the oil (or water) reservoir from the ceramic powder. Figure 13.41 is a schematic design for dry bag isostatic pressing of an automobile spark plug insulator. Here the pressure is transferred from the oil to the rubber to the ceramic powder, which is compacted onto the metal conductor which runs down the center of the spark plug. The method of ramming shown in Figure 13.28 is also a method ofisostatic pressing, where the elasticity of the mold provides the pressure for compaction.

FIGURE 13.41

Schematic of dry bag isostatic mold for an automotive spark plug. Radial normal stress is used to compress the powder against the metal conductor placed at the central axis of the mold.

672

Chapter 13 Ceramic Green Body Formation

In all these isostatic pressing methods, the pressure is applied uniformly to the surface of the green body because the rubber mold deforms to follow the compaction of the powder. In addition there is little or no wall friction between the powder and the rubber mold. As a result, the force balance given by h . r = 0 for cylindrical coordinates gives, for the radial component, 0 =

10(rrrr______~)+ 10(TrO.~_______ ) r Or

r O0

TOO } O(Trz) r Oz

(13.98)

At a particular point, this force balance shows that the radial normal stress, rrr , or applied pressure is related to the angular normal stress, too, and the two radial shear stresses, rrz and fro. Under load, the particles in a volume element will densify, if the Coulombs yield criterion rij = Tii tan 6 + c has been exceeded. Therefore, we find that TrO --~ Trr

tan 6 + c and Trz : Trr tan 6 + c

(13.99)

where a is the angle of internal friction for the powder and c is the powder cohesion. Because the radial normal stress, rrr, at any radius in the isostatic pressed piece, is not a function of 0 or z but a function of only the radius, we find that 7"00

0 = 10(r'rrr) rOr

(13.100)

r

or

O(rT"rr) Or

-

too

(13.101)

in which the radial normal stress is compensated by the angular normal stress at all radii. Upon integration, we have f O(rT"rr) = f 7"00Or

(13.102)

or

Trr

if

r

Too Or + c l / r

(13.103)

where c~ is the constant of integration. This constant must be 0, because at r = 0 this equation predicts that the radial normal stress goes to infinity, which is unrealistic. Because r00 is nearly a constant, we find that Trr : TOO

(13.104)

This constant radial normal stress is also a constant for all radii and therefore results in amazingly uniform packing density of isostatic

13.5 Green Body Formation with Dry Powders--Dry Pressing

673

Values are density in g m / c c

1.91

1.91

1.51

I.

93 ~_!.~ 1.95 I.g3 1.93 a

I.

1.80 b

FIGURE 13.42

Comparison of the uniformity of green density of a thin walled crucible made by (a) isostatic pressing and (b) uniaxial die pressing. The numbers indicated are the densities (gm/cc) of the various zones. Data taken from Gill and Bryne [90].

pressed pieces compared to uniaxial die pressed pieces as is shown in Figure 13.42.

13.5.7 Green Machining Many green pieces made by slip casting, uniaxial die pressing, or isostatic pressing are used as "rough stock." That means they are further shaped by machining techniques before subsequent processing. In the green state, ceramics are easily cut with cutting tools and ground with grinding wheels typically used in the machining of metals because the forces holding the particles together in the green body are weak, being essentially those of the polymeric binder. The surface finish of the machined green piece will have a roughness on the scale of the particles used in fabrication of the ceramic piece. If spray dried powders were used, then the surface roughness could be determined by the agglomerate size or the particle size depending upon whether pressing pressures used in green body fabrication were higher than the agglomerate yield stress, Py, resulting in deformation and internal compaction of the aggregates. For smoother machined surfaces, the binder should be easily deformable, which means that machining should take place above the glass transition temperature of the polymeric binder system. Because machining friction is dissipated in heat, the polymer binder is often above its glass transition temperature during machining. In some cases, the entire green body is heated to above the glass transition temperature of the binder before machining to make it easier to machine. Green machining is much easier than machining after sintering, where it is used to achieve the final dimensional tolerances of the final piece. After sintering, the ceramic is very hard and difficult to machine, as well as very brittle and easy to break.

674

Chapter 13

Ceramic Green Body Formation

13.6 G R E E N B OD Y C H A R A C T E R I Z A T I O N The characterization of the ceramic green body after its fabrication is an important step if problems arise either during fabrication or during subsequent processing steps. These problems are typically warping and cracking of the green body. At this point in the processing, the green body is either still wet, if prepared from suspensions or pastes, or it contains binder, if prepared by dry pressing. At this stage green body characterization is concerned with the u n i f o r m i t y of packing density, particle size distribution, bulk chemistry, binder distribution, and anisotropic particle orientation. Nonuniformity of these green body properties leads to warping and cracking during drying, binder burnout and sintering. To understand the cause of subsequent warping and cracking, characterization of the green body uniformity is absolutely necessary, albeit not performed until after these drying or sintering problems are encountered. To determine the uniformity of the green body, it is cut into sections. Each section is analyzed for the density, bulk chemistry, and percent of binder. The surface of each section can also be examined under the microscope to see if there are any packing flaws and check for particle alignment. The type of microscope used depends on the size of the particles used in the manufacture of the ceramic piece. The scanning electron microscope (SEM) is often used for ceramic green bodies made from ceramic powders which are submicron. The optical microscope is also commonly used for more traditional ceramics, which used ground natural minerals that have larger particle sizes. Particle alignment can also be measured by X-ray peak intensity ratios [89]. The pore size distribution is measured by mercury penetration porosimetry, a technique which combines information about particle size, particle size distribution, packing density, particle alignment, and binder content. As a result, this method can be used to quickly screen the various possible reasons for the lack of homogeneity in the green body which would require many other experimental methods (e.g., particle size and its distribution, density, particle alignment) to isolate. With the green body properties for the various sections of the green body, a map of these properties can be made to visualize the zones of the green body where large variances in the properties in the green body exist in close proximity to one another. Such maps for the density in uniaxial and isostatically pressed green bodies are shown in Figures 13.35, 13.36, and 13.42 [90]. To assess the uniformity of the microstructure in a green body, the variance, O'g2b is used[91]. n

(r~b(x~) = _1 ~ (x~, i - 2~) 2 ni=l

(13.105)

13.7 Summary

675

where xl,i are the measured values of any property (e.g., the density, mean particle size, mean pore size, permeability in one direction, polymer concentration, fraction of one constituent, hardness) at various locations in the green body, n is the total number of samples measured, 21 is the average value of the same property. For an ideally uniform green body, (r~b(x1) = 0. But this is rarely the case because the technique used to measure the property experimentally also incurs errors re2 flected in (rexp(Xl), which alters the total variance measured: 2 ~t2ot(Xl) = ~exp(X~) + (r~b(x1)

(13.106)

When the experimental variance, O"exp z (X 1 ), is larger than the variance in the property, (r~(x~), it is impossible to determine whether stochastic homogeneity exists in the green body. For this case, only an effective homogeneity can be determined. The experimental variance is established by repeated measurements of the property for a single sample. With these methods of characterization, the reasons for problems of warping and cracking at later stages of processing can be established.

13.7 S U M M A R Y In this chapter we have discussed the various methods of green body synthesis from ceramic powders in suspension, paste, and dry powder form. With suspension processing, we have discussed particle filtration in slip casting, drain casting, and filter pressing. With paste processing we have used the momentum balance to determine the velocity profile in the molding operations of tape casting, extrusion, and injection molding. With dry pressing, we have used a force balance and the Coulomb yield criterion to establish when compaction will occur. With all these methods, we have been concerned with the uniformity of packing density, particle segregation, and particle alignment. The characterization of the resulting green body has been discussed with respect to experimental methods which can be used to determine its uniformity. The uniformity of powder packing density, particle size distribution, bulk chemistry, particle alignment, and binder distribution are necessary if the green body is to survive the subsequent processing steps of drying, binder burnout, and finally sintering.

Problems 1. For 10 gms of spherical T120 powder with a geometric mean size by mass of 1.5 tLm and a geometric standard deviation of 1.8 in 1 liter of aqueous suspension at pH 6, determine the particle size segregation that occurs if the suspension is allowed to settle under

676

Chapter 13

Ceramic Green Body Formation

normal gravity in a cylinder with a cross-sectional area of 10 cm 2. Compare the mean particle size on the bottom and top surfaces of the settled cake. Also determine the time required to settle 99% of the powder. Assume settling under laminar flow. Data density of T120 = 9.52 gm/cc, IEP(T120) = pH 8.9. .

.

.

.

For the preceding tallium oxide suspension, determine the time required to filter press the same volume of suspension. The filtration area is 10 cm 2. Assume that the porous plate used for filtration has a negligible permeability compared to that of the powder. How thick is the filter cake? Using a suspension of hard spheres immersed in water at a volume fraction of 45% determine the cast thickness if the suspension is tape cast with a doctor blade 10 cm long and 1 cm wide onto a moving piece of mylar which is 0.2 cm from the doctor blade. The mylar is moving with a velocity of 5 cm/sec and the height of the suspension in the doctor box is 5 cm. No gas pressure is applied to the doctor box. A Bingham fluid has a yield stress of 1 kPa and a viscosity of 1 Pa.sec. It is used in extrusion with a die which forms a green body in the form of a pipe. The pressure used for extrusion is 100 kPa and the length of the die is 5 cm. Determine the velocity profile for a Bingham fluid during extrusion and the wall shear stress on both walls of the die. If platelet particles with an axial ratio are used in the extrusion in problem 4, where in the extruded pipe would you expect to find oriented particles. Explain the method you have used to make these conclusions.

6. Determine the pressure distribution in a cylindrical uniaxially pressed powder with a length to diameter ratio of 1.5. The powder being pressed is cohesionless with an angle of repose of 30 ~ The wall of the hardened steel die is well lubricated with stearic acid, which gives an angle of friction with the wall of 1~ The mold was filled and tapped to a density of 54% of theoretical stress then pressed at 100 MPa. Assume that a parabolic stress profile on the die plunger is applied. 7. With a knowledge of the pressure distribution in problem 6, determine the density distribution in the cylindrical green body. The powder used is MgO with a packing density pressure relationship given in Table 13.1. 8. Calculate the thickness of the coating produced by wash coating a catalytic converter honeycomb structure with an aqueous suspension of A1203 powder at 30% by volume. The catalytic substrate is withdrawn from a well-mixed suspension at a velocity of 2 cm/sec.

References

677

Assume that the monosized spherical A1203 powder is sterically stabilized with polyvinyl butral. 9. A 10% volume SiC ethanol suspension (/)~c = 0.3 ftm ~rg = 2) is slip cast with a conical plaster of Paris mold to make a radome for an airplane. The pores in the mold are 50 ~ m in diameter and the permeability of the mold is negligible compared to that of the powder. Determine the time necessary to slip cast a conical green body with a wall thickness of 1.5 cm. Assume that the ultimate packing density of the SiC powder is 80% theoretical. Data for ethanol:density 0.7893 gm/cc viscosity 1.2 cP and Tie = 24 dynes/cm.

References 1. Onoda, G. Y., and Hench, L. L., "Ceramic Processing before Firing." Wiley, New York, 1978. 2. Hirano, S., Messing, G. L., and Hausner, H., "Ceramic Powder Science IV." Am. Ceram. Soc., Westerville, OH, 1991. 3. Mangels, J. A., ed., "Advances in Ceramics," Vol. 9. Am. Ceram. Soc., Columbus, OH, 1984. 4. Hausner, H., Messing, G. L., and Hirano, S., eds., "Ceramic Powder Processing Science--Proceedings of the Second International Conference--1988-Berchtesgaden, FRG." Dtsch. Keram. Ges., Koln, 1988. 5. Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., "Introduction to Ceramics," 2nd ed. Wiley, New York, 1976. 6. Reed, J. S., "Introduction to the Principles of Ceramic Processing." Wiley (Interscience), New York, 1988. 7. Cheremisinoff, N. P., "Handbook of Ceramics and Composites," Vol. 1. Dekker, New York, 1990. 8. Rumpf, H., "Particle Technology" (translated by F. A. Bull). Chapman & Hall, London, 1975. 9. Richardson, D. W., "Modern Ceramic Engineering," 2nd ed. Dekker, New York, 1992. 10. Ichinose, N., "Introduction to Fine Ceramics: Applications in Engineering." Wiley, New York, 1987. 11. Briggs, J., "Processes for the Production of Technical Ceramics," Teaching Module for the EMMSE-West European Coordinating Committee supported by NATO Science Programs, Copyright 1981, Pennsylvania State University, University Park, 1981. 12. Onoda, G. Y., and Janney, M. A., in "Advances in Powder Technology" (G. Y. Chin, ed.), 1981 ASM Mater. Sci. Semin., p. 53. Am. Soc. Metals, Metals Park, OH, 1982. 13. Roy, R. W., AIChE J. 36(4), 481-510 (1990). 14. Adcock, D. S., and McDowall, I. C., J. Am. Ceram. Soc. 40, 355-362 (1957). 15. Dal, P. H., and Deen, W., Proc. Int. Ceram. Congr., 6th, Wiesbaden, pp. 219-242 (1958). 16. Ergun, S., Chem. Eng. Prog. 48, 89-94 (1952). 17. Haines, W. B., J. Agric. Sci. 17, 264 (1927). 18. Moreno, R., Requena, J., and Moya, J. S., J. Am. Ceram. Soc. 71, 1036-1040 (1988). 19. Hampton, J. H. D., Savage, S. B., and Drew, R. A. L., J. Am. Ceram. Soc. 71, 1040-1045 (1988). 20. Hampton, J. H. D., Savage, S. B., and Drew, R. A. L., in "Ceramic Powder Science IIB" (G. Messing, E. Fuller, Jr., and H. Hausner, eds.), pp. 749-757. Am. Ceram. Soc., Westerville, OH, 1988.

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Chapter 13

Ceramic Green Body Formation

21. Tiller, F. M., and Chow, R., Chem. Eng. Prog. 77, 61-68 (1981). 22. Perry, R. H., and Chilton, C. H., "Chemical Engineers' Handbook," 5th ed., pp. 19-58. McGraw-Hill, New York, 1973. 23. Weymouth, J. H., and Williamson, W. O., Trans. Br. Ceram. Soc. 52, 311-328 (1953). 24. Williamson, W. O., Trans. Br. Ceram. Soc. 54, 413-441 (1955). 25. Aksay, I. A., and Schilling, C. H., in "Ultrastructure Procesing of Ceramics, Glasses and Composites" (L. L. Hench and D. R. Ulrich, eds.), pp. 85, 439-447. Wiley, New York, 1984. 26. Velazquez, M., and Danforth, S. C., Adv. Ceram. 9, 105 (1984). 27. Bonekamp, B. C., Philipse, A. P., and Veringa, H. J., in "Ceramic Powder Processing Science Proceedings of the Second International Conference--1988--Berchtesgaden, FRG" (H. Hausner, G. L. Messing, and S. Hirano, eds.), p. 599. Deutsche Keramische Gesellschaft, Koln, 1988. 28. Schlichting, "Boundary Layer Theory," p. 76. McGraw Hill, New York, 1955. 29. Chou, Y. T., Ko, Y. T., and Yan, M. Y.,J. Am. Ceram. Soc. 70(10), C280-C282 (1987). 30. Tallmadge, J. A., and Tuntibarphol, M., AIChE J. 30, 666 (1984). 31. Sullivan, T. M., and Middleman, S., J. Non-Newtonian Fluid Mech. 21, 13-38 (1986). 32. Ring, T. A., Adv. Ceramics 26, 569 (1989). 33. Ring, T. A., unpublished work. 34. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena." Wiley, New York, 1960. 35. de Kruif, C. G., van Iersel, E. M. F., Vrij, A., and Russel, W. B., J. Chem. Phys. 83, 4717-4725 (1986). 36. Longwell, P. A., "Mechanics of Fluid Flow," p. 145. McGraw-Hill, New York, 1966. 37. Runk, R. B., and Andrejco, M. J., Ceram. Bull. 54, 199-200 (1974). 38. Sanchez, L. E., Int. Symp. Ceram. Substrates Packages, Denver, CO, 1987. 39. Happel, J., and Brenner, H., "Low Reynolds Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, NJ, 1965. 40. Chang, J. C., Velamakanni, B. V., Lange, F. F., and Pearson, D. S., J. Am. Ceram. Soc. 74(9), 2201-2204 (1991). 41. Hunter, R. J., ed., "Zeta Potential in Colloid Science: Principles and Applications," p. 247. Academic Press, New York. 1981. 42. Henry, D. C., Proc. R. Soc. London, Ser. A 133, 106 (1931). 43. Wiersma, P. H., Loeb, A. L., and Overbeck, J. T. G., J. Colloid Interface Sci. 22, 78 (1966). 44. Overbeek, J. T. G., in "Colloid Science" (H. R. Kruyt, ed.), Vol. 1, p. 234. Elsevier, Amsterdam, 1952. 45. Hill, C. G., Lovering, P. E., and Rees, A. L. G., Trans. Faraday Soc. 43, 4507 (1947). 46. Brown, D. R., and Salt, F. W., J. Appl. Chem. 15, 40 (1965). 47. Nass, R., Storch, W., Schmidt, H., Harbach, F., Neef, R., and Nienburg, H., in "Ceramic Powder Processing Science Proceedings of the Second International Conference-1988--Berchtesgaden, FRG" (H. Hausner, G. L. Messing, and S. Hirano, eds.), p. 625. Deutsche Keramische Gesellschaft, Koln, 1988. 48. Clasen, R., in "Ceramic Powder Processing Science--Proceedings of the Second International Conference--1988--Berchtesgaden, FRG" (H. Hausner, G. L. Messing, and S. Hirano, eds.), p. 633. Deutsche Keramische Gesellschaft, Koln, 1988. 49. Furono, N., Kawai, H., and Qyabu, Y., J. Colloid Interface Sci. 55, 297 (1976). 50. Tikhonov, A. P., Senapova, O. V., and Krivoshchepov, A. F., Colloid J. (USSR) (Engl. Transl.) 36, 926 (1976). 51. Kolesov, I. K., Lunina, M. A., and Khachaturyan, M. A., Colloid J. (USSR) (Engl. Transl.) 39, 850 (1977). 52. Petrov, V. A., and Panov, V. I., Colloid J. (USSR) (Engl. Transl.) 37, 359 (1975). 53. Malov, V. A., Kalminskaya, I. A., Bezruk, V. I., Lazarev, A. N., and Lavrov, I. S., Colloid J. (USSR) (Engl. Transl.) 36, 348 (1974).

References

679

54. Hein, M., Muller, G., Peil, H., Ponto, L., Becks, M., Klein, U., and Peiniger, M., J. Appl. Phys. (1989). 55. Deryagin, B. M., and Levi, S. M., "Film Coating Theory." Focal Press, London, 1964. 56. Deryagin, B. V., and Titievskaya, A. S., Dokl. Akad. Nauk S S S R 50, 307 (1945). 57. Werk-Albers, M. L., and Ring, T. A., unpublished manuscript. 58. Van de Ven, T. G. M., in "Colloidal Hydrodynamics" (R. H. Ottewill and R. L. Rowell, eds.), Colloid Sci. Monogr. Ser. Academic Press, Boston, 1989. 59. Perrin, J., "Brownian Motion and Molecular Reality." Taylor & Francis, London, 1910. 60. Perrin, F., J. Phys. Radium 5, 497 (1934); 7, 1 (1936). 61. Deryagin, B. V., Dokl. Akad. Nauk S S S R 39, 11 (1943), reviewed in Deryagin and Levi [55]. 62. Mutsuddy, B. C., Adv. Ceram. 9, 212-219 (1984). 63. Ovenston, A., and Benbow, J. J., Trans. Br. Ceram. Soc. 67, 543-567 (1988). 64. Funk, J. E., Adv. Ceram. 9, 184 (1984). 65. Warner, D. A., and Sorensen, O. T., Proc. Eur. Ceram. Soc., 1991. 66. Holman, R. A., Bigg, D. M., Mutsuddy, B. C., and Kurth, R. E., Adv. Ceram. 9, 259 (1984). 67. German, R. M., and Hens, K. F., Ceram. Bull. 70(8), 1294-1302 (1991). 68. Takahashi, M., and Suzuki, S., in "Handbook of Ceramics and Composites" (N. P. Cheremisinoff, ed.), Vol. 1, p. 65. Dekker, New York, 1990. 69. O. J. Whittemore, Jr., in "Ceramic Processing Before Firing" (G. Y. Onoda and L. L. Hench, eds.), p. 343. Wiley, New York, 1978. 70. Takahashi, M., and Suzuki, S., Am. Ceram. Soc. Bull. 65(12), 1587-1590 (1986). 71. DiMilia, R. A., and Reed, J. S., Ceram. Bull. 62(4), 484-488 (1983). 72. Kawakita, K., and Ludde, K.-H., Powder Technol. 4, 61-68 (1970-1971). 73. Cooper, A. R., and Eaton, L. E., J. Am. Ceram. Soc. 45(3), 97-101 (1962). 74. Schwartz, E. G., and Weinstein, A. S., J. Am. Ceram. Soc. 48(7), 346-350 (1965). 75. Thompson, R. A., Am. Ceram. Soc. Bull. 60(2), 237-243 (1981). 76. Unkel, H., Arch. Eisenhuettenwes. 18, 125-130 (1944). 77. Train, E., Trans. Inst. Chem. Eng. 35, 258 (1957). 78. Perelman, V. E., and Roman, O. V., Proc. Int. Conf. Compact. Consolidtion Particulate Matter, pp. 41-45 (1972). 79. Afaasjev, L. N., Roman, O. V., and Perelman, V. E., "Experimental Determination of Parameters, Characterizing the Densification of Metal Powders," Conf. Perm, No. 6. (1969). 80. Torre, C., Berg and Huttenmannische, Monatsch. B. 93 (1948). 81. Kingery, W. D., ed., "Ceramic Fabrication Processes." MIT Press Cambridge, MA, 1963. 82. Bortzmeyer, D., Powder Technol. 70, 131-139 (1992). 83. Onoda, G. Y., and Janney, M. A., in "Advances in Powder Technology" (G. Y. Chin, ed.), 1981 ASM Mater. Sci. Semin., p. 53. Am. Soc. Metals, Cleveland, OH, 1982. 84. Nagao, T., Trans. Jpn. Soc. Mech. Eng. 33, 229 (1967). 85. Nagao, T., Trans. Jpn. Soc. Mech. Eng. 44, 2967 (1978). 86. Nagao, T., J. Soc. Powder Technol. Jpn. 23, 240 (1986). 87. Strijbos, S., Ceramurgia Int. 6(4), 119-122 (1980). 88. Richardson, D. W., "Modern Ceramic Engineering," 2nd ed., p. 441. Dekker, New York, 1992. 89. Lotgering, F. K., J. Inorg. Nucl. Chem. 9, 113-123 (1958). 90. Gill, R. M., and Byrne, J., in "Science of Ceramics" (G. H. Stewart, ed.), Vol. 4. Br. Ceram. Res. Assoc., London, 1968. 91. Sommer, K., and Rumpf, H., in "Ceramic Processing Science Before Firing" (G. Onoda and L. Hench, eds.), Chapter 20. Wiley, New York, 1978.

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PART

V PRESINTERING HEAT T R E A T M E N T S OF DRYING AND BINDER BURNOUT In Part IV we saw how ceramic green bodies are manufactured. At this point the green body consists of a mass of ceramic powder held together by either a liquid in the pores, if green body fabrication used a ceramic suspension, or a polymer, if green body fabrication used spray dried powder. As we have seen in Chapter 13, the strength of the green body is determined either by the liquid in the pores or the polymer distributed at the particle-particle contacts. Heat treatment before sintering must remove the liquid by evaporation and then remove the polymer by thermal decomposition. Drying the solvent out of the green body is discussed in Chapter 14. Thermal decomposition of the polymer (also surfactants and plasticizers) is discussed in Chapter 15. Both operations require heat to be transferred into the green body to the location of either a chemical reaction or phase transformation. The necessary heat of evaporation for solvent drying and the heat of reaction, either exothermic or endothermic, for polymer thermal decomposition (also called binder burnout) must be supplied for the reaction or evaporation to continue. Heat is transferred in both the boundary layer surrounding the green body and in the porous network of the green body. These heat transfer steps can limit the rate of either

682

Part V

Presenting Heat Treatments

evaporation or thermal decomposition. Once the heat of reaction is provided, the reaction can proceed. Both evaporation and thermal decomposition give off huge volumes of gas for each gram of material reacted. This gas must diffuse through the porous network of the green body and through the boundary layer surrounding the green body. These mass transfer steps can also limit the rate of either evaporation or thermal decomposition. Thus, from the mass and heat transfer points of view, drying and thermal decomposition of the binder have similar fundamentals. Different types of stress act on the green body during these processes. One such is the stress induced by thermal gradients in the green body. Another is the stress due to a gas pressure gradient in the green body, caused by the flow of gas from the inside of the green body to its surface. The last type of stress can also be due to a liquid pressure gradient in the green body caused by the flow of liquid from the inside of the green body to its surface. These stresses are additive and act to warp or crack the green body. When the green body's surface is in tension, it is most vulnerable to cracking because, like all ceramics, a green body is weakest in tension. In drying, the stress acting on the green body will be shown to depend on the flux of vapor leaving the green body. In binder burnout, the stress acting on the green body will also be shown to depend on the flux of gas leaving the green body and on the rate of cool-down from the pyrolysis temperature after the binder has been burnt out.

14

Green Body Drying

14.1 O B J E C T I V E S This chapter discusses the kinetics of drying for various conditions of temperature and solvent partial pressure in the atmosphere. Simultaneous heat and mass transfer in both the boundary layer surrounding the green body and the pores inside the green body are responsible for the drying kinetics. Depending on the kinetics of drying, different stresses are developed in the green body. Critical drying conditions that cause cracking in the green body are elucidated, taking into consideration the strength of the green body at various stages in the drying process.

14.2 I N T R O D U C T I O N To remove the solvent used to suspend the ceramic powder for green body fabrication, the green body is heated or placed in an atmosphere 683

684

Chapter 14 Green Body Drying

where the solvent evaporates. Depending on the type of ceramic powder used and the green body fabrication process different solvents are used. Solvents are typically organics or water. Organic solvents are chosen to easily wet the ceramic powder surface and easily evaporate during drying. This means that a low enthalpy of vaporization and a reasonably high vapor pressure is important for a solvent to be easily evaporated. The weight loss during drying is schematically shown in Figure 14.1. This figure shows a constant rate period where the surface of the green body is always wet by the flow of liquid to the surface. This flow emanates from the rearrangement of particles in the green body, which is caused by the compressive capillary pressure at the surface of the green body. At some point, the particle network becomes rigid and no more shrinkage can take place. This critical point is referred to as the leatherh a r d p o i n t in clay ceramics nomenclature. With particulate ceramics, this rigidity threshold takes place at high volume fraction where the particles come into contact. With gels, the particle network exists at the start of drying--shrinkage occurs as a result of this compliant network stopping at the rigidity threshold of the network. After this critical point, the liquid-vapor interface starts to recede into the pores. At this stage, surface tension driven flows in the direction of the free surface attempt to keep a monolayer of solvent on the surface of all the ceramic powder. Such capillary flow will continue as long as there is a continuous pathway from the liquid front to the green body surface.

FIGURE 14.1 Schematicdiagram of the drying of a ceramic green body showing the

weight loss and shrinkage with time.

14.2 Introduction

685

Once the drying front enters the green body, the drying rate decreases abruptly. This decreasing rate continues until the large pores are essentially free of liquid, except for that which is trapped at the point of contact between the particles as shown in the inset of Figure 14.1. This trapped liquid will be removed only if the partial pressure generated by the radius of curvature, rpore , of the liquid in the capillary is greater than the partial pressure present in the green body in the proximity of the pore [1], that is,

PI; P~exp[2~/LAVLc~ Rg Trpore

> P1

(14.1)

VL

is the molar volume of the liquid, R~ is the gas constant, T where is the temperature, ~/LA is the surface tension of the solvent, 0 is the contact angle of the liquid on the ceramic and po is the saturation partial pressure of the solvent over the liquid in the ceramic. This equation assumes that only solvent is in the pores of the green body. The equation shows that less solvent will remain in the green body if the surface tension is lowered by the addition of surfactants. Instead of an abrupt drying front, there is a diffuse drying front (see Figure 14.2) [2] in which the capillaries (or pores) smaller than those

FIGURE 14.2 The drying front in a green body composed of monosized 0.5 ~m SiO2 particles. In this photo, saturated pores are white and empty pores are black. The drying front is fractally rough on the size scale of the particles (a) but smooth on a larger scale (b). Taken from Shaw [2]. Reprinted by permission of the Materials Research Society.

686

Chapter 14 GreenBody Drying

equivalent to the partial pressure, Pz, at a point in the green body, are filled with solvent and others larger than those equivalent to the partial pressure at that point that are empty. The vapor pressure gradient in the partially drained pores results in a gradient in capillary pressure which drives [3] the liquid along the filled capillary network to the leading edge of the drying front according to Darcy's law (see equation 13.1). This is called the f u n i c u l a r c o n d i t i o n and often persists long after the liquid-vapor interface invades the pores of the green body. Shaw [2,4] has shown that the drying front is fractally rough on the scale of the particle size and smooth on a larger scale. The thickness of the front decreases with the velocity of the front (i.e., width a (velocity) -m with m = 0.48 _ 0.1 for spheres according to Shaw). Throughout the balance of this chapter, we will consider that the drying front is defined by a uniform surface, ignoring its diffuse nature on the size scale of the ceramic particles. As the partial pressure gradient moves into the green body, the pores empty accordingly. For this reason, the green body will not be completely dry when the bulk gas used for drying has a nonzero partial pressure of the solvent. Some liquid will remain in the smallest capillaries where the radius of curvature is sufficiently small. These filled capillaries will contribute to a compressive capillary force which holds the green body together. The evaporation of solvent is endothermic, requiring energy to be transported to the evaporation site. The evaporated solvent must then diffuse (or flow if sufficient pressure drop is available) from the evaporation site through a porous network of ceramic particles to the bulk gas surrounding the green body. Thus, simultaneous mass transfer and heat transfer control the rate of green body drying. A diagram of a green body during drying (Figure 14.3) [5] shows the partial pressure profile and the temperature profile near the surface of a ceramic green body and in its pores. This figure also shows the geometry of the pores with the nomenclature to be used for drying.

14.2.1 Heat Transfer The flux of heat, q, into a green body is given by the boundary layer heat transfer: q = Q/A = h(Ts - Ts)

(14.2)

where Q is the quantity of heat transferred per unit time, A is the geometric surface area of the green body at temperature T s exposed to the bulk gas at temperature TB, and h is the heat transfer coefficient which is a function of the gas flow rate around the green body. The

14.2 Introduction

687

Schematic diagram of evaporation in a porous network (a) geometry of pore and boundary layer, (b) liquid partial pressure profile, and (c) temperature profile. Taken from Castro et al. [5]. Reprinted by permission of the American Ceramic Society.

F I G U R E 14.3

heat flux inside the green body is due to heat conduction in a porous network, given by q = -kp VT

(14.3)

where VT is the gradient of temperature and kp is the effective thermal conductivity of the porous ceramic green body.

14.2.2 Mass Transfer The mass flux, j, is related to the heat transfer flux, q, required to evaporate those molecules at the point of evaporization by the following expression: q

= AHva p

,j

(14.4)

where AHva p is the molar enthalpy of solvent evaporated. This relation requires that the two differential equations for the fluxes be linked for

688

Chapter 14 GreenBody Drying

their simultaneous solution. The mass transfer flux, j, at the surface of the green body due to boundary layer mass transport is given by j = J/A = Kc \ReT s

RgTB ]

(14.5)

where J is the number of moles transfered per unit of time, A is the external area of the green body, Kc is the mass transfer coefficient, p s is the solvent partial pressure at the green body surface, pB is the solvent partial pressure in the bulk gas, T s is the surface temperature, and TB is the bulk gas temperature. Using the ideal gas law, the terms of the type Pl/Rg T are the solvent concentration in the gas at various points. The mass transfer coefficient, Kc, is determined from the molecular diffusion coefficient for the diffusing species in the gas mixture, the geometry of the green body, and the flow rate of the bulk gas. The partial pressure of the solvent over a solution of dispersant, polymer, and other salts was discussed in Section 11.5. The partial pressure is a function of solution, composition and temperature. For the solvent alone, the partial pressure of the solvent, p o, as a function of temperature, T, is given by P~

= [1 atm] exp

Re

( T~1e

(14.6)

where AH~ is the enthalpy of evaporation and TBp is the normal boiling point of the solvent. For a salt solution (or surfactant solution), the partial pressure of solvent, Pl, is given by P~(T) = [1 atm]al exp [ Re

T~e

(14.7)

where a l ( = ~/lXl ) is the activity of the solvent in the solution. For an ideal solution, the value of the activity coefficient ~/~ is 1.0 and a 1 = x~ = 1 - x2, where x~ is the mole fraction of species i. For nonideal solutions, ~1 can be greater than 1, resulting in positive deviations, or less than 1, resulting in negative deviations from ideality. For surfactants, strong negative deviations from ideality are observed. The mass flux inside the green body due to molecular diffusion in a porous network is j = -DpV

P(R~)

(14.8a)

where V(PJRgT) is the gradient of molar concentration (Pt/RgT) given in terms of the partial pressure of the solvent and Dp is the effective diffusion coefficient for the porous ceramic green body. Whenever a

14.2 Introduction

689

continuous filled capillary path exists, capillary flow predominates over vapor phase molecular diffusion.

14.2.3 Flow o f L i q u i d in the Pores A pressure drop, VP, along the pores leads to liquid flow. The mass flux in the pores of the green body due to liquid flow is given by Darcy's law: j = amPV P c

(14.8b)

where ap (cm 2) is the specific permeability (= s3/(clS~)(1 - s)2), cl is a constant = 4.2 [6] or 5.0 [7], So is the surface area of the particles per unit volume of the particles (S0 = 3/rpart for a green body containing spherical ceramic particles of radius, rpart), and s is the void fraction of the green body and V is the viscosity of the liquid. The pressure gradient comes from a gradient in the capillary pressure, Pc [= TLA(1/rl + 1/r2) = 2TLA c o s O/rpore]. At the surface of the green body, there are two radii of curvature, rl and r2, which are characteristic of a pore of the green body that can be determined from the mean pore radius, rpore (i.e., rl = r2 = --rpore/COS 0, where 0 is the liquid-solid contact angle). At the center of the green body, there is effectively an infinite radius of curvature because there is no liquid-vapor interface. Therefore the pressure gradient is the capillary pressure acting to compress the green body, as shown in Figure 14.4. When the pores in the green body are partially filled, capillary pressure will also create a pressure driving force if the radius of curvature at one end of the pore is different than the other. As a result, flow will be induced from the highest pressure corresponding to the lowest radius of curvature to the lowest pressure corresponding to the highest radius

FIGURE 14.4 Compressive stress inside a ceramic green body due to the presence of a polymer after it is dried of solvent.

690

Chapter 14 GreenBody Drying

of curvature. This type of flow is dealt with in exactly the same way using Darcy's law, equation 14.8b, with the capillary pressure gradient used as the pressure gradient. In three dimensions, these simultaneous differential equations are very complex and require numerical solution. For this reason, we will discuss simple geometries in the balance of this chapter with only one rate determining step. By considering the previous equations, we can see that there are four steps in the drying of ceramic green bodies: 1. 2. 3. 4.

Boundary layer mass transfer (BLMT), Pore diffusion (PD) or pore flow (PF), Boundary layer heat transfer (BLHT), Pore thermal conductivity (PTC).

Each of these steps could be the slowest and limit the drying rate of a ceramic green body. The slowest step is called the rate determining step. In general, the rate determining step will change as drying takes place. Initially, the pores are completely full of liquid and boundary layer heat or mass transfer will be rate determining. After some time, the liquid recedes into the pores and either pore diffusion or pore heat conduction will dominate for relatively long pores. An estimate of the total time, Ttot, to dry a green body is the sum of all the maximum drying times, r, considering only one rate determining step at a time: Ttot : (TBLMT + TPD or PF + TBLHT + ~'PTC)/2

(14.9)

These maximum drying times are derived later in the chapter. The factor of 2 is because the maximum time due to one step of mass transfer and one step of heat transfer must be equal since there is an equivalence of heat and mass transfer described by equation 14.4. The determination of the drying kinetics, including the maximum drying times for different rate determining steps, is the subject discussed in the next section for different green body geometries. But first, we must discuss drying shrinkage.

14.2.4 Drying Shrinkage During the constant rate period shown in Figure 14.1, either the boundary layer mass or heat transport is rate controlling. The flow of liquid to the surface of the green body to keep it wet is governed by the permeability equation for the flow of liquid relative to the ceramic particles [8,9], written as Fick's second law, OC/dt = V(DwVC), for diffusion considering (1 - ~)[= e = volume fraction of liquid] to be the

14.2 Introduction

691

"liquid concentration": a(1 - ~b)= V[DwV(1 - ~b)] dt

(14.10)

where ~ is the volume fraction of solids, DWis a flow transport coefficient describing the flow of liquid, which is analogous to a diffusion coefficient. This equation draws on an analogy to Darcy's law,j = ~p/~V(P), which is equivalent to Fick's first law for diffusion, j = D VC, except the pressure gradient, VP, is used. Note that P / R g T has concentration units. Considering the continuity equation for a green body, the pressure gradient can be shown [3, p. 438] to be proportional to the volume fraction liquid, s, or 1 - ~b; thus, allowing VP to be replaced by V(1 - ~). The flow transport coefficient, DW, is a nonlinear function of the volume fraction of solid particles, ~b, as would be expected from permeability considerations: ap

(1 - ~)~RgT

DW=--~R~T=

c~v8~(4~)2

s3RgT =ClVS~(1 - s) 2

(14.11)

where these terms are defined by the specific permeability, ap, in Darcy's law, equation 14.8b. At the surface of the green body, the time derivative, 0(1 - 4~)/dt ISurf, is proportional to the drying flux,j, according to the continuity equation just given. To maintain a flow of liquid to the surface at a constant rate, the green body must shrink, expelling liquid. The compression of a ceramic green body is shown in Figure 14.5. At low r the curve is linear. At high values of ~, there is a critical value, ~b*, where no further shrinkage takes place, corresponding to liquid just filling the pores at the "leatherhard point." This critical volume fraction, ~b*, occurs when the mechanical properties of the particle network is sufficiently rigid to resist the compressive capillary pressure. The liquid expansion of a ceramic green body, ac, is defined by 1 0V ac = V0(1 - ~)

(14.12)

This equation assumes that the green body is isotropic (i.e., acx = acy = acz = 1/3ac, where acx is the linear compressibility in the x direction acx = 1/Lx OLx/O(1 - 4~). With particle segregation and nonspherical particle alignment, anisotropic compressibilities are often observed [10-13] and can be very important [14] in warping and cracking of the green body during drying.

14.2.5 Drying I n d u c e d Stresses When the green body is wet by the solvent, it has a compressive capillary force which holds it together. This capillary induced tension

692

Chapter 14 Green Body Drying

In(V/Vo)

Solids Volume Fraction, FIGURE 14.5 Liquid expansion as a function of solids volume fraction with critical volume fraction where the particles are in contact.

is always present when the surface tension driven flows keep the surface wet. During drying, this capillary force disappears because the liquid disappears. With a polymeric binder present, the compressive capillary force is replaced by the cohesive force of the binder, as shown schematically in Figure 14.4. The cohesive force of the binder is much smaller than the original compressive capillary force of the solvent as we will see later in this chapter. Due to its weakened state, the green body may rupture if undue stress is placed upon it during drying. During drying, the green body is susceptible to nonuniform stresses that may warp or crack it due to (1) the pressure gradient of the flow of liquid during shrinkage in the constant rate period, (2) the macroscopic pressure gradient of the escaping gasses during the decreasing rate period, or (3) the differential thermal expansion of the ceramic due to temperature gradients in the green body. The flow of liquid, the macroscopic pressure gradient, and the temperature gradient are controlled by the drying rate. The rate of green body drying is typically controlled by external conditions (i.e., TB and pS ). To be slow enough that excessive stress, which would warp or crack the green body, do not build up, these external conditions must be gentle. Designing the drying conditions to such slow rates is the objective of successful ceramic green body drying.

693

14.3 Sphere and Cylinder Drying

14.3 S P H E R E A N D C Y L I N D E R D R Y I N G In this section, we will discuss in detail the linked differential equations for mass and heat transfer which describe the drying of a spherical green body. This same analysis can also be used for plate and cylinder green bodies with corrections for the geometry. The equations for cylinder and plate drying are presented in Tables 14.3 and 14.4.

14.3.1 Boundary Layer Heat and Mass Transfer, Giving the Drying Rate for the Constant Rate Period Consider a spherical green body of radius Ro, where all the pores are filled and the surface is initially wet with solution. For this condition, the rate of mass transfer, J, from the surface of the green body is given by

J : 4 R Kc (

\R~Ts

RgTB]

(14.13)

where Kc is the mass transfer coefficient for the boundary layer. The equilibrium partial pressure of the solvent po(T), was given by equation 14.7. The mass transfer coefficient, K c, for a sphere can be determined from the Sherwood number, Nsh (= Kc2Ro/DAB, where DAB is the molecular diffusion coefficient of the solvent, species A, in the drying gas, species B), and the following engineering correlation [15]" Nsh = 2.0 +

0"6N1/2nT1/3Re ~" Sc

(14.14)

where N R e (--- ulpg2R/t~g) is the Reynolds number (based on the drying gas velocity, u~, the gas viscosity, t~g, and the gas density, pg) and Nsc (= t~g/pgDAB) is the Schmidt number. The mass transfer, J, is related to the heat transfer, Q, required to evaporate those molecules Q

=

AHvap * J

(14.15)

and the heat transfer is given by

Q - 47rR~h(Ts - Ts)

(14.16)

where h is the boundary layer heat transfer coefficient. The heat transfer coefficient for a sphere is given by the Nusselt number NNu (= h2Ro/k, where k is the thermal conductivity of the drying gas) and the following engineering correlation [15]" N N u - 2.0 +

0.6N~/2eN1/3Pr

(14.17)

694

Chapter 14 GreenBody Drying

where Npr( = Cpt~g/k) is the Prandt number and Cp is the heat capacity of the drying gas at constant pressure. The temperature at the surface of the sphere is determined by the evaporation rate obtained from simultaneous mass and heat transfer. Under some drying conditions, the heat transfer is the slow step, limiting evaporation, and in others the mass transfer is the slow step. The surface of the green body will continue to stay wet if the green body shrinks, expelling the solution, or surface tension driven flow continues to supply liquid. Shrinkage stops when the rigidity threshold of the particle network is reached. When the boundary layer mass or heat transfer is the rate determining step for the whole drying time even after the liquid recedes into the pores, the following analysis is applicable. By performing a mass balance on the sphere, the rate at which the liquid will recede into the pores (see Figure 14.6), dR/dt, is given by

spl 47rR2dR M~ -~=J-

Q

Agvap

(14.18)

where s is the void fraction initially filled with solvent, M ~ is the molecular weight of the evaporating solvent, and p~ is its density. This

FIGURE 14.6 Recedingcore of spherical green body during drying.

14.3 Sphere and Cylinder Drying

695

expression can be rewritten as

_R2dR _ M~vR~Kc( PS dt ePl \RgTs

pS ~= M wRoh 1 2 RgTB]

~pl AHvap ( TB

Ts )

(14.19)

for mass transfer and heat transfer limited evaporation, respectively. When the surface is wet (i.e., R = Ro), both of these expressions show a constant rate (i.e., not a function of radius, R), which is frequently observed for a certain time initially [16]. During this period, the surface of the green body is always wet by the flow of fluid from within the green body to the surface, and the temperature at the surface, Ts, is essentially constant at the "wet bulb" temperature. If water is the solvent, tables of the wet bulb temperature are given in psychometric charts [17]. Assuming t h a t this rate determining step remains the slow step for the complete drying of the green body, an integration of the preceding equation from R = R0 to Re gives the time, t, needed to dry a porous sphere from its initial size to filled core size, Re (see Figure 14.6). The results of this integration (given in Table 14.1) are of the form

t = rf(Rc/R o)

(14.20)

where r is the m a x i m u m time for this rate determining step and

f(Rc/Ro) is a function of the dimensionless radius, Rc/Ro, of the spherical green body of radius R0.

14.3.2 S h r i n k a g e

during the Constant Rate Period

The flow of liquid in ceramic green bodies of liquid volume fraction, &l [ = 1 - ~b], has been shown to follow permeability models for the flow of liquid relative to ceramic particles [8,9] given in spherical coordi-

T A B L E 14.1 Spherical Green Body Drying Kinetics by Mass Transfer Controlling Steps

Boundary layerPmass transfer

t - 1 - (Rc/Ro) TBLHT

t - 1 - (Rc/Ro)3 TBLMT TBLMT =

Boundary layermheat transfer

epiRo

eplAHvapRo rBLHT = 3M~vh(TB - Ts)

3M~Kc \RgTs

RgTB]

696

Chapter 14 GreenBody Drying

nates by 0(1-~b)_ 1 0 [ 4))] Ot R 2 OR DWR 2 0(1 OR

(14.21)

where DW is a flow transport coefficient describing the flow of liquid. This equation is Fick's second law for diffusion in solids, OC/Ot = V(DVC); written in terms of the concentration (or volume fraction) of liquid, C is replaced by [= 1 - ~].* The flow transport coefficient, Dw, is a nonlinear function of the volume fraction of solid particles, ~b, as would be expected from permeability considerations (see equation 14.11) and for this reason is included inside the differential. However, this flow transport coefficient is assumed to be constant [18], albeit unrealistically, in order to attempt a solution of the preceding equation for a sphere with a constant flux at the surface; that is, 0(1

~)1 dt

jM w 1 PlR~

=

-

Ro

(14.22)

To maintain a flow of liquid to the surface at a constant rate, the green body must shrink. A solution to this equation gives the shrinkage at various radii in the spherical green body as a function of time during the constant rate period. The boundary conditions for this equation are t = 0 ~b = ~bo for all R 0__~_~= 0 at R = 0 for all t OR

(14.23)

= ~R0 = constant at R = Ro for t > 0 This solution is given by ~R~ -- ~ -- E

d~Ro - d~o

[ nO t] o [

An exp -

n=O

R~

~ sinh V~pn r

(14.24a)

where Pn are the eigenvalues given by the roots of the following equation: tan V ' - p n =

3N/--pn

a vro C p ] - s.sC n3

(14.24b)

* An alternate description of the derivation of this equation, given by Brinker and Scherer [3], describes the driving force, VC, in terms of the pressure, VP, and then relates this pressure gradient to the liquid volume fraction, ~bl [= 1 - ~b].

14.3 Sphere and Cylinder Drying

for n = 1, 2, 3, ..., ~ and ary condition

697

the coefficients given by the bound-

A n are

R0 ( R)

1 = n~oAn s i n Rh- :

(14.24b)

k~Pn Roo

A plot of the solution to this equation is given in Figure 14.7. This gives the particle volume fraction at all locations within the sphere during constant rate drying. The flow of liquid to the surface of the sphere will have two effects: (1) it will cause the redistribution of binder and soluble surfactant in the green body, and (2) it will realign anisotropic particles if the shear strain is large enough and the particles are at sufficiently low in volume fraction to allow movement. The redistribution of soluble surfactant and binder can cause surface tension driven flows [19] (the Marongoni effect) because their concentration determines the surface tension of the liquid.

1.o f c ~

Surface of sphere,

of s ~ ,

0

Dft/R 2 = 0.4

0.8

eo- R

~R-(~)Ro ~o_(1)R o

0.6

0.4 I

J

J

0.2

0.4

J fj "

(])Ro-(~o

0.6

O"

0.04 j o

o

0.2

0.4

9R/Ro

---"

0.6

0.8

FIGURE 14.7 Dimensionless solid volume fraction, (~b -

1.0

1.0

( ~ R ) / ( ~ O - - (~R), distribution in a sphere resulting from transient flow of liquid to the surface when there is a fixed volume fraction DR at the surface of the sphere at r = R. Here, ~b0 is the initial volume fraction. The different curves correspond to different values of Dwt/R 2.

698

Chapter 14 GreenBody Drying

14.3.3 Diffusion and Heat Conduction in the Porous Network, Giving the Drying Rate for the Falling Rate Period Whitaker [20,21] has developed an analysis of heat and mass transfer for the drying of rigid materials that gives a description of the falling rate period. This model presented here has been verified by Wei et al. [22,23] using porous sandstone. When pore diffusion becomes important, the diffusive flux of solvent vapor through the product layer, J, is given by

d = 47rR2D1E

dR

R

= Essentially constant

(14.25)

where P is the partial pressure of the solvent, R is any sphere radius between Ro and Re, the point at which the pores are filled with liquid, D u~ is the effective diffusion coefficient of the vaporized solvent 1 in the porous layer. The effective diffusion coefficient for a porous layer is given by D1E = [~-~k+ D--~] -l-e~

(14.26)

where D k (= rp ~v/18RgThrMw) is the Knudsen [24] diffusion coefficient (rpore is the pore radius); DAB is the molecular diffusion coefficient of the solvent, A, through the gas, B, in the pores; s is the void fraction of product layer; ~ is the tortuosity of the pores (typically ~ 2.0 for roughly spherical particles). In the case of large transport, a pressure drop may build up and Darcy's law should be used for the transport through the pores

R~ J = 47rR2Dwd dR

R = Essentially constant

(14.27)

where Dfis the transport coefficient given by Darcy's law as was defined in equation 14.11. Equation 14.25 has the same form as equation 14.27, so if Df is larger than D1E , Df should be used in the place of D1E in the following analysis for the rates of drying. Integrating equation 14.25 or 14.27 from R0 to Re, we obtain P~ J

dR= 47rD (R~Ts d R~T~

P(RgT)

(14.28)

14.3 Sphere and Cylinder Drying

6~

or

(1 J Rc

1) (ps I~o = 4rrD1E \RgTs

pB] (14.29)

RgTB]

This expression represents the conditions of a drying porous sphere for a given Rc at any time. To complete this analysis, we have to understand how the rate of diffusion decreases as Re gets smaller. This is done by writing J in terms of Re, using the mass balance:

j =

spl 2 dRc M~w 4~r Re dt

(14.30)

substitution of J into the preceding equation and integrating gives

splfR~c( M~w o R - ~1 ) R 2 dR = D~E [ , p s \RgTs

If;

pB RgTB/

dt

(14.31)

or

t-

(

_PZ

6MlwD1E\RgTs

1- 3

\Ro/

+ 2

\Ro/ J

(14.32)

RgTB]

This analysis constitutes a pseudo-steady state assumption; that is, the flux is assumed to be constant (i.e., equation 14.25 and 14.27) for a particular core radius of the sphere. This is a reasonably good approximation because the volume of vapor produced is very large compared to the volume of liquid evaporated by a factor of approximately 1000. This results in an equation for the drying time which is of the form t = 9 * f(Rc/Ro). Now let us consider the porous ceramic heat conduction. The heat transport in the porous layer is given by

Q = 47rR 2 k e

dT

- Essentially constant

(14.33)

where k e is the effective thermal conductivity of the empty porous layer. The effective thermal conductivity of this layer is given by

ks

+

(14.34)

where s is the void fraction in the empty porous layer; ks is the thermal conductivity of the solid; kf is the thermal conductivity of the gas in the pores.

700

Chapter 14

Green Body Drying

To complete the picture, the flux of solvent vapor is maintained by the transfer of heat to the liquid surface in the empty pores. ePl

M147r

R2 d R --~ = J -

Q

(14.35)

AHvap

Using these equations for heat conduction in the porous network and the pseudo-steady state analysis described for pore diffusion, the time to dry a spherical green body with pore heat conduction as the rate determining step is given by

8R1AHvapR~ t = 6 M~w k e(TB - Ts)

[1- 3(Rc'~2--l2(Rc' - 3]~ \Ro]

(].4.36)

\Ro]

This analysis does not account for the heat required to heat the liquid filled core to a new temperature which is nearly equal to the liquid surface temperature. This amount of heat is small compared to the heat of evaporation. Again the pseudo-steady state approximation has been used for similar reasons. A summary of the derived equations for the drying time when transport in the pores is the rate determining step are given in Table 14.2. These expressions are good for rate limiting steps of only pore diffusion and pore heat conduction corresponding to the decreasing rate period. Combined with the equations in Table 14.1, all the possible rate controlling steps are established allowing the prediction of the total time, Ttot, to dry a green body: Ttot = (TBLMT + TpD + TBLHT + TPTC)/2

(14.37)

P r o b l e m 14.1 D r y i n g Time C a l c u l a t i o n Determine the total time to dry a 1.0 cm spherical green body with void fraction 40% formed from an aqueous Zr02 paste with polymer at

TABLE 14.2 Evaporative Drying of Spherical Green Body, by Heat Transfer Controlling Steps Pore diffusion

t/TpD = 1 -- 3 \Ro] + 2 \Ro] J

[

(Re? ;Rc?I

tlrPwC = 1 -- 3 \Ro] + 2 \Ro] J

~:plbHvapR~

eplR~ TpD =

6M~cDIE( pS \RgTs

Pore heat conduction

p ~ .~ RgTB]

TpTC = 6M~ke(TB _ Ts )

701

14.3 Sphere and Cylinder Drying

a mole fraction of 0.01 assuming the solution is ideal. The green body with a pore radius of 0.5 t~m is in stagnant air with a relative humidity of 10% at 90~ The following data for water and air will be needed. Water: A H v a p - 539.55 cal/gm, Mw = 18 gm/mole, molecular radius r s = 2.655/~. Air: Mw = 29 gm/mole, molecular radius r A = 3.617/k, C~A = 0.25 cal/gm. DAB(T) = -~

2MA + 2MB]

P(rA + rs) 2

(14.38)

where NA is Advogadro's number. The gas viscosity is given by 1 k/MA ksT ~g(T) - 67r3/2 NA t2Ar~

(14.39)

where ~A is the collision integral for air, t2A = 0.735; kg( T) = ~g( T) CpA( T) yp r

(14.40)

where Npr is the Prandlt number which for air is 0.73. Ideal gas law can be used for density of air and water concentration.

Solution With these data and the equations given in the previous section, the maximum times for the various rate limiting steps (given in Tables 14.1 and 14.2) can be calculated as a function of the temperature, Ts, at the surface of the liquid interface, either at the surface of the green body in the constant rate period or in the pores of the green body in the decreasing rate period. The surface temperature is unknown, and as a result, we must either approximate it or solve for it using t h e equivalence of mass and heat transfer stated in equation 14.15. A plot of these maximum times as a function of T S is given in Figure 14.8.

FIGURE 14.8 Maximum drying times as a function of Ts.

702

Chapter 14

Green Body Drying

The intersection between a mass transfer line (either PD or BLMT) and a heat transfer line (either PTC or BLHT) gives the surface temperature, which corresponds to the equivalence of the mass and heat fluxes given by equation 14.15. As we can see, there are four such intersections, giving four possible surface temperatures. The intersection which gives the largest maximum drying time is the one responsible for the rate limiting step. The largest maximum drying time in this case is that of pore diffusion and boundary layer heat transfer, which gives liquid interface surface t e m p e r a t u r e of 49.5~ For this surface temperature, the maximum times are

~plRo

TBLMT =

- 4.6 • 103 sec

3M~Kc\ReT s

RgTB]

splAHvapR~

rsanw = 3M~ h ( TB - Ts ) TpD

=

1.6 • 104 sec

splR~ =

- -

\ReTs

1.6 • 104 sec

RgTB]

2

SplAHvapRo - 3.7 • 103 sec rPWC= 6 M~ k e ( T B - T S) For a total drying time of Ttot

= (TBLMT ~- TpD + TBLHT ~-

rPWC)/2 = 2.1 • 104 sec or 5.8 hr

This time is long and shows precisely the problem in drying large green bodies. If the green body were 10 cm in diameter r a t h e r t h a n 1.0 cm in diameter then the drying time would be ~20 days. The common solution to long drying times is to dry large green bodies with an increasing drying t e m p e r a t u r e after the constant rate period is over. The idea is to keep the liquid surface temperature high, near that of the boiling point. Care is taken to prevent the liquid surface temperature from exceeding the boiling point so that the gas flow will not exert an internal pressure on the green body and cause it to crack.

14.3.4 Cylinder Drying Similar equations for the time needed to dry a cylindrical green body can be developed using the same methodology just discussed. The results for a cylindrical green body are given in Table 14.3.

14.4 Drying of Flat Plates T A B L E 14.3

703

Evaporative Drying of Cylindrical Green Body

Boundary layer

Pore diffusion

Rate controlling s t e p - - m a s s transfer

t/~rpD= 1 -

t/rBLMT- 1 - \Ro/ sRopl rSLMT =

2M~Kc

\RgTs

In

sR2pl

pS ~

[ pS

\Ro/ +

TPD =

[ pS

4M~D~

RgTB]

\RgTs

pB .~

RgTB]

Rate controlling s t e p - - h e a t transfer

t/~'BLHT-1 - \Ro/

t/rPWC= 1 -- \Ro/ +

In R00

sR2plAHvap

eRoPlAgvap rsanw = 2M~h(Ts - Ts)

rPWC= 4M~k~(TB - Ts)

14.4 D R Y I N G OF F L A T P L A T E S For flat plate geometry, the overall mass trasnfer flux,j (moles/area/ time), resulting from the total partial pressure profile over the boundary layer and across the pores is given by (1

x)-l[PSl

J= -~cc+-~IE \RgT S RgTB]

(14.41)

where x is the length of the open pore over which pore diffusion takes place and D1E is the effective diffusion coefficient for the vaporized solvent 1 in the pores, given by

DiE=

1 +

s/#

(14.42)

where D K is the Knudsen diffusion coefficient for the porous network, DAB is the bulk diffusion coefficient for the gas. The porous network is characterized by the void fraction e and the tortuosity (typically ~: 2.0). If flow is to be considered by Darcy's law, the value of D1E should be replaced by DW(see the discussion surrounding equations 14.25 and 14.27). The partial pressure of the solvent over a solution is given by equation 14.7.

704

Chapter 14 GreenBody Drying

For a plate geometry, the overall heat transfer flux, q (energy/area/ time), resulting from the total temperature profile over the boundary layer and across the pores is given by q =

+

( T ~ - Ts)

(14.43)

where x is the length of the pore over which heat conduction takes place and ko is the effective thermal conductivity of the porous network, given by ks

+

(14.44)

where ks is the thermal conductivity of the solid, kg is the thermal conductivity of the gas in the pores, and s is the void fraction of the porous network. Please note ~b = 1 - s. At steady state, both the mass transfer flux and the heat transfer flux are balanced according to q

= AHva p

*j

(14.45)

giving rise to a "wet bulb" temperature at the liquid interface inside the porous structure. To determine the rate solvent recedes into the pores, d x / d t , an overall mass balance is performed: spl A -dx ~ = _jAM 1

(14.46)

where A is the exposed surface area of the green body, Pl is the density, and Mlw is the molecular weight of the solvent. These expression can be integrated for various rate determining steps. The results of this integration (given in Table 14.4) are always of the form t = r(Xc/Xo )n

(14.47)

where r is the maximum time for this rate determining step and Xo is the half thickness of the plate. With the equations in Table 14.4, all the possible rate controlling steps are established allowing the prediction of the total time, rtot, to dry a green body, which is the sum of all the drying times for the green body, considering only one rate determining step at a time. Ttot-- (TBLMT+ TpD -4- TBLHT+ TPTC)/2

Experimental findings by Castro et al. [25] have shown that the loss of solvent during drying from a compact of silicon powder (D~n = 280 nm (rg = 1.7) heat sunk to give a constant temperature in an atmosphere of N2 flowing at various flow rates follows either a a t or a ~ behavior

705

14.5 Warping and Cracking during Drying TABLE 14.4

Evaporative Drying of a Flat Green Body

Boundary layer

Pore diffusion

Rate controlling s t e p - - m a s s transfer t/rBLMT -- (Xc/XO)

TBLMT =

(

t/'rpD = (Xc/XO) 2

pS

exOPl

M~vKc \RgTs

pf,~ RgT~/

~,X2pl TPD--

\RgVs

RgTB/

Rate controlling s t e p - - h e a t transfer t/rBLHT- (Xc/XO)

SXoplhHvap rBLHT = M~vh(TB - Ts)

t/zPWC = (xc/xo) 2 sx 2oPl A gvap

rPWC = 2M~vkp(T B _ Ts)

Volume fraction profile during constant rate period a (i.e., boundary layer mass or heat transport is the rate determining step)

d)(x , t) = 4'0 -

xojM~v [ x 2 DW 1 - 2 (-1)rico ~ s \(n~xt exp p l D f -~X2oq x 2 6 7r2 ~1 = n x0 /

-

n 27r 2t~

a Cooper, A. R., in "Ceramic Processing Before Firing" (G. Y. Onoda and L. L. Hench, eds.), pp. 261-276. Wiley (Interscience), New York, 1978.

depending on the type of solvent used (see Figure 14.9). These experimental results are in agreement with the equations presented in Table 14.4.

14.5 WARPING AND CRACKING DURING DRYING When the green body is wet by the solvent it has a compressive capillary force which holds it together. During drying this capillary force disappears. With a polymeric binder present, the compressive capillary force is replaced by the cohesive force of the binder, which is usually smaller than the original compressive capillary force of the solvent. In its weakened state during drying, the green body is susceptible to stress that may warp and crack it. Such stress is due to either (1) the capillary pressure gradient, (2) the pressure gradient of the escaping solvent, or (3) the differential thermal expansion of the ceramic because of temperature gradients within the green body.

706 a

__o

'O't

Chapter 14 Green Body Drying

Octan01

10-I

'

'

b

1

o

'0'

I

I

Methanol

.zz~176176 ~ 1 ~ ,~176 10"t

.10-1

10-2

100 101 Time ( minutes ) C tO ~

2

10"10-I

i -

tt) tf) 0

102

.~,-'~_o,t0~ _O _-

i0 0 101 Time (minutes)

I

I

10z

I

Acet0nitrile

~oo

=

u 10t 0

In'2

100

i , i,lilil i 101

i

I , t0 z

1

103

Time (minutes)

FIGURE 14.9

Log-log plot of weight loss versus time for disc shaped Si powder green body made with different solvents and dried at different conditions: (a) solvent-octanol, T = 200~ N2 velocity 0.1 cm/sec; (b) solvent-methanol, T = 80~ N2 velocity 1.3 cm/ sec; (c) solvent-acetonitrile, T = 27~ N2 velocity 0 cm/sec. Taken from Castro et al. [25]. Reprinted by permission of the American Ceramic Society.

Each stage of drying has its own special stresses. During the constant rate period, the flow of liquid induced by a capillary pressure gradient causes stresses to develop in the ceramic green body. During the decreasing rate period, the flow along the pores of vaporized solvent molecules from the interior to exterior of the green body causes a pressure gradient to develop which induces stress. During the cool down after drying the ceramic green body, stress is also built up by contraction of the surface with respect to the center. Each of these forms of stress can be manipulated by the rate at which the green body is dried (or cooled down). If the stress is nonuniform it will warp the green body. The surface of the green body is the most susceptible to

14.5 Warping and Cracking during Drying

707

stress. Tensile stress at the surface is the most difficult to withstand without cracking because ceramic green bodies, even more t h a n fully dense ceramics, are much weaker in tension than in compression. The following analysis of stresses assumes that the green body is purely elastic. Certainly this is not the complete picture because wet, sticky powders or gels are not elastic but plastic, showing deformation of the particle network by the frictional movement of particles against each other. We have discussed these mechanical properties in Sections 12.5 and 13.5 where the plastic nature of cohesive powder packings was discussed. The plastic nature of a green body greatly complicates the analysis of stress and for this reason is not presented here. Two types of stress are important in drying. The first is the total stress, which corresponds to the force per unit area acting on both the liquid and the particle network. When the pores are filled with liquid, the stress is spread evenly over the whole green body, because the essentially incompressible liquid distributes the stress evenly in all directions. The second type of stress is the network stress, which is the force per unit area acting only on the particle network. When we consider the warping and cracking of the particle network, the stress on the particle network is important not the total stress. Stress induces during drying is due to temperature profiles and pressure profiles in the ceramic green body. The effective local stress, (re, is given by [26] O" e - -

O" T %

Pgi - d~l Pci

(14.48)

where o-T is the stress due to the temperature profile within the green body, i is the unit tensor, Pg is the gas pressure, r is the volume fraction of liquid in the green body, and Pc is the capillary pressure. The two pressure terms are multiplied by the unit tensor because pressure is evenly distributed in all directions. The effective local stress is the total stress discussed previously. The net local stress, ~r, arises from the difference between the local effective stress, (re, and the average value of the local effective stress, @, throughout the body [27]; for example, ~r = r - ~

(14.49)

where

~r = ~pp

r dV

(14.50)

The net local stress is the network stress discussed earlier. The stress, ~rT, due to the temperature profile is discussed next and the stress due to the gas pressure, Pg, induced by the flow of volatiles is discussed last. Two types of temperature profile are to be considered:

708

Chapter 14 GreenBody Drying

one is the simple heating or cooling of a green body and the other is due to the endothermic heat of evaporation (or heat of reaction in binder burnout discussed in Chapter 15). The flow of volatiles is caused by a pressure distribution in the green body. The gas pressure distribution can be determined by the flux of volatiles in the green body discussed earlier in this chapter.

14.5.1 Thermal Stress Induced during Drying In this section, we assume that the green body is already dry and the stress is caused by the thermal expansion of the ceramic particles that make up the porous ceramic due to the temperature profile in the green body in either heating or cooling. For an infinite plate of thickness 2x0, the normal stress (r(x) at a position x in the green body depends on the temperature difference between that point, T, and the average temperature, T a . This gives the strain at that point and fixes the net local stress at [28] %(x)

•EoL

= (rz(X) = 1 - v [Ta -

T(x)]

(14.51)

where 6 is the solids fraction, E is Young's modulus of elasticity, with typical values 10 to 50 GPa (Note: the product ~E is the Young's modulus for the porous green body), a is the thermal expansion coefficient, and v is Poisson's ratio, defined as the ratio of the strain in the direction of the applied stress to that perpendicular to the applied stress. For ceramic green bodies that are incompressible v = 1/2, a more typical value is v = 0.28, giving some degree of compressibility. This compressibility can be due to either the ceramic or the polymeric binder within the green body. When the green body is heated, the surface is in compression and the center is in tension, as shown in Figure 14.10. When the green body is cooled, the surface temperature is lower than the average and the surface is in tension ((r+) and the center is in compression ((r-). (Please note that there is a difference in sign convention between pressure, P, and stress, ~, that is, compression, P+, (r-). This condition may cause cracking because ceramic green bodies are very weak in tension. Essentially, the surface, if free, would contract by a certain amount due to cooling, but it is restrained by the center, which remains at the higher temperature, and a tensile stress results at the surface. For stress equilibrium, the surface tensile stress must be balanced by the compressive stress in the interior. For a plate initially at temperature, T i and t = 0, heat is transferred to the surface of the green body from a gas at temperature T~ and the

14.5 Warping and Cracking during Drying

FIGURE 14.10

709

Schematic of temperature and stress distributions for a plate.

temperature profile within the plate is given by the differential equation O T _ ~, 02T c~t C~X2

(14.52)

where ~' (= kp/pC~) is the thermal diffusivity of the green body, p is the density, and C v is the heat capacity of the porous ceramic green body. Using the boundary conditions T ( x , O) = T i uniform,

0T(0, t) = 0 at center line x = 0 0x 0T(xo, t) Ox

(14.53)

h + y-[T(xo, t) - T 0] = 0 a t x = Xo av

where h is the heat transfer coefficient and kp is the effective thermal conductivity of the porous ceramic green body. The solution to this equation is given by T ( x , t) - T b = ~ An exp(- h2 ~' t ) COS(~,nx ) Ti - Tb n= l

(14.54)

where A n are constants given by An _

2 ( T i - T b) sin ~,nXO ~nX o + sin ~nXO COS~nXO

(14.55)

710

Chapter 14 Green Body Drying of slab

atlx~ =

Surface of

~.---.-'~"

1.0.

O~

,~.~

~..,..,.,,,,.~j

0.2

- - ~--

0.6

. . . . .

0.4

rn - 7'0

0.4

-

0,2 0

0.2

~

. . . .

i

J

0

j

9

jy

I

~

0.4

0.5

0.6

0.8

l.O

0.8

l.O

X/Xo

FIGURE 14.11

T r a n s i e n t t e m p e r a t u r e profile in a slab of finite thickness initially at t e m p e r a t u r e T 0. T 1 is the surface t e m p e r a t u r e for t > 0. T a k e n from C a r s l a w and J a e g e r [29, p. 101] by permission of Oxford U n i v e r s i t y Press.

and

An a r e

eigenvalues corresponding to the roots of the equation ~knkp

cot ~nXo = h

(14.56)

This equation gives a temperature profile which is shown in Figure 14.11. The temperature profile in an infinite cylinder and a sphere are shown in Figures 14.12 and 14.13. These graphs come from Carslaw and Jaeger [29] and Heisler [30]. To determine the average temperature as a function of time, the following analysis is performed. For a plate initially at temperature T/ whose surface temperature is changed to T s at t = 0, the average temperature, T a, as a function of time is given by [15, p. 486] T a - T s - 8 ~~ exp[ -(2n+l)27r2~'t] Ti - Ts 7T2 n= O( 2 n + 1)2 4x02

(14.57)

This expression allows the calculation of the difference between the surface temperature and the average temperature, which allows the

~.of

Axis of cylinder

Surface of cylinder

I

~F~~~I ,.

/

0.2

0.8

,!

T-

t~

0.6

0 2 ~ ~

i0.4

~-

,!

"]-" 7 ~

'r,-

7~

i

0.4 ~

,

0.2 ~ - ~ - "

....

:

m~--i 0.04

~

'

0;2

i:

0.8

0.01

0.4

0.6

O.8

l.O 1.0

r/R Transient temperature profile in a cylinder initially at temperature To. T1 is the surface temperature for t > 0. From Carslaw and Jaeger [29, p. 200] by permission of Oxford University Press.

F I G U R E 14.12

Surface of

Center o! sphere

1.0

0 at

0.8

=0.4

-

.~

_._.2~~

t

/i

j r -'~

02

0.6

TI-T

r,-ro

/

.....

0.4

! /

/

0.2

0

02

0.4

~ 0.6

0~ 0.01/

r/R

0.6

0.8

1.0

1.0

Transient temperature profile in a sphere initially at temperature T0. T1 is the surface temperature for t > 0. From Carslaw and Jaeger [29, p. 234] by permission of Oxford University Press.

F I G U R E 14.13

712

Chapter 14

Green Body Drying

calculation of the surface stress, equation 14.51, during heating or cooling. The tensile surface stress during cooling is the most difficult stress for the green body to withstand because it is weakest in tension. During cooling, this stress will cause cracking when it exceeds the strength of the green body, (re. When the temperature distribution is nonsymmetric, the plate will warp during heating or cooling. The curvature of the plate, p, is given by

to- 1

3~T4) fXo T(x ) x dx

~

2x~

(14.58)

-xo

where OLT is the thermal expansion coefficient. For other geometries, the stress and the temperature differences between the surface and the average temperature are given in Table 14.5.

P r o b l e m 14.2 T e m p e r a t u r e D i f f e r e n c e I n d u c e d Tensile Stress Determine the temperature required to induce cracking in a mullite green body in the form of a plate with a void fraction of 0.4 that has a strength of only 200 KPa caused by interparticle van der Waals interactions. The following properties of mullite are needed: O~T -- 5 X 10-~/~ E = 69 GPa [28, pp. 593, 770, 777], v = 0.25.

T A B L E 14.5

Surface S t r e s s e s a and T e m p e r a t u r e Differences b for Various S h a p e s

Stress

Temperature difference

(bEaT

Infinite plate

O'y =

Infinite cylinder

c% = 0

o" z - -

(T a -

~

T s)

Ta-Ts

Ti-

Ts

8~ ~ 1 [-(2n + 1)2~r2a't] 77"2 (2n + 1)2 exp 4x~ n=

Ta-Ts ~4 (--~na't) c T i - T s --n= ~n2exp Ro2

dPEaT 0"0 -- ~ - V (Ta -- Ts) Sphere

Va- Vs T i - Ts

O'r= O

~bEaT

or~ = ~-- v (Ta

-

6 ~01 ( -n2zr2a't ) ~r2n= ~--~exp \ Ro2

Ts)

a Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., "Introduction to Ceramics," 2nd ed., p. 819. Wiley, New York, 1976. b Geiger, G. H., and Poirier, D. R., "Transport Phenomena in Metallurgy," p. 486. Addison-Wesley, Reading, MA, 1973. c ~:n are roots of the equation Jo(x) = 0, where Jo(x) is the zero-order Bessel function.

14.5 Warping and Cracking during Drying

Solution

713

S u b s t i t u t i n g these values into the equation (re = 200 KPa -<- (ry = ~ 1E~ - v AT = 0.6 *69 GPa* 5 • 10-~/~ * 1/0.75 *AT

and solving for A T = ( T a - T s ) , we find the critical h T to be only 0.7~ Therefore, cooling the ceramic green body m u s t be very slow so t h a t no t e m p e r a t u r e gradients larger t h a n this critical value are experienced. This is very difficult to do, but most green bodies contain polymer also. We now consider t h a t the green body contains a polymer (polymethyl methacrylate) at 1% by volume, which enhances its s t r e n g t h to 300 KPa with the mechanical properties of the polymer of ap = 70 • 10-6/~ Ep = 2.24 GPa [31], vp = 0.2. S u b s t i t u t i n g these values into the equation (re = - 3 0 0 KPa -< % = ~p I - vp = 0.01 *2.24 GPa* 70 x 10-~/~ * 1/0.8 *AT and solving for AT = ( T a - T s ) , we find the critical AT to be 38~ So we can see t h a t the polymer present in the green body after drying is very useful in allowing for larger critical values of AT during the cool down.

14.5.2 Flow Stress during Drying 14.5.2.1 C o n s t a n t Rate D r y i n g P e r i o d The w a t e r expansion of a ceramic green body is similar to t h a t of a t h e r m a l expansion of a glass system. The compression of a ceramic green body is shown in Figure 14.5. At low ~, the curve is linear. At high values of~b, there is a critical value, ~b*, where no f u r t h e r shrinkage takes place, corresponding to the volume fraction of liquid j u s t filling the pores where the particle network is rigid. The liquid expansion of a green body, ac, defined by

1( v)

ac = ~

0(1 - ~b)

(14.59)

can be approximately described by ac

= 0 for ~ -< ~b*

ac =

1 r - ~for6 >

or

(14.60)

714

Chapter 14

Green Body Drying

These equations assume that the green body is isotropic (i.e., C~cx = Scy = Scz = 1/3 Sc, where Sex is the linear compressibility in the x direction Sex = 1/Lx [OLx/O(1 - 4) )], for example. With particle segregation and nonspherical particle alignment, anisotropic compressibilities are often observed [10-13] and can be important [14] in fracture and warping. Using an analogy [18] with the thermal stress, liquid flow stress can be determined from a replacement of 1. The solids volume fraction, ~b, for the temperature, T. 2. The liquid expansion coefficient, ac, for the thermal expansion coefficient, S T 3. Thermal diffusivity, ~', with the liquid flow diffusivity, Df. For an infinite plate of thickness 2x 0, the normal stress ~r(x) at a position x in the green body depends on the volume fraction difference between that point, ~b, and the average value, ~a, in the green body. This gives the strain at that point and fixes the net local stress at [28] Oz(X) = O-y(X) - EGBSc [ (~a

1-v

--

~(X)]

(14.61)

where EGBis Young's modulus of elasticity for the green body in this case and v is Poisson's ratio. Furthermore, the flow of liquid to the surface of the green body to keep it wet is governed by the permeability equation for the flow of liquid relative to the ceramic particles [8,9], given by* 0(1 - ~b) _ D [ 02(1 - 4~)) dt

-

f~

-0~

(14.62)

where ~b is the volume fraction of solids and Df is the flow transport coefficient. This flow transport coefficient is a nonlinear function of the volume fraction of solid particles, ~b, as would be expected from permeability considerations, such as Df(dp, x ) = (1 +

1 - ~b)aD(~b = ~b,,x )

but we will not consider this functionality in detail and will use a constant value of DW for further calculations. Cooper [18] has shown that the constant Dw = D(rb = ~b*, x) solution for the tensile strength at the surface of the green body is an upper bound of this value. * See the discussion surrounding equation 14.10.

14.5 Warping and Cracking during Drying

At

a(1

-

dt

the

surface

6)

of the

green

body,

the

time

71

derivative,

, is equal to the drying flux, - j l x o , giving Surf

(d)a- dps)=

(14.63)

jxo 3Dw

To maintain a flow of liquid to the surface at a constant rate, the green body must shrink, expelling liquid. Assuming that the flow diffusivity is a constant for simple analysis of this problem, Cooper [18] was able to determine the drying stress at the surface as a function of the drying flux: E~c (d)~ - d)s) EGB~c ( jXo~ (rz = (ry - l _ v -- 1 - - v - 3Dw]

(14.64)

For other shapes the drying stresses are given in Table 14.6. Scherer [32] has determined that the capillary pressure in the pores of a viscous gel network is not uniform during drying in the constant rate period. This capillary pressure distribution is analogous to the solids volume fraction distribution in particulate green bodies, which is not uniform during the constant rate period. The stress at any point can be determined from the difference in the pressure at that location, P, and the average pressure,
Drying Stresses for the Constant Rate Period

Shape

Surface stress

Infinite plate

Er z --

Er y = ~

Infinite cylinder

Er r =

0

~o=

~

O" r - -

0

Sphere

E~c

~o= ~

( ~) a -- ~)s ) -- ~

Ec( jxo

E~c (

--9Df]

(~a -- ~ s ) = ~

--

(~a -- ~s) = ~

--

l_~r )

716

Chapter 14 Green Body Drying

14.5.2.2 Decreasing Rate Period When the constant rate period is over, vapor phase pore diffusion plays an important role in the cracking of ceramic green bodies. The capillary pressure difference, hPc, between the surface and interior of the green body during drying can be estimated from the flux of solvent, j, giving AR C -

DJ f * R g T* Ax

(14.65)

The capillary pressure difference, APC, can be quite large if the flux, j, is large or if the flow diffusivity, Dw, is small. It should be kept in mind that each cc of H20 in the voids of a ceramic green body is converted into 1357 cc of vapor at 298~ Thus a huge volume of gas must be removed from even small green bodies, creating flow stress. The difference between the pressure, P(x), at a point x and the average pressure, (P), gives the stress at that point; that is, (r(x) = P(x) - (P). If this stress is larger than the strength of the green body (or the critical stress holding the green body together), that is, cr(x) > (re, it will rupture, leaving cracks in the microstructure of the green body. These cracks will release the pressure build-up. If we assume that the partial pressure of the solvent is 0 at the surface of the green body during drying and at the other end of the pore the partial pressure is at its equilibrium value (determined by the temperature and the solution concentration using equation 14.7), then the stress will start to be important at temperatures at the end of the pore above the boiling point of the solution filling the pores, because the partial pressure of the solvent is exponential in temperature. This situation gives a tensile stress at the surface, which may rupture the green body if it is larger than the green body strength, (re: [1 atm] el {exp [AH~ [ Rg

( T~

p

T)]-l}=(r>(rcforT>TBp

(14.66)

Because one atmosphere corresponds to - 100 KPa, a rather large stress for a ceramic to withstand, temperatures exceeding the boiling point of the solution can be expected to cause rupture. As a result temperatures below the boiling point of the solvent are always used for the gentle drying of ceramic green bodies.

14.5.3 Capillary Stress The local effective stress [27] is given by the sum of the gas pressure, P~, the capillary pressure, Pc, and stress within the solid, (rs"

(re = (rs + Pg + Pc

(14.67)

14.5 Warping and Cracking during Drying

71

The gas pressure takes into account the partial pressure of the solvent as well as the pressure of any inert gases like air in the green body. The capillary pressure, Pc = (2~/LAc o s 0 / r p o r e ) , is related to the liquid-solid surface tension, ~/LS(= ~/LVCOS0) and the radius of curvature of the pore, rp, where ~/LVis the surface tension of the solvent at the liquid-vapor interface, 0 is the contact angle of the liquid on the ceramic surface, and ~ is the void fraction of the ceramic. If the liquid fills the pores, it distributes the capillary pressure throughout the green body. If the pores are essentially empty, with the smaller pores filled in relation to the partial pressure profile within the green body, the pore liquid will give a capillary pressure gradient within the green body. This capillary pressure gradient produces a stress which is supported by the particle network. From this analysis, the net stress is given by [26] the difference between the effective stress and the average stress: (r = (re - ((r)

(14.68)

where ((r) is the average stress integrated over the pore volume of the green body, Vp given by ((r) -- ~pp

(9re

dV

(14.69)

The stress within the solid is often assumed to be constant over the green body, thus it disappears from the net stress. If, however, there is a temperature profile in the green body, as during the later stages of drying when the pores are essentially empty, the stress in the solid is important and can be calculated from the elastic modulus, E, of the ceramic solid as shown in equations 14.71 and 14.72 for a sphere. In the initial stage of drying, when the liquid fills the pores and there is a temperature profile in the green body, the thermal expansion of the liquid will add another stress, not listed in equation 14.67, t h a t can be substantial. However, this liquid expansion stress is either (1) negligible because the liquid filling the pores transfers heat relatively fast and eliminates thermal gradients or (2) relieved by flow if the permeability of the particle network is high. For this reason, this liquid expansion stress is not listed in equation 14.67. From the gas pressure which includes the partial pressure profile, the temperature profile, and local capillary pressure which is a function of the volume fraction profile and the liquid radius of curvature profile inside the green body, the average stress can be determined. The partial pressure profile is determined by the flux, j, and the effective pore diffusion coefficient, D1E, a s follows: V P(R~) =

J D1E

(14.70)

718

Chapter 14 Green Body Drying

Higher fluxes and lower effective pore diffusion coefficients will give a higher partial pressure gradient and result in a larger stress due to capillary pressure. The stress in ceramic film during drying have been measured by Chiu and Chima [19]. They have shown that the tensile stress is at its maximum value when the drying front penetrates the surface of the film, and this maximum stress is dependent on the size of the ceramic particles in the film. The total stress on the ceramic green body can be estimated from the temperature stress, pressure stress, and the capillary stress. The total radial and tangential stress at radius r in a spherical green body of radius R is given by [27] ~

-- 1 - v ~

Oro(r) = 1 - v

-if5

~ ( r ) r z d r - -~

(r(r)r z d r

(14.71)

(r(r)r 2 d r + -~

(r(r)r 2 d r - ~ ( r )

(14.72)

Again the product ~bE is the Young's modulus for the dry porous green body. If the tensile stress induced at the surface during drying is larger than strength of the ceramic green body, cracks will develop during drying. In this section of Chapter 14, we have looked at the individual stresses caused by temperature, flow, and capillarity. In a ceramic green body undergoing drying all of these stresses will be operating continuously. As a result, the preceding linearization of stress is applicable for only elastic green bodies.

14.6 CHARACTERIZATION GREEN BODIES

OF

CERAMIC

Ceramic green bodies are typically characterized in detail after drying because it is not necessary to take extraordinary precautions to assure that the green body lose no further weight. Green body characterization is typically done by observing the microstructure of a fracture surface with a scanning electron microscope, measuring the green body strength and determining its average density. In some cases the pore size distribution is also measured. An observation of the microstructure shows 1. How well the particles are packed, 2. The uniformity of packing, 3. The degree of orientation of nonspherical particles,

14.6 Characterization of Ceramic Green Bodies

719

4. The degree of segregation of different sized particles or particles with different shape or density, 5. Whether cracking has been caused by drying the green body. As a result, a picture of the microstructure is a valuable tool in diagnosing drying problems in green body manufacture and predicting problems during binder burnout and sintering.

14.6.1 Green Density The most frequently measured property of a ceramic green body is the average density. In its simplest form, the density is the weight of the green body divided by its volume, including the pore volume. Sometimes the pore volume is measured by the weight gain associated with filling the pores with a liquid of known density. The average density is the usual measure of how well the ceramic powder is packed into the green body. For this reason, the average density is often reported as a fraction of theoretical density, which is also the solids volume fraction, ~, related to the void fraction, e; that is, (b = 1 - ~. The average density is certainly an important property, however, the uniformity of the density and uniformity of the microstructure in the green body are among the most important properties because anisotropic green bodies warp and crack during drying, binder burnout, and especially sintering.

14.6.2 Uniformity of Microstructure Mixedness To assess how uniform the microstructure in a green body is, the variance, (r~b, is used [33]. (rgeb(xl) = 1 ~ (xl,/- 21)2

(14.73)

ni=l

where Xl,i are the measured values of any property [1] (e.g., the density, mean particle size, mean pore size, permeability in one direction, polymer concentration, fraction of one constituent, hardness) at various locations in the green body, n is the total number of samples measured, 2~ is the average value of the same property. To obtain several samples, a green body is often broken or cut into pieces that are then measured separately. For an ideally uniform green body, (r~b(xt) = 0. But this is rarely the case, because the technique used to measure the property experimentally also incurs errors reflected in O"exp 2 (X 1 ) which alters the total variance measured: 2 O'2ot(Xl) = O'exp(Xl) -~ (r~b

(14.74)

720

Chapter 14 GreenBody Drying

2 1 ), is determined by multiple meaThe experimental variance, Gr'exp(X surements of the property Xl on the same sample. When the experimental variance is larger than the variance in the property, (r~b(x1), it is impossible to determine whether stochastic homogeneity exists in the green body. For this case, only an effective homogeneity can be determined.

14.6.2.1 Particle Segregation and Its Uniformity For a two-component system of equal sized spheres mixed in a proportion, X~, the variance of the proportion is given by o-2(Zl) =X~(1 - X l )

(14.75)

when the two components are completely separated. For two-component systems where the two particles are not the same size but the sampling volume Vs is the same, the variance for complete segregation is given by

- Zl)

-

ys

(14.7 )

where j is a number of fractions, defined by Vl = i

i*j = 1, 2, 3 , . . .

(14.77)

v2 j and v i is the volume of particles 1 and 2, respectively. The added term accounts for the sampling probability of a uniform random mixture. This formula can be generalized to multicomponent suspensions. For the special case where the particle volumes vi are integral multiples of the particle volume vl , j v l = v2, as in the aggregation ofmonodisperse spheres, the stochastic homogeneity of a two-component green body is given by el

o'2(-'~1)- -'YI( 1 - X1) ~

(14.78)

which is plotted in Figure 14.14 as a function of the ratio of particle volume to the sample volume, v l / V s , for the case X1 = 0.5. As the ratio of the particle volume to the sample volume goes to a value of 1.0, the variance becomes the value corresponding to complete segregation. As this ratio gets small, the variance goes to 0, the uniform random mixture value. As a result, the sample volume must be considered so that a large number of particles are in each sample volume for accuracy. Again experimental errors may occur. If the experimental variance is larger than the mixing variance, it is impossible to determine whether stochastic homogeneity has been reached. Only the effective state of mixing can be determined under these conditions.

14.6 Characterization of Ceramic Green Bodies

FIGURE 14.14

721

Variance of random mixtures as a function of the sample volume, V.

14.6.3 Green Body Strength 14.6.3.1 Strength of a Wet Green Body When the green body is wet, the compressive stress holding the green body together is given by cos0 ~rc = ~ P c ~

- ~TLA

+

= --

(14.79)

rpore

where TLA is the surface tension of the liquid air interface, r 1 and r 2 are the radii of curvature of the solution in the pores with radius, %, of the ceramic green body as shown in Figure 14.15, 0 is the contact angle between the ceramic and the liquid, and ~ is the void fraction of the green body. This compressive stress must be overcome to break the green body, and for this reason, it is also called the c r i t i c a l s t r e s s intensity

for failure,

~rc .

For a ZrO2 green body composed of 1/zm particles with 8 = 0.4 and a mean pore size of 0.1 ftm filled with water at 25~ (TLA = 72 dynes/cm, 0 = 0~ the stress is - 0 . 5 7 6 MPa (compressive).

14.6.3.2 Strength of a Dry Green B o d y - - t h e van der Waals A p p r o a c h When the green body is dry, it is held together by van der Waals forces, Fvdw, between the ceramic powder particles in the green body. A useful approximation for the van der Waals force between two spherical

722

Chapter 14 Green Body Drying

FIGURE 14.15 Compressive stress caused by the capillary action of liquid filling the pores of a ceramic green body.

particles of radii a l and a2 t h a t can be used for all separation distances, h, is [34] Fvdw = - dVA(h_____))= _

dh

Allala2

6h 2(al + a2)

f(p)

(14.80)

where A is the H a m a k e r constant for the interaction of two particles of the same t y p e over a gap of air h thick and

f ( p ) = 1 + 3.54p

forp < 1

(14.81)

1 + 1.77p

f(P) =

0.98 P

0.434 0.067 2 ~ 3 P P

forp > 1

(14.82)

where p (= 27rh/k) and ~, is the London r e t a r d a t i o n wavelength, which is usually on the order of 100 nm. This expression for the van der Waals force is an approximate form of the derivative with respect to h of the van der Waals interaction energy, VA(h), given in Table 10.1 for two

14.6 Characterization of Ceramic Green Bodies

~

unequal spheres. Rumpf [35] has developed a summation of the van der Waals forces for spherical particles (i.e., al = a2) packed into a generalized structure with a void fraction s. The compressive stress (or green body strength) that results is 1.1(1 - s ) A l l f ( p ) ~c =

(14.83)

12salh 2

This equation shows the important part played by particle packing in the strength holding the green body together. When the void fraction, 8, is small, the green body is strong. It should be noted that the void fraction is lowest when there is a broad distribution of particle sizes such that the fine particles fill the voids between the larger particles. The green body is also strengthened by smaller particles, a l, and shorter distances between particles, h. The distance between particles after drying a ceramic compact will never be exactly 0 (which would give an infinite green body strength) because either some small amount of solvent will likely remain (one monolayer of water gives h ~ 0.2 nm) or some degree of packing imperfections will prevent particle contact. To decrease the interparticle distance, high pressures are used in filter and dry pressing ceramic powders. For a ZrO2 (A~ ~ 10 x 10 -20 Joules) green body with a void fraction of 0.4 with equally sized particles 1.0 ~m in diameter separated on average by a gap of 0.5 nm, the green body compressive stress can be approximated by the preceding equation giving a value of only - 0.116 MPa (compressive). This value is much smaller than the compressive force holding the green body together when it is wet. To compensate for this weakness, a polymeric binder is often used. As a result of the lack of precision on h, equation 14.83 is not generally used. The following work by Kendal is generally used for dry / green bodies. 14.6.3.3 S t r e n g t h of a D r y G r e e n B o d y m t h e Griflith A p p r o a c h A large distribution of green strengths, however, can be found in "identical" dry green bodies. This situation is normal for all materials with defects in their structure. Kendal [36] has developed a model of green strength in accord with Griffith fracture analysis. The surface energy required to separate two deformed spherical particles in contact is rr d2~/gb 4

(14.84)

The spheres of equal radii, r, are deformed by attractive forces and are in contact with grain boundary energy, ~/g~, and d is the diameter

724

Chapter 14

Green Body Drying

of the contact circle given by d =

[187r

11/a p2)2]

--if-r2(1 _

(14.85)

where E is Young's modulus and v is Poisson's ratio of the ceramic powder (typical value 0.28). As a result the surface energy, Vs, to separate the two spheres is

Vs= [5.06Ir5 E2

5 4

Tg~r ( 1 -

p2)2] 1/3

(14.86)

Once the two spheres are separated, they return to their spherical shape. The elastic energy, V~, released to restore them to their original spherical shape is [0.324rr5 ]1/3 V~- b E2 T5br4(1_ p2)2

(14.87)

Subtracting the elastic energy from the surface energy gives the total energy to separate two spheres, VT,

[474 E2

T~br4(1-- v2)2

(14.88)

Considering a green body to be a cubic array of equal spheres, we can calculate the fracture energy, VW,across the (001) plane with its (2r) -2 contacts per unit area: r0.0741r

Vf= L ~

5

Tgsb(1-

/22)2]1/3

(14.89)

Recalculation in terms of the volume fraction solid, 6, gives

Vf =

2.2764

Lr2E2j11/ r T5b

(14.90)

Using Griffith analysis [28, pp. 797, 819] for the crack propagation in the green body, the green strength, (re , is given by

/ (r c = \ ~

/

=

5 \1/3- 1/2 r2E 2

_K1c

(14.91)

where c is the flaw length and K1c is the fracture toughness. This equation shows that, as either the flaw size, c, or the particle size, r, decreases, the green body strength increases; and as the volume fraction of solids increase, the green body strength increases. For a ZrO2 green body (E - 150 GPa [28, p. 777] and ~/g~ - 2.0

14.6 Characterization of Ceramic Green Bodies

725

joule/m 2 [35, p. 13]) made up of 1.0 ftm particles packed at a void fraction of 0.4. The green strength has the following values for various flaw sizes: Flaw size (t~m) Green Strength (MPa) 1 10 100

6.75 2.13 0.67

As the flaw size decreases, the green body becomes stronger. Also note, these Griffith type calculations give higher dry strengths than the van der Waals approach discussed previously. The preceding equation also highlights certain material properties that are important for strong green bodies; namely, Young's modulus and the grain boundary energy. Covalent ceramics (i.e., SiC, Si3N4, B4C) have high values of the grain boundary energy. Flaws in the green body cause weakness. In any ceramic powder process, flaws are invariably introduced [37] by dirt and dust, large aggregates in the ceramic powder, insoluble polymer pieces, drying too fast, or simply rough handling of the green body. These flaws have a distribution of size and, as a result, give rise to a distribution of green body strengths, (re, given by 2

k

f((r c) = ((rclm

(14.92)

\O'o/

where (ro is the characteristic strength and m is the Weibull modulus of the strength distribution. A broad distribution of flaw sizes (low m) gives a broad distribution of green body strengths. In many cases, the Weibull modulus determined on ceramic green bodies made by one process is the same as that determined on the final sintered pieces [35, p. 145] produced suggesting that the flaws present in the green body are not removed or healed during sintering.

14.6.3.4 Strength of a Dry Green Body with Polymeric Binder After drying, a polymeric binder is highly concentrated at the point of interparticle contact in the ceramic green body as shown in Figure 14.4. In this configuration the polymer gives a compressive stress holding the green body together. This compressive force can be estimated by

2d~pTPA

(re = ~

(14.93)

rpolymer

where rpolymer is now the radius of curvature of the polymer surface at the point of contact between two ceramic powder particles, ~/PA, is the

726

Chapter 14 GreenBody Drying

surface tension of the polymer air interface and &pis the volume fraction of the polymer in the green body assumed to be uniformly distributed. Indeed, the effect of polymer volume fraction on compressive green body strength has been shown to be linear for small amounts of polyacrylic acid in BaTiO3 green bodies [38]. For a ZrO2 green body with v = 0.4 and a mean particle size of 1.0 ftm, the polymer will have a much smaller radius of curvature, say, 0.01 ftm. With a polymer surface tension of 0.150 joule/m 2 and a polymer volume fraction of 0.02, the compressive stress is -0.6 MPa (compressive), which is five times the compressive stress without binder and similar to the compressive stress holding the green body together when it was wet with water. This strength with polymer gives a green body that is easily green machined to a net shape. Green body machining is commonly performed in industry with dry pressed forms of simple shape (i.e., cylinders or slabs). The machined green body is of a more complicated shape and is near net shape before it is sintered. The sintered shape requires a minor amount of finishing to meet the final tolerances for the part but that is all.

14.7 S U M M A R Y The drying of ceramic green bodies has been examined in terms of its kinetics. Simultaneous mass and heat transfer equations must be solved to give a complete kinetic picture of the drying process. An analysis of the possible rate determining steps is presented which allows the estimation of the drying time for ceramic green bodies of idealized shapes. After a discussion of drying kinetics, its effects on internal streses in the green body was discussed, giving specific conditions to be avoided during drying to prevent warping and cracking. A short discussion on ceramic green body characterization and strength complete this chapter.

Problems 1. Determine the temperature difference necessary to prevent cracking of a spherical green body when it is cooled down from 90 to 25~ in air if the green body strength is only 500 KPa. 2. Determine the time to dry in 10% relative humidity air at 100~ a spherical green body of radius 10 cm with a void fraction of 0.4 filled with H20. Assume that boundary layer mass or heat transfer is the rate limiting step. The temperature of the surface is the wet bulb temperature. This time is the maximum time for the constant rate period. Data: mean particle size = 1.0 ftm; mean pore diameter = 0.1 ftm. See Problem 14.1 for drying data on water and air.

References

727

3. For the constant rate period in the preceding problem, determine the tensile stress at the surface of the green body. 4. Determine the Griffith strength of an absolutely dry A1203 green body without binder. E = 100 GPA and Tgb = 6.0 • 10 -s joule/m 2, s = 0.3, r = 1.0 ftm, c = 20.0 fern. 5. A two-component ceramic green body made up of 5% by volume 0.1 ftm A1203 platelets and 0.5 ftm TiN spherical powder (chosen because these two particles have the same volume) breaks into seven equal pieces during drying. Each of the pieces (10 cc) is analyzed for its density. The densities are 2.10, 2.30, 2.03, 2.13, 2.31, 2.22, and 2.26 gm/cc. With one piece you perform the density measurement five times, giving values of 2.10, 2.11, 2.13, 2.07, and 2.09 gm/cc. Are the differences in density in the green body significant and, if so, could the differences in density be the reason that the green body broke during drying? Give the reasoning for your conclusions. Data: pA1203 = 3.97 gm/cc, pTiN = 5.22 gm/cc.

References 1. Hiemenz, P. C., "Principles of Colloid and Surface Chemistry," 2nd ed., p. 303. Dekker, New York, 1986. 2. Shaw, T. M., Mater. Res. Soc. Symp. 73, 215-223 (1986). 3. Brinker, C. J., and Scherer, G. W., "Sol-Gel Science," p. 472. Academic Press, San Diego, CA, 1990. 4. Shaw, T. M., Phys. Rev. Lett. 59(15), 1671-1674 (1987). 5. Castro, D., Ring, T. A., and Haggerty, J. S., Adv. Ceram. Mater. 2, 162-166 (1987). 6. Ergun, S., Chem. Eng. Prog. 48, 93 (1952). 7. Carmen, P. C., Trans. Inst. Chem. Eng. 15, 150-166 (1937). 8. Philip, J. R., Soil Sci. 117, 257-262 (1974). 9. Philip, J. R., and Knight, J. H., Soil Sci. 117, 1-13 (1974). 10. Macey, H. H., Trans. Br. Ceram. Soc. 38, 464 (1939). 11. Macey, H. H., and Wilde, F. G., Trans. Br. Ceram. Soc. 43, 93 (1944). 12. Packard, R. Q., J. Am. Ceram. Soc. 50, 223 (1963). 13. Macey, H. H., Trans. Br. Ceram. Soc. 46, 207 (1947). 14. Ford, R. W., "Institute of Ceramics Textbook. Series 3. Drying." McLaren, London, 1964. 15. Geiger, G. H., and Poirier, D. R., "Transport Phenomena in Metallurgy." AddisonWesley, Reading, MA, 1973. 16. Comings, E. W., and Sherwood, T. K., Ind. Eng. Chem. 26, 1096 (1934). 17. Perry, R. H., and Chilton, C. H., "Chemical Engineer's Handbook," 5th ed., pp. 12-13. McGraw-Hill, New York, 1973. 18. Cooper, A. R., in "Ceramic Processing Before Firing" (G. Y. Onoda and L. L. Hench, eds.), pp. 261-276. Wiley (Interscience), New York, 1978. 19. Chiu, R. C., and Cima, M. J., in "Ceramic Powder Science IV" (S. Hirano, G. L. Mesing, and H. Hausner, ed.), p. 347. Am. Ceram. Soc., Westerville, OH, 1991. 20. Whitaker, S., Adv. Heat Transfer 13, 119-203 (1977). 21. Whitaker, S., in "Advances in Drying" (A. S. Mujumdar, ed.), Vol. 1, pp. 23-61. Hemisphere, New York, 1980.

728

Chapter 14

Green Body Drying

22. Wei, C. K., Davis, H. T., Davis, E. A., and Gordon, J., AIChE J. 31(8), 1338-1348 (1985). 23. Wei, C. K., Davis, H. T., Davis, E. A., and Gordon, J.,AIChE J. 31(5), 842-848 (1985). 24. Knudsen, M., "The Kinetic Theory of Gases." Methuen, London, 1934. 25. Castro, D., Ring, T. A., and Haggerty, J. S., Adv. Ceram. Mater. 3, 162-166 (1988). 26. McTigue, D. F., Wilson, R. K., and Nunziato, J. W., in "Mechanics of Granular Materials: New Model and Constitutive Relations" (J. T. Jenkins and M. Stake, eds.), pp. 195-210. Elsevier, Amsterdam, 1983. 27. Timoshenko, S. P., and Goodier, N. J., "Theory of Elasticity." McGraw-Hill, New York, 1970. 28. Kingery, W. D., Bowen, H. K., and Uhlmann, D. R. "Introduction to Ceramics," 2nd ed., p. 819. Wiley, New York, 1976. 29. Carslaw, H. S., and Jaeger, J. C., "Conduction of Heat in Solids," 2nd ed., p. 491. Oxford Univ. Press, Oxford, 1959. 30. Heisler, M. P., Trans. ASME 69, 227-236 (1947). 31. Callister, W. D., "Materials Science and Engineering: An Introduction," 3rd ed., p. 769. Wiley, New York, 1994. 32. Scherer, G. W., J. Non-Cryst. Solids 109, 171-182 (1989). 33. Sommer, K., and Rumpf, H., in "Ceramic Processing Science Before Firing" (G. Onoda and L. Hench, eds.), Chapter 20. Wiley, New York, 1978. 34. Schenkel, J. M., and Kitchener, J. A., Trans. Faraday Soc. 56, 161(1960). 35. McColm, I. J., and Clark, N. J., "Forming, Shaping and Working of High-Performance Ceramics," p. 142. Blackie, London, 1988. 36. Kendal, K., AIP Conf. Proc. 107, 78-88 (1984). 37. Chapell, J., Ring, T. A., and Birchall, J. D., Proc. Br. Ceram. Soc. 38, 49-57 (1986). 38. Chen, Z-C., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Fed6rale de Lausanne, Lausanne, Switzerland (1992). _

15

Binder Burnout

15.1 O B J E C T I V E S The polymers that are used to give a ceramic green body strength when the green body is dry must be burned out before sintering because they would decompose in an uncontrolled manner at sintering temperatures, giving off huge volumes of gas at high pressure that will cause the green body to crack. Binder burnout is performed at temperatures between 300 and 700~ much below the temperatures used for sintering. During binder burnout, the polymer undergoes a controlled thermal decomposition reaction that can take several forms. In general, thermal decomposition of polymers form both volatile and solid residues as products of the reaction. The solid residues react further at higher temperatures to give subsequent volatile products and other solid residues. The kinetics of binder burnout is discussed in terms of the thermal decomposition reaction kinetics, as well as the kinetics of mass transfer for the volatiles and the heat transfer required to supply the heat of reaction. 729

730

Chapter 15 Binder Burnout

A small amount of polymer can produce an enormous volume of gas, which must be removed from the porous green body. The flow of this gas in the porous network of the ceramic green body can create a pressure build-up which puts stress on the green body. The stress induced by flow and temperature gradients in the green body are also discussed, so that binder burnout conditions can be selected to prevent cracking of the green body.

15.2 I N T R O D U C T I O N Binders play several roles in green body manufacture. In the simplest case of suspension casting, the binder is used to hold together the ceramic green body when it has been dried. For dry pressing, the binder is also used to give green body strength after pressing, as well as particle lubrication during pressing. For injection molding and tape casting of flexible tape, more complex binder systems are used, consisting of a polymeric binder and a plasticizer. The plasticizer is essentially a solvent for the polymer to give interparticle movement (i.e., flexibility) and lubrication and flow ability in injection molding. The movement of plasticizer during binder removal is similar to the movement of solvent during drying. As a result capillary flow and stresses develop in the green body. Plasticizer capillary flow and the resulting stresses are not discussed in this chapter. Only the simple case of the removal of one polymeric binder at a time is considered in this chapter because it in itself is sufficiently complex. To remove the polymeric binder, the green body is heated often in an oxidizing atmosphere. During this process the polymer degrades along many possible degradation pathways, which include (1) scission of the main chain, (2) reaction with side chains and substituents, and (3) cross-linking and cyclization ending in carbon formation. In some cases, the oxidizing atmosphere plays a role in the reaction (e.g., oxidative cross-linking and oxidation of residual carbon) and in others it does not. These degradation reactions are either exothermic or endothermic, requiring energy to be transported to or from the reaction site. The degradation reactions produce volatile products which must diffuse from the reaction site through a porous network of the ceramic particles to the bulk gas in the furnace. Deep within the green body, the oxygen partial pressure is frequently small because it often cannot diffuse fast enough to keep up with the reactions taking place. For this reason, even under an oxidizing atmosphere, the center of the green body often undergoes degradation reactions with reducing conditions operating. All specific degradation reactions can be written in the following gen-

15.2 Introduction

731

eral way: bP(s) + nO2(g)--~ vV(g) + sS(s)

+ Agrx n

where b moles of polymer, P, react with n moles of 02 to produce v moles of volatile(s), V, possibly of different types, and s moles of solid residue(s), S, possibly of different types; AH~x~ is the enthalpy of reaction. Under reducing conditions, the number of moles of 02 is zero and V is typically fragments of the original polymer including monomers, low molecular weight polymers, and H 2 and CO as well as other more complex molecules. For oxidation reactions, AHrxn is often exothermic (-). For depolymerization and other chain scissions under reducing conditions, AH=~ is often endothermic (+). At equilibrium, this polymer degradation reaction has the following partial pressure equilibrium constant: (Pv) v

gp -[Po2]n

(15.1)

where Pv is the partial pressure of volatiles and Po2 is the partial pressure of oxygen. Under nonequilibrium conditions, the kinetics of this polymer degradation reaction will depend upon the polymer, the ceramic, the atmosphere, and the temperature. The kinetics of polymer degradation are discussed in terms of thermal and oxidative degradation later in this chapter. For now we will discuss the overall heat and mass transfer occurring in the green body during binder burnout.

15.2.1 H e a t Transfer The flux of heat, q, to a point in a green body is given by the flux at the surface due to boundary layer heat transfer: q = Q / A = h(TB - Ts)

(15.2)

where A is the area of the green body at a surface temperature T s exposed to the bulk gas at temperature TB and h is the heat transfer coefficient, which is a function of the gas flow rate. The heat flux inside the green body is due to conduction in a porous network: q = -kp VT

(15.3)

where VT is the temperature gradient and kp is the effective thermal conductivity of the porous ceramic, given by kp =

[

(1 - e) + e ks

(15.4)

732

Chapter 15 Binder Burnout

where s is the void fraction of the ceramic particles, ks is the thermal conductivity of the solid and kg is the thermal conductivity of the gas in the pores.

15.2.2 Mass Transfer The mass flux, j, is related to the heat transfer flux, q, required to react those molecules at the reaction front by the following expression:

q = hHrxn *j

(15.5)

where AHrxn is the enthalpy of reaction. This relation requires that the two differential equations for the two fluxes be linked for their simultaneous solution. The mass transfer flux, j, at the surface of the green body is given by

J = KC\RgTs

RgTB]

(15.6)

where K~ is the mass transfer coefficient, p s is the partial pressure of the diffusing species, i, at the green body surface, pS is its partial pressure in the bulk gas. The mass transfer coefficient, Kc, is determined from the molecular diffusion coefficient for the diffusing species in the gas mixture, DAB, the geometry of the system and the flow rate of the bulk gas. This mass flux equation can be used for a volatile species, leaving the green body as well as that of oxygen entering the green body. The mass flux inside the green body is due to diffusion in a porous network: J = -Dp V P(RJ)

(15.7)

where V(Pi/RgT) is the gradient of the concentration (Pi/RgT), where Pi is the partial pressure of the diffusing species, i; Dp is the effective diffusion coefficient for the porous ceramic given by [1] Dp =

1 +

~/~:

(15.8)

where DK(= rp X/18R~T/TrMw) is the Knudsen diffusion coefficient for the porous network [2] with pores of radius rp, DiB is the molecular diffusion coefficient for the diffusing species in the bulk gas inside the pore; and ~ is the tortuosity of the porous network (typically ~2.0). Such an equation (5.7) was also derived by Calvert and Cima [3]. Sometimes the flow of large volumes of gas limits the mass flux inside

15.3 Thermal Degradation of Polymers

733

the green body. In this case, a pressure drop may build up and Darcy's law should be used for the gas flow through the pores

j = -DfV P

(15.9)

where D f is the flow diffusivity given by Darcy's law [= e3/(cl ~ S~ (1 e)2), cl is a constant (= 4.2 [4] or 5.0 [5]), So is the surface area of the particles per unit volume of the particles (So = 3/r for a spherical particles), e is the void fraction of the green body, and ~ is the viscosity of the gas in the pores]. The equation for pore diffusion and flow have the same mathematical form (i.e.,j = - D V P). For multiple species in three dimensions, these simultaneous differential equations for heat and mass transport are very complex and require numerical solution [6,7]. As a result of this complexity, we will discuss only simple geometries in the balance of this chapter and determine the rate limiting steps for these geometries. But, first, we present a detailed discussion of polymer degradation reactions.

15.3 T H E R M A L D E G R A D A T I O N OF P O L Y M E R S In general, many mechanisms are available for thermal decomposition of polymers used for binders in ceramics. These general mechanisms are outlined in Figures 15.1 and 15.2. One of these main thermal

Decrease M w Breaking

Monomer Volatile Formation i

Main Chain

n-mers

Reaction Cyclization Increase Mw Cross - Linking

Carbon Formation

Gel Formation ii

FIGURE 15.1 Main chain reaction p a t h w a y s [14].

734

Chapter 15 Binder Burnout

Volatile Formation Elimination

Main Chain Scission

Side Chain (or

Main Chain

Cyclization then

Substituent) Reaction

Cross -Linking

Carbon Formation

Unsaturation Cyclization then Carbon Formation FIGURE 15.2

Side chain reaction pathways [14].

d e g r a d a t i o n r o u t e s c o n c e r n s r e a c t i o n s of t h e m a i n chain. T h e m a i n c h a i n c a n e i t h e r b r e a k by c h a i n scission or c r o s s - l i n k to a n o t h e r chain. S c i s s i o n l e a d s to a d e c r e a s e in t h e m o l e c u l a r w e i g h t , M w , of t h e p o l y m e r r e s i d u e a n d volatile f o r m a t i o n . T h e v o l a t i l e s c a n be m o n o m e r s , dimm e r s , a n d so forth. P o l y m e r s t h a t u n d e r g o c h a i n scission (or d e p o l y m e r ization) a r e d e s i r a b l e for c e r a m i c b i n d e r s b e c a u s e t h e y b u r n o u t c l e a n l y w i t h little c a r b o n r e s i d u e . A p a r t i a l list of p o l y m e r s t h a t u n d e r g o depol y m e r i z a t i o n u p o n t h e r m a l d e g r a d a t i o n a r e g i v e n in T a b l e 15.1. Crossl i n k i n g c a u s e s a n i n c r e a s e in m o l e c u l a r w e i g h t , w h i c h l e a d s to cyclization, u n s a t u r a t i o n of c a r b o n b o n d s , g r a p h i t i z a t i o n , a n d finally c a r b o n f o r m a t i o n . V o l a t i l e s m a y well be f o r m e d d u r i n g t h e s e s t e p s b u t t h e

TABLE 15.1 List of Polymers That Depolymerize during Thermal Degradation a

Polymer

Volatiles Monomers (%)

Methylmethacrylate Methyl-a-phenylacrylate n-Butylmethacrylate Styrene a-Methylstyrene Acrylic acid

100 45 50 45b 45 45

T(~

Mechanism

275 + 340 -250 >300

EI + CS RI + CS EI + CS WLS RI + CS RI + CS

350

Notes: EI = end initiation, RCS = random chain scission, WLS = weak link scission. FRI = free radical initiation, RI = random initiation, CS = chain scission. a David, C., in "Comprehensive Chemical Kinetics--Degradation of Polymers" (C. H. Gamford and C. F. H. Tipper, eds.), Vol. 14, Chapter 1. Elsevier, New York, 1975. b The balance of the volatiles are dimer, trimer, tetramer and traces of pentamer in decreasing amounts.

15.3 Thermal Degradation of Polymers

~~

molecular weight of the residue increases. In a reducing atmosphere, carbon is stable to 3652~ where it sublimes, making carbon very difficult to remove before sintering of most oxide ceramics. This leads to carbon impurities in the final ceramic. In an oxidizing atmosphere, carbon will readily react with oxygen at temperatures <800~ For this reason, oxidizing atmospheres are often used for binder burnout. The other major mechanism of polymer thermal degradation is by side chain (or substituent) reaction, shown in Figure 15.2. These reactions can either cause the elimination of the side chain or the substituent or cyclization with the main chain. Upon elimination, volatiles are formed and the new polymer can react further by (1) main chain scission producing more volatiles or by (2) main chain cross-linking, which leads to cyclization, unsaturation, graphitization, and finally carbon formation. A reaction of the side chain with the main chain causes cyclization, which leads to graphitization and finally carbon formation. A complete picture of polymer thermal degradation is clouded because multiple reaction mechanisms can be operable for a single polymer at a single temperature, leading to a host of volatile products and residues. Therefore, one has to consider the relative rates of these competing reactions to establish an optimized and controlled binder burnout. With common ceramic binder systems, there are dispersants and plasticizers as well as the binder. Even though the dispersant and plasticizer are used in small concentrations compared to the binder, they often play a significant role in binder burnout. The plasticizer is a low molecular weight material which lowers the glass transition temperature of the polymeric binder. It will tend to soften the polymer, causing it to flow in a way similar to the flow of solvent during the drying of the ceramic green body discussed in Chapter 14. This flow is a result of the organization of the binder system due to capillary forces present in the ceramic green body. As a result, this flow will be caused by the surface tension of the binder system and slowed down by the viscosity of the binder system. Both the viscosity and the surface tension of the binder system depend on the composition of the binder system and this composition is likely to vary from point to point within the green body due to polymer degradation and volitilization. In addition the plasticizer, due to its low molecular weight, will tend to evaporate from the binder system at elevated temperatures like the solvent during drying. The partial pressure of the plasticizer is a complicated function of temperature and composition. The partial pressure of plasticizer, p~, is given by

(

[AH~ 1 Rg ~p

p0 ( T ) = [1 atm] a l exp [

T)]

(15.10)

736

Chapter 15 Binder Burnout

where AH0ap is the enthalpy of evaporation and TBp is the normal boiling point of the plasticizer and a l [= Yl x l] is the activity of the plasticizer in the binder system. For an ideal solution, the value of the activity coefficient y~ is 1.0 and a~ = x~ = 1 - xz. For nonideal solutions y~ can be greater than 1, resulting in positive deviations, or less than 1, resulting in negative deviations from ideality. Plasticizer evaporation can be treated in a way similar to solvent removal by drying (as discussed in Chapter 14), using simultaneous heat and mass transfer to establish the rate determining step for plasticizer removal. The removal of solvent-type plasticizers was found to be a function of their vapor pressure in the system [8]. Another aspect of binder burnout is the flow of polymer that results when the polymer melts during binder burnout. As the temperature is increased, some binders (or more specifically binders with plasticizers) will melt. The melting point is a function of the composition of the binder system. When liquid, capillary forces will cause the binder to redistribute itself within the ceramic green body. This flow is caused by the pressure difference, AP, due to differences in the radius of curvature, ri, of the binder at various locations within the porous ceramic green body (i.e., A P = 2y (1/rl - l/r2)). The surface tension of the binder, y, depends on the composition of the binder (i.e., the amount of plasticizer and other additives in the binder), which is also depends on the location within the ceramic green body due to partial volatilization of the plasticizer. Cima et al. [9] argued that enhancement of capillary forces during binder burnout may provide certain processing advantages. The binder can be removed homogeneously from the green body when capillary forces draw melt to the surface; thus, large gradients in stress are avoided. In a related method of binder removal, German [10] suggests that the binder may be removed from the green body by wicking the binder into a porous mold in the same way that suspensions are solid cast as discussed in Section 13.3.1. Here the green body is placed on a porous plate and the temperature is increased until the binder system melts. With binder wicking there are two resistance to flow: the green body and the porous plate [11]. Considering only one porous medium, the flow rate, Q, through porous media of area, A, and thickness, x, caused by a pressure drop, AP, is given by Darcy's law [1,4]:

AP

L -~?aVo

(15.11)

where a is the specific resistance to flow equal to the reciprocal of the permeability (equation 13.3 defines a in terms of particle size and void fraction, equation 14.8b defines the permeability, ap(= 1/a), ~ is the

15.3 Thermal Degradation of Polymers

73 7

viscosity of the melted binder system, and Vo (= Q/A) is the superficial velocity of the flowing binder. Using this filtration concept with two resistance to flow, Darcy's law becomes AP T L - V(ac + am)Vo

(15.12)

where ac is. the specific resistance of the cake and a~ is the specific resistance of the mold. The total hydrostatic pressure, APT, is given as a sum of the applied pressure, AP, and the suction pressure of the mold, as follows:

1)] where p is the binder density, g is the acceleration due to gravity, h is the height of binder above the surface of the mold, T is the interfacial surface tension between the pore wall and the melted binder with all of its additives including dispersants and wetting agents, and ri is the radii of curvature. Subscript 1 refers to the mean pore radius of the mold, and subscript 2 refers to the mean pore radius of the green body. Several different polymer loading microstructures could develop [13], as burnout proceeds in a ceramic green body containing low molecular weight plasticizer and polymer binders. Initially, the organic content can range from a fully loaded system, in which the organic completely fills the interstices between the ceramic particles. This is the case for injection molded ceramic parts. At the other extreme, after binder removal, we have a system in which the organic phase has been completely removed. The intermediate microstructures can show the transition from isolated or closed porosity to interconnected or open porosity. Alternatively, other possible microstructures [3] might appear for polymers which do not melt during burnout. Long connected pores may develop, and the polymer may break up into "islands" and "peninsulas," which are distributed throughout the green body. These inhomogeneities lead to nonuniform packing which may cause the ceramic green body to crack during sintering. Extensive porosity develops throughout the green body in the early stages of burnout.

15.3.1 R e a c t i o n K i n e t i c s This section draws upon the excellent work by C. David [14] on the thermal degradation of polymers. A general kinetic expression for the

738

Chapter 15 Binder Burnout

rate at which a bulk polymer degrades is given by [15] dPndt

- k s ( n - 1)Pn -- kEPn - k l ( R * / V ) ( n - 1)P n

(15.14)

oo

+ kl ( R * / V )

~ Pj + k l ( R n / V ) ~ n P n § kT~ n j=n+l n=L

where the terms on the right-hand side show the rate of polymer and 1. 2. 3. 4. 5.

disappearance by random scission initiation: - k s ( n - 1)Pn disappearance by an end group initiation: - k E P n disappearance by transfer initiation: - k l ( R * / V ) ( n - 1)P n appearance by transfer o f P j , + k ~ ( R * / Y ) + 1 Pj appearance by termination of a radical by transfer of

~j~=n

Pn" + k l ( R n / V ) ~:=L nPn

6. appearance by other termination reactions: +k T a n . Pn is the number of polymer molecules of degree of polymerization n, R* is the number of radicals found in a volume V, R n is the number of

polymer radicals with degree of polymerization n found in a volume, V. For other definitions, please use the nomenclature associated with Table 15.2. Noting equation 15.14, the kinetics of polymer degradation are very complex. Only the most simple mechanisms have been thoroughly researched. These simplified reactions presented in Table 15.2 are sometimes zero order, more frequently first order, and infrequently second order in polymer mass. These simplified rate expressions are typically used to model binder burnout.

15.3.2 P o l y m e r R e s i d u e s a n d Volatiles Depending on the mechanism of polymer degradation, the volatiles and residues will vary. The average molecular weight of the residue as a function of time is given in Table 15.2. For degradations that increase the molecular weight of the residue, the volatiles will be lower molecular weight species, including alcohols, ketones, carbon dioxide, carbon monoxide, hydrogen, and water. The final residue will be carbon under reducing conditions. For degradations that decrease the molecular weight of the residue, the volatiles will be n-mers of the polymer due to chain scission and other low molecular weight species.

15.4 OXIDATIVE P O L Y M E R D E G R A D A T I O N The exposure of a polymer to an oxygen atmosphere at room temperature is characterized by an induction period in which the polymer shows

739

15.4 Oxidative Polymer Degradation

no signs of oxygen absorption. This period is nonetheless an important step in polymer oxidation because small amounts of hydroperoxides are formed and initiate subsequent rapid polymer auto-oxidation. Increasing the temperature reduces the induction period and accelerates the auto-oxidation. When the polymer contains trace amounts of peroxide, metallic salts, or metal oxide particles, the induction period is often too short to be observed and the process of catalytic oxidation begins immediately. The reaction of hydroperoxides (also called free radicals) is commonly recognized as the process responsible for the further rapid oxidation [16]. A general mechanism of polymer oxidation is the following.

Initiation

Initiation by the formation of polymer radicals is P H --> P* + H*

where P H is the polymer and P* is the polymer radical which is necessary for subsequent rapid oxidation of the polymer. This initiation process is induced by oxygen at increased temperatures and enhanced by catalysis from ceramic surfaces. Initiation by reaction with free radical, R*, is P H + R * - - . P* + R H

plus free radical reaction with oxygen to produce highly active peroxy radicals, ROO*: R* + 02 --* ROO*

Propagation Reactions Forming Oxidized Polymers action proceeds by a host of pathways: P* + O2--* POO* POO* + PH--* P O O H + P* POOH ~ PO* + HO* POOH + RO* --* POO* + ROH POOH + HO*--* POO* + H20 PO* + PH--* POH + P* POOH--* PO + HO* 2 PO* --* 2 PO

Chain Scission PnO* --> Pn-,nO + P*

The re-

T A B L E 15.2

Summary of Thermal Decomposition Reactions and Kinetics for Polymers at Long and Short Zip Lengths

z~ length

Termination

1. R a n d o m initiation First order

Short 1/7 0 < x ~

Disproportionation

Initial molecular weight distribution

Insensitive to initial dist.

Insensitive to initial dist.

Weight loss rate

dM 1 (1) ~ = -2ks Ml"

First-order only until x no

Early stages

longer > 1/7 (early stages only) First-order until x no longer > 1/7 (early stages only) First-order only until x no longer > 1/7 Fairly wide range

d~

(1) = -2ks

Early stages Recombination

Most probable

_~=_4ks(1)M1 Over fairly wide range

Long >x o

Mono

Most probable

2. E n d initiation Short First order 1/7 o < x 0

Mono

Most probable

Order

d M l _ ksxOM1 dt First order dM 1 = -2ksxM 1

dM1 dt dM1 _ dt

k_E1M x 7 1 k___E1 M x ~7 1

Ml"

Dependence of initial rate on molecular weight

Rate constant for initial weight loss / 1 dMl~ "~'~1 - - ~ ]t=ol

Independent

~ 2ks

Independent

2ksk P kT

/ ksm o 2k s l = 2kP ~ ] ~

Independent

Molecular weight against time

Molecular weight vs. conversion

l_l=kst x x0

Drops rapidly until x ~ 1/7

1 _ 1 = ks t x x~

Drops rapidly until x ~ 1/7

In x/x ~ = k s t

~ o ) 2 = M__~ 1 Relative d.p. always greater than rel. wt Stays constant

MO

~] 2kTdo

First order

Proportional to x ~

ksx ~

x=x 0

x changes slowly, not quite first order

Proportional to x ~

2ksx ~

1 x

Zero order until x no longer > 1/7

Inversely proportional to x ~

kE 1 x~7

kEkp X~

x - x 0 = _k E 1 t

First order

Inversely proportional to x ~

kE 1 xoT

kEkp x~

x = x0

1 x o - kst

Falls slowly

7

x xo

M1 M o'

while x > 1/7 Stays constant

Disproportionation

Mono

Most probable

Recombination

Long 1/70 > x ~

Most probable

Insensitive to initial dist.

dM1 kE 1 M dt x 7 1 Until x no longer > 1/7 dM 1 dt 1 M1

dM 1 dt

kE 1M x7 1 dM1 dt

kE/2

x changes, not first order

Inversely proportional to X/x-6

kE 1 /. kEmo -~ ~ = kp ~]2kTd~176

~r

V~x ~

First order

Inversely proportional to k/rx"5

kE 1 /i kEmo -~-~=kp ~]2kTd~176

x = x~

Stays constant

First order (short zip is maintained, x increases) First order

Independent

kE/2

In x/x ~ = 1/4kEt

x increases rapidly

Independent

kE

x = x~

-

-kE =

while x > 1/7

t 27~

x M1 -- = --, x~ M ~ while 1/7 < x

x"-'~ = \M-T] Stays constant

kEM1

Taken from David, C., in "Comprehensive Chemical Kinetics--Degradation of Polymers" (C. H. Gamford and C. F. H. Tipper, eds.), Vol. 14, Chapter 1. Elsevier, New York, 1975. Terminology is attached List of Symbols: an parameter specifying type of termination. parameter specifying type of termination. 7, 7(x) reciprocal of the average zip length between initiation and termination or transfer, which is a function of x through R(x) in the presence of end group initiation. r k l R / k E , transfer parameter. do sample density. ks rate coefficient for chain scission. kE rate coefficient for end group. kI rate coefficient for intermolecular chain transfer. kp rate coefficient for propagation (unzipping). kT rate coefficient for termination. L smallest degree of polymerization of molecules not volatile in sample. m0 molecular weight of a repeat unit. Mo total number of polymer molecules ~: Pn (zeroth moment of molecular weight distribution). M1 total number of repeat units in sample ~: Pn ; when multiplied by m 0 it is the same weight; first moment of molecular weight distribution. Pn number of polymer molecules of degree of polymerization n. Rn number of polymer radicals of degree of polymerization n. R total number of radicals. R, R(x) total radical concentration which may be a function ofx in the presence of end group initiation. V sample volume. x number-average degree of polymerization. Superscript zero (o) indicates initial values; that is, x ~ is the initial number average degree of polymerization.

742

Chapter 15 Binder Burnout

Termination Reactions Giving Cross-Linking or Cyclization The reaction sequence finishes by a terminal reaction which eliminates two radicals. P O 0 * + P* PO0* + R* PO* + R* PO* + RO* P* + R O 0 *

~ ~ ~ --. --.

POOP POOR POR POOR POOR

These products may be either volatile or a residue.

Metal Catalyzed Oxidation of Polymers The effects of metals and metal oxides on the oxidation is summarized in a review paper by Babek [16]. Metals are often present as impurities in the polymers as a result of catalysts used in their manufacture. In other cases, the metal oxide surfaces of the ceramic powders in the green body are present. In polymers, metal oxides and metal hydroxides are used as fillers for flame r e t a r d a n t additives. Metals and metal oxides play an important role in the degradation mechanism. Hansen et al. [17] and Meltzer et al. [18] found t h a t copper is an active catalyst for the oxidation of polypropylene. In the presence of copper, the induction period is absent and the oxidation is autocatalytic. These experiments suggest that cuprous oxide is responsible for the catalysis, not metallic copper or cupric oxide. A number of workers [19-22] have reported that TiO2 greatly increases the rate of oxidative photo degradation of nylon-6,6 in the presence of oxygen and moisture. TiO2 may enhance the free radical-peroxide formation in these systems. It was also found that the ZnO catalyzes the formation and decomposition of peroxides under the influence of UV radiation [23]. Metals can also accelerate the rate of propagation in oxidation. Ozawa et al. [24] has shown that powdered polypropylene oxidizes in the presence of metallic salts in the following order of catalytic reactivity. Co > Mn > Cu > Fe > V > Ni > Ti > A1 > Mg > Ba The effect of various metals on the rate of oxidation and on the induction time is shown in Table 15.3. This table suggests the following general rules: 1. Metals in which a one electron transfer occurs during a redox reaction are more active (e.g., Co, Cu, Mn, Ce, and Fe). 2. Metals in which a two electron transfer occurs during a redox reaction are less active (e.g., Pg and Sn). 3. Metals in which no electron transfer occurs in a redox reaction are normally inactive (e.g., Na).

15.4 Oxidative Polymer Degradation T A B L E 15.3 Polymers a

~4~

C a t a l y t i c A c t i v i t y of V a r i o u s O x i d e s on t h e O x i d a t i o n of D i f f e r e n t

Type of polymer cis-Polyisoprene Polybutadiene, 7.5% v i n y l i s o m e r , 32-35 cis i s o m e r cis-Poly-l,3 butadiene

Relative activities of metal, by induction period M n 2§ > Co 2§ > C u 2§ > F e 3§

C o 2+ ~ M n 2§ > C u 2+ ~> F e 3+

Co 2§ = F e 3§ F e 3+ > p b 2+

C o 2+ ~ F e 3+ > C u 2+ ~ M n 2+ >

M n 2§ > C u 2+ > > P b 2§ > Ce 4+ Co 2+ > Ce 4+ > C u 2+ > M n 2+

Styrene butadiene rubber

M n 2§ = Co 2§ = F e 3§ > C u 2§ > S n 2§

Copolymer isobutene and isoprene Ethylene-propene terpolymer

Co 2§ > t h e o t h e r s Co 2§ = F e 3§ > C u 2§ > M n 2§ > Ce 4§

Poly(vinyl butyral)

Pb 2+ > Ce 4+ F e 3+ > Ce 4+ > p b 2+ > S n 2+ > Co 2+ > M n 2+ > C u 2+

C o 2+ ~ C u 2+ ~ M n 2+ :> p b 2+ >

F e 3+ > Ce 4+ CeO2 > A1203 > TiO2 > SiO2 > Cr203 > ZrO2 Co 2+ > Ce 4+ > C u 2+ > S n 2+ = N i 2+ > p b 2+ = Mn2+ > Zn2+ = F e 3+ B a 2+ > Ti +4

P r o p e n e oxide r u b b e r

P o l y a c r y l i c acid Polyacrylic rubber

Relative activities of metal, by rate

C a 2 + _ Fe3+ > Ce 4+ =

p b 2+ > S n 2+ Butadieneacrylopnitrile rubber

F e 3+ > M n 2§ = Ni 2+ > P b 2+ > Co 2§

a M o s t d a t a t a k e n f r o m Lee, L. H., Stacy, C. L., a n d E n g e l , R. G., 10, 1699 (1966). O t h e r d a t a p r o v i d e d by t h e a u t h o r .

J. Appl. Polym.

Sci.

Vink [25] found that Fe+3 and Cr § ions slightly affected the course of oxidative degradation of (1) hydroxyethylcellulose, (2) methylcellulose (both frequently used as dry press binders), (3) polyoxyethylene, (4) polyvinyl alcohol, and (5) polymethacrylic acid (frequently used as steric stabilizers and binders for aqueous ceramic suspensions). The effects of oxides on the binder burnout of poly(vinyl butyral) (PVB) was studied by Masia et al. [26]. The binder decomposition kinetics in air was shown to be a strong function of the ceramic oxide present. In all cases, the oxide decreased the temperature necessary for thermal decomposition compared to PVB alone. The order was the following: CeO2(AT = 200~

> A1203 > TiO 2 > SiO 2 > Cr203 > ZrO2(AT = 40~

with their lowering of the decomposition temperature given in parenthesis. The decomposition temperature for argon decomposition (i.e.,

744

Chapter 15 Binder Burnout

60 ppm O2) showed a much reduced catalytic effect, indicating that oxygen plays a role in catalysis. The PVB decomposed over a relatively narrow range of temperature in the presence of all oxides studied. The residue of carbon at 600~ after thermal decomposition in air was also shown to depend on the oxide present. The residual carbon content correlated with the isoelectric point of the oxide. For oxides with an isoelectric point near 6 to 7, the residual carbon content was the lowest. For oxides with isoelectric points lower and higher than 6 to 7, the residual carbon increased substantially. These results are valid for decomposition of PVB in the presence of air and argon and suggests that both surface acidity and surface basicity are responsible for the catalysis. Shabtai [27] has found that SO43 ions attached to A1203 surfaces act as a super-acid catalysts for the depolymerization of polyethylene. The effect of a A1203 surface on the binder burnout of poly(methyl methacrylate) (PMMA) [28], polymethacrylic acid (PMAA) [28], and polyvinyl butyral (PVB) [29,30] were studied by Sacks et al. The chemical formula for these polymers is shown in Figure 15.3(c) and 15.4(d). The effect of atmosphere on the degradation of PVB on A1203 is shown in Figure 15.3(a). In air, the decomposition has two steps: one at 300~ and the other at 500~ During the 300~ decomposition the volatiles are butyraldehyde and water, leaving behind a residue which further oxidizes at 500~ In N2 atmosphere the degradation of PVB on A1203 also has two steps: one at 300~ as in air, and another very broad step starting at 450~ and lasting to beyond 1000~ This second step is the cyclization and the formation of carbon. The effect of atmosphere on the degradation of PMAA on A1203 is shown in Figure 15.3(b). In air, the decomposition has three steps: one at 200~ one at 400~ and the other at 475~ In N2 atmosphere, the degradation of PMAA on A1203 also has three steps: one at 200~ as in air, another at 430~ and another very broad step starting at 450~ and lasting to beyond 1000~ This third step is the cyclization and the formation of carbon. The thermal decomposition of PMAA alone was studied by Grant and Grassie [31] and Geushens et al. [32]. They have shown that PMAA decomposes at 200~ where two reactions occur. The minor reaction is depolymerization yielding volatile monomer. The major reaction is the elimination of water as shown in Figure 15.5. A kinetic study of this reaction gave an activation energy of 37 _ 3 Kcal/mole. From this cyclic product, a head-to-tail polymerization reaction of PMAA was confirmed. This structure breaks down during the second stage, cross-linking and cyclization apparently occur during this process. These degradation-resistant cyclic and cross-linked structures can be broken down oxidatively at higher temperatures in air. In contrast, there is a large amount of residue at high temperatures with nitrogen pyrolysis [30].

15.4 Oxidative Polymer Degradation

a

100

.....

-- ' -'

9

,

_~

745

,i

PVB

90 80

.~ 70 "1-

(9 ,'n

AIR ----'~ il

60

I"--

50

NITROGEN

40

30 20 10 0

100

200

300

400

Ij%,,._,

~

500

600

700

800

900

1000 1100

TEMPERATURE (~

b lOO

P,,

90 80

~60 -1,-n 50

A,.

40

NITROGEN

30 20 10 0

0

100

200

300

400

500

600

700

800

900

1000

TEMPERATURE (~ C ~(CHz

ICH

CH

.o

I

~

CH 3

I

CH2 ~ ' ~ n

0

/%OH

C3H? Polyvinyl butyral

Polymethacrylic Acid (PMAA)

FIGURE 15.3 Thermal gravimetric analysis for (a) Polyvinyl butyral (PVB) in air and nitrogen atmosphere. Taken from Shih et al. [29]. (b) Poly methacrylic acid (PMAA) in air and nitrogen atmosphere. Taken from Sun et al. [28]. (c) Structure of polyvinyl butral (PVB) and polymethacrylic acid (P AA). Reprinted by permission of the American Ceramic Society.

746

Chapter 15

a 100

B i n d e r Burnout

HI I

A~

.

.

.

.

.

.

.

.

.

.

.

.

,w

AIR

90 80 70 A

o~

60

I

.

40 ~

30

PMMA

i

20

I ~

AI203/8"3 wt% PMMA

10 0

0

100

200

300

400

500

600

700

800

900

1000

TEMPERATURE (~

PMMA

AIR

0.8 O C=O(1730cm "1) t

0.6

C-O(1147cm

"1)

Q C-H(2993cm'l)

0.4

0.2

0

100

200

300

400

500

600

700

TEMPERATURE (~

(a) Thermal gravimetric analysis for poly(methyl methacrylate) (PMMA) and PMMA/A1203 mixture in air atmosphere. Taken from Sun et al. [28]. (b) Fourier transform infrared (FTIR) peak intensity versus pyrolysis temperature for poly(methyl methacrylate) (PMMA) in air atmosphere. Taken from Sun et al. [28]. (c) Fourier transform infrared (FTIR) peak intensity versus pyrolysis temperature for poly(methyl methacrylate) (PMMA)/A1203 mixture in air atmosphere. Taken from Sun et al. [28]. (d) Structure of polymethyl methacrylate (PMMA). Reprinted by permission of the American Ceramic Society.

F I G U R E 15.4

15.4 Oxidative Polymer Degradation C

74

1 Calcined AI203/PMMA AIR

>.1-09 Z LM

IZ D

0.8 0

C - O (1147 cm "1)

0.6

-< I..U 12..

o

C - H (2993 cm "1)

o

/ C ~ ( 1 5 7 5 0 m "1)

LU 0.4 > II W

n'-

C = O ( 1 7 3 0 c m "1)

O

O

0.2

0

100

200

300

400

500

600

700

TEMPERATURE (~

CH s

I I

~

oaXo

CH 3

Polymethyl methacrylate (PNNA) FIGURE 15.4 (continued)

The pyrolysis of polymers of alkali and alkaline earth metal salts of PMAA was studied by McNeill and Zulfigar [33]. The first pyrolysis reaction is the elimination of water as in Figure 15.5. Then, two distinct processes may be discerned in the breakdown of the alkali metal salts of PMAA; namely, chain scission and carbonate formation. Chain scission leads to monomer and metal isobutyrate. The metal carbonate formation occurs by intramolecular reaction of adjacent salt units in the chain, resulting in the elimination reaction of unstable four-membered ring structure species which undergo various transformations to cyclic or acrylic ketones. For PMMA, a polymer that undergoes unzipping depolymerization [34,35], the atmosphere plays no role in thermal decomposition but the

748

Chapter 15 Binder Burnout CH 3

I 1

'x~'CH 2

C

0

/\

CH3

I 1

CH2

OH

CH 3

C ~

0

/\

~

OH

"A~CH 2

0

CH 2

CH3

\/\/ i I /\ /\ "C

C

0

+

OH

H20

FIGURE 15.5 The elimination of w a t e r as PMAA decomposes, cyclizing the structure.

A1203 surface had an important effect on the thermal decomposition as seen in Figure 15.4. For PMMA alone, there is only one step in the thermal decomposition at 325~ as seen in Figure 15.4(a). Following the infrared spectra, we see that all the various PMMA peaks of the residue decrease to 0 at 450~ as in Figure 15.4(b). Based on the conversion to monomer, it was established that there are two types of chain initiation for PMMA. One was chain-end initiation, which takes place at slightly lower temperature, and the other was random initiation, which takes place at slightly higher temperature [36]. With A1203 present, the same decomposition is broadened as seen in Figure 15.4(a). The decrease to zero for the infrared peaks is prolonged to 480~ as seen in Figure 15.4(c). An additional peak is also observed in the infrared spectrum corresponding to the CO0 bond, which is present in the residue only when an A1203 surface is present. This is a result of the chemical reaction of PMMA with the A1203 surface. The effect of BaTiO3 catalysis on the thermal degradation of polyacrylic acid (PAA) is shown in Figure 15.6(a) and (b) [37]. Here we see that the decomposition of PAA in air alone is rather complicated, giving a broad and complex peak between 150 and 300~ and distinct peaks at 400 and 475~ Both ketone and anhydride volatiles were found [37,38] at different temperatures, which are the result of cyclization and cross-linking reactions. All peaks greater than 150~ are exothermic peaks as measured by differential scanning calorimetry (DSC). The reaction sequence for the decomposition of PAA alone is shown in Figure 15.7 [39]. It is found that the reaction sequence is similar to that of PMAA. First, by water elimination, a difficult to volatilize cyclic structure is developed. Then, other reactions which eliminate CO2 and CO take place at higher temperature. The decomposition of surface BaCO3 takes place at 600~ In the presence of BaTiO3 powder, the decomposition of PAA occurs by a single-stage reaction at 300~ [38], as shown in Figure 15.6(b). (The second peak centered at ~600~ in Figure 15.6(b) is the decomposi-

15.4 Oxidative Polymer Degradation

749

tion of BaCO3, which is a surface impurity in the powder.) At 300~ the volatiles measured were acrylic acid, monomers of PAA. No anhydrides or ketones were detected by mass spectrometry. The residues measured by FTIR between 250~ and 350~ were lower molecular weight polymers of PAA. The presence of the BaTiO3 surface was found by Chen [37] to catalyze the PAA depolymerization reaction. Scanning calorimetry of this depolymerization reaction gave a heat of reaction of-809.5 Kcal/mole PAA ( M w = 5000)(or 13.0 Kcal/mole of monomer) for the peak centered at 300~ for PAA decomposition with BaTiO3 in air (Figure 15.6(b)). A kinetic study of this depolymerization [37] using a constant heating rate gave an activation energy of 41.3 +_ 2.0 Kcal/mole. To summarize pyrolysis of different acrylic polymers, the O-H bond in pure PMAA and PAA is the most unstable linkage and appears to determine the mechanism of decomposition. Reaction of the O-H bond leads to esterification of adjacent carboxylic groups and forms more thermal-resistant cyclic and cross-linked structures. However, for PMMA and metal salts of PMAA, stronger bonding in the side-chains increases the first weight-loss temperature where the main chain scission takes place. Catalysis by metals, metal salts, and the oxide surfaces play an important role in the reaction mechanism of the thermal decomposition of polymers. Searching for polymer ceramic systems which undergo thermal depolymerization leads to cleaner binder burnout.

15.4.1 R e a c t i o n K i n e t i c s These multiple reaction mechanisms can operate for a single polymer at a single temperature, leading to a host of volatile products and residues. A general kinetic expression for the rate of autocatalytic oxidative polymer degradation is given by [45] d(PH) dt

= kAc(PH)~(R*) r

(15.15)

which can be used for initiation 02 + P H - - . P O * + HO*

and propagation P H + PO*---~ 2P* + OH*

Because two or more radicals, R*, are produced by the reaction of one initial polymer molecule, P H , the reaction speeds up as the reaction proceeds, as shown in Figure 15.8. (Note PH = A in this figure.) Autocatalytic oxidation kinetics are often prevalent with hydroperoxide and metal catalyzed oxidation of polymers.

750

Chapter 15 Binder Burnout a o.o

(O

-0.1

E d

Q.

o

if)

-0.2

-0.3 50

150

250

350

450

550

Temperature

0.00

O ED E d f:L

1

BaCO3-->BaO + CO2 -0.01

mO CD

-0.02, '50

-

' .

.

. . . 250

,

'-,'

450

-

-'

. . . . . . 650

Temperature, ~

FIGURE 15.6 Differential thermal gravimetry of pyrolysis for polyacrylic acid (PAA; Mw - 5,000) in (a) oxygen at a heating rate of 2.5~ and (b) mixed with BaTiO3 in air at a heating rate of 2.5~ Comparing the two, we see different peaks, suggesting a different pyrolysis mechanism catalyzed by the BaTiO3 surface. (c) Schematic of the multiple mechanism for the thermal decomposition of polyacrylic acid used as a binder in a green body of BaTiO~. Cyclization of the polyacrylic acid occurs for a small fraction of the polymer. The major fraction of the polymer undergoes depolymerization, giving a relatively clean binder burnout. M is either a metal ion attached to the BaTiO3 surface or the metal ion that is the metal salt of polyacrylic acid. From Chen [37].

15.4.2 Polymer Residues and Volatiles When oxygen plays a role in the degradation of a polymer, the structure of the polymer is often altered. The volatiles and residues will vary depending on the mechanism of oxidative polymer degradation.

751

15.4 Oxidative Polymer Degradation

c

O~c/O. CH 3

~

I I

Z-

O

CH 3

c

~

/\ I i

C,

C

~

//\

I I

2

C ~

C

OM

CH 2

OM

/

CH 3

I I

i

CH z ------C

CH3

CH 3

O

CH 3 ~,v.CH ~

I I

2 I

i I

OOOM

Depolymerization H Abstraction

9 + .CH z----C

OM

0

OM

9C H

z

C

CH3

\/

C"

.C

I I

CH

Isobutyrate

(~H3 +

+

O3OM

Z Cydization Carbonate formation

CH3

Monomer

Ot 3

CH 3 0

I I

C

CH 3 CH z + ICHz

\ /

I

C-----CH z

\/

C O + OM + COOM

(A)

I

O . +OM

0 +

(S) F I G U R E 15.6

(continued)

OM

(c)

For degradations that increase the molecular weight of the residue, the volatiles will be of a lower molecular weight species including alcohols, ketones, carbon dioxide and monoxide, hydrogen, and water. In some cases, the volatiles will further react with 02 in the atmosphere to chemically alter the volatiles. This can alter the measured heat of reaction drastically. The final residue will be carbon under reducing conditions. For temperatures near 900~ carbon is oxidized to CO2 by a highly exothermic reaction. For degradations, that decrease the molecular weight of the residue, the volatiles will be of a low molecular weight species, typically containing oxygen. The residues under these conditions are oxygen containing carbon ring structures. Much less

752

Chapter 15

Binder Burnout

Reaction Sequence

Decomposition Step

Reference

- - - CH2---- CH---" CH2"---"C H - ' -

A, B

40

A, B

41

A,B,C

42

A, B

43

A, B

44

I

I

COOH

COOH

120_150Oc ~ A ---- CH2---- CH---- CH2---- CH'---

I

O~C---"

I

+ H20

0"--- C ~ O

250Oc ~ B ----CH2---" C ~ , , ~ C H ~ C C

H ---+ CO2

II 0

300~

I C

-----CH2---- C H ~ i ~ C H - - - HC

+ CO

I

CH2

I 350~ FIGURE 15.7

Reaction sequence for the decomposition of PAA alone.

carbon is formed under oxidative conditions than under reducing conditions.

15.5 K I N E T I C S OF B I N D E R B URNO UT Fundamental studies in the literature of the kinetics of binder burnout are scarce. This material has been developed by synthesis of several articles [3,10,46,47] and the application of principles of gas-solid reac-

15.5 Kinetics of Binder Burnout

-Rate=-r

753

A

R* in Feed

ression of Reaction 0

CA

A+R* --->

Initial C

A

R*+R*

F I G U R E 15.8 Typical rate versus concentration curve for the type --rA = kc~crR, A is the polymer.

autocatalytic reactions of

tions discussed in Levinspeil [48]. The reaction kinetics of binder burnout will be discussed in terms of the following generalized reaction: bP(s) + n O 2 ( g ) ~ vV(g) + sS(s) + AHrxn Binder burnout can be kinetically limited over several steps: 1. Chemical reaction, 2. Mass transfer of oxygen in the boundary layer surrounding the green body, 3. Diffusion of oxygen in porous green body, 4. Diffusion of volatiles, V, out of the porous green body, 5. Mass transfer of volatiles, V, out of the boundary layer surrounding the green body, 6. Heat transfer in the boundary layer surrounding the green body, 7. Heat conduction in the porous body. These rate determining steps are shown in Figure 15.9. The chemical reaction discussed in the preceding sections of this chapter can be used to establish the kinetic expressions for the decomposition of the polymer. Mass transfer of oxygen through the boundary layer to the green body surface is followed by the diffusion of oxygen in the pores to the reaction surface. This reaction surface is thought to be a plane for simplification but the polymer may not completely fill the pores.

754

Chapter 15 Binder Burnout Pore f Gas

Velocity

Diameter

.Ceramic

i ~ @

Powder

olymer

Pro i e

V Bulk

gas

X

9

PO2

Reaction

Surface P(volitiles)

Partial

Pressure

0

Profile

x

Reaction

Surface

TB

v

Temperature

0

Profile

Schematicof the rate determiningsteps for the thermal decomposition of a polymericbinder.

FIGURE 15.9

Under these conditions the reaction front may be diffuse (i.e., fractally rough [49] like that of the drying front see Figure 14.2). Also the polymer often undergoes several decompositions at various temperatures. The first decomposition leaves behind a residue in the porous green body which further reacts at a higher temperature. Once the reaction takes place, the volatiles must diffuse through the pores and through the boundary layer into the bulk gas to be removed from the furnace. In the pores, gaseous oxygen may react with the volatiles and produce other volatile species, but this is not discussed in this chapter. Any of the preceding steps can be rate determining. Heat must also be transferred from the bulk gas through the boundary layer and through

15.5 Kinetics of Binder Burnout

755

the porous layer to supply the heat of reaction. Either of these heat transfer steps can also be rate determining. This same reaction sequence can be used to describe the thermal decomposition of polymers under reducing conditions. In this case, the value ofn is equal to 0 and b is usually set equal to 1.0 in the generalized reaction. Under these conditions, the mass transfer is limited to the removal of the volatiles from the porous green body. This mass transfer can be limited by the pore diffusion or the boundary layer. We still must consider that the surface reaction or the steps of heat transfer in the boundary layer or heat conduction in the porous body could also be rate controlling in this case of the thermal decomposition of polymers under reducing conditions.

15.5.1 Kinetics of B i n d e r Oxidation To discuss binder burnout in detail, let us consider a spherical green body of radius R with a polymer completely filling its pores undergoing an oxidative thermal decomposition. Figure 15.9 is a schematic of this process. We will use the case where the number of moles of oxygen, n, is 1; that is, bP(s) + 1.0 O2(g)--* vV(g) + sS(s) + AHrx~ The mass flux associated with boundary layer mass transfer is given by

J~ = 47rR2K~( Co2B - C02R ) ~ 47rR 2Kg Co2B

(15.16)

where Kg is the mass transfer coefficient (given by the Colburn analogy for a sphere),* C02B is the concentration of 02 in the bulk gas (C02B = PO2B/RgT), C02R is the concentration of 02 at the surface of the particle of radius R (C02R = PO2B/RgT). If the concentration of 02 at the surface of the sphere, r = R, is near 0, the boundary layer mass transfer is the rate determining step. Simultaneously, there is a diffusive flux of 02 through the pores, J2, given by dCo 2

J2 = 47rr2Do2e dr

= constant

(15.17)

r

where D02~ is the effective diffusion coefficient of 02 in the pores. This flux is essentially constant because it does not change quickly with

* Colburn analogy: Sh = 2Kg ~R = 2.0 + 0.6 ReX/2 Scl/3 , where Sh is the Sherwood number, Re is the Reynold's n u m b e r for flow around the sphere, Sc is the Schmidt number, DA is the diffusion coefficient of oxygen in the gas.

756

Chapter 15 Binder Burnout

time. (i.e., pseudo-steady state assumed). The effective diffusion coefficient for a porous layer is given by Doze =

+

(15.18)

_s

where DK (=rp X / 1 8 R g T / I r M w ) is the Knudsen [2] diffusion coefficient for a pore of radius rp, Dos is the molecular diffusion coefficient through the gas in the pores, s is the void fraction of product layer, ~ is the tortuosity of the pores (typically ~- 2.0). For the oxidative thermal decomposition of the polymer given by the reaction bP(s) + nOz(g)--* vV(g) + sS(s) + AHrxn the equilibrium concentration of oxygen at the reaction surface, r, is given by

CozE = (RgTr) -1

[

( P v ) v PVT-nexp

(

AGrxn]] 1/n

+ RgTr ]

(15.19)

where h Grin is the standard Gibbs free energy of reaction at the temperature of the reaction plane and PT is the total pressure. The flux due to surface thermal degradation reaction is given by J3 = 4Ir r z ks(Coz r -

Co~) -~ 47r r e k s Co2 r

(15.20)

where Co2r is the concentration of Oz at the reaction plane r and ks is the pseudo-first-order rate constant for the surface reaction. Note: 1

dNp _

Area d t

1

dr

dr

- Are----~4 r 2 4)p pP - ~ = - rbp p p - ~ = ks Cos

(15.21)

where +p is the volume fraction of polymer in the green body and p~ is the molar density of the polymer. If the reaction is far from equilibrium, the concentration of Oz in equilibrium with the polymer P is essentially 0, allowing the simplification at the right of equation 15.20. The total mass transfer is related to the total heat transfer, making sure that the amount of mass reacted is equal to the heat available for reaction. This balance gives J =

Q hH~

(15.22)

The heat flux can be composed of two parts: (1) the heat flux in the boundary layer, Q1 = 47r R 2 h ( T s -

TR)

(15.23)

15.5 Kinetics of Binder Burnout

757

where h is the boundary layer heat transfer coefficient (given by the Colburn analogy* for a sphere), T s is the bulk gas temperature, TR is the temperature at the surface of the sphere; and (2) the heat flux through the product layer: dT Q2 = 47r r 2 k e - ~ r

=

constant

(15.24)

r

where k e is the effective thermal conductivity of the product layer. If the product layer is porous, the effective thermal conductivity is given by kTs

+

(15.25)

where s is the void fraction in the porous product layer, kTs is the thermal conductivity of the ceramic powder and any polymer residue present, and k ~ is the thermal conductivity of the gas in the pores. To complete the picture, the flux of gas, 02, must be related to the degradation of P and the size of shrinking core radius, r. This can be accomplishedby considering that the change in moles of P is equal to b times the change in moles 02, from the reaction stoichiometry, which is also equal to the flux J described by the flux equations for different rate determining steps already described: dNo2 dNp _ _ b dt

dt

- - bJo2 *Area

(15.26)

assuming n = 1.0. The change in the number of moles of polymer is related to the change in the volume of P in the core: -dNe

= -rbp OR d V = - O R rb~ 47r r 2 d r

(15.27)

where 4~p OR is the molar density of the polymer in the green body (i.e., moles of polymer per unit volume of green body). Using the relation between the flux J and the change in core radius, r, given in equations 15.26 and 15.27 with the definitions of the fluxes for mass (equations 15.16, 15.17, and 15.20) and heat transfer equations (equations 15.23 and 15.24), with equation 15.22 for the intereonversion of heat and mass flux, it is possible to determine the time dependence of conversion, X p :

* Colburn Analogy: Nu = 2hR/kTf = 2.0 + 0.6 Re1/2Pr 1/3,where Nu is the Nusselt number, Pr is the Prandlt number, and kTfis the thermal conductivityof the fluid.

758

Chapter 15 Binder Burnout

for the shrinking core model for a sphere as given in Table 15.4. This approach is similar to that applied to fluid particle reactions by Levinspiel [48] in Chapter 5. The kinetic expressions for other geometries (cylinder and plate) undergoing a shrinking core burnout are also given in Table 15.4. The results of these various models for a sphere are plotted Figure 5.7 (Chapter 5). All these models have similar trends with respect to conversion, Xp versus dimensionless time. When the time scale is dimensional, it is easy to determine the rate controlling mechanism. Pore diffusion (with wgt. loss a X/t) has been shown to be the rate limiting step for the binder burnout of wax from a stainless steel compact [50, 51], where heat transfer is fast. The surface reaction has been shown to be the rate limiting step in the oxidative removal of binders from multilayer ceramic capacitors [47]. The effect of sample size and heating rate on the rate of binder (polystyrene) burnout from a cylinder of BaTiO3 made by injection molding was investigated by Shukla and Hill [52]. h c r e a s i n g the sample size shows a lower maximum binder evolution rate and further increases the temperature of the exothermic decomposition. Pore diffusion was found to be the rate determining step for low heating rates and small samples. Pore heat transfer was found to be rate determining for large samples with high heating rates. The decomposition of polystyrene is highly exothermic.

15.5.2 Kinetics of Volatiles Loss These results just presented, correspond to the rate determining steps 1, 2, 3, 6, and 7 given at the beginning of this section. To account for the transport of volatiles being rate controlling (steps 4 and 5), the following analysis is applied. The mass flux associated with boundary layer mass transfer is given by J1 = 47r R 2 Kg(CvB - CVR) ~- --41r R 2 Kg CVR

(15.29)

where Kg is the mass transfer coefficient (given by the Colburn analogy for a sphere),* CVB is the concentration of volatiles in the bulk gas (CvR = PvB/RgT), CVR is the concentration of V at the surface of the particle of radius R (CvR = PvR/RgT). If the concentration of V at the surface of the sphere, R, is near that at the reaction plane, r, boundary layer mass transfer is the rate R = 2.0 + 0.6 Re1/28cl/3, where Sh is the Sherwood * Colburn analogy: Sh = 2 k~ DAA number, Re is the Reynold's number for flow around the sphere, Sc is the Schmidt number, and DA is the diffusion coefficient of the volatiles in the gas.

15.5 K i n e t i c s o f B i n d e r B u r n o u t

TABLE 15.4 Model

759

Binder Oxidation Conversion, Xp, versus Time for Shrinking Core

B o u n d a r y layer

Pore d i f f u s i o n or c o n d u c t i o n

S u r f a c e reaction

The sphere: Xp = 1 - (r/R)3 Rate Controlling Step--Mass Transfer t/r = 1 -

t/r = Xp CbpppR

3(1-Xp)2/3+ 2(1-Xp)

t/r = 1 -

d~pppR 2

r = 3bKgCo2B

r = 6bDfpCo2 s

r-

( 1 - Xp)l/3 (bpppR bksCo2 s

Rate Controlling S t e p - - H e a t Transfer t/r = Xp t/r = 1 - 3(1 - Xp)2/3 + 2(1 - Xp) Ag~

c~pppR

r = 3bh(Ts-

Ts)

A H~

2

~ = 6 b K e ( T B - Ts)

Plate: Xp = 1 - (x/L) Rate Controlling Step--Mass Transfer t/T = Xp 6pppL

t/~ = X 2 6pppL 2

t i t = Xp d~pppL T--

r-

bKgCo2B

r = 2bDeCo2B

bksCo2s

Rate Controlling S t e p - - H e a t Transfer t/r

=

t/r = X 2 h H ~ dppppL 2

Zp

h H ~ dppppL r = bh(Ts-

Ts)

r = 2bke(T s _ Ts )

Cylinder: Xp = 1 - (r/R) 2 Rate Controlling Step--Mass Transfer t/T = Xp CbpppR T

--~ ~

t/r = Xp + (1 - Xp) ln(1 - Xp) CbpppR2 T

2bKgCo2s

tit = 1T

=

4bDeCo2B

( 1 - Zp)l/2 d~pppR

--

bksCo2s

Rate Controlling S t e p - - H e a t Transfer t/r = Xp AH~ r r = 2 b h ( T B - Ts)

t/~ = Xp + (1 - Xp) ln(1 - Xp) AH~ r 2 r = 4bke(T s _ Ts)

Note" The model is O2(g) + bP(s)-o sS(s) + vV(g), assuming the m a s s t r a n s f e r o f oxygen is rate controlling. Volatile, V, transfer is not rate controlling.

determining step. Simultaneously, there is a diffusive flux of V through the product layer, J2, given by dCy J2 = 4 7r r 2 DyE -~r

= constant r

(15.30)

760

Chapter 15 BinderBurnout

where DyE is the effective diffusion coefficient of V in the product layer. This flux is essentially constant because it does not change quickly with time (i.e., a pseudo-steady state assumption). The effective diffusion coefficient for a porous layer is given by (15.31) where D g is the Knudsen diffusion coefficient, Dv is the molecular diffusion coefficient of the volatile, V, through the gas in the pores, is the void fraction of porous burnout layer, and ~ is the tortuosity of the pores. In the case of large fluxes, a pressure drop may build up and Darcy's law should be used for the flux of gas through the pores (15.32)

j = -DF VP

where DWis the flow diffusivity given by Darcy's law [= s 3 RgT/(cl S] (1 - s)2), cl is a constant = 4.2 [4] or 5.0 [5], S Ois the surface area of the particles per unit volume of the particles (So = 3/r for a spherical particles), s is the void fraction of the green body, and ~ is the viscosity of the gas]. This flow equation has the same mathematical form as that for pore diffusion. So, ifDf is larger than DYE, Of should be used in the place of DyE in the following analysis for the rates of volatile loss.

P r o b l e m 15.1 From a 10 cc ZrO2 green body with 0.001 gm of PMMA per cc, determine the volume of volatile monomer. The PMMA used has a molecular weight of 10,000 gin/mole. S o l u t i o n A 10 cc green body will contain 0.01 moles of PMMA, which decomposes into 1.0 mole of methyl metacrylate (MMA = 100 gm/mole) that has a gaseous volume of 24,451 cc at 298 K and 1 atm pressure. The mass transfer of such huge amounts of gas volatiles is somtimes the rate determining step for depolymerization. For the oxidation thermal decomposition of the polymer given by the reaction bP(s) + n02(g)--~ vV(g) + sS(s) + hHrxn the equilibrium gas phase concentration of V at the reaction surface, r, is given by CVE = (RgTr) -1

(Po2)n(PT)n-v exp

R~r /

(15.33)

761

15.5 Kinetics of B i n d e r B u r n o u t

TABLE 15.5 Model

Binder Oxidation Conversion, Xp, versus Time for Shrinking Core

Boundary layer

Pore diffusion

Sphere: Xp = 1 - (r/R) 3 Rate Controlling StepmMass Transfer t/r

t/r = Xp r =

= 1 -

cbpppR

3(1

-

Xp) 2/3 "4- 2 ( 1 d~pppR2

--

Xp)

T----

3 b KgCvE

6 b DeCyE V

Plate: Xp = 1 - (x/L) Rate Controlling StepmMass Transfer t/r = X ~ d~pppL2

t/r = Xp d~pppL

~K~C~

T ~

2 bDeeyE V

Cylinder: Xp = 1 - (r/R) 2 Rate Controlling Step--Mass Transfer t/r = Xp r =

t/r = Xp +

d~pppR

Xp) l n ( 1 d~pppR2

(1 -

-

Xp)

T--

2 vb KgCvE

4 b DeCyE V

Note: The model, where volatile loss is rate controlling: O2(g) + bP(s)---> sS(s)

+ vV(g)

where hGrx~ o is the standard Gibbs free energy of reaction at the temperature of the reaction plane, r. Applying this analysis for the shrinking core with the mole balance, dNp_ -b dt

dNo2 b dNy b - § - + -Jy*Area dt v dt v

(15.34)

we find the kinetic expressions for volatiles loss given in Table 15.5. The results of these various models for a sphere are also plotted Figure 5.7. All the preceding models assume that the volatile loss is rate determining. If heat transfer or surface reaction is rate controlling, the equations in Table 15.4 should be used. 15.5.3 Kinetics

of Binder

Pyrolysis

without

Oxygen

For reducing atmospheres or reactions where oxygen is not a reactant, the reaction is

bP(s)--* vV(g) + sS(s)

+

~/rxn

762

Chapter 15 Binder Burnout

with the equilibrium gas phase concentration of V at the reaction surface, r, given by

CVE = (RgTr) -1 [(PT) v

exp(

AGOn~ 1/v

R-~Trr] ]

(15.35)

where the exponential quantity in parentheses is the pressure equilibrium constant, Kp = (Pv) v. For depolymerization reactions (where the stoichiometric coefficient s for the reaction residue S(s) is 0, b is 1.0, and v is the number of monomers in the polymer also called the degree of polymerization), the temperature which gives a partial pressure of monomer of 1 atm is given in Table 15.6. As a result of there being only volatiles to be removed from the green body, the only mass transfer steps are those of the volatiles. The surface reaction in this case is not dependent on the oxygen concentration as before but solely on the polymer being present. For simplicity, we will assume that the surface reaction is zero order in polymer concentration with a rate constant, ks. The heat transfer steps are the same as those of the preceding analysis. As a result of all these considerations, the kinetics of binder burnout for reducing conditions can be summarized as in Table 15.7. This table shows different values of the maximum times for various rate determining steps, the values of ~. These r values are consistent with the oxygenless reactions of the binder in a porous ceramic body.

15.5.4 Kinetics o f Carbon R e m o v a l Carbon removal is a particularly important case of binder burnout. It takes place at high temperatures and is highly exothermic. As a

TABLE 15.6 Temperature at Which the Partial Pressure of Monomer is 1 atm. for the Depolymerization Reaction P(s) ~ vV(g)

Polymer Poly(~-methylstyrene) Poly(oxymethylene) Poly(methyl methacrylate) Poly(ethyl methacrylate) Poly(teramethylene oxide)

T(~ 61 119 164 173 83

Data taken from Masia, S., Calvert, P. D., Rhine, W. E., and Bowen, H. K., J. Mater. Sci. 24, 1907-1912 (1989).

15.5 Kinetics of B i n d e r B u r n o u t

TABLE 15.7 Core Model

763

Pyrolysis Without Oxygen Conversion, Xp, versus Time for Shrinking

B o u n d a r y layer

Pore d i f f u s i o n or conduction

Surface reaction

Sphere: Xp = 1 - (r/R) 3 Rate Controlling Step--Mass Transfer t/r = Xp 6pppR

t/r = 1 -

3(1-Xp)2/3 + 2 ( 1 - X p )

T =

T ='

6 b DeCvE

V

AH~ =

b ks

V

Rate Controlling S t e p - - H e a t Transfer t/r = Xp t/r = 1 - 3(1

Xp)2/3

-

V

+ 2(1

5H~ T--

3 b h ( T B - T s)

(1-Xp)1/3 6pOpR

T'-

3 b KgCvE

T

t/r = 1 -

6pppR 2

-

Xp)

2

6bKe(T B - T s)

Plate: Xp = 1 - (x/L ) Rate Controlling Step--Mass Transfer t/T = Xp

t/~ = X 2

~.p~L

t i t = Xp dppppL

~p~L ~

T ~

T =

b Kgc~

2

T =

b_DeCvE

b__ks O

V

Rate Controlling S t e p - - H e a t Transfer t/r = Xp AH~ T =

t/z = X~ AH~ T--

bh(Ts - Ts)

2

2bke(Ts - Ts)

Cylinder: Xp = 1 - (r/R)2 Rate Controlling Step--Mass Transfer t/~ = Xp r

t/~ = Xp + (1 - Xp) ln(1 - Xp) ~ppR 2

T'-

T ~-

2 b ggevE

tit = 1 - (1-

4 b DeCvE

V

Zp)l/2

"r = d~pppR

V

b-k s O

Rate Controlling S t e p - - H e a t Transfer t/r = Xp A H ~ d~pppR ~'~

t/r = Xp + (1 - Xp) ln(1 - Xp) A H ~ dppppR2 T--

2 b h ( T B - T s)

4bke(Ts - Ts)

Note: The model is bP(s) --* sS(s) + vV(g), assuming the volatile, V, transfer is rate controlling.

result, run away reactions can take place which lead to hot spots and high stresses in the green body. Specifically the reaction is given by O2(g) + C(s) ~ CO2(g)

+ Ag~

=

-94.0518 Kcal/mole [53]

(15.36)

764

Chapter 15 Binder Burnout

Thus for the shrinking sphere analysis, the value of the stiochiometric coefficient of carbon, b, in the previous analysis is 1.0 and no solid residues are formed. This reaction can be kinetically limited by the same steps as noted previously: 1. 2. 3. 4. 5.

Surface reaction, Mass transfer in the boundary layer surrounding the particle, Diffusion in the product layer, Heat transfer in the boundary layer surrounding the particle, Heat conduction in the product layer.

The reaction equilibrium is established by the free energy of formation, hG~ = -94.2598 Kcal/mole [53, p. 212], which gives an equilibrium constant at 298 K of [-hG~ Pco2 (Kp)29s = exp L / ~ - ~ ~ j - Po2

(15.37)

For another temperature, T, the equilibrium constant can be determined by the Clausius-Claperon equation: In Kp = ln(Kp)29s +

o(

hHrxn 1 Rg 268

(15.38)

From the plot of the equilibrium constant as a function of temperature, as given in Figure 15.10(a) [54-56], we see that the reaction is spontaneous to the right for all temperatures but less so at higher temperatures. The reaction kinetics are given by [57] ldNc_4.32• dt V~

S

[ 44Kcal/mole] exp RgT Co2 = ksCo2

(15.39)

where N c is the number of moles of carbon reacted, ks (cm/sec) is the surface reaction rate constant, T is the absolute temperature (K), Co2 is the molar concentration of 02 at the carbon surface of area S. A plot of the rate constant, ks is given in Figure 15.10(b). Here we see that the rate constant is low for temperature less than 1000 K(~700~ but increases drastically above this temperature. This equation results in the change in the core size, r, as follows: 1 dN c _ 1 47rr2p c d r dr S dt - - 47rr 2 - ~ = - P c - ~ = ksCo2

(15.40)

where p~ is the molar density of carbon in the green body (i.e., moles o f c a r b o n p e r u n i t v o l u m e o f the g r e e n body).

15.5 Kinetics of Binder Burnout

765

FIGURE 15.10

(a) Equilibrium constant and (b) reaction rate constant as a function of temperature for the reaction C(s) + 02 ---> CO2 [54-56].

P r o b l e m 15.2 Determine the carbon burnout kinetics for a spherical S i O 2 green body 1 cm in radius, R, with a void fraction of 0.3, pore radii of 1 tLm, and 0.01 moles of carbon per cc of green body being burned out in air at 400~ The following data are needed: air is DAB = 0.2 cm2/sec, Kg = 2*DAB/(2*R), h = 2*kTw/(2*R), kTf = 6"10 -5 cal/cm/sec; and SiO2 has tortuosity ~ = 2.0, kT~ = 0.002 cal/cm/sec. In addition, equations for (1) the concentration of oxygen in air from the ideal gas law, (2) the Knudsen diffusion constant, (3) the heat of reaction, and (4) the reaction rate constant for the oxidation of carbon have already been used.

~66

Chapter 15

Binder Burnout

S o l u t i o n The only data missing is the temperature at the surface, r, where the carbon oxidation is taking place. This can be obtained by equating the flux due to mass transfer to the mass flux due to heat transfer, that is, equation 15.5. By plotting the maximum times for the various rate limiting steps (given in Table 15.7) as a function of the surface temperature, Ts, as shown in Figure 15.11 for TB = 400~ this flux equivalence is assured by an intersection of mass and heat transfer lines for values of r as a function of surface temperature. Here we see the intersection between a mass transfer line (either PD, BLMT, or R) and a heat transfer line (either PTC or HTBL) gives the surface temperature which corresponds to the equivalence of the mass and heat fluxes by equation 15.5. As we can see, there are six such intersections, giving six possible surface temperatures. However, the intersection between PTC and PD is not observed in the temperature range shown. The intersection which gives the largest m a x i m u m burnout time in this case is that of surface reaction (R) and boundary layer heat transfer (HTBL), which gives a surface temperature of 600~ higher t h a n the furnace t e m p e r a t u r e of 400~ because the oxidation of carbon is exothermic. For this surface temperature, the maximum times are as follows: Boundary layer

Pore diffusion

Surface reaction

Rate controlling stepmmass t/r = 1-

3(1-Xc) + 2(1 - X c )

t/r = X C

TBL - -

pcR _ 3ggCo B 1.1 • 103 sec

TpD

_

t/r = 1-

pc R2 6DeCoB - 4.5 x 103 sec

Rate controlling stepmheat t / r = 1 - 3(1

t/r = X C

transfer

~a

-

rR

_

pcR

-- ksCo s

(1-Xc)

m

=2.6•

transfer

Xc) 2/3

+ 2(1 - X c ) hH~x~pcR

rHTBL = 3 h ( T s - T s ) = 2.6 x 104 sec

AHrxnPc R 2 rpwc = 6 K e(TB _ T s )

= 4.2 x 10 a sec

Here we see that the m a x i m u m value of r is that for surface reaction and boundary layer heat transfer. This maximum time corresponds to 2.6 • 104 sec or 7.25 hr. As the bulk temperature is increased, the surface reaction curve moves to lower temperatures compared to the other curves, eventually causing a three-curve intersection (R, HTBL, PTC). This gives a new set of rate determining steps, where we have a mixed resistance for the transfer of heat. In addition, sufficiently high surface temperatures, for example, initially, can cause a r u n a w a y of the surface temperature to near 2000~ These carbon burnout condi-

15.6 Stress Induced during Binder Burnout

F I G U R E 15.11 for TB = 400~

767

Various rate limiting effects as a function of surface temperature, Ts,

tions give rise to a massive t h e r m a l shock for the green body t h a t can cause cracking.

15.6 S T R E S S I N D U C E D D U R I N G BINDER BURNOUT Stress induced during binder b u r n o u t are due to t e m p e r a t u r e profiles and pressure profiles in the ceramic green bodies. The effective local stress, (re, is given by [58] (re : (rT + PG i

(15.41)

where (rT is the t h e r m a l stress, i is the unit tensor, and PG is the gas pressure. The net local stress, (r, arises from the difference between the local effective stress, (re, and the average value, ~, t h r o u g h o u t the green body [59]; for example, (r = (re - (~

(15.42)

where

~r = ~

O'edV

(15.43)

If m a t e r i a l properties are constant, this condition assures t h a t the elastic properties of the solid, t h a t is, stress is proportional to strain.

768

Chapter 15 Binder Burnout

Thermal stress, O'T, is discussed next and the stress due to the gas pressure, PG, induced by the flow of volatiles is discussed thereafter. Two types of temperature profile are to be considered: one is the simple heating or cooling of a green body and the other is due to the heat of reaction of the decomposition reaction for the binder. The flow of volatiles is caused by a pressure distribution in the green body. The gas pressure distribution can be determined by the flux of volatiles in the green body.

15.6.1 T h e r m a l Stress I n d u c e d d u r i n g Binder Burnout The thermal stress in binder burnout is the same as those discussed in Section 14.5.1. For a detailed treatment, please refer to that section. Here we give a brief description of the stress for an infinite plate, to specify the nomenclature. Table 15.8 gives the details of thermal stresses for plate, cylinder, and sphere geometries. For an infinite plate of thickness 2Xo, the normal stress cr(x) at a position x in the green body depends on the temperature difference between that point, T, and the average temperature, T a . This gives the strain at that point and fixes the stress at [12, p. 819] %(x)

= ~z(X)=

r

1--v

[T a _

T(x)]

(15.44)

where ~bis the solids fraction, E is Young's modulus of elasticity (typical values 10-50 GPa), ~ is the thermal expansion coefficient, and ~ is Poisson's ratio (defined as the ratio of the strain in the direction of the applied stress to that perpendicular the applied stress; for ceramic

TABLE 15.8 Surface Stresses a and Temperature Differences b for Various Shapes Infinite plate

d~Ea % = Crz = ~ ( Ta - Ts)

Ta-Ts 8~ 1 [-(2n + 1)27r2a't] Ti - Ts - ~ -_0 (2n + 1)2 exp 4x~

Infinite

o- r - -

Ta -

cylinder Sphere

0

~bEa

zo = y=--~_v ( T a - Ts) O"r ---- 0

~Ea

~o = i-=--~_~ ( Ta - Ts)

~

4 exp[-~2n~'tl c " R02 J

T S = ~= l~n T i - Ts

T a - T~ 6 n ~ 1 [-nU~r2a't.) Ti-T,--~ =0 ~-'2exp[ ~00

a Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., "Introduction to Ceramics," 2nd ed., p. 819. Wiley, New York, 1976. b Geiger, G. H., and Poirier, D. R., "Transport Phenomena in Metallurgy," p. 486. AddisonWesley, Reading, MA, 1973. c Ca are roots of the equation Jo(x) = 0, where Jo(x) is the zero under Bessel function.

15.6 Stress Induced during Binder Burnout

~~

green bodies that are incompressible v = 1, a more typical value is v = 0.28, giving some degree of compressibility). When the green body is heated, the surface is in compression ((r-) and the center is in tension (~+). When the green body is cooled, the surface temperature is lower than the average and the surface is in tension ((r+) and the center is in compression ((r-). This condition may cause cracking because the surface is in tension and ceramic green bodies are very weak in tension. For a plate initially at temperature T i and at t = 0, heat is transferred to the surface of the green body from a gas at temperature TB, the temperature profile within the plate is given by the differential equation OT c~t

-

a

,02T c~X2

(15.45)

where ~' is the thermal diffusivity = kv/p Cp. Using the boundary conditions T(x, O) = T i uniform OT(O, t_____= ~) 0 at center line z = 0

0x

OT(xo, t) + h Ox ~ [T(xo, t) - T B] = 0 at the surface x o

where h is the heat transfer coefficient and kp is the effective thermal conductivity of the porous ceramic green body. The solution to this equation is given in Section 14.5.1. The temperature difference, (Ta Ts), responsible for the stress is given in Table 15.8. When the temperature distribution or the solids volume fractions is nonsymmetric, the plate will warp during heating or cooling. The curvature of the plate, p, is given by

p-1

30~T~bf~0 T(x) x dx 2x~

x0

(15.46)

where o~T is the thermal expansion coefficient. For other geometries, the stress and the temperature differences between the surface and the average temperature are given in Table 15.8. The temperature profiles inside these geometries which are responsible for the stress are also shown in Figures 14.11 to 14.13. This behavior is similar to that with green body drying. When a polymer thermal decomposition reaction is endothermic, the center of the green body is cooled with respect to the surface. This cooling will keep the surface of the green body in compression. But when there is an exothermic thermal decomposition reaction (i.e., oxidation of carbon), the center of the green body is heated with respect to the surface. This heating will place the surface

770

Chapter 15 Binder Burnout

of the green body in tension and lead to cracking if the stress is large enough. The stress induced by the exothermic reaction can be estimated from the temperature difference between the surface temperature and that at the reaction surface (AT = TB - Ts) by setting the AT equal to that necessary for the ~ of the pore conduction heat transfer to be equal to the r for the actual rate determining step. This then gives the temperature profile in the pores from which the average temperature can be determined, allowing the surface stress to be determined from the equations in Table 15.8. This phenomena has not been adequately investigated in the literature to date.

15.6.2 S t r e s s D u e to V o l a t i l e F l o w Pore diffusion plays an important role in the cracking of ceramic green bodies. The pressure difference, AP, between the surface and interior of the green body during binder burnout can be estimated from the flux of volatiles, j, giving AP = - ~ J.

R g T . Ax

(15.47)

/

The pressure difference, AP (bulk pressure), can be quite large if the flux, j, is large or if the flow diffusivity, Dw, is small. It should be kept in mind that huge amounts of volatile gas are produced from very small amounts of solid polymer. So, a huge volume of gas must be removed from even small green bodies creating flow stress. The difference between the pressure, P(x), at a point, x, and the average pressure, (P), gives the stress at that point, that is, (r(x) = P ( x ) - (P)

(15.48)

If this stress is larger than the critical stress holding the green body together, that is, (r > (re, it will rupture, leaving cracks in the microstructure of the green body. These cracks will release the pressure build-up, thus allowing the volatiles to flow easier. If we assume that the partial pressure of the volatiles is 0 at the surface of the green body during binder burnout, and at the length of the pore, the partial pressure is its equilibrium value (determined by the Gibbs free energy of reaction and the temperature using equation 15.19), then the stress will start to be important for cracking at temperatures at the end of the pore where the volatile has a volatiles partial pressure greater than PT 1 atm (~100 KPa). This situation leads to compressive stress at the end of the pore and tensile stress at the surface. As a result, temperatures below these corresponding to the "boiling point" for the volatiles (see Table 15.6) are always used for the gentle binder burnout of ceramic green bodies. Numerical solutions [6,46] to the mass transfer and heat transfer =

15.7 Summary

771

equations have been coupled to the flow equations to calculate the stress on the green body caused by binder removal. These results show that the tangential stress at the surface of ceramic green bodies is tensile and becomes large in magnitude at a particular time, depending on burnout conditions. Stangle and Aksay [46] have shown that control of the volatile flow with the temperature is a viable method to prevent cracking. Tsai [6] has shown that, using pressurization during binder burnout, cracking can be prevented. The reason why higher ambient pressure prevents cracking is that the "boiling point" of the volatiles is increased when the ambient pressure is increased. The effect of sample size and heating rate on the rate of binder (polystyrene) burnout from a cylinder of BaTiO3 made by injection molding was investigated by Shukla and Hill [52]. A strong exothermic reaction with a 100~ thermal runaway takes place at around 350 to 400~ The exothermic decomposition of polystyrene takes place at 350~ for low heating rates of 0.5~ (the resulting hT is also low) and increases to 387~ for a 5~ heating rate (where the AT is 100~ Increasing the sample size shows a lower maximum binder evolution rate and further increases the temperature of the exothermic decomposition and AT. This AT (or the resulting pressure build-up or both) caused cracks to develop in the microstructure even at low heating rates of 0.28~ in the larger samples. Control of the binder evolution rate was found to be essential to making crack-free green bodies. Thus temperature programmed burnout is often used industrially.

15. 7 S U M M A R Y

Binder burnout has been shown to have several possible rate determining steps" 1. Chemical reaction, 2. Mass transfer of oxygen in the boundary layer surrounding the green body, 3. Diffusion of oxygen in porous green body, 4. Diffusion of volatiles, V, out of the porous green body, 5. Mass transfer of volatiles, V, out of the boundary layer surrounding the green body, 6. Heat transfer in the boundary layer surrounding the green body, 7. Heat conduction in the porous green body. Accounting for these rate determining steps the kinetics of binder burnout can be established for simple decompositions of polymers, such as depolymerization and the oxidation of carbon. For polymers with

772

Chapter 15 Binder Burnout

complex decomposition routes, this approach will have to be applied for each step of the decomposition. As a result, for most practical cases, polymer thermal decomposition kinetics can be predicted by only numercial methods. During binder burnout large amounts of volatiles of different chemical composition are produced, as are various forms of solid residues, depending on the polymer decomposition route. The flow of these volatiles leads to pressure differences in the green body that leads to stress. Another type of stress induced in the green body during binder burnout is caused by the temperature profile in the green body. When the surface of the green body is in tension, cracking is likely because ceramics are weak in tension. Tension at the surface can be generated by cooling the green body, exothermic decomposition reactions inside the green body, and flow of volatiles form the interior of the green body to the surface. This chapter has given the relevant equations to perform order of magnitude calculations for (1) the thermal decomposition kinetics for the various rate determining steps and (2) the stress at the surface for green bodies of simple geometry.

Problems 1. A mullite green body (~b = 0.7) in the form of an infinite plate of thickness 5 mm filled with polystyrene binder is fired in air at 450~ As a result, an exothermic reaction with a temperature runaway of 150~ inside a mullite green body occurs. Estimate of the surface stress. Assume that E = 144 GPa, O~T ---- 5 X 1 0 - 6 / ~ and v = 0.25. 2. Calculate the maximum time for all the rate determining steps for the binder burnout of PMMA from a spherical green body of radius R = 1.0 cm filled to a volume fraction of 0.8 with spherical 0.5 ftm BaTiO3 particles and 0.001 gms PMMA per cc. PMMA (Mw = 10,000) undergoes depolymerization forming MMA (Mw = 100) at 350~ (hHrxn = +5 Kcal/mol [28]) by a first-order reaction [3] with an activation energy of 41.8 Kcal/mole [3].* 3. The thermal decomposition of polyvinyl butyral in air has been studied by Sacks et al. [29,30]. They have found that at 350~ a major weight loss is observed where a residue is produced in addition to volatile products, which are butyraldehyde and water. The thermal gravimetric data for this thermal decomposition of PVB in an air atmosphere is given in Figure 15.3 and the Figure 15.12. Write the chemical reaction responsible for the weight loss at 350~ and specify the residue. This residue is further reacted with oxygen in * Also confirmed by private communication [M. Sacks, 3/4/93] giving 30-40 Kcal/ mole for the apparent activation energy for the first step of thermal decomposition.

773

15.7 Summary cE

10

E

8

v UJ >

4

E.

o

I.U

6

- -

PVB

i"1 I I

DTG /

AIR

~,

1

DTA " " - - -

I--

m x

O -H 1

100

9

!

9

200

FIGURE 15.12

t.

300

9

!

400

,

.g

500

,

"1" m

t

600

TEMPERATURE (~

700

800- 9-oo ~_ O

DTG and DTA results from PVB in air.

the air at 500~ Assuming that the thermal decomposition is rate limited by the surface reaction, estimate the rate constants for the decompositions at both 350~ where PVB decomposes, and 500~ where the resulting solid residue is oxidized. The vinyl butyral mer unit is [--CH2CHOCH(C3H7)OCHCH2-] 4. Using the surface rate constant calculated in problem 3. Calculate the time necessary for PVB binder burnout from a spherical ZrO2 green body 10 cm in diameter. Assume that the polymer fills the pores of the green body, which has a porosity of 30%. The ZrO2 powder has a 0.3 t~m geometric mean diameter and a geometric standard deviation of 1.15. Will the green body be likely to crack during binder burnout due to the temperature difference caused by the heat of reaction? 5. The thermal decomposition of polymethyl methacrylate in air has been studied by Sun et al. [28]. They have found that at 300~ a major weight loss is observed without residue formation. Only one volatile product was observed, which was methyl methacrylate monomer. The thermal gravimetric data for this thermal decomposition of PMMA in an air atmosphere is given in Figure 15.4(a) and Figure 15.13. Write the chemical reaction responsible for the weight loss at 300~ Assuming that the thermal decomposition is rate limited by the surface reaction, estimate the rate constant for this depolymerization reaction.

774

Chapter 15 Binder Burnout 20

-

1o

-

PMMA~

AIR

- - - - DTG

s

a

o

m

z c9

o

"Ira 3o 0

o

loo

200

ao0

4oo

soo

TEMPERATURE F I G U R E 15.13

6oo

7oo

aoo

9oo

i000

+

(~

T h e r m a l decomposition os P ~

i n air.

Methyl methacrylate (MMA) has the formula CH2 = C(CH3) (COOCH3). The vapor pressure of MMA due to its vaporization is estimated by [3] In P(Atm) = 11.79 - 4410/T(K), assuming a constant heat of vaporization. For the depolymerization reaction, the temperature which gives 1 atm of MMA (the product) is 164~ [26]. This temperature establishes the pressure equilibrium constant and therefore the Gibbs free energy for the depolymerization reaction. Answer: The depolymerization of PMMA rate constant given by [26] rate constant = 3.87 • 1011 exp(-175 kJ/mole/RgT) gm/cm 3 sec. 6. Using the surface rate constant calculated in Problem 5, calculate the time necessary for PMMA binder burnout from a cylindrical A1203 green body 15 cm in diameter. Assume that the polymer fills the pores of the green body, which has a porosity of 25%. The A1203 powder has a 0.5 t~m geometric mean diameter and a geometric standard deviation of 1.2. Will the green body be likely to crack during binder burnout due to the temperature difference caused by the heat of reaction? 7. What is the temperature under which we should operate for carbon burnout from a large porcelain green body? Hint: use data for the equilibrium constant to establish this temperature.

References

775

References 1. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., "Transport Phenomena," p. 544, Wiley, New York, 1960. 2. Knudsen, M., "The Kinetic Theory of Gases." Methuen, London, 1934. 3. Calvert, P., and Cima, M., J. Am. Ceram. Soc. 73(3), 575-579 (1990). 4. Ergun, S., Chem. Eng. Prog. 48, 93 (1952). 5. Carmen, P. C., Trans. Inst. Chem. Eng. 15, 150-166 (1937). 6. Tsai, D. S., AIChE J. 37(4), 547-554 (1991). 7. Strangle, G. C., and Aksay, I. A., Chem. Eng. Sci. 45, 1719 (1990). 8. J. Crank and G. S. Park, "Diffusion in Polymers." Academic Press, New York, 1968. 9. M. J. Cima, J. A. Lewis, and A. D. Devoe, J. Am. Ceram. Soc. 72(7), 1992 (1989). 10. German, R. M., Int. J. Powder MetaU. 23(4), 237-245 (1987). 11. Dal, P. H., and Deen, W., Proc. Int. Ceram. Congr. 6th, Wiesbaden, 1958, pp. 219242 (1958). 12. Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., "Introduction to Ceramics," 2nd ed., p. 385. Wiley, New York, 1976. 13. Sproson, D. W., and Messing, G. L., in "Ceramic Powder Science II" (G. L. Messing, E. R. Fuller, Jr., and H. Hausner, eds.), p. 528. Am. Ceram. Soc., Columbus, OH, 1988. 14. David, C., in "Comprehensive Chemical Kinetics-Degradation of Polymers" (C. H. Gamford and C. F. H. Tipper, eds.), Vol. 14, Chapter 1. Elsevier, New York, 1975. 15. Boyd, R. H., in "Thermal Stability of Polymers" (R. T. Conley, ed.), Dekker, New York, 1970. 16. Rabek, J. F., in "Comprehensive Chemical Kinetics-Degradation of Polymers" (C. H. Gamford and C. F. H. Tipper, eds.), Vol. 14, Chapter 4. Elsevier, New York, 1975. 17. Hansen, R. H., Russell, C. A., DeBenedictis, T., Martin, W. M., and Pascal, J. V., J. Polym. Sci., Polym. Chem. Ed. 2, 587 (1964). 18. Meltzer, T. H., Kelly, J. J., and Goldey, R. N., J. Appl. Polym. Sci. 3, 84 (1960). 19. Taylor, H. A., Tincher, W. C., and Hamner, W. F., J. Appl. Polym. Sci. 14, 141 (1970). 20. Schurz, J., and Windish, K., Faserforsch. Textiltech. 14, 485 (1963). 21. Marek, B., and Lerch, E., J. Soc. Dyers Colour. 81, 481 (1965). 22. Kroes, G. H., Recl. Trav. Chim. Pays-Bas 82, 979 (1963). 23. Vesolovskii, V. I., and Shub, D. M., Zh. Fiz. Khim. 26, 509 (1952). 24. Ozawa, Z., Shibamya, T., and Matsuzaki, K., Kogyo Kagaku Zasshi 71, 552 (1968). 25. Vink, H., Macromol. Chem. 67, 105 (1963). 26. Masia, S., Calvert, P. D., Rhine, W. E., and Bowen, H. K., J. Mater. Res. 24, 19071912 (1989). 27. Shabtai, J. S., private communication (1994). 28. Sun, Y. N., Sacks, M. D., and Williams, J. W., in "Ceramic Powder Science II" (G. L. Messing, E. R. Fuller, and H. Hausner, eds.), pp. 538-548. Am. Ceram. Soc., Westerville, OH, 1988. 29. Shih, W. K., Sacks, M. D., Scheiffele, G. W., Sun, Y. N., and Williams, J. W., in "Ceramic Powder Science II" (G. L. Messing, E. R. Fuller, and H. Hausner, eds.), pp. 549-558. Am. Ceram. Soc., Westerville, OH, 1988. 30. Scheiffele, G. W., and Sacks, M. D., in "Ceramic Powder Science II" (G. L. Messing, E. R. Fuller, and H. Hausner, eds.), pp. 559-566. Am. Ceram. Soc., Westerville, OH, 1988. 31. D. H. Grant and N. Grassie, Polymer 1, 126 (1960). 32. G. Geuskens, E. Hellinckx, and C. David, Eur. Polym. J. 7, 561 (1971). 33. McNeill, I. C., and Zulfigar, M., J. Polym. Sci., Polym. Chem. Ed. 16, 3201 (1978); McNeill, I. C., and Zulfigar, M., Polym. Degradation Stabil. 1, 89 (1979).

776

Chapter 15 Binder Burnout

34. Grassi, N., and Melville, H. W., Proc. R. Soc. London, Ser. A 199, 14; 24 (1949). 35. A. Brockhaus and E. Jenckel, Macromol. Chem. 18/19, 262 (1956). 36. N. Grassie and E. Vance, Trans. Faraday Soc. 49, 184 (1953). 37. Chen, Z-C., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992). 38. Chen, Z.-C., and Ring, T. A., unpublished results. 39. Maurer, J. J., Eustance, D. J., and Ratcliffe, C. T., Macromolecules 20, 196 (1987). 40. Roux, F. X., Audebertm, R. A., and Quivoron, C., Eur. Polym. Lett. J. 9, 815 (1973). 41. Eisenberg, A., Yokoyama, T., and Sambalido, E., J. Polym. Sci., Part A-1 7, 717 (1969). 42. McGaugh, M. C., and Kottle, S., Polym. Lett. 5, 817 (1967). 43. Kabanov, V. P., Dubnitskaya, V. A., and Khar'kov, S. N., Vysokomol. Soedin., Ser. A 17, 1604 (1974). 44. Girard, H., Monjol, P., and Audebert, R. C., C. R. Hebd. Seances Acad. Sci., Ser. C C279, 597 (1974). 45. Boyd, R. H., in "Thermal Stability of Polymers" (R. T. Conley, ed.). New York, 1970. 46. Stangle, G. C., and Aksay, I. A., Chem. Eng. Sci. 45, 1719 (1990). 47. Verweij, H., and Bruggink, W. H. M., J. Am. Ceram. Soc. 73(2) 226-231 (1990). 48. Levinspeil, O., "Chemical Reaction Engineering," 2nd ed., Chapter 12. Wiley, New York, 1972. 49. Cima, M. J., Dudziak, M., and Lewis, J. A., J. Am. Ceram. Soc. 72(6), 1087-1090 (1989). 50. Waikar, R. J., and Patterson, B. R., in "Horizons of Powder Metallurgy, Part II" (W. A. Kaysser and W. J. Huppmann, eds.), p. 661. Verlag Schmid, Freiburg, West Germany, 1986. 51. Angermann, H. H., Yang, F. K., and van der Biest, O., Eur. Ceram. Soc. Conf., 2nd Augsburn FRG, Abstr. A32 (1991). 52. Shukla, V. N., and Hill, D. C., J. Am. Ceram. Soc. 72(10), 1797-803 (1989). 53. Castellan, G. W., "Physical Chemistry," p. 120. Addison-Wesley, Reading, MA, 1964. 54. Yagi, S., and Kunii, D., Symp. Combust., 5th, p. 231 (1955). 55. Yagi, S., and Kunii, D., Chem. Eng. Sci. 16, 364, 372, 380 (1961). 56. Yagi, S., and Kunii, D., Chem. Eng., Jpn. 19, 500 (1955). 57. Parker, A. L., and Hottel, H. C., Ind. Eng. Chem. 28, 1334 (1936). 58. McTigue, D. F., Wilson, R. K., and Nunziato, J. W., in "Mechanics of Granular Materials: New Model and Constitutive Relations" (J. T. Jenkins and M. Satake, eds.), pp. 195-210. Elsevier, Amsterdam, 1983. 59. Timoshenko, S. P., and Goodier, N. J., "Theory of Elasticity." McGraw-Hill, New York, 1970.

PART

VI SINTERING AND FINISHING After binder burnout, we have a ceramic compact that consists of an assembly of ceramic particles giving a porous ceramic green body. At this stage the ceramic green body is at its most fragile state and must be handled with care or, better yet, simply not handled at all. Often this is the case as the binder burnout and the next step, sintering, are performed either (1) in the same furnace but at different t e m p e r a t u r e s (and sometimes different atmospheres) or (2) in the same kiln albeit in different sections. In this type of kiln, a long tunnel kiln, the green bodies are stacked on a car and the car is transported from the section of the kiln where binder burnout is performed to the section of the kiln where sintering is performed. Sintering is a process whereby the porosity is removed from the ceramic green body, giving a fully dense ceramic piece. The driving force for sintering is the reduction of surface area. The flow of material to fill the pores can take place by different methods from diffusion (by several different processes) to viscous flow. Each of these processes is speeded up by increasing the temperature. These mechanisms will be discussed in detail in Chapter 16. As the ceramic green body sinters it decreases in volume or densities. Nonuniform shrinkage causes stress to build up, resulting in warping and cracking. Sintering takes place in a kiln. There are many types of kilns for the different types of sintering used in industry. Two types of kiln are typical. One is the box kiln and the other is a tunnel kiln. The box kiln is simply an insulated box in which the ceramic green bodies are piled in their

778

Part VI Sintering and Finishing

FIGURE VI.1

Woodblock print of a "dragon kiln" with several chambers. These kilns were so named because they snaked their way at a constant gradient of the hillside and because they breathed fire when operated for several days and nights. Taken from T'ien kun K'ai Wu, 1637, print from "Description of Pottery and Porcelain" (translation of T'ao Shuo) by S.W. Bushell, Oxford University Press, Oxford, 1977. Reprinted with the permission of Oxford University Press.

saggars. Saggars are pieces of already fired ceramics which separate the green bodies so they do not sinter together and allow them to be piled compactly one on top of the other. The tunnel kiln is a long and narrow insulated room into which cars loaded with green bodies piled in their saggars are placed. In some cases, the kiln is fired under batch conditions; that is, the kiln is loaded with several cars and the insulated doors are closed for firing. In other cases, the loaded cars are constantly entering and leaving the kiln in a continuous fashion. Continuous tunnel kilns are used for large-scale production of materials like bathroom fixtures and bricks and tiles. Small box kilns are frequently fired by electricity. Larger box kilns are fired by natural gas or other fuel. N a t u r a l gas is the most desired because it burns cleanly. Tunnel kilns are most often fired by natural gas due to their size and the cost of energy. With tunnel kilns a great deal of effort is spent to recover and

Part VI Sintering and Finishing

~

recycle the exhaust heat and use it for heat, thus making the kiln more thermally efficient. Two things must be considered when engineering a kiln. One is the kiln: its dimensions, design, burner placement, insulation, thermal efficiency, structural integrity, and so forth. The other is the heat transfer to the ware. The heat transfer to a ceramic piece buried in a pile of other green bodies on a car inside a tunnel kiln is a complicated problem due to the flow of combustion gasses around and into the loaded car and the radiation from the walls of the kiln and between pieces in the pile of ceramic green bodies. For electric heat, the source of energy is nearly all radiation. For combusion heat, the convective heat transfer is less t h a n 15% even at low temperatures, with the balance of the heat from radiation [1]. Ideally, all the ceramic green bodies should experience the same t i m e - t e m p e r a t u r e profile to give uniform microstructures in all of the pieces. But, as you can imagine, this is difficult to achieve. In some cases, the ceramic green body is composed of particles of different chemical composition. These different particles will undergo solid state reaction during sintering and create a new phase after reaction. This is called reactive sintering. Such reactions can occur at the same time as sintering, adding further complication. These reactions are also often associated with a change in density and cause stress to develop in the green body. Sometimes, a part of the ceramic material melts and provides a liquid phase during sintering. This liquid phase provides a medium for faster flow of material to fill the pores because the liquid can (1) act as a flux and dissolve the solid ceramic materials of the porous ceramic green body, (2) provide a pathway for fast diffusion compared to solid state diffusion, and (3) precipitate the final ceramic material in the pore thereby filling it. As a result sintering is a complex phenomena. After the ceramic has been sintered, it must be cooled to room temperature and removed from the kiln. This cool-down can be very important to warping and cracking because the temperature gradient in the ceramic piece can cause stress to build up. In some cases, the solid undergoes phase transformations during cool-down. If these phase transformations incur large changes in molar volume, then they can cause stress to build up. Such a problem occurs in the cool-down of porcelain bodies which contain the phase crystoballite.

780

Part VI Sintering and Finishing

After the ceramic has cooled to room temperature, it is ready to be inspected and tested. Often the ceramic is slightly out of shape and must be ground and polished to meet the dimensional tolerances of the application. In some cases, a glaze or decoration is added to the surface of the sintered ceramic piece and then retired. For electronic ceramics, the active component is soldered to wires and packaged in a hermetically sealed material, which may be either a ceramic glaze or a polymer. Joining ceramics to metal is often required for different applications, and this joining requires the use of a glaze which can cope with the differences in the thermal expansion coefficient of the metal and the ceramic. Finishing in all of its ramifications is discussed in the short Chapter 17, which terminates this book.

Reference 1. Harman, C. G., Jr., Ceram. Bull. 72(1), 14 (1993).

16

Sintering

16.1 O B J E C T I V E S During sintering, the ceramic green body is heated to very high temperatures~approaching the melting point of the ceramic. Changes occur during the sintering of a ceramic: (1) changes in grain size and shape and (2) changes in pore size and shape. The initial porous network changes shape, breaking up into individual pores, which can be either spherical and located at the interior of a grain or cylindrical and located at the grain boundaries between grains. The cylindrical pores will break up into nearly spherical pores if the length of the cylinder is much longer than its diameter; that is, Rayleigh instability. The elimination of porosity leads to an increase in density of the ceramic piece, which is referred to as densification or sintering. When the pores change shape without an increase in density, coarsening takes place. After sintering, the grain structure, upon which many but not all the final ceramic properties are based, may be refined by grain growth during prolonged heat treatment at similarly high temperatures. Two types of grain 781

782

Chapter 16 Sintering

growth, normal and abnormal, are discussed in this chapter. In addition to solid state sintering, reactive sintering involving a sintering solid's interaction or reaction with a liquid, gas, and another solid will be discussed. The main aims of this chapter are to understand the microstructural evolution and the effect of green density upon sintering.

16.2 I N T R O D U C T I O N For maximum properties of sintered ceramics such as strength, translucency, thermal conductivity, and electrical properties, it is desirable to eliminate as much porosity as possible. For some other applications like filters, it is necessary to increase the strength without decreasing the gas permeability. Therefore, depending upon the type of ceramic and its application, different objectives are desired during sintering. Nonetheless, all of these results are achieved by sintering the ceramic at temperatures approaching the melting point of the solid. As formed, a powder compact after binder removal is composed of individual grains with 25-60% porosity, depending on the particular material and the green body forming process used. The ceramic particles that compose the green body are in direct contact. If the green body was produced by isostatic pressing, then the particles may be deformed and have a relatively large contact area. Without pressing little deformation is found. This assembly of particles has a porous network which allows gases inside the pores to exchange with the atmosphere of the kiln during firing. As the temperature is increased, material flows from various sources within the ceramic green body to fill the pores between particles. This process is referred to as sintering and the now deformed particles are referred to as grains. Sintering takes place is three primary stages: an initial stage, an intermediate stage, and a final stage. The distinction between initial and intermediate stage sintering is somewhat cloudy. Figure 16.1 [1] shows the sintered density obtained after 4 hr at each temperature for two log-normal sized distribution A1203 powders. In this figure, the finer powder, which has the lower density initially, sinters to a higher density at a lower temperature because it has a higher specific surface area. In its most simple form, the driving force for sintering is the reduction of the surface energy of a powder compact by the replacement of solid-air interface with lower energy solid-solid interfaces. As an order of magnitude estimate, the net decrease in free energy for a 1 tLm particles size material is about 1 cal/gm [2]. This driving force for sintering is small and should not be squandered on mechanisms (such as surface diffusion and evaporation-condensation) that do not lead to densification.

16.2 Introduction

Initial

Int~atc

783

Final

100

AI203 9O

FinePowder

CoarsePowder

80

i I f~

a 0

70 dIntermediateq,=,~ I J 60

800

.

.

I

1000

9

9

I

1200

9

9

I

1400

9

9

I

1600

Final Stage 9

I

1800

9

2000

Temperature (~C) FIGURE 16.1 Densification of two log-normal alumina powders (dgm = 1.3 and 0.8 /~m) showing the initial, intermediate, and final stages of sintering. Sintering is under constant heating rate conditions of 7.3~ Data from Reed [1].

In Figure 16.1, the initial, intermediate, and final stages of sintering are noted. The various stages of sintering are described next"

Initial Stage: Particle surface smoothing and rounding of pores Grain boundaries form Neck formation and growth Homogenization of segregated material by diffusion Open pores Small porosity decreases <12%

784

Chapter 16 Sintering

Intermediate Stage: Intersection of grain boundaries Shrinkage of open pores Porosity decreases substantially Slow grain growth (Differential pore shrinkage and grain growth in heterogeneous material) Final Stage: Closed pores--density >92% Closed pores intersect grain boundaries Pores shrink to a limiting size or disappear Pores larger than the grains shrink very slowly

These stages are the typical progression of events that take place during sintering. However, sometimes the density does not increase even though the pores change shape. This is called coarsening. The difference between sintering and coarsening is schematically shown in Figure 16.2. If the interparticle separation distance remains the same, coarsening takes place. If the interparticle separation distance decreases, sintering takes place. Only sintering leads to an increase in density, which is referred to as densification. Almost all density variations in the green compacts tend to be amplified during sintering. These density variations lead to warping and cracking during sintering. For this reason, we have devoted an enormous effort in the preceding chapters to the subject of density uniformity in the ceramic green body. Rejecting the ceramic part after sintering is tantamount to throwing away the raw material as well as all the energy and labor used to make the part. Its rejection at this stage is a very expensive loss. In some cases, the rejection rate can be as

Changes in pore shape

r

Change in shape and shrinkage

AL FIGURE 16.2 The differences between coarsening and sintering. Taken from Kingery et al. [2, p. 469]. Copyright 9 1976 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

16.3 Solid State Sintering Mechanisms

785

high as 50%. Only in some cases can the faulty sintered ceramic part be recycled by grinding it into a ceramic powder raw material. This grinding incurs yet a higher energy cost for this raw material but this may be cheaper t h a n buying fresh raw materials.

16.3 S O L I D S T A T E S I N T E R I N G M E C H A N I S M S After burnout of the binder, we are left with a green body that is composed of ceramic particles in contact as shown in Figure 16.2. As the temperature is increased, material flows from various sources within the ceramic green body to the neck at the intersection between particles, as shown in Figure 16.3 [3]. This neck has a negative curvature, compared to the positive curvature of the spherical ceramic particle, and results in a energetically more favorable location for the material. A tabulation of the possible sources of material is given in Table 16.1 with the transport pathway used to transport the material from the source to the neck. These pathways are drawn on Figure 16.3 and consist of surface, lattice, and grain boundary diffusion, as well as vapor phase diffusion. These pathways give rise to different mechanisms of

6

.L-L.

J

/;

"~\4

FIGURE 16.3 Pathways for the transport of material during the initial stage of sintering; see Table 16.1 for details on paths. Reprinted from Ashby [3], copyright 1975 with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

786

Chapter 16 Sintering

TABLE 16.1 Pathways for the Transport of Material during the Initial Stage of Sintering a

Pathway (on Figure 16.3)

Transport path

Source

Sink

Process

1 2 3 4 5 6

Surface diffusion Lattice diffusion Vapor transport Boundary diffusion Lattice diffusion Lattice diffusion

Surface Surface Surface Grain boundary Grain boundary Dislocations

Neck Neck Neck Neck Neck Neck

Coarsening Coarsening Coarsening Sintering Sintering Sintering

a Reprinted from Ashby, M. F., Acta Metall. 22, 275, copyright (1975) with kind permission from Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

sintering. Each of these mechanisms is used to establish the kinetics of sintering. Before we can discuss sintering kinetics, the driving force for sintering must be defined.

16.3.1 Driving Force for Sintering The driving force, DF, for sintering is the reduction in the total free energy of the system AG T -

AGv +

AGg b +

AGs

(16.1a)

where AGe, AGg~, and AGs represent the change in free energy associated with the volume, boundaries, and surfaces of the grains, respectively. If we consider that sintering removes only the solid-vapor inter-

TABLE 16.2 Surface Energy for Ceramics Using the Equation T = A + BT(K) J/m 2a

Ceramic

Plane

A (J/m 2)

A1203 BeO fl-SiC CaO MgO SiO2

[1034] [001] [110] [001] [001] [110]

2.6 2.44 3.0 2.2 2.6 0.93

B (J/m2/K) -0.48 -0.47 -0.55 -0.38 -0.48 -0.19

• • • • • •

10 -3 10 -3 10 -3 10 -3 10 -3 10 -3

a Data taken from McColm, I. J., and Clark, N. J., "The Forming, Shaping and Working of High Performance Ceramics," p. 19. Blackie, London, 1988.

16.3 Solid State Sintering Mechanisms

787

face, the change in the surface free energy is given by

hGs = 7sv (Final surface area - Initial surface area)

(16.1b)

where 7sv is the solid-vapor interfacial energy, which is typically ~0.1 J / m 2. Typical values of the interfacial energy are given in Table 16.2. These values of the interfacial energy are a function of the crystallographic plane. In equation 16.1b the surface area is usually given per unit of volume. Thus, for a sphere, the initial surface area per unit volume is given by 3/Ro, where R0 is the mean radius of the ceramic particles composing the ceramic green body. As a result the free energy difference given in equation 16.1b can be related to a difference in the reciprocal of the radius of curvature before and after sintering. Diffusion during sintering is also driven by a difference in free energy, and as a result, diffusion takes place from a region of positive surface curvature to a region of less positive, zero, or negative surface curvature. Recently, the sintering driving force has been modified to the socalled sintering stress [4,5] (or, in other terminology, sinteringpressure [6], sintering force [7], and sintering potential [8]), which is defined as the equivalent mechanical stress, that is, force per unit area the same units as pressure, necessary to stop densification. This mechanical stress to stop sintering is caused by a tensile force. A compressive force will speed sintering at a particular temperature or cause sintering at a lower temperature. This implies that, when the sintering stress is changed, the system of ceramic particles or grains is forced through a different microstructural path. Such a different microstructural path can be observed in Figure 16.4, where the same fine A1203 powder is sintering with and without pressure. The hot pressed sample shows densification at lower temperatures, that is, a different microstructural path, t h a n the sample without pressure. The effect of an applied compressive force or applied pressure on the driving force, DF, for sintering is described by*

DFa[eapplied-2[Tsv~] \Ro]

(16.1c)

where Ro is the mean size of particles composing the ceramic green body. Without an applied pressure, the sintering stress is given in a simplified form as 2(7~v/Ro). For sintering with different starting particle sizes, the sintering stress increases with decreased particle size.

* Analogy to the analysis of the sintering of a string of spheres by Cannon and Carter [9].

788

Chapter 16 Sintering 100

9O

A

~e m

Hot Pressed

[ Ordinary Sintering

~,

80

m o

a 70

60 800

1000

1200

1400

1600

1800

2000

Temperature (~ Sintering of a fine A1203powder ( d ~ = 0.8 tim) with and without pressure. Sintering is under constant heating rate conditions of 7.3~ Data from Reed [1].

FIGURE 16.4

For sintering with the same starting particle size, the sintering stress changes only with a variation in starting green density [10,11]. The compact with higher green density has a higher sintering stress. These concepts will be discussed quantitatively in Section 16.3.2.1 (subsection Sintering Stress and the Effect of Pressure on Sintering) and in Section 16.6 (Pressure Sintering).

16.3.2 Sintering Kinetics by Stage Since the introduction of a mathematical model for sintering by Kuczynski [12] numerous other models have been proposed. Reviews of these sintering kinetic models are given in references [13-19]. This description of sintering kinetics is organized into initial, intermediate, and final stage kinetic models.

16.3.2.1 Initial Stage Sintering The initial stage of sintering [20-23] is frequently referred to as the neck formation stage, as is shown in Figure 16.1. The sintering driving force for the initial powder compact is due to the curvature difference between particle surface and that of the neck, see Figure 16.5. The six

16.3 Solid State Sintering Mechanisms

789

FIGURE 16.5

Schematic representation of the contact area between two partically sintered spheres: (a) center-to-center distance is constant, (b) decreasing center-to-center distance. The sphere radius is a, x is the radius of the neck, 2h is the decrease in the center-to-center distance, and K is the radius of curvature (negative) for the neck.

distinguishable diffusion-controlled paths of material transport (vacancies diffuse normally), considered to be the most probable sintering mechanisms, are shown in Figure 16.3. Any one of these pathways increases the neck size, but not all these mechanisms decrease the center-to-center distance giving densification or sintering. To clearly see this distinction, we will discuss two transport mechanisms: (1) vapor transport and (2) bulk diffusion.

Vapor Transport The surface curvatures in Figure 16.5(a) introduces an excess pressure according to the Kelvin equation In

(~oo) = pNAkBT~/SvMw (1+

1)

(16.2)

where ~/svis the interfacial energy of the solid at the sintering temperature, T, Mw is the molecular weight of the vapor, NA is Advogadro's number, and p is the density. We can see initially the radius of curvature in the neck, K, is much smaller than that of the particle, a, therefore the pressure difference AP = Po - P1 is small allowing the approximation In (PJPo) ~" AP/Po, giving

AP= ~/P~ p~]'Aks T

(16.3)

This result is the Kelvin equation. In this approximation, we have further assumed that the particle is essentially a flat particle (with a vapor pressure of P0) compared to the radius of curvature of the neck. We can calculate the rate at which the neck increases by equating the rate of material transfer to the surface of the lens between the spheres with the increase in its volume. The rate of condensation, m, is proportional to the difference in equilibrium vapor pressure, AP, as given by

790

Chapter 16 Sintering

the Langmuir analysis"

( Mw

m=aAP\27r~BNAT]

(16.4)

g m/cm2/sec

where a is the accommodation constant which is nearly 1.0. This rate of condensation is equal to the increase in the volume of the lens between the particles:

mA p

=

dv dt

~

(16.5)

For the neck region the area, A, volume, v, and the radius of curvature, K, have the following definitions: A

7r2x3 - - ~

a

'

U --

fl'x4

2a

x 2 ,

K -

2~( a -~ x )-

x 2

2af~

x

~0"3"

(16.6)

The reason for the limitation ofx/a ~ 0.3 on these definitions of A, v, and K is that a simplified spherical geometry has been used for these calculations. This limits the approach to small simtering times. Substituting these geometric definitions and the condensation rate, m, into the earlier equation, we obtain a relationship for the growth rate of the neck diameter, x, with time:

x _ ~3~/-~?svMwPo~ 1/3 _2/3tl/3 \-~-~2/3T3/-------~2/ a a -

-

(16.7)

This equation is given in terms of a constant (3X/~?svMwPo/ (~f2k~/3T3/2K2)) 1/3, a function of particle size, a -2/3, and a time function, t 1/3. This neck size is an important factor in determining the strength of the ceramic and also its conductivity. The radius of the neck, x, will continue to increase until it has become essentially flat. An experimental study of the sintering of spherical NaC1 particles [2, p. 472] shows that this relationship for the neck size is indeed followed for this evaporation condensation mechanism. If we consider the microstructural changes to the particles, it is clear that the spherical particles will alter their shape with time to that of a prolate ellipsoid of revolution by this transfer of material from the spherical surface to that of the neck. In fact, the distance between the centers of the particles is not affected by vapor phase transport and only the shape of the pores is changed. Without a decrease in the centerto-center distance there is no densification or sintering. Therefore, this is one example of coarsening during the initial stage sintering. The difference in free energy or chemical potential between the neck area and the surface of the particle provides a sintering driving force which causes the transfer of material by the fastest means available.

16.3 Solid State Sintering Mechanisms

791

If the vapor pressure is low, then the materials will transfer by solid state processes. Several processes can be imagined, as are shown in Table 16.1. The transfer of material from the surface to the neck by surface or lattice diffusion does not lead to decrease in the center-tocenter distance between particles and does not contribute to sintering.

Only transport of matter from the particle volume or from the grain boundary between the particles causes shrinkage and a decrease in porosity, resulting in densification. The following analysis will discuss the diffusion of vacancies from the neck surface region, where they are in high concentration due to the high curvature, to the bulk either at the center of the neck or the bulk of the particle. The excess concentration of vacancies, AC, at the neck surface with curvature, K, over that in a plane surface, Co, is given by

AC = ~/sv~Co kBTK

(16.8)

where ~ is the atomic volume of the diffusing vacancy and Tso is the interfacial energy of the solid. Bulk Diffusion surface is given by

The flux of vacancy diffusion away from the neck J = 4D o AC moles/cm2/sec

(16.9)

where Do(= D*/(~Co)) is the diffusion coefficient for vacancies, D* is the self-diffusion coefficient for vacancies. Using the substitution of J for m, the condensation rate in the previous analysis, in addition to the definitions of area, volume, and radius of curvature of the neck, as well as the conservation of mass expressed as a differential equation, we can write an equation for the neck radius, x, as a function of time: x

a

= ~40~'~oD-D*~~/5 _3/5t~/5 \ k---~ ] a

(16.10)

This equation is given in terms of a constant, (40TsvD.D*/(kBT)) 1/5, a function of particle size, a -3/5, and a time function, t ~/5. This functionality has been observed experimentally in a number of metal and ceramic systems [2, p. 472]. With bulk diffusion, in addition to the increase in the neck radius, there is a decrease in the center-to-center distance. The rate of decrease in the center-to-center distance is d(x2/2a)/dt. Using the preceding neck radius growth, equation 16.10, and taking the appropriate derivative, we find that the center-to-center shrinkage, AL, is given by

AL

~20Tso~D*] 1/5 tn / a-6/5t2/5=Cj~-~

~-0 = [ ~ k - ~

(16.11)

792

Chapter 16 Sintering TABLE 16.3 Initial Stage Sintering Parameters for the Equation AL/Lo = Cj(tn/a m)

Mechanism

n

m

Cj

Lattice diffusion a

2 5

6 5

(10yso~ Dv)"~5 kBT

Lattice diffusion b

1 2

3 2

f(2~/~vaD ~ 1/2 \ kBT ]

Grain-boundary diffusionb

1 3

4 3

~37svt)Db~1/3 \ ksT ]

a Kingery, W. D., and Berg, M., J. Appl. Phys. 2{}(10), 1205 (1955). b Coble, R. J., J. Am. Ceram. Soc. 41, 55 (1958).

where Lo is the initial center-to-center distance. The shrinkage of the spheres is proportional to the two-fifths power of time. The decrease in densification rate with time gives rise to an apparent endpoint density, if experiments are carried out for similar periods of time. Log-log plots correct this behavior. For other diffusion pathways, the constant Cj and n and m of the generalized expression are given in Table 16.3. For a compact of spheres the volumetric shrinkage, AV, can be estimated from the linear shrinkage, AL, as follows, assuming that the shrinkage is the same in all three dimensions: ~V AL =3 Vo Lo

(16.12)

This is valid for all sintering mechanisms. Table 16.4 show the formula for initial stage sintering for all the possible mechanisms. These mechanisms use the previous formalism with different material sources and sinks and different transport pathways. These mechanisms all have the general formula: ~V

Vo = Cj(tn/a m)

(16.13)

where Cj is another collection of constants, now dependent on the number of interparticle contacts per sphere, z. Included in these mechanisms is the transport of material by viscous flow. The interfacial energy of the surface is a rather weak function of temperature but can be strongly affected by impurities. These impurities can be added to the solids used for sintering or they can adsorb or dissolve from the atmo-

16.3 Solid State Sintering Mechanisms

793

TABLE 16.4 Initial Stage Sintering P a r a m e t e r s for the E q u a t i o n AV/Vo = Cj(tn/a m)

Mechanism

n

m

Cj

Lattice diffusion

4 5

12 5

3z(40~/svt2Dv~4j5 --8\ k s T ]

Viscous flow

2

2

a

(

27ZT~v~ 1672 ]

Note: z = n u m b e r of particle contacts per sphere. a Kingery, W. D., a n d Berg, M., J. Appl. Phys. 26(10), 1205 (1955).

sphere. Examples include (1) H2 and 02 are soluble in A1203 and used to enhance the sintering of MgO doped A1203 (N2 is insoluble) [1, p. 459] and (2) H20 vapor used to enhance the sintering of pure SiO2 (without H20 vapor SiO2 sintering is essentially prevented at a temperature of 1000~ [24]. Both the diffusion coefficient of crystalline materials and the viscosity of glassy materials are functions of composition (especially impurity concentration) and strong functions of temperature. The impurities added to increase the sintering rate are referred to as sintering aids. Thus sintering kinetics are manipulated by temperature and sintering aid impurities to a major degree. Some subtilities are not incorporated in the models given so far. The shape of the neck is very important in establishing the curvature, which is in turn responsible for the sintering driving force. Coblenz et al. [25] have developed models that account for the effects of small neck versus large neck, K

=

X 2

2(a - x)'

X

fOr-alarge,

X

2

X

K ~ ~2a, fOr-a~ 1

dihedral angle, 0, x2

for x large a

K --

2a

1-Xsin a x2

K~

2a ( 1 - cos ~ ) '

-cos forX~ 1 a

(16.14)

794

Chapter 16 Sintering

a n d n e a r neck p i t t i n g or u n d e r c u t t i n g [26], K~0.26

x,

for x ~ l a

K~0.58

x,

for x ~ 0 . 3 a

(16.15)

The effects of neck shape on the neck g r o w t h rate was t h e n predicted by Coblenz et al., a s s u m i n g t h a t t h e r e is conservation of solid volume a n d the d i h e d r a l angle is large. These r e s u l t s are t a b u l a t e d in Table 16.5. The effect of u n d e r c u t t i n g the sphere surface j u s t at the edge of t h e neck h a s been shown to have a m a j o r effect on the m e c h a n i s m s of surface diffusion. In t h e c o m m o n cases, m u l t i p l e t r a n s p o r t m e c h a n i s m s m a y well be o p e r a t i n g at the same time. Simply a d d i n g the contributions to the

TABLE 16.5 Neck Growth Rates for Various Mechanisms with the Initial Stage Geometry Assumed a Mechanism (source ~ sink)

Geometry assumption

X

Evaporation-condensation (total sphere surface neck) Surface diffusion (near neck sphere surface neck) Grain-boundary diffusion (grain boundary neck) Lattice diffusion (grain boundary 9 neck) Lattice diffusion (near neck surface of sphere neck)

2

2a

K~ 0.58

I_

X m

=\

kBT

(O sv O,

2

4a

K~ 0.58

k//2--~w(R~)3/2j (Ka)-

k~

xb

X2 K~4a

K~

Neck growth rate ~ = dx dt

x

] (K3a)-I

] (Kx

2)1

=\

kBT ](KX

=\

kBT ] (aK2)-I

a Taken from Coblenz, W. S., Dynys, J. M., Cannon, R. M., and Coble, R. L., in "Sintering Processes" (G. C. Kuczynski, ed.), pp. 141-157. Plenum, New York, 1980. b Numerical solution with x/a < 0.3 predicts this radius of curvature by Nichols, F. A., and Mullins, W. W., J. Appl. Phys. 36(6), 1826-1835 (1965).

16.3 Solid State Sintering Mechanisms

795

neck size, x, from the various mechanisms, however, is not reasonable because each mechanism will alter the radius of curvature of the neck and thus alter the driving force. Therefore, the evolving neck geometry will depend on the relative fluxes from the various operating diffusion mechanisms.

T e m p e r a t u r e Gradient Up to this point, the driving force for sintering has been considered the curvature of the surface in a constant temperature environment. Searcy [27] argues that a temperature gradient in the ceramic may also provide a driving force for sintering in metals. In this case, the driving force, AP, is given by

.

Po - exp [NAk B

1)]

T1 T2

(16.16)

where /~L~vap is the enthalpy of vaporization and T1 and T2 are two different temperatures within the green body. Searcy compared the driving force due to curvature to that for a temperature gradient for spherical silver particles sintering at 1175 K, where vapor transport was responsible for sintering. Figure 16.6 shows this comparison. The temperature difference of I~ would create a driving force about the same as the driving force for particles 1 ftm in radius, a 0.1~ temperature difference would create a driving force about the same as the driving force for 10 ftm radius particles. To simplify sintering, isothermal conditions are used for many industrial processes and also in experiments to determine sintering mechanisms.

S i n t e r i n g Stress and the Effect of P r e s s u r e on S i n t e r i n g The following analysis comes from the work of Cannon and Carter [9]. A more accurate assessment of the sintering driving force is given by the Gibbs free energy of sintering. At constant temperature without chemical reaction or change of phase, we have only the change in the Gibbs free energy associated with the different surfaces within the green body: AG = h(T~4g~)+ h(TsvAsv)

(16.17)

where Tgb and Tsv are the interfacial energies and Ago and ABe are the areas of the grain boundary (subscript gb) and the solid-vapor (subscript sv) surfaces, respectively. The delta refers to the final state minus the initial state. In the simple case of a string of touching spheres initially, essentially all the area is the solid-vapor area, see Figure 16.7(a). After either sintering or coarsening, the shape will be altered. The equilibrium shape is determined by minimizing the energy, AG,

796

Chapter 16 Sintering

10

100

10

Z~T, ~

1.0

0.1

1.0

o. 10-1 o. O. 2 o I

o~-

o .G . m

,~

10-a

10 -4

lO-S[

1.0

1

l,.,

0.1 0.01 Particle Radius, ram.

0.( t01

FIGURE 16.6 Sintering driving force as a function of temperature difference (top scale) and particle radius (bottom scale). Taken from Searcy [27].

with respect to the shape of the system while maintaining the volume of the solid constant. The results of such a minimization is shown in Figure 16.7(b), where the center-to-center distance, L = L o = 2Ro, between the spheres was forced to be a constant (i.e., coarsening by transport of material by various means from the particle surface to the neck region) and Figure 16.7(c) where the center-to-center distance, L, was allowed to decrease (i.e., sintering by transport of material by diffusion from the bulk to the neck region). In this situation, Figure 16.7(c), the dihedral angle is 120 ~ The dihedral angle has the definition cos O = 7g___kb 2 2%v

(16.18)

which suggests that it is dependent on only the ratio of the interfacial energies. The initial and the final curvature of the system of spheres gives the sintering driving force as defined by AG. Sintering potential [9] (or sintering pressure [6] or sintering stress [4,5]) (symbol 9 defined later) is the tensil stress (or pressure) required to halt densification.

16.3 Solid State Sintering Mechanisms

797

a

L

v

L C

L Schematic of sintering and coarsening: (a) row of initially spherical particles of size R0, (b) equilibrium structure where coarsening takes place, (c) equilibrium structure when sintering or densification takes place without an imposed force.

FIGURE 16.7

The sintering potential is analogous to the zero creep stress [28]. Applied stress less t h a n the sintering potential permit densification and even accentuate densification, if the stress is compressive, because the driving force for sintering is increased. Tensile stress larger t h a n the sintering force pulls the system apart. The sintering potential is determined by the derivative of the Gibbs free energy with respect to the center-to-center distance, L, at constant volume of solid (v~); t h a t is, Fsinter =

d-L s

(16.19)

This derivative d e m o n s t r a t e s a force acting on the string of spheres. To calculate a pressure or stress, this force m u s t be divided by the cross-sectional area of the spheres:

=~

7rR 2o

(16.20)

798

Chapter 16

Sintering

This expression is the sintering stress, ~I'. This is analogous to the thermodynamic definition of the pressure, (0A/0V)T = - p , where A is the Helmholtz free energy, V the system volume in which the solid volume, Vs, remains constant, and T the system temperature (a constant in both analyses). The sintering potential is a function of the centerto-center distance, L, for a particular size particle, R0, and the dihedral angle, ~ [9]. For the string of spheres the sintering force is given by

Fsinter--

m - 1) 7rR~

= TsvKTr

m 2

0 < m < (sin $/2) -1

,

(16.21)

where m = (sin ~/2) -1 for the initial state. (Note: K in this equation is the average radius of curvature for a grain (meter) not the curvature (meter -1) as used in the Cannon and Carter [9] nomenclature.)

Bulk Diffusion Transport in Sintering a String of Spheres The initial sintering stress is dependent upon only the dihedral angle. The application of stress on a string of particles will affect the chemical potential at the grain boundary and, more important, determine the sintering rate. The chemical potential of a surface atom is given by [9] ft = ft 0 + ~---~12

(16.22)

K

(111)

where K in this equation is the average radius of curvature for a grain with two radii of curvature, r 1 and

+ , and the chemical r~ potential of a grain-boundary atom by analogy to equation 16.22 is given r2,

=-

f t -- f t 0 -- O ' n n a

(16.23)

where O'nn is the normal stress at the grain boundary and 12 is the atomic volume. The diffusive flux in the boundary is given by jgb = -

D g b V~ lz ksTl2

(16.24)

where Dgb is the diffusion coefficient in the grain boundary and V~/z is the divergence of the chemical potential along the grain-boundary interface. The flux continuity within the grain-boundary region of width, 6, requires that OC

0--t-= - V~Jgb 6 - 2J~ - 2fl

(16.25)

where J~ is the lattice flux. At steady state, that is, O C / O t = 0, the atomic concentration, C, and the grain-boundary thickness, 8, remain constant. The accumulation of mass results in a deposition flux of 2fl

16.3 Solid State SinteringMechanisms

~

onto the adjoining lattices, which is uniform along the boundary. As a result, the grain centers move at a rate dL d--t-= 2fit2

(16.26)

Assuming lattice diffusion (i.e., J~ = 0) is slow compared to grainboundary diffusion, the shrinkage rate is given by d L = _VsJg b 8 t2 = - 8Vs(Dgb Vst.t)/(ksT) - - S D ~ V21.t/(ks T) dt

(16.27)

For a grain boundary diffusion coefficient which is not a function of position, we find the later equality. The Laplacian of the chemical potential can be rewritten Vs2/~ -'- V s2( T a n , if the preceding definitions of the chemical potential are used and if the radius of curvature of the neck, K, and the interfacial energy, ~/sv, are assumed to be constant. Furthermore, the normal stress on the grain boundary corresponding to a neck thickness, x, can be written as [9]

Crnn_ 2(eapplied_ ~) [l _ (r) 2]

TsvK

(16.28)

Upon substitution of this expression into the shrinkage expression and taking the Laplacian, we find the shrinkage rate to be

dt

= 4 \kBTR~]

I

L

(~/sv~ \R3o]

(16.29)

where the term (~/sv/R~) is used to render the pressure (or sintering stress) dimensionless and the term (t2 8 DgJ(kBTR4)) = r~ 1 has the dimensions of time -1 and is used to render the time coordinate dimensionless. Here the sintering stress is described = 2 ~/s___~sin v 0 x 2

~/sv K

(16.30)

in terms of the local radius of curvature, K, and the radius of the neck, x, and the solid-vapor interfacial energy. As sintering progresses, x increases and K decreases, thus altering the sintering potential. As a result, the densification rate is a complex function of time. Cannon and Carter [9] have performed this calculation and give theoretical instantaneous densification rates as a function of dihedral angle at various degrees of shrinkage, as shown in Figure 16.8. The neck growth of the initial stage will be accompanied by interparticle shrinkage of only several percent. After the neck has become

800

Chapter 16 Sintering

FIGURE 16.8

Instantaneous sintering rate during the progress of sintering (with an applied force) as a function of dihedral angle: rb = (~SDg/(ks T R4)) -1. Taken from Cannon and Carter [9]. Reprinted by permission of the American Ceramic Society.

blunted by neck growth, grain growth becomes possible. The point at which grain growth first occurs is considered to terminate the initial stage of sintering. Equation 16.30 is expected to be valid for the densification up to 2%, as in the case in Figure 16.7(b).

16.3.2.2 Intermediate Stage Sintering The intermediate stage of sintering [29-32] begins after grain and pore shape changes caused during the initial stage produce a pore and grain-boundary matrix consisting of equilibrium dihedral angles formed on the solid-vapor (pore) surface at the intersections with the solid-solid (grain-boundary) interfaces. At this stage, the pore shape approximates a continuous cylindrical channel coincident with three grain edges throughout the matrix as shown in Figure 16.9. During this intermediate stage of sintering, the cylindrical pore simply shrinks. Assuming all grains have the same size and shape (cube, dodecahedron, or tetrakaidecahedron) and all pores are cylindric and of the same size, the porosity function, f(~b) was derived by Johnson [29] as f(dp) = K \ R 2 o k s T ] ( t f - t)

(16.31)

where ~(= 1 - P/Pth) is the porosity, K and m are constants, Tsv = surface energy, D is the diffusion coefficient, and t2 is the volume occupied by a single vacancy, Rg0 is the initial grain size, and tf is the extrapolated time for disappearance of the cylindrical pores. The values

801

16.3 Solid State Sintering Mechanisms

F I G U R E 16.9 The microstructure model used for intermediate stage sintering. This model consists of tetrakaidecahedra with cylindrical pores at the triple junctions of grains. Face-to-face grain junctions have no pores in this model. Please note that, at the beginning of intermediate stage sintering, the pores are much larger than drawn in this figure. Taken from Coble [30].

of f(~), K, and m from different models are given in Table 16.6. None of these models accounts for the effect of applied pressure. An attempt to make this conversion would be to replace the ratio (~/sv/R~) in the preceding expression, noting m >- 3, with the t e r m (Papplied -- xI~). How-

TABLE 16.6 Intermediate Stage Sintering Models: Parameters for Simplified Models Represented by the Equation f((b) = K[(TsvflD )/(R ~oks T )](tf - t) Model Volume diffusion: Coblea, b Model A in Coble and Gupta c Model B in Coble and Gupta c Johnson d Grain boundary diffusion: Coble a,b Coble and Gupta c Johnson d

f (dp)

K

m

(b ~b ~b~2[1 - ~ ln(1.2~b)] - l n (1 - (b)

720 335 1190 378

3 3 3 3

2~2

80b 860b 1000b

4 4 4

2~3/2

Note: d~ = 1 - P/Pth, b is grain boundary thickness. a Coble, R. L., J. Appl. Phys. 32(5), 787 (1961). b Coble, R. L., J. Appl. Phys. 32(5), 793 (1961). c Coble, R. L., and Gupta, T. K., in "Sintering and Related Phenomena" (G. C. Kuczynski, N. A. Hooton, and C. F. Gibbon, eds.), p. 423. Gordon Breach, New York, 1967. d Johnson, D. L., J. Am. Ceram. Soc. 53(10), 574 (1970).

802

Chapter 16 Sintering

ever, the sintering stress, ~, would have a different definition than that here due to a different geometry of the system. A definition for the sintering stress is not available for intermediate stage sintering but may be assumed to be approximately, 9 ~ Ysv/Rp, where Rp is the cylindrical pore radius. The cylindrical pore will break up into a string of spherical pores, as shown in Figure 16.10, when the length to radius ratio exceeds a critical ratio. This breakup is analogous to the Rayleigh instability of a cylinder of liquid discussed in Chapter 8 and would have a critical ratio of L/Rp > 1.5. This critical ratio can also be affected by the presence of surface diffusion [33]. With this pore breakup, the end of the intermediate stage of sintering is triggered, allowing the calculation of tf. The time required to pinch off the cylindrical pore into a string of spherical pores can also be significant, so that tf would be composed of two terms, one consisting of the time to diminish the pore radius to its critical unstable size and another consisting of the time required to pinch off the cylindrical pore. At this point, the pores are closed and exchange of gases with the kiln atmosphere stops. The resulting spherical pores are located at the grain boundaries within the green body. Open and closed porosity can be determined by density measurements with and without impregnation by an inert liquid. Using this technique, Chen [34] showed that open porosity decreases with time in the sintering of BaTiO3. A major amount of closed porosity appears at 88% density and is slowly removed as final stage sintering progresses. At a certain time, the pores become closed as is shown in Figure 16.11.

Grain ,ylindrical Pore

/

Grain

ica'/

Grain 9

~

Grain FIGURE 16.10 Breakup of a cylindrical pore into a string of spherical pores.

16.3 S o l i d S t a t e S i n t e r i n g M e c h a n i s m s 45.1

30

,

,I

I

803

I

-

o,

L 15 -

0

1

7O

"

I

80

90

1{~

Density, % Theoretical Volume fraction for open and closed pores as a function of BaTiOa sintered density. Pores are closed at densities above 88% theoretical density. Taken from Chen [34].

FIGURE

16.11

16.3.2.3 Final Stage During the final stage [30], the removal of these closed pores takes place. The final stage densification is dependent on the association of pores with grain boundaries and the rate of grain growth. The string of pores at the grain boundary will migrate to the point of lowest energy. The energetically favored pore location is at the intersection of three grains in two dimensions, four grains in three dimensions. As shown in Figure 16.12, the movement of the string of pores to a single pore at the four grain intersection is a complicated process. In some cases, it occurs but in others it is prevented from occuring by the movement of grain boundaries caused by discontinuous (or abnormal) grain growth, which is discussed in detail later. When pore migration does occur boundary diffusion and lattice diffusion from sources on the boundary are responsible for the pore reorganization. For nearly spherical pores, the flux of material to a pore can be approximated by [2, p. 481] J = 47rDvhC \Ro - R J

(16.32)

804

Chapter 16 Sintering

Grain

O

Spherical Pores

/

Grain

Grain

Spherical/ ~.f.,.~res ~ - ~

Grain FIGURE 16.12 The mean diffusion distance for material transport is smaller when

there are more of the same size of pores on the grain boundary.

where Dv is the volume diffusion coefficient, Rp is the pore radius, and Ro is the effective radius of curvature of the grain. The distance between spherical pores plays a role in the time required to reorganize the structure. For a system where there are a number of equal sized spheres on a grain boundary, the mean diffusional distance is smaller. As a result, the pores are more effectively moved to the four grain intersection. A detailed analysis of this problem has not been made to date. Primarily because it is often complicated by simultaneous grain growth. The pores located at the intersection of four grains can then shrink continuously to zero size in a stable fashion. "Discontinuous" grain growth can also occur at this stage. Instead of pores shrinking while the grains remain the same size, some of the grains grow at the expense of others and trap the pores inside a grain. The final density is then limited to less than the theoretical density. Kingery and Franqois [35] have made model calculations which predict pore growth until the dihedral angle at the grain boundary is in equilibrium with the structural configuration. A large pore can be thermodynamically stable depending on the value of the dihedral angle and the pore size/grain size ratio as shown in Figure 16.13. For a given dihedral angle and pore size, there is a critical grain size, above which the pore is unstable and below which it is stable with respect to its elimination, leading to further densification. Lange and Davis [36] proposed that the sinterability of a pore depends on the coordination number of the pore, instead of size. Only when the coordination number, n, is lower than a critical number,

16.3 Solid State Sintering Mechanisms

805

180 160

Pores

shrink

Pores

C

e

grow

40

0

2

4

6 8 10 12 14 16 18 20 22 24 26 28 30 Number of grains surrounding a pore (in space)

FIGURE 16.13 Conditions for pore stability as determined for the ratio of pore radius to grain radius, rJRg = 0. When this ratio is negative, the pores will tend to grow; and when this ratio is positive, the pores will tend to shrink. Taken from Kingery et al. [2, p. 488]. Copyright 9 1976 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

nc, will the pore shrink. If n > nc, the pore will grow. Those large pores

with higher coordination number, after a long time, can shrink or be trapped by nearby grains. During discontinuous grain growth, the grain boundary can move as is shown in Figure 16.14 [37]. Here we see that the grain boundary velocity is perpendicular to the grain boundary and in the direction

50

FIGURE 16.14 Cross-section view ofpolycrystalline solid showing grains with a different number of sides given for some grains. Arrows indicate the direction of grain boundary movement during grain growth. Taken from Burke and Rosolowski [37].

806

Chapter 16 Sintering GRAIN

Up

BOUNDARY

~

--- ~

b

" A

D s ~ rs

rb %.0 2--

Up,4.4 ii

Up= 11.2

FIGURE 16.15 Predicted pore shapes for several values of the normalized pore velocity, %, at r = zr/3. Reprinted from Hsueh et al. [38], copyright 1982, with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

from the larger grain to the smaller grain. The grain boundary movement causes the pores at the grain boundary to be dragged along with the grain boundary as shown in Figure 16.15 and 16.16. Sometimes the grain boundary moves too fast and leaves behind the pore within an enlarging grain. Pore dragging has been studied by Hsueh et al. [38]. They found that pores attached to the grain boundary exhibit a maximum steady state pore velocity, Vp,~ax, that varies with the dihedral angle, r

(D Ssa sv]

vp,~a~= \ 7rR3oksT / [

: 2 cos($/2) I

for ~ < ~r

(16.33)

where Dsas is the surface diffusion parameter and %v is the interfacial energy of the solid-vapor interface. The factor [17.9 - 6.25/(2 cos(S/ 2))], is caused by the change in pore shape due to its movement. This factor is always greater than 1. At low velocity, the pore sitting on a grain boundary has a lens shape that is determined by the dihedral angle. If the dihedral angle is 7r, the pore is initially spherical. As the

16.3 Solid State Sintering Mechanisms

807

FIGURE 16.16 Pores subject to motion with a grain boundary in the sintering of MgO. Reprinted from Hsueh et al. [38], copyright 1982, with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

velocity increases the lens shape of the pore with dihedral angle, ~, changes into a spherical cap where the trailing surface is spherical, as shown in Figure 16.15. At a particular pore velocity, a steady state pore shape develops. At very high grain boundary velocities, the pore may detach from the grain boundary. Detachment occurs by a movement of the leading grain boundaries, moving from the edge of the pore toward the center of the nearly flat pore surface on the leading surface. Once the two grain boundaries make contact inside one grain, the pore detaches from the grain boundary. An analysis of the critical grainboundary velocity* responsible for pore detachment, identifies a maxim u m pore size which will ensure attachment. The maximum pore size, Rp,max, is a function of dihedral angle*: Rp,ma x

n

~ 1 . 2 8 1 ~ - 1.42 ~1/3

~ 0.5 \i:27-~ - 0 ~ - 6 ]

(16.34)

where L/2 is the distance along the grain boundary between the centers of two pores. * Here, Vgb = V~TgbMgbi2~3/Rg, where Tgb is the grain boundary interfacial energy, Rg is the radius of curvature of the grain, and Mgb is the grain-boundary mobility.

808

Chapter 16 Sintering

FIGURE 16.17 Microstructure of sintered A1203 showing a string of pores detached from the grain boundary and left within a large grain, ~3 ~m in size.

Because of long diffusion distances, those entrapped pores are very difficult to eliminate. It is believed [38-41] that only the pores situated at the two grain junction can be trapped inside the grain. Figure 16.17 shows a line of pores that have detached from a grain boundary and are left to reside within a large grain. A mechanism of transferring a pore from three to two grain junctions was proposed by Fang and Palmour [41], as shown in Figure 16.18. As a result of the complicated microstructure of grains and pores and the long diffusion distances for a pore imbedded in a grain and the grain deep within the microstructure of the polycrystalline ceramic material, the last vestiges of pore removal are slow. Final stage sintering kinetics is further complicated because some pores are stable within microstructure, depending on where they are located. Coble [30] has developed a simplified lattice diffusion model for the removal of all pores centered at the 24 corners of a tetrakaidecahedron-shaped grain. The shrinkage is expressed in terms of the pore volume, V~, given by

Vp = Vo (4.247r Dlygba~ (tf- t)/L 3 \ ksT /

(16.35)

a

FIGURE 16.18 Mechanism for pore detachment from the grain boundary consisting of (a) a pore at a three grain junction, (b) a pore at a two grain junction, and (c) a pore detached from the grain boundary. Taken from Fang and Palmour [41]. Reprinted by permission of the American Ceramic Society.

16.3 Solid State Sintering Mechanisms

809

where V0 is the volume of the polyhedron with edge length, L; [4.247r(D1Tg~II/(kBT))] is a collection of constants; Dl is the lattice diffusion coefficient; ~/g~is the grain-boundary interfacial energy; and tf is the time necessary to reach theoretical density, which is possible because all pores are unstable. The polyhedron volume is approximated by the grain volume, which is constant with time, simplifying the expression. This neglect of pore volume limits the validity of the equation toYp < 0.4 Yo.

16.3.2.4 Determining the Sintering Mechanism from Sintering Data Sintering data consist of either AL or AV or sintered density data as a function of time for either isothermal sintering or constant heating rate sintering. In addition, the mean grain size is sometimes measured at various times at temperatures in the isothermal sintering experiments. If isothermal, the data can be fit with the various expressions given for the initial, intermediate, and final stages of sintering. These expressions are generally of the form i V _ 3 AL _ h(1 - r _ 1/Ap Vo Lo 1 - ~o 1/po -- C j [ ( t - t f ) n / R g ]

(16.36)

where Rg is the size of the grains in intermediate and final stage sintering or the particle size during initial stage of sintering. For the initial stage, tf = O. Dilatometer experiments for the shrinkage as a function of time are plotted as the log of the shrinkage versus the log of the sintering time; this will be done later in Figure 16.26. The data are best fit to a line, and the slope is n. Comparing the slope with the values of n for the various sintering mechanisms given in Tables 16.3 and 16.4 allows the mechanism to be established. Plotting the y intercept versus 1/T for other experiments for the same powder with the same green density allows the determintion of the apparent activation energy, Eact, for the diffusion coefficient (i.e., D = D O e x p ( - E a c t / k B T ) , which is responsible for the sintering mechanism. This implicitly assumes that a single mechanism is responsible for sintering. If viscous flow is responsible for sintering, a similar plot of y intercept versus 1/T gives the activation for the viscosity. The diffusion coefficient or the viscosity is found in the definition of Cj given in the tables just cited. If the sintering time is kept constant and the temperature is varied for a green body made with a given powder and different green densities, this analysis can also be used. Using a plot of A V / V o versus T is shown in Figure 16.19, we replot the data as AV/Vo versus 1/T and obtain the apparent activation energy from the slope of the curve. In another test of the sintering mechanism, the grain size as a function of sintering time is used. A plot of the log of the shrinkage

810

Chapter 16 Sintering

FIGURE 16.19 Effect of green density on the sintering ofBaTiO3: (a) shrinkage versus temperature, (b) fired density versus temperature, (c) densification rate versus temperature, and (d) temperature at highest sintering rate and highest sintered density versus initial green density. Taken from Chen [34].

versus the log of the grain size (both at the same sintering time) is used to make a plot which is best fit by a line. Using equation 16.36, we see that the slope corresponds to the value m. To use constant heating rate sintering data from a dilatometer, we have to integrate the sintering kinetic expression with respect to time because the temperature increases linearly with time; that is, q = d T / d t . Therefore,

16.3 Solid State Sintering Mechanisms

811

we start with d [hL(t)l L~ J dt

= Cj(T)[(t- tf ) n - 1 /R~m )]

(16.37)

inside the C i ( T ) term is either a temperature dependent diffusion coefficient (i.e., D = Do e x p ( - E a c t / k s T ) ) o r a temperature dependent viscosity (i.e., ~ = ~o e x p ( - E a c t / k s T ) ) 9Upon substitution for the temperature corresponding to constant heating rate (i.e., T = qt - To, where q is the heating rate and TO is the initial temperature for heating) and integrating with respect to time, we obtain expressions for the shrinkage given by hL(t)

=

Cj(T = qt-

T o ) ( t - tf)n-1/R~ d t

Lo

(16.38)

The resulting expression depends upon the temperature/time dependence of the collection of constants, C j ( T = qt - To). The resulting equation will have a particular time dependence, which is different for each sintering mechanism.

16.3.3 Effect of Green Density of Sintering Kinetics Chen [34] studied the effect of green density upon shrinkage of pure BaTiO3. Figure 16.19(a) shows the dilatometer shrinkage versus temperature of BaTiO3 sample heated to 1350~ with constant heating rate of 3.3~ Different samples with different green densities (51-61%) were used. Below ll00~ virtually no shrinkage was observed after correction for thermal expansion. At 1100~ most samples start to shrink. The specimens with lower green density shrank more. From the data in Figure 16.19(a), the density versus temperature can be calculated, which is shown in Figure 16.19(b). At about 1290~ the highest sintered densities are reached for all green densities. The highest density was 99% for the sample with highest initial density of 61%; for the sample with initial density of 51%, the highest density achieved is 97.5%. The densities of all the specimens decreased with temperatures beyond 1290~ which is consistent with other sintered density measurements by Chen. The reason for this "desintering" is due to BaCO3 surface impurities, which thermally decompose creating CO2 pores in the microstructure [42]. The shrinkage rates of these samples were calculated and plotted in Figure 16.19(c). Samples with lower green density have a shrinkage rate peak at a higher temperature than ones with a higher green

812

Chapter 16 Sintering

density. The densification rate at any temperature increases slightly with decreasing green density. Allemann, Hofmann, and Gauckler [43] studied the relation between the green and sintered states of TZP (tetragonal zirconia polycrystal). The shrinkage rate was found to directly depend on the pore size distribution of the compact. Lange [44] also found that sintering depends on the distribution of pore sizes and the homogeneity of porosity. In Figure 16.19(c), for the lower green density samples, the densification rate has a maximum at higher temperatures, suggesting a larger proportion of large pores. This is unlike the case for the sintering of ZnO [45], where the densification rate was dependent more on the temperature than on the density. The significance of the effect of green density can be summarized in Figure 16.19(d), where the maxima in densification rate and highest sintered density were plotted with green density, showing (1) the temperature for the maximum densification rate is lower for higher green density and (2) the maximum sintered density is higher for the higher green density samples. The highest achievable densities are drastically higher for the green density higher than 58%. It is suggested that the population of the pores whose coordination number n is higher than n c , the critical value, is drastically reduced for green densities larger than 58% giving higher portion of sinterable pores.

16.3.4 Effect of P a r t i c l e Size D i s t r i b u t i o n on Sintering Kinetics As the average particle size in green compact is reduced, the specific surface area increases and the diffusion distance is reduced. Both effects produce higher densification rates. According to the Herring scaling law [22], t 2 = (r2/rl)ntl

(16.39)

where t / a n d ri are sintering time and grain size and n is a constant depending upon the sintering mechanism (for volume diffusion, n = 3). Implicitly assumed in this equation is that the rate controlling mechanism remains the same for both particle radii. The Herring scaling law clearly shows that, as the particle size is reduced, so is the sintering time. Therefore, the sintering rate is higher for smaller grain sizes. The effect of the particle size distribution on shrinkage rates has been considered previously for the initial stage sintering [46]. The problem was treated for the linear shrinkage of a pair of different size spherical particles in contact. These results were extended to distributions of a few discrete sizes by a weighting of all possible contacts. For a continuous distribution in size, this pairwise interaction can be

16.3 Solid State Sintering Mechanisms

813

applied by involving some rigorous calculations.* In the work by Chapell et al. [47], the effect of nearest neighbors is neglected and the shrinkage is approximated for any sintering stage by considering individual contributions of all sizes. This approach is based on generalized relative shrinkage expression:

hG(t) - k ( r ) t n Go where k(r) is introduced as a particle size dependent rate constant of the form k ( r ) = Cj rm

(16.41)

This expression describes the shrinkage for a particle of a given size when its nearest neighbors are of the same size. This "individual" shrinkage may be regarded as a first-order term in estimating the contribution of a given size to the shrinkage for a compact composed of a distribution of sizes. The length (or volume) of a particular particle is related to the radius by

Go = Ar i

(16.42)

where i is the dimension of interest and A is the shape factor [i = 3, A = (4/3)rr for a spherical volume; i = 1 A = 2 for the length of a sample composed of spheres]. This particle provides a time-dependent shrinkage with the form hG(t)

=

ACjri-mt n

(16.43)

As we have seen before, n and m depend upon the sintering mechanism shown in Tables 16.3 and 16.4. The weighting of some size distribution function f(r) gives the shrinkage contribution by particle size as AG(t)

=

A C j t n fo ri-mf(r) d r

(16.44)

The length of the compact or volume of the powder is described according to the contributions by particle size

Go = A fo rif(r) dr

(16.45)

which ensures that the length or volume of the compact is conserved (normalization). The relative shrinkage for the compact is given the

* The calculation involves a double integration.

814

Chapter 16 Sintering

form, as before, hG(t) - kd tn Go

(16.46)

with the rate constant kd defined for a size distribution by

firi-mf(r)drt

(16.47)

This expression implicitly assumes that the rate controlling mechanism is not changed by the size of the particles. This formulation allows the effect of the size distribution on the shrinkage to be estimated in the rate constant k d for either the particles of the initial stage or for the grains of the intermediate and final stages. However, this result is based on a perturbative approach, where a particle acts independent of its neighbors but is assumed to be near in size to them. The expression is then most applicable to narrow size distributions of a single mode, and its accuracy suffers as the size distribution becomes broader. If the initial particle size distribution is log-normal (see Chapter 2) with a geometric mean size, rg, and a distribution width paramter, (rz, the integrals can be evaluated [48], giving

kdffrz) = kd((rz -- 0)exp[m(rz2(m - 6)/2]

(16.48)

where kd((rz = 0)(= CjRg ] is the sintering rate constant for a monosized particulate system. Figure 16.20 is a plot of the relative sintering rate constants for the initial and intermediate stage sintering as a function of the width, ~z, of the log-normal particle size distribution. Here we see as the distribution increases in width the sintering rate constant decreases for both the initial and intermediate stages of sintering. This theory for the sintering of a powder compact describes the pores as shrinking at a rate according to the particle size, and independent of the pore size. The compact shrinks at an average rate and involves all of the particles contributing to the shrinkage. The average rate of shrinkage is valid then only so long as all particles have some porosity in their neighborhood. When porosity is eliminated within some locally dense region, those particles can no longer contribute to the overall shrinkage rate of the compact. This requires the pores within the compact to be of comparable size and to shrink at a rate near that of the average for the expression to be valid throughout the sintering stage. TM

This is synonymous with having the compact homogeneously packed with the particles randomly distributed. The particles are also assumed to retain a coordination and packing geometry similar to those originally described by the models. This dictates that the coordination number should be nearly the same for all particles in the initial stage.

815

16.3 Solid State Sintering Mechanisms

,o

- -

'"

v

|

\

b 1.0

"

i.,I.' " ~ ~

g-b. d.

0.3

0.3

'. v. f.

k (o"z)

k(o-z) .. o.i

k(o) o.,

k(O}

I.d.

0.03

0.03

0.01

0.01 0

0.2

0.4

O-z

0.6

0.8

1.0

0

0.2

0.4

O-z

0.6

0.8

1.0

F I G U R E 16.20 Sintering rate constant for the (a) initial stage and (b) intermediate s t a g e a s a function of the log-normal size distribution width parameter, orz (1.d.: lattice diffusion, g-b.d, grain-boundary diffusion, v.f. viscous flow). Taken from Chappell et al. [47].

Similarly, the grain/pore geometry of the intermediate stage should be reasonably approximated by Kelvin's tetrakaidecahedron with three coordinate cylindrical pores along its edges. The process of shrinkage in the intermediate stage is complicated by the coincident phenomenon of grain growth. Intuitively, an effect which increases the size of grains will slow the shrinkage process, according to theories discussed in Section 16.4. For a distribution of grain sizes, l, this effect can be described by a distribution function, f(/), which becomes time dependent, f(l, t). Thus, the sintering rate constant also becomes time dependent:

(f:li-mf(1, t) dl~ kd(t) : C j \ ]: li-~l: ti d 1 ] Chappell

et al.

(16.49)

[47], used the generalized grain growth law: /(t) g= lgo+ kgt

(16.50)

816

Chapter 16 Sintering

where l0 is the grain size at t = 0, g is the grain growth exponent, and kg is the grain growth rate constant. Upon accounting for simultaneous growth and densification, the rate constant is given by

'

+ k t)

o)]

(16.51)

Other theories for intermediate stage sintering have been developed in order not to restrict the grain/pore geometry [29] and to explicitly allow for grain growth [49]. They conventionally treat shrinkage and grain growth as independent processes, as this approach does, but grain growth and geometric changes in the microstructure must be known a priori to predict the shrinkage behavior. This requires the microstructure to be closely observed throughout sintering with measurements made on several geometric factors or that these factors be approximated with averages. These statistical theories have been used [50,51] in an effort to account accurately for the variability of the microstructure. The resulting density expression was then reevaluated to be kd(O-z)

-[ .t'

t n-1 d t

p(t')- p(t' : O)+ nCj Lkd-(-~z~z-o)J Jo (lgo+ kgt) m/g

(16.52)

This approach is taken to simulate the effect that an evolving grain size distribution has on the shrinkage of a powder compact. The expression regards the shrinkage and grain growth as independent processes and is insensitive to changes in the radial pore size brought about by pore coalescence. The formulation is subject to the same packing concerns as before, only now the grains should meet these conditions as they evolve. This seems to be reasonable for a random arrangement of grains, provided that exaggerated grain growth does not occur. Similarly, the grain/pore geometry should be maintained as the grain size distribution changes. The results of this model is shown in Figure 16.21, where the relative density is plotted versus time for various distribution width parameters. Here, we see that the relative density reaches a value of 1.0 very quickly if (rz = 0 (monosized). For very broad size distributions, a longer and longer time is needed to reach a relative density of 1.0. This theory can also be applied to predict the grain size as the density increases for various distribution widths as shown in Figure 16.22(a). Here, we see that the final grain size (at p = 1.0) is small if the width of the particle size distribution is narrow ((rz -o 0). When the (rz is larger than 0.5, the grain size goes to ~ before reaching p = 1.0. This behavior has been experimentally observed [52] with the sintering of narrow lognormal particle size distributions of TiO2 and A1203, as seen in Figure 16.22(b) [53-55].

817

16.3 Solid State Sintering Mechanisms

a LO

b

o,

1.0

I/o,-o/ '

,

'

,

K_oI

o.,i/

.

,o.

0450

;

4

6

06 j "O

I 2

I 4

tK Iot

6

tK3 i0

F I G U R E 16.21 Relative density versus time for the g r a i n g r o w t h exponent: (a) g = 2 (with Klo/Ci = 1) a n d (b) g = 3 (with K/Ci = 1) for various side distribution width p a r a m e t e r s , ~z. T a k e n from Chappell et al. [47].

16.3.5 The Effect o f F r a c t a l Aggregates on S i n t e r i n g Kinetics Fine ceramic powders made by spray drying, by sol-gel powder synthesis, and sometimes by precipitation usually have very low packing densities. A possible reason for these low packing densities is that the basic packing units are porous agglomerates. In sol-gel synthesis, a

12

-

'

h

'

~20

Kl~ --E--~

I

] O"Z -0.4 O'z=O.~!

8 I

~-, ~o CO (D

,

4

/

I I I I I I I I I I I I I I I !

I

/

,/ o__

F I G U R E 16.22

'

0.7

I

0.8

I

0.9

,.o

0

i , 0.4

i

i 0.6

PIP,.

i

! /

I

o

"

./ J i 0.8

i

9 1.0

(a) Relative g r a i n size versus relative density for a g r a i n g r o w t h expon e n t g = 2 as a function of the log-normal size distribution width p a r a m e t e r , (r z . T a k e n from Chappell et al. [47]. (b) E x p e r i m e n t a l grain size (GS) divided by initial particle size (PS0 ~ 0.3 tLm) versus relative density for - - monodisperse TiO2 [53] ((r z = 0.1), - . . . . TiO2 [54] (~z = 0.3), a n d . . . . A1203 ((rz = 0.5) [55].

818

Chapter 16 Sintering

FIGURE 16.23 Fractal aggregate with three generations.

the aggregates grow by random attachment of particles in Brownian motion around the aggregate. It has been demonstrated that such aggregates have a fractal structure with fractal dimensions smaller than 3.* Onoda [57] has detailed a succession of hard agglomerate generations each at larger scales of size as shown in Figure 16.23. Structures having multiple generations of agglomerates of ever-increasing size have fractal dimensionality under certain conditions where a self-similar character is developed. This occurs when each successive generation of agglomerates is larger than the previous by a factor S, and when the packing faction ~b of each generation of agglomerates is the same from generation to generation. When this occurs the fractal character of the ceramic green body extends from the size of the individual particle to the size of the green body. Onoda's article goes on to show that the factal dimensions D is given by D =3 +ln+/lnS

(16.54)

As a result of fractal geometry, the relative density is P/Pth = ~ g

(16.55)

where K is equal to the number of generations. With a value of S = 10 and a packing density of 50% for each generation, the fractal dimension is 2.7. If there are three size generations, (e.g., 0.005 t~m, 0.05 t~m, and 0.5 t~m) then the relative density would be 0.125. The processing of fractal ceramic powders is severely complicated by the evolving nature of the microstructure throughout the course of sintering. Changes in microstructure are influenced by a host of factors, * Please note that the packing of fibers also has a fractal structure. The packing fractal dimension is obtained by measuring the number of holes in the fiber mat of various sizes. A log-log plot of number of holes versus their size gives a line whose slope is the fractal dimension of the fiber packing. See Kaye [56].

819

16.3 Solid State Sintering Mechanisms

which are determined by the chemical and physical properties of the material as well as the packing configuration in the green body.

16.3.5.1 Fraetal Sintering Model Generalizing the sintering model discussed in the previous section to a fractal ceramic compact requires that equation 16.46 be used for the shrinkage of each generation of particles. Equation 16.55 can be rewritten as a product of the packing fractions of each generation, i, as a function of sintering time" K P(t)/Pth = I-[ dPi(t) (16.56) i-1

(hi(t), are described by ) }

The sintering of the individual packing fractions,

rbi(t) = r

{CjtnB(~

1 + [rg~~r=i]

m

(16.57)

where ~o is the initial packing fraction of a generation, S [= re(i + 1)/ re(i)] is the size ratio between generations of aggregates (i + 1 and i), and rgl is the geometric mean size of the particles in the first generation. The powers n and m depend on the sintering mechanism as given in Tables 16.4 and 16.6 along with the values of Cj. In this equation, rbi(t) goes from ~bo to 1.0 maximum. As you can see from equation 16.57, each generation of particles has a time constant associated with the sintering of that generation, where (hi(t) = 1.0. The time constant, is given by

{ [rgl(rb~/3S)i-1 ]m}l/n Ti =

~jB-(~;)

(16.58)

and the ratio of time constants between generations is equal to Ti+1

Pi

-- (S~/)o)m/n

(16.59)

Using this analysis, we can plot the relative density versus time for the sintering of a fractal ceramic compact. This is shown in Figure 16.24 [58,59] for an isothermal viscous sintering, where n = 2 and m = 2. The results are compared to the isothermal sintering of a ceramic compact consisting of solid ceramic particles of a size r rgl S(K-1). (Note: one generation is not a fractal geometry.) Results show faster sintering for the fractal ceramic compact than for the traditional ceramic compact because it is composed of smaller particles. For this reason, sol-gel ceramic powders sinter much faster t h a n traditional ceramics. Also shown in Figure 16.24 is the stepwise sintering that fractal ceramics undergo. For a three generation (K = 3) fractal ceramic compact, the first generation sinters to its full density of 0.25 at a =

820

Chapter 16 Sintering 1.0

0.5

f

0 10-3

I

,

0.01

0.1

TIME

10

FIGURE 16.24

R e l a t i v e d e n s i t y v e r s u s d i m e n s i o n l e s s t i m e for t h e s i n t e r i n g of a f r a c t a l c e r a m i c c o m p a c t w i t h S = 10, n = 2, m = 2 (viscous s i n t e r i n g ) : (A) K = 1, (B) K = 2, (C) K = 3. T a k e n f r o m R i n g [58].

dimensionless time of 0.01. The second generation sinters to its full density of 0.5 at a dimensionless time of 0.08, and the third generation sinters to its full density of 1.0 at a dimensionless time of 0.62. For a two generation (K = 2) fractal ceramic compact, the first generation sinters to its full density of 0.5 at a dimensionless time of 0.1, and the second generation sinters to its full density of 1.0 at a dimensionless time of 0.79. In this figure, the time constant, Tmax, is the time to sinter a nonfractal compact with a packing fraction of 0.5 and a particle size r rglS (k-l). This is the size of the largest aggregate when K was 2 or 3. This maximum time is given by =

(K-1)]m~ 1/n

Tmax=

{[rglS

C-jjB-~zi J

(16.60)

Such a nonfractal ceramic compact sinters to a full density of 1.0 at a dimensionless time of 1.0. The results of fractal sintering of three generations (K = 3) with a 0.5 packing fraction for each generation and a size ratio of 10 between generations are shown in Figure 16.25 [60,61], where the effects of various rate controlling diffusion mechanisms are compared. Each rate controlling diffusion mechanism shows stepwise sintering. However, the time when the step takes place is different for each sintering mechanism. For some mechanisms (grainboundary diffusion, lattice diffusion, both initial and final stage it takes times from 10 -4 to 1.0 times tmax to complete the sintering of the last generation. For other mechanisms (viscous diffusion) it takes only 0.1 to 1.0 times tmax. All other mechanisms fall between these two extremes.

16.3 Solid State Sintering Mechanisms

821

1.0

o i 6a ,

l

,6 4

L

,6"

~

I6'

1

1.0

TIME

F I G U R E 16.25 Relative density versus dimensionless time for the sintering of fractal ceramic compacts with K = 3, S = 10, ~bi0 = 0.5: (A) grain boundary diffusion, initial, n = ~, m = 2; (B) grain boundary diffusion, intermediate, n = 3, m = ~; (C) lattice diffusion, initial, n = }, m = ~; (D) lattice diffusion, final (or intermediate), n = 1, m = 3, also lattice diffusion, initial [60], n = 4, m = ~; (E) lattice diffusion, initial [61], n = 89m = 3.2,and (F) viscous flow, n = 2, m = 2. Taken from Ring [58].

T h e i s o t h e r m a l s i n t e r i n g o f a d r i e d TiO2 p o w d e r [53] is s h o w n i n Figure 16.26. These sintering results showed enhanced sintering kinetics c o m p a r e d t o c o n v e n t i o n a l TiO2 c e r a m i c p o w d e r s . T h i s p o w d e r w a s amorphous by X-ray diffraction but showed diffuse electron diffraction s u g g e s t i n g ~ 4 n m c r y s t a l l i t e s [62]. N i t r o g e n a d s o r p t i o n o n a s a m p l e

I

I

I

I

I

I

I

_.oa--~_-',,.,,."~"

I

,-~'~" ^'~~'~--

I

o.~176

I/"/o,/

- 2.0 F

I.~

/~/

o"~1

.o-o'~

o.'~

-5.0 o

/

Ti 0 2 o 1020" C

<] v (-.~

'~

/

~ 1060 ~ C

-4.0

o II O0 ~ C

,', l 160 ~ C

_/~ -5.0

0

I 1

! 2

I 5

I I 4 5 ,ln t (min)

I 6

I 7

8

F I G U R E 16.26 Relative linear shrinkage versus sintering time for a dried sol-gel TiO2 powder: (a) 1020~ (b) 1060~ (c) 1100~ and (d) 1160~ Taken from Barringer [53].

8~

Chapter 16 Sintering

dried at 300~ for 18 hr gave 160.8 m Z / g m [63], which translates to an aggregate size of 0.01 t~m and the scanning electron microscope showed the particles to be spherical and monodisperse at 0.3 t~m in diameter with a distribution width parameter (r~ = 0.1 [53]. Measurements of the powder density gave 1.49 gm/cc as precipitated [63], 2.55 gm/cc after drying at 100~ for 18 hr [63], and 4.26 gm/cc after calcining at 1000~ overnight [53]. The powder particles shrunk 30% during drying [63]. These observations suggest that the powder is porous with two internal generations of aggregates, each with a packing density of 0.67. In addition, the scale factor S of the first generation of aggregates is 2.5 and that for the second generation is 30. Edelson and Glaeser [64] have also observed multistep sintering of alkoxide derived TiO2. Internal aggregate sintering took place at 425-800~ Analysis of the sintering kinetics shown in Figure 16.26, along the traditional lines of initial stage, intermediate stage, and final stage, each corresponding to a linear section of the log-log A L / L o versus time curve, does not seem appropriate with such a powder. A more reasonable explanation for this sintering data is that the first knee in the curves corresponds to the densification of the second generation of aggregates in the ceramic powder and the second knee corresponds to the sintering transition from "intermediate" to final stage. This would give a knee in the densification curves at a packing density of 0.6 (or A L / L o ~ 0.06) and another at a packing density of 0.36. The last value of packing density is lower than that of the starting powder used in the sintering experiments. Therefore, the sintering of this generation is not observed in the sintering experiments but in drying where a 30% shrinkage in particle size was observed. The knee in the sintering curves at A L / L o values of 0.16 corresponding to a packing density of 0.92 is the transition from "intermediate" to final stage sintering. In addition, the slopes of the linear sections in Figure 16.26 are the same. The slope of the "intermediate" stage is 1/3 suggesting grain boundary diffusion as the rate determining diffusion mechanism. Using this analysis, Ring [58] determined that grain-boundary diffusion was the rate determining step and that the activation energy for grain-boundary diffusion was 83.9 Kcal/mole. Using this temperature dependence for the grain-boundary diffusion, it is reasonable to expect the first generation of 4 nm crystallites to sinter at 100~ during the extended periods of drying. As a result of this analysis, we find that a fractal ceramic powder will sinter at very low temperatures and to theoretical density in shorter times than a nonfractal ceramic powder. Yan [65] has shown that the sintered grain size distribution is similar to the initial particle size distribution. In the proceeding chapters, we have seen that green bodies may have density variations due to various sources. Such packing defects include (1) macroscopic inclusions of organic material which burns out to leave a void, (2) macroscopic voids

16.3 Solid State Sintering Mechanisms

823

FIGURE 16.27 Orderedpacking of monodisperse SiO2 particles. From Ring [66].

due to occluded gas arising from bubbles in the suspension or too fast drying and binder burnout, (3) foreign bodies, and (4) aggregates Of primary particles. These packing defects tend to produce large flaws and can be decreased in number by clean working conditions. If monosized size particles are used and packed into ordered arrays, packing defects also arise due to (1) packing dislocations, (2) packing vacancies, and (3) domain boundaries between packed grains see Figure 16.27 [66]. Rhodes [67] and Barringer [52] have shown that the domains formed during green packing produce intradomain pores, different shrinkages were found for the inter- and intradomain pores. These domain boundary packing defects are large encompassing 100 to 1000 particles, depending on the conditions associated with green body formation. During sintering these grains tend to pull away from one an-

824

Chapter 16 Sintering

FIGURE 16.28 Partially sintered ordered packing of monodisperse particles showing the pulling away of an ordered domain. From Lange [69]. Reprinted by permission of the American Ceramic Society.

other [68], causing large flaws (see Figure 16.28) [69] in the ceramic, leading to low strength and Weibull modulus. For this reason, work on truly monodisperse particles as ceramic precursors has not been successful. Enhanced sintering kinetics has been observed as well as decreased tendencies for abnormal grain growth. But improved strength has not been demonstrated with these monosized ceramics

[7O]. 16.4 G R A I N GR O W T H Grain growth is an integral part of sintering during the final stage because the movement of the grain boundary can be looked upon as that of a broom sweeping out microstructural features such as pores and inclusions. Cahn [71], Atkinson [72], and Brook [73] have reviewed grain growth. There are two types of grain growth: normal and discontinuous or abnormal, also called secondary recrystallization or canabalistic grain growth. In abnormal grain growth, a few large grains develop and eventually consume all the smaller grains, see Figure 16.29. In

16.4 Grain Growth

FIGURE 16.29 ity, Vgb.

825

Schematic of abnormal grain growth showing the grain boundary veloc-

normal grain growth, the grain size distribution is relatively narrow and has a fixed distribution shape throughout growth, see Figure 16.30. The control of the final grain size is absolutely necessary because the final grain size and its distribution establish (1) the strength of the

normal

abnormal

t2 > t 1

eo

~9

t2 > t 1

t3 > t2

t3 > t2

e~ 0

---) Grain Size ---)

FIGURE 16.30 Schematic of grain size distribution at various times for normal and abnormal grain growth. For abnormal grain growth a bimodal size distribution is produced from initially a monomodal size distribution.

826

Chapter 16

Sintering

TABLE 16.7 The Dependence of Creep, ~, on Grain Size, Rg, for Diffusion Controlled Processes According to the Equation = f(Rgm) a Mechanism

Diffusional creep: Lattice Grain boundary Grain boundary sliding: Grain elongation No grain elongation

m

2 3 1 3

a Taken from Gifkins, R. G., J. Am. Ceram. Soc. 51, 69 (1968). Reprinted with permission of the American Ceramic Society.

ceramic piece by Griffith's fracture law (rca 1/V~c, where c, the flaw size and can be equated with R~,~ax, is the maximum grain size; (2) the creep of the ceramic ial/Ry where m is constant between 1 and 3 and its value depends on creep mechanism (see Table 16.7 for details); (3) electrical properties; and (4) magnetic properties. Many electrical and magnetic parameters are affected by grain size. The most work has been done in this area, deliberately manipulating the microstructure of ceramics to prepare materials with suitable properties for specific applications [74-76]. Generally, for ferromagnets and ferroelectrics where hysteresis loops are observed, the size of the domain (either magnetic or electric) and its interaction with the grain is the key to these properties. Below a certain grain size (depending on the type of material) domains cannot form, giving hard magnetic properties in ferrites and high dielectric constants in ferroelectrics. Hard magnetic properties in ferrites include high coercive field and high remanent magnetization. When the grain size is above this critical size, domains are formed inside the grains, giving soft magnetic properties in ferrites and lower dielectric constants in ferroelectrics. In this case, grain boundaries are regarded as impediments to domain wall movement. A direct relation between permeability and grain size has been observed by Perduijn and Peloschek [77]. Grain boundaries have a pronounced effect on electrical conductivity in ceramics. Grain boundaries have the largest concentration of vacancies in a ceramic. The degree of disorder at the grain boundary can cause them to act as relatively conducting regions for anions as well as oxygen. These grain boundaries can adjust more quickly to changes in either atmosphere and temperature than the interior of the grains and thus alter their electrical conductivity more quickly. In cases where

16.4 Grain Growth

827

Microstructure of a PbO doped SrTiO 3 barrier layer capacitor showing the Pb segregation to the grain boundary as analysed by a microprobe. Photo from Advances in Ceramics Vol I, 1980, p. 268, "Grain Boundary Phenomena in Electronic Ceramics," American Ceramic Society.

F I G U R E 16.31

additives have been added during sintering a liquid phase or a second solid phase will be associated with the grain boundary. These phases serve to isolate one grain from another and prevent the passage of charge carriers from one grain to the other. Such is the case with barrier layer capacitors of SrTiO3 doped with PbO or Bi203 . An example of such a microstructure is shown in Figure 16.31.

16.4.1 N o r m a l G r a i n G r o w t h In a classic paper, Burke and Turnbull [78] presented a model of the migration of a grain boundary due to atom transport under a driving force caused by the difference in surface curvature. A conceptual idea of the change in free energy associated with the diffusion of an atom from one grain to the next is shown in Figure 16.32. Due to the pressure of the surface curvature, the boundary tends to migrate toward its center of curvature, as this reduces the grain boundary area and hence its energy. For a pure material, the velocity of grain boundary migration, %b, is given by the product of the grain-boundary mobility, Mg~, and the force vgb = MgbF~

(16.61)

For various mechanisms of grain boundary migration, the mobility will have various definitions, as given in Table 16.8. The driving force for grain-boundary migration is assumed to be due to the difference in grain curvature:

828

Chapter 16

Sintering

F I G U R E 16.32 The movement of an atom from one side of the grain boundary to the other involves a change of Gibbs free energy. Taken from Kingery et al. [2, p. 453]. Copyright 9 1976 by John Wiley & Sons, Inc. Reprinted with permission of John Wiley & Sons, Inc.

(16.62) where ~/gb is the interfacial energy of the grain boundary between two adjacent grains and K1 and K2 are the average radius of curvature of the grain boundaries 1 and 2. Here, Ki is given by 1 tr i

-

1 ril

1

t

(16.63)

ri2

where ril and ri2 are the principle radii of curvature of the surface i. Burke and Turnbull now assume t h a t (1) the interfacial energy is not a function of grain size, grain orientation, or time; (2) the average radius of curvature, Ki, is proportional to the average grain size, Rg; (3) the distribution of grain sizes remains constant during growth; (4) the grain boundary thickness is not a function of grain size; and (5) the driving force is due to surface curvature only. (This author suggests t h a t an applied pressure can be added to the driving force as follows: AP = Papplied -4- "Ygb [ ( l / K 1 ) -- ( l / K 2 ) ] . The driving force F for an atom to migrate is the chemical potential gradient, Vt~, caused by the pressure difference across the boundary: F = Vt~ = V(VAP)= t2~/eb

[(1) (1)]1 +

-w

(16.64)

where Vis the atomic volume, t2, and w is the grain boundary thickness. To proceed with their analysis, the instantaneous average rate of grain growth, dRg/dt, is now assumed to be proportional to the instanta-

TABLE 16.8 Kinetics of Grain Growth for Different Mechanisms, Using the mn --n Equation R g -Rgo = Cit, Obtained by Integration of dRJdt = MF, Where Both M and F Are Defined Here for Each Grain Growth Mechanism

Mechanism Pore control, vp = MpFp

n

Mgb

Maximum drag force on spherical pore,

F~ = APTrR~ = ~rRpTgb Dsw~ kB TIrR 4

Surface diffusion

4

Lattice diffusion

3

Gas phase diffusion (P = constant)

3

Dgpgt2 kB Tps27rR

Gas phase diffusion (P =

2

Dgpg~ kB Tps27rR ~

2Tgb/Rp )

Grain boundary control,

Force on pore free curved boundary,

vgb = MgbAP

Pure system

Fgb = AP~ 2/3 _ 2Tgbt2~3 KR~ 2

Da

kB T 1 (1

Impure systems Coalescence of second phase by lattice diffusion Coalescence of second phase by grain boundary diffusion Solution of second phase Diffusion through continuous second phase Impurity drag (low solubility) Impurity drag (high solubility)

fkBT1rR~

k sT ~ + 3

4~pRgwQCot2~3) -1 DD rp

4

1 3 3 2

Notes: The pore control kinetics are given for the situation where pore separation is related to grain size. Changes in distribution during growth would change the kinetics. f is the correlation factor for diffusion. pg is the density in the gas phase of the rate controlling species. Ps is the density in the solid phase of the rate controlling species. K is defined by Rg - Rg0 o~ 2Kt/r~o, where r i o is the inclusion radius at time zero. Q is the impurity partition function.

829

8~

Chapter 16

Sintering

neous average grain boundary velocity, v~b, giving with all the previous assumptions taken into account,

dRe dt ~

Vgb :

MgbFe~ Mgb 2~/gb

(16.65)

which may be integrated to give - 2 - Rg0 - 2 = 2 Mg~ ~/g~t Rg

(16.66)

which is the so-called parabolic growth law starting at an initial average grain size, Rgo. Experimental measurements of normal grain growth in metals and ceramics has shown that the parabolic growth is rarely obeyed. For this reason other theories have been developed which account for (1) impurity drag by various rate determining diffusion steps and (2) pore drag by various rate determining steps. These theories follow the same generalized expression: mn mn Rg - Rgo = Ci t

(16.67)

where n is an integer and Ci is a collection of constants which are given in Table 16.8. The term t2 in this table is the molar volume of material transferred. The diffusion coefficient in the various mechanisms correspond to the slower moving species. Experiments have shown that the typical value of n is 3 for grain growth in metals [72]. 16.4.1.1 S e c o n d a r y P h a s e s a n d T h e i r Effect o n Grain Growth

When we have an impure single phase system, the impurities can give rise to an impurity drag effect which impedes boundary motion [79]. This effect rises from any preferred segregation of impurity either to or from the grain boundary. Impurities (subscript b) are preferred at the grain boundary due to their charge or size differences compared with the major component (subscript a). The movement of the grain boundary implies that the impurities must diffuse along with the boundary or that the impurity must be left behind, thereby raising the energy of the grain. Analysis of the impurity drag effect for low grain boundary velocity situations leads to the following analysis:

(ksT

vg~ = Fg~ \ - ~ a +

4r

Db

]

(16.68)

where Co is the bulk impurity concentration, Q is the impurity partition coefficient, w is the grain boundary width, D e and D~ are the diffusion coefficients for the host and the impurity, respectively, and q) is the volume concentration of atoms. This equation is a series of resistances

16.4 Grain Growth

831

equation for the reciprocal of the mobility--the first term being the mobility of the pure system and the second term being the mobility of the impurity. From this equation it can be seen that whenever the impurity concentration or the degree of segregation is important, the rate of grain boundary migration is reduced. It should also be noted that the bulk impurity concentration is a function, f(~/~b/Rg), of the grain size [80]: Co =

CT

(16.69)

1 + f\-z-/% ( O - 1)

where C T is the total impurity concentration which remains constant. Under the conditions where Q > 1, Rg is small, and CT is large, a grain growth law of the form results

dRg dt - (Rg)-2

(16.70)

(Rg)3 _ (R~0)3 a t

(16.71)

giving

In solution drag, the width of the grain boundary, w, is the zone over which impurities interact. The grain boundary thickness is taken to be independent of grain size. The diffusion coefficients of the host, D e , and the impurity, Db, will depend on the structure in the grain boundary. Impurities can therefore have an effect on the grain boundary velocity, %~, by either their effect on the Co term or an effect on the diffusion coefficients for host and impurity. If a liquid is the second phase, the simplest idealized case is when it is continuously distributed throughout the ceramic body due to complete wetting of the grains; that is, dihedral angle zero. In this case, grain growth takes place by migration of a dissolved atom through the liquid to the growing grain. For a constant total volume of liquid, the growth law takes the form

(Rg) 3 - (Rgo) 3 ~ t

(16.72)

Cases of varying dihedral angle and large quantities of liquid phase treated annalogously to Ostwald ripening have been described quantatively [81] to follow this cubic growth law. When the second phase is discontinuous, like that of an inclusion, the grain boundaries are pinned as shown in Figure 16.33 [82]. The total energy of a grain boundary with inclusions is given by the energy of the grain boundary without the inclusion minus the energy of the grain-inclusion interface. If the inclusion is relatively small and mobile

832

Chapter 16 Sintering

FIGURE 16.33 Inclusions segregated to the grain boundary of A1203-doped SrZrO3. From Scott [82].

it may move along with the grain boundary. If the inclusion is large and immobile, then the only way the grain boundary can move is by detaching from the inclusion [83]. 16.4.1.2 I m m o b i l e S e c o n d P h a s e

To detach the grain boundary from a spherical inclusion requires a force equivalent to Fd = 2zrRi "Yi-gb

(16.73)

where R~ is the radius of the inclusion and ~/~_g~is the energy of the inclusion-grain boundary interface. This force can also be considered

16.4 Grain Growth

833

a grain boundary drag. When this force is equal to the driving force for grain-boundary migration, the grain-boundary migration ceases giving a limiting grain size [84], R g . l i m i t . (16.74)

Ri

Rg-limit ~ (~i

where ~i is the volume fraction of inclusions. This means that the grain size can be stabilized by the presence of inclusions. Further grain growth can occur if (1) the inclusions coalesce by Oswald ripening, giving (Rg_limit) 3 o~ t

(16.75)

(2) the inclusions gradually dissolve into the host grain, giving Rg - Rg0 a

t

3

Ri

(16.76)

or (3) if abnormal grain growth commences as discussed later. 16.4.1.3 Mobile S e c o n d P h a s e - - P o r e s

When pores are the second phase, they are mobile and move at a velocity equal to the velocity of the grain boundary, thus vgb - vp MpFp, where the pore mobility, Mp, and the drag force on the pore, Fp, is given in Table 16.8. Separation of the pores from the grain boundary occurs when vg~ > vp [85]. Thus the pores exhibit a drag on the grain boundary velocity, giving [86] Vgb -- (Fg b - N F p ) M g b

(16.77)

where N is the number of pores on the grain boundary. Making substitutions, we find that the grain boundary velocity can be expressed as

vg~ = Fgb

MpMgb NMgb + Mp

(16.78)

which has two cases: when NMgt, >>Mp, giving

vg~ = Fg~(Mp/N)

(16.79)

where pore parameters control the grain boundary movement; and when NMg~ ~ Mp, giving

vg~ = FgbMgb

(16.80)

where grain-boundary parameters control its movement. Using the case where the pores located at the grain boundary corners (i.e., N a R~ 2) control the grain boundary velocity, the grain growth rate is

834

Chapter 16 Sintering

given by

dRg Fgb (Mp/N ) dt ~ vg~ =

(16.81)

a(Rg) -~ (Rg) -n (Rg) 2

(16.82)

where n depends on the diffusion mechanism (i.e., n = 4 for surface diffusion, n = 3 for lattice and gas phase diffusion [73]). After integration, this results in the grain growth expression:

(Rg) 2-n - (Rgo) 2-n o~ t

(16.83)

This approach, however, does not account for the eventual disappearance of grains. In fact if all the grains are the same size, as Burke and Turnbull [78] assumed, then any growth in grain size must come from the annilation of grains. Hunderi and Ryum [87] formulated a theory to show that, during normal grain growth, there is an increase in grain size due to the decrease in the number of grains. This process can be viewed as the change in the grain size distribution, v(Rg), with time. 16.4.1.4 P o p u l a t i o n B a l a n c e M o d e l s of Grain G r o w t h

Feltham [88] and Hillert [89] assumed that the grain size distribution is determined by the equation

0 ~?(Rg, t) Ot

+

0 [Vgb ~(Rg, t)] =0 ORg

(16.84)

where it is implicity assumed that grain boundary mobility is responsible for growth. This equation is a simple population balance for the grains without birth and death terms (see Chapter 3). Hillert assumed that the velocity of the grain boundary is given by

Vgb:OLiMgbTgbi(Rlcrit

Rg)

(16.85)

where a~ is a geometric factor and Rg.crit is the critical grain size for growth, which is a function of time. When the grain is larger than the critical size, the grain will grow; and when its is smaller than the critical size, it will shrink. Using this population balance, the kinetics becomes identical to Oswald ripening discussed in Chapter 6. In this particular case of Oswald ripening, the particles are dispersed in a second phase and mass transfer to the surface of the particles is the rate determining step. The resulting grain growth kinetics are not surprisingly parabolic~the same as for the Burke and Turnbull kinetics. Feltham's analysis [88], using the same population balance equation,

16.4 Grain Growth

835

suggests that the population is log-normal and time invariant when the grain size is scaled with respect to the average grain size, Rg. The resulting grain boundary velocity, %~, is given by (16.86) giving the following grain growth kinetics:

Making the additional approximation that R e -~ Re.m~ = 2.5 Re, the grain growth kinetics can be shown to be parabolic, like that of Burke and Turnbull. 16.4.1.5 G r a i n D e a t h R a t e

All the pseudo-population balance approaches given to this point have neglected that grains actually are annihilated in order for other grains to grow. This annihilation can be accounted for in the population balance given in Chapter 3: 0 ~(Rg, t) a [%~ ~(Rg, t)] + =B(R~)-D(Rg)=O 0t 0Rg

(16.88)

where B (Rg) and D(R~) are the birth and death functions. This equation has the initial condition given by V(Rg, t = 0 ) = Vo(Rg)

(16.89)

Since annihilation of grains occurs because the grains shrink to zero size, it is not necessary to account for annihilation with a death rate term, although one could, but by a boundary condition; thus, D(Rg) = 0: 0 v(Rg = 0, t) = constant [91,92] Ot

(16.90)

Experiments by Hunderi [93] suggests that this constant is ~1.7 grains when the grain velocity sweeps through a volume equivalent to one average grain. The other boundary condition is ~(Rg, t = 0) = ~0(Rg). In the population balance, the birth rate term, B(Rg), is 0, because there is no process in grain growth whereby grains can nucleate. Essentially all growth (and shrinkage) is accounted for with the grain boundary migration velocity in this analysis, as it was in the analysis of Hillert, with vg~ = al Mgb T~b (1/Rg-crit - 1/Rg). The solution to this population balance is greatly simplified if the velocity of grain growth is not a function of grain size. In this case, the

8~6

Chapter 16 Sintering

grain size distribution is simply a transformation in time of the initial grain size distribution (see Figure 6.26). But the approximation that the velocity of grain growth is not a function of grain size is contrary to the observation that some small grains shrink and other larger grains grow during grain growth. Thus, the grain size distribution will depend essentially on the function used for the grain boundary velocity as a function of grain size and the initial grain size distribution. Using the Hillert expression for the grain boundary velocity, the population balance becomes

O ~(Rg, t) + O~lMgb~/g b c~t

1 RScrit

(16.91)

O v(Rg, t )

+ a~Mgb ~/gb(1R-~g2)~(Rg, t) = 0

The solution for the grain size distribution is given by v(Rg, t) =

Bx exp(hr) Sx(x___))

~ X=O,1,2 ....

(16.92)

X

where = t (al Mgb ~/g~)

(16.93)

g-crit

x=

Rg

(16.94)

g-crit

S~(x) = (x - 1)(x+l) exp(-kx)

(16.95)

The coefficients Bx are determined from the initial conditions and can be calculated from ~cx~

B~ = j ~o(X) S~(x) dx o

(16.96)

x

When the initial grain size distribution is smaller than R g _ c r i t the mean grain size increases with time while some of the small grains die. When the initial grain size distribution has grains larger than Rg-crit (i.e., x > 1), we have a situation where the coefficient B1 dominates the solution, giving a bimodal grain size distribution where the minimum between the two modes occurs at Rg = Rg_critor x - - 1 . The larger particles increase in size with time and the smaller particles decrease in size with time as shown in Figure 16.30 for abnormal grain growth. This solution also shows that the death rate of particles, a~(R~ = 0, t)/Ot, is a constant, but only long after abnormal grain growth has been initiated.

16.4 Grain Growth

83 7

This is a simplistic view of grain growth in which we have assumed that the energy of each crystal face is the same and the grains have a single shape. In a real grain assembly, there are many grain shapes, and each face results from the termination of a particular crystal orientation with a particular interfacial energy. Abbruzzese and Lucke [95] found that each crystal orientation at the grain boundary gave a different critical radius for growth, with the value depending on all other grain boundary orientations. As a result they showed that textured microstructures typically found in ceramics give grain size distributions with pronounced differences and grain growth kinetics which do not even approximate the parabolic law. In fact, the grain growth kinetics may be stepped, with different time dependences applying to different stages of grain growth. In addition, topological requirements play a role in grain shape. Smith [96] states, "Normal grain growth results from the interaction between the topological requirements for space filling and the geometrical needs of surface tension equilibrium." The topological requirements in three dimensions (3-D) are # Faces - # Grains - # Edges + # Vertices = 1

(16.97)

and in two dimensions (2-D), # Faces - # Edges + # Vertices = 1

(16.98)

These topological requirements correspond to Euler's theorum for two and three dimensions. The number of edges joined to a given vertex is its coordination number, z. For topologically stable structures, z = 4 (in 3-D) and z = 3 (in 2-D)

(16.99)

Therefore, a uniform 2-D array will consist of hexagonal grains giving the number of faces per grain, n = 6, and an angle at the vertices of 120 ~ For nonuniform arrays, we have a mixture of polygons. In a grain assembly, if even one five-sided polygon is introduced into an array (and it is balanced by a seven-sided one maintaining (n) = 6), then the sides of the grains must become curved to maintain 120 ~ angles at the vertices. Due to this curvature, grain boundary migration will take place reducing the interfacial energy. Grains with n > 6 will tend to grow at the expense of grains with n < 6. The shrinkage of the fivesided polygon will decrease until it produces a four-sided polygon then a three-sided polygon, and so on until it disappears as shown in Figure 16.34. For a 3-D array where all vertices have z = 4, the average number of faces is 12 (# Faces) = ~ 6 - (n)

(16.100)

838

Chapter 16 Sintering

0

FIGURJE 16.34 Unstablegrain structure during normal grain growth. Reprinted from Hillert [89], with kind permisison of Elsevier Science Ltd, copyright 1965, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK..

where (n> is the average number of sides per face in the cell. Most random structures have {# Faces> ~ 14 giving a value of 6 12/14 ~ 5.153. No polygons have 5.153 sides per face, the nearest is a 14-sided polygon, the Kelvin tetrakaidecahedraon, spaced on a bodycentered cubic lattice (see Figure 16.35) [96,97], but the angles are not exactly those required, causing the boundaries to be curved. Deviations from the Kelvin tetrakaidecahedraon spaced on a body-centered cubic lattice structure in the random grain structure of a ceramic will lead to grain shrinkage if the number of faces of a polyhedron grain has less than 14 sides. All grains with more than 14 faces are stable with respect to grain growth. Much of this topological work draws heavily on the analogous system of a soap bubble froth [96,98]. Even statistical mechanics has been used to aid in prediction of the structure and distribution of cellular networks [99]. Considering a maximum entropy for the grains, the grain size distribution is predicted to be an exponential distribution, f(R~) ~ V'Rg/(Rg> e x p ( - VRJ(Rg>). Such an exponential distribution was seen experimentally for MgO [100]. Thus the ensemble average or mean field growth laws developed earlier are not applicable to individual grains. An individual grain will grow or shrink depending on (1) its own geometry and (2) the geometry

16.4 Grain Growth

~: h

s

c~,/~

--a~ '~

839

s-squore'(lO0)' faces

h- hexagonal '(111)',aces a = }25 ~ 16' ~ k ~ f 109~ 28'

Kelvin'stetrakaidecahedra on a body-centered cubic lattice, showing angles of the two types of facesmsquare and hexagonal. Taken from Smith [96] and Doherty [97].

FIGURE 16.35

of its nearest neighbors. Thus, a polycrystalline ceramic must be separated into individual grains and the growth of individual grains studied within their own particular localized environments to better understand grain growth. In essence, this individualized approach is valid if the vacancy concentration profile responsible for grain growth is limited to grain boundary between the grains. If, however, the grain boundary represents a continuous network where the vacancy concentration is constant, then all the grains will feel the effects of the other

840

Chapter 16 Sintering

grains, validating the ensemble grain growth laws presented previously. The situation is caused by a very high grain boundary (or surface) diffusion coefficient compared to that of the lattice diffusion coefficient. Furthermore, considerable work is still to be done by grain growth theoreticians in this field to assess the validity of the mean field approach, random motion of configurational space, soap froth equivalent for polycrystals, and whether topology should be adhered to at each step along the grain growth. However, one thing is certain: two-dimensional computer simulations are not convincing simulations for grain growth in ceramics.

16.4.2 A b n o r m a l G r a i n G r o w t h In abnormal grain growth, a small number of grains in the population grow rapidly to a very large size, typically several orders of magnitude larger than the average in the population at the expense of the smallest particles in the distribution. The populations that result for normal and abnormal grain growth are shown in Figure 16.30. Abnormal grain growth is important because it leads to extremely large grains. The grain boundary velocity is so large that the grain boundaries can pull away for any restraining second phase (i.e., pores or precipitates) leaving these second phases inside the large grains. Pores that are left inside large grains are trapped, and densification is essentially limited. Once abnormal grain growth starts, the grain boundary velocity is given by vgb - dRg2 d t - Mg~ (2t22/3 Tg~/Rgl )

(16.101)

where the grain-boundary velocity has increased, owing to a larger driving force, because the radius curvature of the large grain, Rg2, is very large in comparison to the small grain, Rg~. To determine the grain size as a function of time, it is commonly assumed that the difference in driving force is sufficient so that the rate of change of the matrix grains is negligible. As a result the abnormal grains increase lineraly with time as Rg2 - Rg2,o ~ t

(16.102)

This argument is more convincing when the matrix grains have reached a limiting size due to the presence of inclusions. The linear growth means that abnormal grain growth can rapidly consume the original grain structure to the point where only the large grains remain. In addition, pores which have been detached from the grain boundary

16.4 Grain Growth

841

FIGURE 16.36 SEM micrograph of 0.71 _+ 0.5 tLm BaTiO3 sintered at 1280~ for 6 minutes (density = 95.4%). Abnormal grain growth is prevalent, giving large grains 50 tLm in diameter in addition to small grains near 1 t~m in diameter. Photo courtesy Z.-C. Chen (EPFL).

during abnormal grain growth are left behind, inside grains. These abnormal grains mixed with very small grains and the pores left behind in the large grains are observed in the microstructures of sintered pure BaTiO3 shown in Figure 16.36. Xue [101] found abnormal grain growth in pure commercial BaTiO3 at all practical sintering temperatures above 1200~ Even trace impurities give profound effects on grain boundary mobility. The abnormal grain growth of BaTiO3, once initiated, has a classical linear relation with time, whether or not a liquid grain boundary phase is present. During the sintering of pure A1203, abnormal grain growth is also observed. In essence, the population balance should be able to predict the conditions where a bimodal grain size distribution is initiated. Rearranging the population balance for the derivative of the population with respect to size, Rg, gives 0 ~(Rg, t) = _

OVg~ O~(Rg t)] ot '

~(Rg, t) ~-R-~g+

(16.103)

Chapter 16 Sintering

842

If this derivative has multiple zeros, then a bimodal grain size distribution is predicted. Dividing by ~?(Rg, t) we find 01n[v(Re , t ) ] _

ORg

-

-

~Ovgb Oln[~(Rg t)]} [0R e + Ot ' Vgb

(16.104)

where the roots now correspond to 0 ln[~(Re, t)]/ORg = -~. If the initial grain size distribution, ~?o(Rg), is used to see if a ceramic is predisposed to abnormal grain growth, we find that

Oln[~(Rg,t)] ORe

_[~o(Rg)O(Vgb)] =

vg---b ~

(16.105)

because the initial grain size distribution is not a function of time, so its time derivative is 0. A bimodal grain size distribution is therefore expected for certain grain size distribution functions and certain grain boundary velocity functions. Substitution for the grain boundary velocity, veb = ozI M g b ~/gb (1/Rg-crit - 1/Rg), gives

Oln[~(Rg, t)] ORg

=

~o(Re)

_

0~1 Mgb'ggb (Re--crit

~o(Rg) R g - c r i t

Re(R e

-

Rg.crit)

O[~ Mgb ~/gb(Rg?er i t

Rg)

Rg)] ,

ORg (16.106)

which has two values where the left-hand side becomes - ~ : Rg = Rg_crit and Rg = 0. Thus, if the initial grain size distribution has grains larger than Rg.crit , then abnormal grain growth will take place. This shows that abnormal grain growth easily occurs if the initial particle size distribution is too broad. Abnormal grain growth can be decreased or eliminated by the use of narrow particle size distributions, so that all the grains are of similar size and none of them has a tendency to grow at the expense of the others [73]. In addition, donor dopant additives can be used [101] to supress the grain boundary velocity to a point where abnormal grain growth cannot occur. Table 16.9 gives a list of additives that can be used for various ceramic materials to either enhance or supress the grain boundary mobility. Another method of inhibiting abnormal grain growth is to alter the microstructural path by (1) liquid phase sintering, (2) adding seed grains, (3) hot pressing, or (4) fast sintering. Hennings [102] used a reactive liquid phase (i.e., CuO with TiO2) at the grain boundary to reduce the sintering temperature of

16.4 Grain Growth

843

TABLE 1{}.9 Effect of Additives on Grain Boundary Migration a Additive Host

Enhancement

Suppression

A1203 BeO Cr203 HfO2 MgO ThO2 UO2 Y203 ZnO ZrO2 BaTiO3 TIG YIG Pb(Zr,Ti)O3 MgCr2Oa CoO

H2, Ti, Mn

Zn, Mg, Ni, Cr, Mo, Ni, W, BN, ZrB2, B/Mg C Mg Cr, Mo, W, Ni, Ti, BN MgFe, Fe, Cr, Mo, Ni, Ti, V, BN, ZrB2 Ca V, H2 Th 02, K H2, Cr, Mo, W, Nm, Ni, Ti, BN, ZrO2 Ta, Nb, Ti, A1/Si/Ti, Ba

Mn, ZrB2 Mn, B Ti

02 Y A1, Nb, Fe, Ta, Bi, La O Li

Taken from Brook, R. J., Treatise Mater. Sci. Technol. 9, 331-364 (1976), with additions by this author. a

BaTiQ to 1020~ to 1150~ Seed grains [103,104] modify the grains size distribution and create more nucleation sites for recrystallization. By applying an external pressure during firing (i.e., hot pressing [105,106]), the driving force for sintering increases without substantially altering the driving forces for surface diffusion or grain growth, producing high density small grained samples. Fast firing, developed by Mostaghaci and Brook [107,108], tends to enhance lattice or grain boundary diffusion relative to surface diffusion, and as a result, densification is promoted relative to grain growth and other microstructural changes. A rate-controlling sintering scheme was proposed by Fang and Palmour [41]. They argued that during the intermediate stage of sintering, grain growth is limited by the interconnected cylindric pores. Slow heating at the intermediate stage allows the sample to densify by eliminating pores, during the final stage of sintering faster heating prevents abnormal grain growth. Based on Fang and Palmour's model, Hsieh [109] examined the morphological evolution of BaTiO3 during sintering. When a four-step heating schedule was used, a translucent sample with 99.8% density and grain size of 1.2/zm was successfully obtained.

844

Chapter 16 Sintering

16.5 R E A C T I V E S I N T E R I N G In some cases, sintering in composite systems is performed while phase transformation or chemical reaction takes place simultaneously. This type of sintering is called reactive sintering, where the most common type is liquid phase sintering in which one component melts during sintering. In addition, chemical reactions between two solids, a liquid and a solid, or a gas and a solid are other types of reactive sintering discussed in this section.

16.5.1 Sintering with a L i q u i d Phase Liquid phase sintering is frequently used in ceramics processing to lower the sintering temperature of the ceramic. For ceramics with melting temperature near 3000~ or higher, the sintering temperatures are very high, nearly 90% of the melting temperature expressed in degrees Kelvin. Few kilns can achieve these temperatures, for these materials. To produce these materials at lower temperatures, additives are used to aid in their sintering. Such sintering aids usually melt at much lower temperatures and form a liquid phase. This liquid phase provides a vehicle for fast diffusion of a solid which, often has a small solubility in the liquid phase. Another use of liquid phase sintering is in the manufacture of metal matrix composites, where a metal is used to cement together the ceramic phase. In fabrication of these metal matrix composites, powders of the metal and ceramic are mixed and sintered at temperatures where the metal melts. This type of manufacture is used for cutting tools, where the metal is a hard metal like Co or Ni alloys and the ceramic is diamond, B4C or WC. A key factor in these formulations is the wettability of the ceramic surface by the metal. An example of the liquid phase sintering of a WC-Co mixture is shown in Figure 16.37. As we can see, the densification starts at temperatures of about 1200~ (1473 K), which is 92% of the low-melting eutectic temperature of 1325~ (1598 K) in K. Thus the 6% Co produces, by interdiffusion, a small amount of eutectic with composition 40% WC (and 60% Co). This eutectic material sinters (probably by viscous sintering) and starts the densification process. When a liquid is formed at 1325~ the system becomes much more fluid, and particle rearrangements can take place due to viscous flow. During the latter sintering stages at temperatures above the eutectic temperatures, dissolution of the WC in the liquid and reprecipitation of WC with a small amount of Co in solid solution takes place until all the liquid is consummed. There are two variants of liquid phase sintering.

845

16.5 Reactive Sintering T,C 1600 L+WC

1400

i

1200

-y+WC

1000

800 ~ Co

i

I I I I I' I I b e

I I I I I I I' i I I I I ~.

WC+6%Co

,

WC composition

0

a 10

-

20

shrinkage, %

FIGURE 16.37 (a) The WC-Co phase diagram. (b) The liquid phase, WC + 6% Co, sintering shrinkage versus temperature for WC + 6% Co.

1. Heterogeneous systems: As the ceramic is heated to sintering temperatures, a liquid phase is formed which persists through out sintering. During cooling, the liquid is solidified. 2. Homogeneous systems: As the ceramic is heated to sintering temperatures, a liquid phase is formed which gradually disappears as it is soluble in the matrix. There are four stages to liquid phase sintering. 1. Particle rearrangement stage: After melting, the solid particles are drawn together by the capillary action caused by the liquid. This leads to rapid shrinkage and pore elimination. 2. Dissolution-reprecipitation stage: In many cases the solid particles are soluble to some extent in the liquid. Dissolution is aided by the curvature of the solid particles and pressure at solid-solid contact points. Once solubilized, the solute will diffuse to a point of negative curvature within the microstructure and precipitate, thus growing grains to larger sizes. The precipitated species may not be the same species as the initial solid, but a new species which have components of both the solid and the liquid phases. Such a precipitation will decrease the quantity of liquid present as the precipitation proceeds. 3. Liquid assimilation: In some cases, the liquid is incorporated into the solid phase directly by a liquid attack (either chemical or physical) of the solid. The resulting solid can be a solid solution caused by liquid absorption or be a new phase crystallized from the melt.

846

Chapter 16 Sintering

4. Solid state grain growth stage: When the liquid is either squeezed out of the compact or dissolves into the solid, grain boundaries appear. Further sintering is controlled by grain growth related phenomena discussed previously. Formation of a liquid phase takes place as the ceramic body is heated to the sintering temperature. Heat Q, is transfered from the furnace at T~, to the surface of the ceramic body of surface area, A, at T s, by convection and radiation: (16.107)

Q = ho A ( T ~ - Ts)

Here we use a simplified overall heat transfer coefficient, ho, to approximate these heat transfer processes. If the heat transfer is primarily by convection, the heat transfer coefficient will depend on the velocity of the gas over the green body [110]. If, however, the heat transfer is primarily by radiation, then the heat transfer coefficient will depend on the temperature of the furnace [110, p. 387]. This heat will be used to heat the ceramic to the temperature where a portion of the material will melt. The effective heat of melting is given by (16.108)

g } = Hr + C.s Ts

where HW is the enthalpy of melting (in units of energy per mass), C~s is the heat capacity of the solid ceramic mixture (including pores) before melting, and A T s is the amount of supercooling the liquid component of the ceramic has during heating. Assuming a flat plate geometry, the heat transfer is related to the depth (or volume, V) to which melting occurs as is shown in Figure 16.38" dx Q = p' H} V = p' H} A -dt -

(16.109)

T. Unmelted Zone Furnace

Ts

Tm ATs

Melted Zone h ~ v

Green Body

FIGURE 16.38

Melting of a liquid phase in a composite ceramic during sintering.

16.5 Reactive Sintering

84 7

T A B L E 16.10 Time to Melt the Liquid Phase in a Ceramic Green Body

Plate of thickness x0

Cylinder of radius R 0

Sphere of radius R0

t =

p 'H~xo

ho(T~- T~)

p'H~R-~~ 2 t = ho(T~- Tin)

t =

p'H~ R~ 3 ho(T~- Tin)

where p' is the mass of the component which melts per unit volume of ceramic, x is the depth of the zone which has melted. If we assume that the surface temperature is equal to the melting temperature, then the time to melt a slab of thickness Xo is given by

t=

P' Hf x~ ho(T~- Tin)

(16.110)

For other geometries, the time to melt is given in Table 16.10. This analysis assumes that the heat transfer is very fast in the melted zone, giving a flat temperature profile in this zone. If this is not the case, then the expression becomes more complicated. The melting temperature will depend on the phase diagram for the system being used. Three types of phase diagrams are typical for liquid phase sintering systems. One is the case (Figure 16.39) where we have

Liquid A and Liquid B Immiscible

Liquid A and Solid B

Temperature

Solid A and Solid B 100% B

100%A

FIGURE 16.39 Schematic of a phase diagram for a ceramic B and a metal A without any solubility of the solid ceramic A in the liquid A. When both A and B are liquid we have an immiscibility gap.

848

Chapter 16 Sintering I

1

I

I

--,=..~......~.~.~ 9 ~..,

2OOO

I

1

w

I

Liquid

....

1800

1600

-

- 1400 (D

-

(Ni CoJO solid solution

1200

1000 800

-

-

-

-

--

6~176

o.2

.4 Mole

.6

.8

coo

fraction

FIGURE 16.40

Binary phase diagram for NiO-CoO system, showing complete solid solution. This is also the case for NiO(Tm = 2000~ = 2800~ [111] and A1203 (Tin = 2045~ = 2260~ [ll2].Copyright 9 1976 by John Wiley & Sons, Inc. Taken from Kingery, et al. [2]. Reprinted with permission of John Wiley & Sons, Inc.

a metal and the ceramic is essentially insoluble in the liquid metal. In this case, the liquid metal acts as a binder for the ceramic. Little ceramic sintering or grain growth is expected to occur during this type of liquid phase sintering. Another is the case (see Figure 16.40) [111,112] where we have a complete solid solution. As soon as the low melting solid melts, it provides a vehicle for the diffusion and dissolution of the other solid. Absorption of the liquid into the higher melting solid can also occur, producing a homogeneous system at the end of liquid phase sintering. The last type of phase diagram is that shown in Figure 16.41 [113], where we have two ceramic materials with their melting points (i.e., 2800~ for MgO and 2600~ for CaO). In between there is a lower melting point (2370~ eutetic composition at 70% CaO. As soon as the eutetic temperature is reached, a small amount of liquid is formed at the junction between the two types of grains in the ceramic. This liquid provides a vehicle for the dissolution, diffusion, and reprecipitation of higher melting solid (MgO) into a solid solution with its limited solubility at ~ 10% CaO. Thus, CaO is gradually used up by the reprecipitation of this solid solution. If there is an excess of CaO compared to the solid solution with MgO, then a heterogeneous system will be produced after sufficient time for complete dissolution and reprecipitation. Also other, more complicated phase diagrams with intermediate compounds AxBy undergo congruent melting with two eutectics per compound and non-

16.5 Reactive Sintering

849

2800

2600

2400

~ 221XI

2000

1800

16000 MgO

I

,

I

20

t

I 40

t

60

Weight % CaO

80

100

Ca0

Binary phase diagram for MgO-CaO, showing a low melting temperature eutectic at 2370~ Taken from Doman et al. [113]. Reprinted by permission of the American Ceramic Society.

F I G U R E 16.41

congruent melting without another eutectic. These complicated phase diagrams, an example of which is shown in Figure 16.42 [114], contain similarities with the simple eutectic system shown in Figure 16.41 but only over zones of composition. Now that the liquid has been formed, it can pull together the unmelted ceramic particles. The first action that a liquid undergoes is to wet and spread on the surface of the ceramic. Wetting is governed by the equation [115] SL/S = ~ / S V - ~ / L V - ~/SL

(16.111)

where SL/S is the spreading coefficient and ~/is the surface tension for the solid-vapor (SV), liquid-vapor (LV), and solid-liquid (SL) interfaces. SL/S is positive when spreading occurs. See Section 9.2 for further details. This equation assumes that the contact angle, 0, between the liquid and the solid is small, near 0. If the spreading coefficient is 0 or negative the liquid will be expelled from the ceramic compact, forming droplets at the surface. In this situation, the contact angle is large, approaching 180 ~ Additives in liquid phase sintering are usually of the type to lower the contact angle between the liquid and the solid.

850

Chapter 16 Sintering 2300~ i 2100

rj 1900

2

1700

E 1500 1300 1:6 + 20 3"1

40

1"1

60

80 1:3 1:6

Nb205

La203 Mol % Nb20 5 FIGURE 16.42 Phase diagram for the binary system La203-Nb2Os, containing four congruently melting intermediate compounds [114].

An example of such an additive is S or Sn in Fe liquid [116]. This liquid will enter a horizontal cylindrical capillary with radius rp with a velocity given by [117] V=

TLV

COS0

(16.112)

If the liquid-forming particles in the green body are large, then when the melt and the melt is wicked away by the capillaries, a large pore will be left behind (see Figure 16.43). These large pores can lead to Griffith flaws in the final piece. The height, h, of the liquid with density, p, will rise in a capillary is given by AP = p gh =

2TLvcOS 0

(16.113)

rp Another consequence of capillarity is that a wetting liquid exerts an force on the solid grains. Once the liquid is in a long cylindrical pore, it will tend to be unstable, due to Raleigh instabilities when the length to diameter ratio is >3.0. If dissolution and reprecipitation can take place, then the cylinder will neck off into droplets of liquid along the reconstructed pore at the grain boundary, as is shown in Figure 16.44.

16.5 Reactive Sintering

851

F I G U R E 16.43 Pore formation at the site of a particle which melted. Fe Ti sintered at T > 1085~ the eutectic temperature. The liquid is wicked into the pores of the nonmelting particles. Taken from German [116, p. 71].

F I G U R E 16.44 Rayleigh instability of liquid initially filling cylindrical pores in a metal-metal composite Fe-7%Ti (b). The droplets formed (a) occupy the grain boundary region. Taken from German [116, p. 47].

852

Chapter 16 Sintering

The size, shape, and frequency of the droplets are a result of the dihedral angle, O[=2arccos(Yss/(2YSL)], and the volume fraction of liquid present. The dihedral angle will depend on the composition of the solid-solid (SS) grain boundary and the solid-liquid (SL) interface. During the course of dissolution the composition of the liquid will change with important consequences on the dihedral angle. The dihedral angle and the volume fraction liquid will also dictate the equilibrium grain-liquid configuration, as shown in Figure 16.45 [118]. The pressure within the liquid is less than the external pressure, producing pressure difference AP = 2YLV COS0 d

(16.114)

where d(=2R cos 0) is the distance between the grains (see Figure 16.46). For a surface tension of 1 J m -2 and a radius of curvature, R, of 0.1 t~m, the AP is 10 MPa, which is substantial. This pressure difference is responsible for a compressive force acting on the ceramic body. If there is sufficient liquid to fill all the spaces between the nonmelting particles, this compressive force leads to particle rearrangement. Particle rearrangement is identical to drying shrinkage discussed in detail in Chapter 14. The particle rearrangement will continue until the liquid volume fraction is a constant throughout the ceramic piece. If there is insufficient liquid, particle rearrangement will take place to fill the pore volume originally occupied by the particles which melted. This

i

ii

3O

iii I

I

iv I

v I

vi

I

'//////.~ai$=~176

iii

.,c ZO C O~

u 10

."tO ..

-

vi

iv

II

O"

0

0

20

40

dihedral

60

angle,

80

100

120

degrees

FIGURE 16.45 Calculated grain-liquid configuration as a function of volume fraction of liquid and dihedral angle. Upper drawings show liquid geometry which corresponds to various regions in the lower plot. Reprinted from Wray [118], copyright 1976, with kind permission of Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

16.5 Reactive Sintering

FIGURE 16.46

853

Liquid bridge between two grains during liquid phase sintering.

causes particle rearrangement limited to the volume of the melting particles. Particle rearrangement that is either large or small leads to a high densification rate just as the liquid is formed. This particle rearrangement gives rise to shrinkage rate given by [116, p. 74]. AL d~ Lo dt

-

AP d 2 Re

(16.115)

where R e is the grain size and V is the viscosity of the melt. This equation is usually an overestimate of the shrinkage rate. In essence, this equation assumes that particle rearrangement is friction free. Accounting for friction, we have the experimental observation [119] that AL ~a(FL0

F o)

(16.116)

where F is the capillary force acting on the body and F 0 is the force which resists particle rearrangement. Particle rearrangement takes place within 3 min after melt formation [120-122]. Subsequent densification takes place by dissolution-reprecipitation as well as grain growth. 16.5.1.1 D i s s o l u t i o n a n d R e p r e c i p i t a t i o n

For the SiO2-A1203 system with the phase diagram shown in Figure 16.47 [123], we have a low temperature melting eutectic at 5% A1203, which will drastically influence the sintering temperature. In this phase diagram cristobalite melts at 1726~ and alumina melts at 2045~ and the mullite phase is an incongruent melting intermediate. The dissolution of cristobalite quartz (100% SiO2) and corrundum alumina (100% A1203) to form the eutectic at 1587~ means that the sintering will start at about 90% of 1860 K (-1587~ + 273) or 1640 K ( - 273 = 1400~ Further dissolution of the refractory phase, A1203, in the liquid gives higher and higher melting points and higher and higher A1203 concentrations in the liquid. When the liquid becomes supersaturated in the temperature range 1587-1828~ mullite (3A1203.2SiO 2)

Chapter 16 Sintering

854

Ai203i~ by weight) 23000

2200

10

20

~ I ~ I

30

t

_ 40

I

~

1

~

50

60

1'

~

*-------- Stable equilibrium diagram

--

70 1

x

'-'I

80

90

~

100

I-

MetastabIe extension of liquidus and solidus lines

. . . . .

._

1-

2100 --

2054 ~ _+ 6 ~

2000 --

Liquid

1900-

/

-

,..-

.

~ 1800'--

.

=.- .

1700

.

,1726 +--5~ _~Sy

, .

.

. .

_ .

.

SiO=

i

~

.

.

-4

.

.

.

.

.

.

.

.

i

____,._z._,,

15oo ~

-

"~.-"

Mullite (ss) + liquid

_ - ixarist~

P

......

~

Ill

14001

_..-.--:~"'~7 ~1890~ -+ 10" --

. _

I

I . I 10

Y+ l 20

~

liquid

I

I_ 30

=' I ! I L [ I J | l l, 40 50 60 AIzO~ (mole %)

1828 ~ +_ ~

10'

-

_

i

-

I J I j

--

Alumina + mullite (ss)

-

I

. I. 70

i I I i I

I 80

~

l 90

-

Alz03

FIGURE 16.47 Binary phase diagram for the SiO2-A1203 system. Reprinted with permission from Aksay and Pask Science 183, 69 (1974). Copyright 1974 American Association for the Advancement of Science.

not alumina will be precipitated. An example ofmullite crystals precipitated in a silica matrix is shown in Figure 16.48. This type of precipitation gives a completely different crystal structure and microstructure from the initial SiO2-A1203 mixture. Alumina will continue to dissolve and precipitate mullite until all the alumina is exhausted because only liquid and mullite are stable phases in this temperature range. From 1828 to 2054~ alumina is the phase precipitated from the melt. In this temperature range, alumina reprecipitation will occur due to surface curvature differences~dissolution from small particles with small positive curvatures and reprecipitation onto necks with small negative curvatures or flat surfaces. Liquid phase sintering in the SiO2-A120 3 system is used to manufacture refractory silica brick (< 1.0% wgt A1203), clay products, high alumina brick used in making steel (60-90% wgt A1203), fused mullite, and fused >90% wgt. alumina. In the case of the CaO-MgO phase diagram shown in Figure 16.41, the MgO refractory phase will dissolve in the near eutectic liquid. Upon

16.5 Reactive Sintering

855

FIGURE 16.48 Mullite crystals in silica matrix (3700x). Taken from Kingery et al.

[2, p. 306]. Copyright 1976 by John Wiley & Sons, Inc. Reprinted by permission John Wiley & Sons, Inc. precipitation, an MgO solid solution (ss) will be precipitated with a composition of 10% CaO. This type of reprecipitation is similar to one of the original starting materials. If a small amount of CaO was used initially, then the CaO phase can disappear into the MgOss precipitated. With larger amounts of CaO, a liquid phase will persist even after all the MgO has been dissolved and reprecipitated. Upon cooling the eutected liquid will solidify into a mixture 21% MgO~ and 79% CaO~, which can be seen in the final crystal structure and microstructure. The solubility of the solid in the liquid, C~-SL, and the solubility of the liquid in the solid, C~-LS, play roles in densification. Limited densification takes place by particle r e a r r a n g e m e n t with an insoluble system. When the solubility of the solid in the liquid increases, other processes, like dissolution, lubrication, and surface smoothing, contribute to the densification. Extensive densification is observed with a highsolid solubility in the liquid. In contrast, a high solubility of the liquid in the solid coupled with a low solubility of the solid in the liquid gives

856

Chapter 16 Sintering

swelling not densification. The solubility ratio, SA, defined as (16.117)

SA = C~-sL/C~-LS

gives the tendency for swelling when SA < 1 and densification when SA > 1 as is schematically shown in Figure 16.49. The problem of swelling during melt formation has been treated by Savitskii et al. [124,125]. For a solubility ratio which is small, the porosity, e, will vary with the volume fraction of liquid forming additive, r and the fraction reacted, fR, as follows: s = So + fR (bL(1 - So) = fR d~L + So(1 -- fR d~L)

(16.118)

where So is the initial void fraction. For a solubility ratio which is high, the porosity, s, after dissolution will vary with the volume fraction of liquid forming additive, r and the concentration, on a volume fraction basis, of solid dissolved in the liquid, C~_SL , a s given by [126]" = So , C~-SL r (1 -- SO)~(1 -- r

-- C~-SL)

(16.119)

The consequence of this volume conservation equation is that the porosity after dissolution varies linearly with the initial porosity. Increasing the volume fraction of liquid forming additive and increasing the solubility will lead to lower porosity and higher densification. The minimum volume fraction of liquid forming additive necessary for maxim u m densification is given by (~L-min---- s0(1 - C~-SL)/(C~-sL -- C~-SL So -- So)

(16.120)

FIGURE 16.49 Schematicof the effects of solubility on densificationor swellingduring

liquid phase sintering. Taken from German [116, p. 69].

16.5 Reactive Sintering

857

This minimum liquid volume fraction should be built into the ceramic green body to take maximum advantage of liquid phase densification. The dissolution process takes place by mass transfer from the surface of the dissolving grain. The flux of material, J, is given by

J = Kc(C s - C~)

(16.121)

where Kc is the mass transfer coefficient, Coois the concentration in the liquid far from the dissolving surface, and Cs is the surface concentration determined from the phase diagram and the temperature. The mass transfer coefficient can be calculated from the Chilton-Colburn analogy for a sphere of radius, r, in a motionless liquid, giving the Sherwood number (=Kc 2r/DAB) equal to 2.0. Here DAB is the diffusion coefficient of A in liquid B. The solubility of the system is enhanced by the size of the particles dissolving. The increase in solubility is given by the Kelvin equation Coo exp ~2~/SL~ C~o \ r ksT ]

(16.122)

where C~0 is the solubility of flat material and ~/SL is the solid-liquid surface free energy, t2 is the atomic volume, and r is the radius of the particle. If the radius of curvature is negative, like that of a neck, the solubility is decreases. A difference in curvature from one surface to another in the ceramic body can be used for dissolution-reprecipitation liquid sintering. The rate of dissolution of small grains, radius r, in a system of small and large grains is given by [81]

dr ( 2 D A B C A ~ T S L ~ ) ( r - - R g ~ dt = ks T \ r2Rg ]

(16.123)

where Rg is the large grain size, with the assumption that the mean grain size is approximately equal to the large grain size. This type of Oswald riping leads to a shrinkage given by [127] (AL~3 6 t2 TSL DABCA~ t Lo ] = R3gks T

(16.124)

This type of densification is enhanced by temperature, which increases the diffusion coefficient and the solubility. Higher temperature decreases the value of ~/sn slightly.

16.5.1.2 Reprecipitation In Chapter 6, precipitation from a solvent was discussed. This subject, which includes nucleation and crystal growth, has the same fundamentals as precipitation from the melt. Crystal growth mechanisms are summarized in Table 16.11. These mechanisms include diffusion,

8~8

Chapter 16

Sintering

TABLE 16.11

Crystal Growth Rate, d R / d t = C*f(S)*g(R)

Growth mechanism

C

f(S)

g(R)

Diffusion bulk Monosurface nucleation Polysurface nucleation Screw dislocation Heat conduction Chemical reaction

vDCeq flAD d -1 D d/(Ceq) ~3 Dsnsefl/(y2p) vkHRT2/AHw ~vDCeq

S - 1 exp[hGs/ksT] d ( S - 1)y3 exp[hG*/3ksT] d $2/$1 t a n h ( S J S ) e In S f S - 1

1/R R2 1 1 1/R 1/R

Reference a

b b c b b

a Volmer, M. M., "Kinetic der Phasenbildung," p. 209, Steinkopff, Dresden, Leipzig, 1939. b Nielsen, A. E., "Kinetics of Precipitation," Pergamon, Oxford, 1964. c Elwell, D. and Scheel, H. J., "Crystal Growth from High-Temperature Solution," Academic Press, London, 1975. d AG* = /3 L2~/ea 2 2/(lflAks T I n S) e S 1 __ (Yo/Ys)S f ln S = f T AHw/(RT2 ) d T For nomenclature, see Chapter 6.

monosurface nucleation, polynuclear surface nucleation, screw dislocation growth, heat conduction, and chemical reaction. The growth mechanism presented in this table occur in the following order as supersaturation increases: screw dislocation, monosurface nucleation, polysurface nucleation, bulk diffusion. Chemical reaction and heat transfer give growth mechanisms which are slower than bulk diffusion. The supersaturation driving force, S - 1 = (Cs - C ~ ) / C ~ , given in Table 16.11, can be replaced by S - S (R*) when the particles are small, see Chapter 6 for details. Here, 213a

S(R*) = exp 3 R * fiv 9N A k s T

(16 125)

comes from the Kelvin equation and describes the saturation ratio at which particles of size R* will dissolve. When S - S ( R * ) is positive, particles will precipitate from solution, when S - S ( R * ) is negative, particles smaller than size R will dissolve and particles larger than size R* will grow. This dissolution of fines and reprecipitation on larger particles is referred to as Oswald ripening, and it occurs in many batch crystallization systems because the supersaturation ratio, S, decreases with time as the crystallization proceeds. Initially, at high supersaturation, nucleation produces large numbers of fine particles. This decreases the supersaturation, preventing further nucleation and leading to slow growth, which further decreases the supersaturation ratio. When the saturation ratio falls below the critical value, S ( R * ) , for the fine parti-

16.5 Reactive Sintering

859

cles previously precipitated, these particles will dissolve, holding the supersaturation ratio constant. At this constant supersaturation ratio, only particles larger than R* will grow or ripen at the expense of all smaller particles present in the suspension. In addition, the radius of curvature of neck regions will play a role in dissolution and reprecipitation. Densification during dissolution-reprecipitation has two rate limiting steps: dissolution and diffusion. If diffusion is rate limiting, the shrinkage is given by [128] (AL~3 12 6 t2 ~/SL DAB CA~ t -~0 ] -R4 ks T

(16.126)

where 6 is the thickness of the liquid layer between grains of size Rg, t2 is the atomic volume of the solid A, CA~ is the solid concentration in the liquid B. If dissolution is the rate limiting, the shrinkage is given by

(AL~ 2 4 k r a T s L C A ~ t Lo] = R~kBT

(16.127)

where k r is the dissolution reaction rate constant. During dissolutionreprecipitation liquid phase sintering, densification is enhanced by a high solubility of the solid in the melt. Decreasing the particle size further enhances the solubility. The role of temperature is to increase the diffusivity and solubility and decrease the surface energy. 16.5.1.3 G r a i n G r o w t h

During the last stage of liquid phase sintering for homogeneous systems, no liquid is left over. Densification can continue at this stage if there are pores in the ceramic and if no pores in the ceramic grain growth can take place upon further heating. Grain growth at this stage is exactly the same as that discussed in Section 16.4. The densification rate developed in Section 16.3 can also be applied here. For heterogeneous systems where, after dissolution-reprecipitation, there is still a liquid phase, due to either equilibrium or just slow dissolution kinetics, this liquid will solidify upon cooling. The liquid, depending on its composition, will precipitate out the constituents that are supersaturated by the cooling (see Table 16.11, Heat Conduction). In the case of the A1203-SiO 2 system with a eutectic liquid, the species to be precipitated upon cooling are crystobalite and mullite (see Figure 16.48). Depending on the cooling rates, the supersaturation is controlled and the resulting nucleation and crystal growth rates controlled. With high cooling, the supersaturation is high, giving rise to a high nucleation rate, which produces a large number of crystals. Assuming all

860

C h a p t e r 1 6Sintering

the material solidifies, the growth of the crystals is therefore limited to a small size. With slow cooling, the supersaturation is lower, giving rise to a lower nucleation rate and fewer crystals. The growth of the same amount of materials onto fewer nuclei gives rise to a larger crystal size. This solidified melt completes the ceramic microstructure of these heterogeneous liquid phase sintered ceramics and acts as a bonding agent between the solid grains.

16.5.2 Solid State Reactive Sintering During sintering of multicomponent green bodies, solid state reactions can take place. Examples of solid state reactive sintering are 1 3 TiO2(s) § 4 AIN(s) --~ 3 TiN(s) + 2 A 1 2 0 3 ( s ) + ~ N2(g) NiO(s) + A1203(s)--* NiA]204(s) MgO(s) + Cr203(s)~ MgCr204(s) ZrO2(s) + CaO(s)--~ ZrCaO3(s) In each case, interdiffusion of the two solids gives an intermediate compound found on the phase diagram. This type of sintering takes place at temperatures below the lowest melting point in the phase diagram, typically one of the two eutectics associated with the incongruently melting intermediate compound. This type of sintering is very complex. In principle, the reaction is taking place by interdiffusion, and simultaneously each of the individual types of reactant particles as well as the product of the reaction are sintering. Because the reactants typically have much higher melting temperatures than those of congruently melting intermediate compounds, the sintering of the intermediate compounds will be much faster than that of the reactants. The slow steps in sintering are therefore limited to the solid state reaction and the sintering of the intermediate compound, assuming that no liquid phase sintering can take place due to the temperature of the system. The intermediate compound often has a higher diffusion coefficient than either reactant material in solid state reactive sintering, due to a larger number of defects in its structure than the original materials. The solid state reaction was discussed in detail in Chapter 5. The volume fraction reacted is given by [1 - (1 - ~)1/312 =

2K -rTt

(16.129)

where r is the particle size, K, given by equation 5.59, (2K/r 2) is essentially a reaction rate constant which is characteristic of reaction condi-

861

16.5 Reactive Sintering

tions (typically temperature). K has the temperature dependence K - K ~ exp

(NA-Q k B T)

(16.130)

where Q is an activation energy of the solid state diffusion coefficient. This kinetic relationship has been found to hold for many solid state powder reactions, including silicates, titanates, and ferrites. There are two simplifications in equation 16.129, 1) same molar volumes of product and reactant and 2) small reaction thicknesses. When corrections are made for these two simplifications, Carter [129, 130] has shown the following equation to be applicable: [1 + (Z - 1)a] 2/3 + (Z - 1)(1 - a)2/3 = Z + (1 -

2K Z)-~t

(16.131)

where Z is the ratio of equivalent volumes-product to reactants. Another type of solid state reactive sintering is combustion sintering, which takes place when the reaction between the two solids is sufficiently exothermic to maintain the reaction. Because high temperatures, near 3000~ are encountered as a result of these reactions, the product particles are bonded together to some degree. Examples of materials prepared by combustion sintering are [131] borides, carbides, carbonitrides, cemented carbides, chalcogenides, nitrides, silicides, selenides, and intermetallics. The interparticle bonding is not complete, so further sintering is required. Depending on the material system, this sintering will take place by liquid phase or solid state sintering. Solid state sintering was discussed previously in this chapter. The sintering kinetics depend upon the rate determining step, which can be either viscous flow, grain boundary diffusion, or lattice diffusion. These sintering kinetics are summarized in Tables 16.4 and 16.6 for the initial and intermediate stage and Section 16.3.2.3 for the final stage.

16.5.3 G a s - S o l i d R e a c t i v e Sintering In some cases, the reaction between a gas and a solid gives a ceramic of interest, and this reaction can be used to densify the green body. The classic example is reaction sintered Si3N4, also called Si3N4. The gas-solid reaction used to make reaction bonded silicon nitride [132] is between silicon metal powder and nitrogen:

reaction

bonded

3 Si(s) + 2 N 2 ( g ) ~ Si3N4(s) + AH = - 7 2 4 kJ/mole As the reaction takes place the pores in the initial silicon powder compact are filled by fibers of a-Si3N 4 at 1200~ At 1400~ silicon melts and the reaction produces fl-Si3N4. The reaction causes a 21.7% change in molar volume, filling the pores in the initial powder compact.

862

Chapter 16 Sintering

FIGURE 16.5{} Schematic of the reactive sintering of Si(s) + N2(g)~ Si3Nt(s).

The initial green density of the Si compact is, therefore, tailored to be just t h a t necessary to give the theoretical density of Si3N 4 after the reaction is complete. The pore diffusion with first-order chemical reaction can be modeled by considering a conceptual pore open to the reactive atmosphere as shown in Figure 16.50. The mass balance on a differential section of the pore gives O( OCN2~ 2k___~Cs s 2 _ (~CN2 ~x D eff Ox ] - rp at

(16.132)

where ks is the first-order surface reaction rate constant. This reaction occurs at the surface of the pore of radius rp:

1 dNN2 CN2 27rrphx dt - ks

(16.133)

where NN2 is the number of moles of N2. The effective diffusion coefficient for a porous layer is given by Deft =

1 +

_s

(16.134)

where DK(=rp~/18RgT/(TrMw)) is the Knudsen [133] diffusion coefficient, DAB is the molecular diffusion coefficient through the gas in the

16.5 Reactive Sintering

863

pores, s is the void fraction of product layer, ~ is the tortuosity of the pores (typically ~ 2.0). Assuming that the effective diffusion coefficient is constant (i.e., the porosity, s, and the pore radius, rp, do not change) and that we have pseudo-steady state, the governing equation becomes

Deft\ 0x2 ] - Zksc rp

=0

(l

.laS)

with the boundary conditions CN2 = CN2-Sat x = 0

deN2 dx

(16.136) -Oatx=L

The solution to this differential equation is [134] CN2 cosh m ( L - x) = CN2-S cosh m L

(16.137)

where m = k / 2 k s / ( D e f f rp)" m L is called the Thiele m o d u l u s . For an arbitrary shaped green body [134, p. 476], L = Volume/(exterior surface area). This analysis assumes that the pore dimensions remain constant with extent of reaction and that the reaction rate is not changed by a product layer that is formed. This later assumption is not a bad assumption when the silicon is liquid because the fl-Si3N4 product is continuously wet by the molten silicon. A more complete analysis of this problem should account for changes in pore geometry and simultaneous product layer diffusion and reaction. This simplified analysis gives the initial concentration profile along the pore. It is highest at the inlet, and as a result, the highest surface reaction rate at the mouth of the pore. This high surface reaction will tend to close off the mouth of the pore because the molar volume of Si3N4 is larger than that of silicon. Because the nitrogen must diffuse in from the outside of the compact, the pores can be plugged off before the interior of the compact has reacted. As a result the initial void fraction usually used is larger than that necessary to accomodate the molar expansion due to reaction and the excess pores are removed by natural solid state sintering or hot pressing. The solid state sintering of Si3N4 is complicated by evaporation at temperatures higher than 1850~ SiaN4(s)--* 3 Si(1) + 2 N2(g) This evaporation of nitrogen will prevent sintering. If the sintering is performed with an overpressure of nitrogen, then the preceding

864

Chapter 16 Sintering

equilibria will be displaced to the left, allowing Si3N4 to be densified rather than volatilized. Using an overpressure of the gas volatilized is a common way to prevent evaporation during sintering [135,136]. Another problem in the sintering of silicon nitride is the pressure of oxygen in the sintering atmosphere or as an impurity in the powder. The presence of oxygen gives rise to SiO(g) being formed at a certain. pressure, depending on the sintering temperature, that can lead to parasitic reactions. Volatilization also occurs with the sintering of Cr203 in air. Chromic oxide can be densified only at low oxygen partial pressures (i.e., 10 -1~ to 10 -11 atm), because in an oxidizing atmosphere Cr203 sublimates as CrO2 or CrO3 [137]. The oxygen partial pressure as well as the partial pressures of PbO and ZnO vapor play a role in the sintering of lead titanates and zinc ferrites, because of incongruent vaporization of PbO and ZnO. The sublimation rate is lower when the partial pressure of PbO and ZnO is maintained at a high level and the partial pressure of oxygen is high [1, p. 459]. Another type of reaction sintering is that of SiC by the reaction [138] Si(g) + C(s)--* SiC(s) where silicon vapor is infiltrated into a carbon compact to produce silicon carbide by a gas-solid reaction. Complete carburization usually leads to a ceramic body with pores remaining, which are removed by solid state sintering. During solid state sintering of SiC, a nitrogen atmosphere is used to inhibit the sublimation of SiC (and B4C dopant). Gas-solid reactive sintering can also be used in the oxidation of most metals or, more important, mixtures of metals. After oxidation, solid state sintering takes place. With mixtures of metals, a mixed oxide is produced which can be sintered by either solid state or liquid phase sintering. One example of alloy oxidation to produce a ceramic* is in the synthesis of the superconductor YBa2Cu3Ox.

16.6 P R E S S U R E S I N T E R I N G Pressure sintering is reviewed by Spriggs and Dutta [139]. The application of pressure during the high temperature stage of sintering is called hot pressing or pressure sintering. Hot pressing is similar to cold pressing of ceramic powders, where a die is used, but the die is heated to temperatures approximately half the absolute melting temperature of the material. Hot pressing is one of the methods used to obtain densification without the abnormal grain growth seen in solid state * A n A m e r i c a n S e m i c o n d u c t o r s ' process.

16.6 Pressure Sintering

865

sintering. This is because other densification mechanisms, like aggregate fragmentation, particle rearrangement, plastic flow, and diffusional creep, are important in hot pressing. Pressure sintering is usually not used when a liquid phase is present because the liquid is squeezed out of the ceramic. Dies of graphite, carbide, boride, and molybdenum are often used, but they are expensive and subject to excessive~wear and oxidation at temperatures greater than 12001500~ Hot pressed shapes are limited as a result of die geometry. The effect of die wall friction leads to density variations similar to those seen in the cold pressing of ceramic powders discussed in Chapter 13. It is also quite common to have grain orientation resulting from slip planes in the ceramic body which are parallel to the die walls. Other variants of pressure sintering are hot isostatic pressing and pseudoisostatic pressing. In hot isostatic pressing (HIP), an electric furnace is place inside a pressure vessel. The pressurizing fluid is a gas like argon or helium. To achieve densification of a ceramic green body, the body must be evacuated and sealed in a gas-impermeable vessel. If there are any leaks in the seal, the ceramic body cannot be pressure sintered. Early gas-impermeable vessels were metallic tantalum cans. Later metal powder or glass encapsulation have been used. The encapsulation layer is first sintered into a gas-impermeable layer before hot isostatic pressing. With glass encapsulation, a glass preform is used as a receptacle for the ceramic. The glass vessel is evacuated and sealed. It then deforms to the ceramic body's shape during hot isostatic pressing. Recently, a glass particulate coating has been applied to ceramic bodies to encapsulate them [140]. In the pressure vessel, evacuation takes place and the temperature is raised to soften the glass until it forms a continuous layer on the ceramic body. The temperature and pressure are then increased to densify the ceramic body. A list of the temperatures, pressures, and times necessary for HIP some ceramics is given in Table 16.12. As with cold pressing, HIP resolves the problems of density variation inside the ceramic, which is prevalent in hot pressed ceramics. When solid state sintering is performed before hot isostatic pressing, the combined process is called sinter-HIP or post-HIP. In pseudo-isostatic pressing [111, p. 561], an inert powder like hexagonal BN or graphite is used as the "pressurizing fluid," because these powders are self-lubricating. The ceramic green body is placed in the inert powder within the die of the hot press. The temperature and pressure are increased, and densification takes place because the pressure is transmitted through the inert powder to the ceramic body. A true isostatic pressure distribution is not achieved, however, but densification takes place albeit with some degree of distortion. As discussed in Section 16.3.2.1, pressure alters the driving force

866

Chapter 16

Sintering

TABLE 16.12 Hot Isostatic Pressing Conditions for Various Ceramicsa Material

T (~

BeO UO2 A1203 TiB2 Si3N4 SiC Ferrites TiC-Co

P(MPa)

Time (hr)

Process

1i00 1150

200 70

0.75 2

HIP HIP

1400 1700 1700 1950 1200 1325

i00 200 200 200 100 i00

1.5 1.5 2.5 2.3 1.5 0.5

Sinter-HIP Sinter-HIP Sinter-HIP Sinter-HIP Sinter-HIP HIP

McColm, I. J., and Clark, N. J., "The Forming, Shaping and Working of High Performance Ceramics," p. 251. Blackie, London, 1988. a

for sintering. Without pressure, the driving force for sintering is the sintering stress, ~I'. With applied p r e s s u r e , Papplied, the driving force for sintering a string of spheres is ~kP-- (Papplied- ~t~r)

(16.138)

where the sintering stress is described by = 2 Tsv sin ~ - Tsv

x

2

(16.139)

K

where x is the neck radius and Kis the radius of curvature of the neck. For the green ceramic compact, the sinter stress can be simplified to a first order by = 2 Tsv

(16.140)

Ro where R0 is the m e a n size of the particles composing the ceramic green body. The sintering driving force is therefore simplified to ~jO =

eapplied -- 2 R--oo]

(16.141)

to a first approximation. Typical pressures used in pressure sintering are 20 MPa (compression) and for HIP are 200 M P a (compression). Comparing this to the sintering stress o f - 1.82 MPa (tension) for a 1 t~m A1203 powder compact at 1850~ ( T s y = 0.91 J / m 2 [141]), we find t h a t the driving force is increased from 11 to 110. This increased driving force for pressure

16.7 Cooling after S i n t e r i n g

867

sintering reduces the dependence of sintering kinetics on the particle size because Papplied is SO much larger than ~. The application of pressure allows pressure sintering to take place at temperatures of half those of the absolute melting temperature in contrast to 90% of the absolute melting temperature in solid state sintering. As a result, it is common never to enter the grain growth region, so t h a t fine equiaxed microstructures are produced. Many times these microstructures are so fine that they are translucent to ordinary light.

16.7

COOLING

AFTER

SINTERING

After sintering the ceramic is cooled. The stress caused by cooling a homoegeneous material has been discussed in Chapter 14. The tensile stress at the surface during cooling is the most severe because ceramics are weakest in tension. The stress at the surface for various shapes is given in Table 16.13 using the same nomenclature as in Chapter 14. Such surface stress is tensile and can lead to fracture of the sintered body if cooling is too abrupt. The cooling rate that gives a particular surface tensile stress can be calculated from these formulas with material properties and initial and average ceramic temperatures. If the

TABLE 16.13

Infinite plate

Surface Stresses a and Temperature Differences b for Various Shapes EaT

Cry = (rz = ~

Infinite o"r cylinder

-

( T , - Ts)

0

T a - Ts Ti-Ts Ts T i - Ts

Ta-

-

8 7r

1 [ - ( 2 n + 1)21r2a't] 2 n~ (2n + 1)2 exp 4x02 =o

n l ex [ _

R20 'tlc

Ea T

~o = ~-7_ ~(Ta - T~)

Sphere

O" r - -

0

6 n~ 1 [--n2~r2a't) Ti - Ts - ~ :on--~exp[ ~202

Ta-Ts

Ea T

868

Chapter 16

Sintering

surface stress is greater than the strength of the ceramic, cracking will result; if it is less than the strength of the ceramic, warping will result. During cooling, several transformations can take place. If a liquid phase is present during the later stages of sintering, it will solidify into the resultant solid phases, as discussed in heterogeneous liquid phase sintering. Even after the complete system is solid, further crystal transformations can take place as the solid is cooled to room temperature. A material will have different crystal structures which are stable at different temperatures. This phenomena is called polymorphism. Polymorphic transformations occur during heating and cooling of a material. During heating, the different crystal structures increase in molar volume abruptly at particular transformation temperatures. This is generally not a problem, as the individual grains are not constrained. But on cooling after sintering, the sudden decrease in volume of either one component of a multicomponent polycrystalline ceramic or one grain of a polycrystalline ceramic with random grain orientations can cause high internal stresses which may cause cracking. A good example is ZrOe. At room temperature, zirconia is monoclinic. At 1000~ it abruptly transforms to a tetragonal crystal structure. This transformation involves a volume change of -~15%. Additives to maintain the cubic ZrOz crystal structure at lower temperatures and prevent this transformation on cooling to room temperature have been used to overcome this problem. These additives include CaO, MgO, or YeO3. Many ceramics undergo polymorphic transformations that decrease their strength during either cooling or thermal cycling during their utilization. Many other materials important to ceramics (C, BN, SiO2, TiO2, AszO3, ZnS, FeS2, CaTiO3, AlzSiO4, etc.) exhibit polymorphism. Silica is particularly rich in polymorphic transformations. At room temperature,

Low Quartz

Low Cristobalite

Low

Tridymite

Displacive 573~

Middle

Displacive

Tridymite

2o0-27o~

II Displacive160~

High Quartz

r--~

High Tridymite

Reconstructive 867 *C

F I G U R E 16.51

= ~

High Cristobalite

Reconstructive 1470*C

Polymorphie forms of SiOz.

16.8 Summary

869

quartz is the stable crystal structure and cristobalite is the high temperature structure. Both quartz and cristobalite have displacive polymorphic transformations (see Figure 16.51) on cooling which involve substantial volume changes. As a result, the cooling of clay bodies, porcelain, high silica brick, and fused silica are subject to large internal stresses and cracking unless precautions are taken to hold the temperature constant near the transformation temperature as the ceramics are cooled.

16.8 S UMMAR Y This chapter has provided a description of sintering of ceramics by solid state sintering, reactive sintering, and pressure sintering. In addition, grain growth and cooling have been discussed. In solid state sintering, there are three stages: initial, intermediate, and final. During the initial stage, necks are formed between the particles in the ceramic green body. During the intermediate stage, the cylindrical pores are located at the three grain junctions between grains. These pores shrink until they become unstable and form spherical pores at the same three grain junction. The spherical pores move to the grain intersections during the final stage of sintering, where they can be eliminated. If the grains grow relatively fast (i.e., abnormal grain growth), the pores may become separated from the grain boundary and left within a grain, where they are difficult to remove by further sintering. Grain growth plays an important role in manipulating the grain size in the final microstructure. Reactive sintering includes liquid phase sintering, solid state reaction sintering, and gas-solid reactive sintering. Liquid phase sintering and pressure sintering are used to prevent abnormal grain growth from occuring, so that high densities can be obtained maintaining fine grained microstructures.

Problems 1. Determine the isothermal initial stage shrinkage at 1200~ of a BaTiO3 sample composed of spherical 0.71 tLm particles during initial stage sintering. Assume that grain boundary diffusion is the rate determining sintering mechanism [34]. Also determine the intermediate stage sintering shrinkage, assuming that the mechanism is unchanged. Use the Coble kinetic expression given in Table 16.6. with a grain boundry thickness of 5/~. Data: p = 6.01 gm/cc; D~b = Do exp (-262 Kcal/mole/NAkBT), Do = 1.0 cm2/sec; ~/gb = 2 . 2 - 0.4 • 10 .3 T(K)J/m 2.

870

Chapter 16 Sintering

2. For the same BaTiO3 sample as in problem 1, determine the initial stage shrinkage for a constant heating rate of 10~ starting at room temperature. 3. Problem 1 assumes that the BaTiO3 powder is monosized. This is not actually the case, the powder used has a geometric mean size of 0.71 t~m and a geometric standard deviation of 1.6. Determine the isothermal shrinkage at 1200~ of this BaTiO3 sample during both initial and intermediate stage sintering. 4. Glass beads are to be used to form a porous plate. The glass beads are 100 t~m in diameter and are to be sintered until the neck between the particles is 0.1 times their diameter, where sufficient strength and porosity are obtained. In the sintering of glass by viscous flow, the viscosity follows the following relationship: 7o = 100 poise,

Qvis~ 140 exp - ( ~ o o ) - Z . 2 0 K c a l / m o l e where Z is the valence of the metal oxide added to the SIO2, X is the mole fraction added, and Xo is 0.125. Compare the times needed to sinter the necks in this porous plate at 1000~ if 20 mole % CaO or 10 mole % K20 is added to the silica. Assume that the solid-vapor surface tension of glass is ~/sy = 0.93 - 0.19 • 10 .3 T(K) J / m 2 and not a function of composition. 5. A mixture of 20% CaO and 80% MgO powders is mixed and pressed into a green body. From this mixture, determine the temperature at which densification will start. What is the temperature at which a liquid phase will form? Note that the melting temperature is higher than the temperature at which densification starts. Due to liquid phase sintering, will the ceramic be homogeneous or heterogeneous? And, finally, will the green body shrink or swell during liquid phase sintering? 6. In the sintering of A1203 a small amount of SIO2, some 5%, is used to lower the sintering temperature from near 1600 to 1400~ due to a low melting eutectic at 1587~ After sintering is complete, calculate the amount of mullite present in the sample. Explain why cooling this sample will be problematic. 7. A zinc ferrite ceramic after hot pressing has a 0.5 t~m mean grain size. This ceramic has hard magnetic properties. To get soft magnetic properties in this sample, the grains have to be grown larger than the size of a magnetic domain in this material, 3 t~m. Using an

References

871

isothermal heat treating cycle, the grains are grown at 1500~ In 40 min the grains have grown from 0.5 ~m to 0.7 ftm. How long must the heat treating cycle be to produce soft magnetic ferrite samples? 8. What is the maximum grain boundary velocity before the 0.13 ftm diameter pores will detach from its grain boundary shown in Figures 16 and 17. in the sintering of pure A1203 at 1550~ Assume that surface diffusion is responsible for the grain boundary movements. Data: Tg~ = 1.2 - 0.23 • 10 -3 T(K) j/m2; Dsw = 10 -12 cm3/sec. 9. If a pressure of 100 MPa is applied during HIP, how long will it take to sinter the BaTiO3 powder described in problem 1.

References 1. Reed, J. S., "Principles of Ceramic Processing," p. 449. Wiley, New York, 1988. 2. Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., "Introduction to Ceramics," 2nd ed., p. 470. Wiley (Interscience), New York, 1976. 3. Ashby, M. F., Acta Metall. 22, 275 (1975). 4. De Jonghe, L. C., Chu, M. Y., and Lin, M. K. F., J. Mater. Sci. 34, 4403 (1989). 5. De Jonghe, L. C., and Rahaman, M. N., Acta Metall. 36, 223 (1988). 6. Raj, R., and Bordia, R. K., Acta Metall. 32, 1003 (1984). 7. Rhines, F. N., and DeHoff, R. T., Mater. Sci. Res. 16, 49 (1984). 8. Hsueh, C. H., Evans, A. G., Cannon, R. M., and Brook, R. J., Acta MetaU. 34, 927 (1986). 9. Cannon, R. M., and Carter, W. C., J. Am. Ceram. Soc. 72(8), 1550-1555 (1989). 10. Rahaman, M. N., and De Jonghe, L. C., J. Am. Ceram. Soc. 67(10), c205 (1984). 11. Rahaman, M. N., De Jonghe, L. C., and Brook, R. J., J. Am. Ceram. Soc. 69(1), 53 (1986). 12. Kuczynski, G. C., Trans. AIME 185, 169 (1949). 13. Johnson, D. L., in "Ultrafine-Grain Ceramics" (J. J. Burke, N. L. Reed, and V. Weiss, eds.), p. 173. Syracuse Univ. Press, Syracuse, NY, 1970. 14. Johnson, D. L., Mater. Sci. Res. 13, 97 (1978). 15. Brook, R. J., Sci. Ceram. 9, 57-66 (1977). 16. Kolar, D., Mater. Sci. Res. 13, (1980). 17. Kellett, B. J., and Lange, F. F., J. Am. Ceram. Sci. 72(5), 725 (1989). 18. Lange, F. F., and Kellett, B. J., J. Am. Ceram. Soc. 72(5), 735 (1989). 19. Johnson, D. L., J. Appl. Phys. 40(1), 192 (1969). 20. Johnson, D. L., and Culter, I. B., J. Am. Ceram. Soc. 46(11), 541, 545 (1963). 21. Kingery, W. D., and Berg, M., J. Phys. 26(10), 1205 (1955). 22. Herring, C., J. Appl. Phys. 21, 301 (1950). 23. Bannister, M. J., J. Am. Ceram. Soc. 51(10), 548 (1968). 24. Prof. Kovats, private communication. 25. Coblenz, W. S., Dynys, J. M., Cannon, R. M., and Coble, R. L., in "Sintering Processes" (G. C. Kuczynski, ed.), pp. 141-157. Plenum, New York, 1980. 26. Nichols, F. A., and Mullins, W. W., J. Appl. Phys. 36(6), 1826-1835 (1965). 27. Searcy, A. W., in "Sintering Processes" (G. C. Kuczynski, ed.), pp. 591-599. Plenum, New York, 1980. 28. Udlin, H., Shaler, A. J., and Wullf, J., Trans. AIME 186, 186-190 (1949). 29. Johnson, D. L., J. Am. Ceram. Soc. 53(10), 574 (1970). 30. Coble, R. L., J. Appl. Phys. 32(5), 787 (1961).

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31. Coble, R. L., J. Appl. Phys. 32(5), 793 (1961). 32. Coble, R. L., and Gupta, T. K., in "Sintering and Related Phenomena" (G. C. Kuczynski, N. A. Hooton, and C. F. Gibbon, eds.), p. 423. Gordon & Breach, New York, 1967. 33. Gupta, T. K., J. Am. Ceram. Soc. 61, 191 (1978). 34. Chen, Z.-C., Ph.D. Thesis, Materials Science Department, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland (1992). 35. Kingery, W. D., and Francois, B., in "Sintering and Related Phenomena" (G. C. Kuczynski, N. A. Hooton, and C. F. Gibbon, eds.), p. 471. Gordon Breach, New York, 1976. 36. Lange, F. F., and Davis, B. I., Adv. Ceram. 12, 699 (1984). 37. Burke, J. E., and Rosolowski, J. H., in "Treatise on Solid State Chemistry" (N. B. Hannay, ed.), Vol. 4, Chapter 10. Plenum, New York, 1976. 38. Hsueh, C. H., Evans, A. G., and Coble, R. L., Acta Mettal. 30, 1269-1279(1982). 39. Brook, R. J., J. Am. Ceram. Soc. 52(1), 56 (1969). 40. Fang, T. T., and Hsieh, H. L., J. Mater. Sci. 27(17), 4639-4646 (1992). 41. Fang, T. T., and Palmour, H., II, Ceram. Int. 16, 1 (1990). 42. Lemaitre, J., and Carry, P., private discussions. 43. Allemann, J., Hofmann, H., and Gauckler, L., Ber. Dtsch Keram. Ges. 6'/(10), 434 (1990). 44. Lange, F. F., J. Am. Ceram. Soc. 67(2), 83-89 (1984). 45. Rahaman, M. N., De Jonghe, L. C., and Chu, M.-Y., J. Am. Ceram. Soc. 74(3), 514 (1991). 46. Coble, R. L., J. Am. Ceram. Soc. 56, 461 (1973). 47. Chappell, J. S., Ring, T. A., and Birchall, J. D., J. Appl. Phys. 69(1), 383-391 (1986). 48. Harrett, T., J. Appl. Phys. 61, 5201 (1987). 49. Rosolowski, J. H., and Greskovich, C., J. Am. Ceram. Soc. 58, 177 (1975). 50. Kuczynski, G. C., Z. Metallkd. 67, 606 (1976). 51. Ikegami, T., Tsutsami, M., Matsuda, S., Shirasaki, S., and Suzuki, H., J. Appl. Phys. 49, 4238 (1978). 52. Barringer, E. A., and Bowen, H. K., J. Am. Ceram. Soc. 65, C-199 (1982). 53. Barringer, E. A., Ph.D. Thesis, Materials Science Department, MIT, Cambridge, MA (1984). 54. Yan, M. F., Mater. Sci. Eng. 48, 53-72 (1981). 55. Wang, D. N. K., Ph.D. Thesis, Materials Science Department, University of California, Berkeley (1976). 56. Kaye, B. H., Kona (Hirakata, Japan) 9, 218-236 (1991). 57. Onoda, G., and Tover, J., J. Am. Ceram. Soc. 69, C278-C279 (1986). 58, 59. Ring, T. A., in "Ceramic Powder Processing Science" (Hausner, H., Messing, G. L., and Hirano, S., eds.), p. 681. Dtsch. Keram. Ges., Koln, 1989. 60. Kingery, W. D., and Bird, M., J. Appl. Phys. 26, 1205 (1955). 61. Coble, R. L., J. Am. Ceram. Soc. 41, 55 (1958). 62. Jean, J.-H., Ph.D. Thesis, Materials Science Department, MIT, Cambridge, MA (1986). 63. Mates, T. E., and Ring, T. A., Colloids Surf. 24, 299-313 (1987). 64. Edelson, L. H., and Glaeser, A. M., J. Am. Ceram. Soc. 71(4), 225-235 (1988). 65. Yan, M. F., Cannon, R. M., Chowdhry, U., and Bowen, H. K., Mater. Sci. Eng. 60, 275 (1983). 66. Ring, T. A., M R S Bull. 12(7), 34-38 (1987). 67. Rhodes, W. H., J. Am. Ceram. Soc. 64, 19 (1981). 68. Evans, A. G., J. Am. Ceram. Soc. 65, 497 (1982). 69. Lange, F., J. Am. Ceram. Soc. 72, 3-15 (1989). 70. Kendall, K., McNalford, N., and Birchall, J. D., Br. Ceram. Proc. 37, 255-265 (1986).

References

873

71. Cahn, R. W., Nature (London) 250, 702 (1974). 72. Atkinson, H. V., Acta Metall. 36(2), 469-491 (1988). 73. Brook, R. J., Treatise Mater. Sci. Technol. 9, 331-364 (1976). 74. Grahm, H. C., and Tallan, N. M., in "Physics of Electronic Ceramics A" (L. L. Hench and D. B. Dore, eds.), p. 491. Dekker, New York, 1971. 75. Jaffe, B., Cook, W. R., Jr., and Jaffe, H., "Piezoelectric Ceramics." Academic Press, New York, 1971. 76. Standley, K. J., "Oxide Magnetic Materials." Oxford Univ. Press, London and New York, 1972. 77. Perduijn, D. J., and Peloschek, H. P., Proc. Br. Ceram. Soc. 10, 263 (1968). 78. Burke, J. E., and Turnbull, D., Prog. Met. Phys. 3, 220 (1952). 79. Cahn, J. W., Acta MetaU. 10, 789 (1962). 80. Brook, R. J., Scr. Metall. 2, 375 (1968). 81. White, J., in "Sintering and Related Phenomena" (G. C. Kucyznski, ed.), pp. 81-108. Plenum, New York, 1973. 82. Scott, C., M. S. Thesis, Alfred University, Alfred, NY (1974). 83. Ashby, M. F., Scr. MetaU. 3, 843 (1969). 84. Woolfrey, J. L., Aust. A.E.C., Res. Estab. [Rep.] AAEC/E AAEC/E170, (1967). 85. Speight, M. V., and Greenwood, G. W., Philos. Mag. [8] 9, 683 (1964). 86. Nichols, F. A., J. Am. Ceram. Soc. 51, 468 (1968). 87. Hunderi, O., and Ryum, N., J. Mater. Sci. 15, 1104 (1980). 88. Feltham, P., Acta Metall. 5, 97 (1957). 89. Hillert, M., Acta Metall. 13, 227 (1965). 90. Louat, N. P., Acta MetaU. 22, 721 (1974). 91. Doherty, R. D., Metall. Trans. A 6A, 588 (1975). 92. Rhines, F. N., and Craig, K. R., Metall. Trans. A 6A, 590 (1975). 93. Hunderi, O., Acta Metall. 27, 167 (1979). 94. Rhines, F. N., and Craig, K. R., Metall. Trans. A 5A, 413 (1974). 95. Abbruzzese, G., and Lucke, K., Acta Metall. 34, 904 (1986). 96. Smith, C. S., "Metal Interfaces," p. 65. Am. Soc. Metals, Cleveland, OH, 1952. 97. Doherty, R. D., J. Mater. Educ. 6, 841 (1984). 98. Glazier, J. A., Gross, S. P., and Stavans, J., Phys. Rev. A 36(1), 306-312 (1987). 99. Rivier, N., Philos. Mag [Part B] B 52(3), 795-819 (1985). 100. Aboav, D. A., and Langdon, T. G., Metallorg 1, 333 (1969); 2, 171 (1969). 101. L. A. Xue, Ph.D. Thesis, University of Leeds, UK (1987). 102. Hennings, D. F. K., Ber. Dtsch. Keram. Ges. 55(7), 359 (1978). 103. Hennings, D. F. K., Sci. Ceram. 12, 405 (1983). 104. Hennings, D. F. K., Janssen, R., and Reynen, P. J. L., J. Am. Ceram. Soc. 70(1), 23 (1989). 105. Carry, C., and Mocellin, A., J. Am. Ceram. Soc. 69(9), c215 (1986). 106. Mostaghaci, H., and Brook, R. J., Trans. J. Br. Ceram. Soc. 4, 203 (1985). 107. Mostaghaci, H., and Brook, R. J., Trans. J. Br. Ceram. Soc. 80, 148 (1980). 108. Mostaghaci, H., and Brook, R. J., Trans. J. Br. Ceram. Soc. 82, 167 (1983). 109. H. L. Hsieh, Ph.D. Thesis, National Cheng Kung University, Tainan, Taiwan (1990) (in Chinese). 110. Geiger, G. H., and Poirier, D. R., "Transport Phenomena in Metallurgy," Chapter 8. Addison-Wesley, Reading, MA, 1973. 111. Richardson, D. W., "Modern Ceramic Engineering," p. 85. Dekker, New York, 1992. 112. Bunting, E. N., J. Res. Natl. Bur. Stand. (U.S.) 6(6), 948 (1931). 113. Doman, R. C., Barr, J. B., McNally, R. N., and Alper, A. M., J. Am. Ceram. Soc. 46(7), 314 (1963). 114. Savchenko, E. P., Godina, N. N., and Keler, E. K., in "Chemistry of High-Temperature Materials" (N. A. Toropov, ed.), p. 111, Consultants Bureau, New York, 1969.

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Chapter 16 Sintering

115. Adamson, A. W., "Physical Chemistry of Surfaces," 4th ed., p. 434. Wiley, New York, 1982. 116. German, R. M., "Liquid Phase Sintering," pp. 50-51. Plenum, New York, 1985. 117. Washburn, E. W., Phys. Rev. Scr. 2(17), 273 (1921). 118. Wray, P. J., Acta Metall. 24, 125-135 (1976). 119. Huppmann, W. J., and Reiger, R., Acta Metall. 23, 965-971 (1975). 120. Magee, B. E., and Lund, J., Z. Metallkd. 67, 596-602 (1976). 121. Pejovnik, S., Kolar, D., Huppmann, W. J., and Petzow, G., in "Sintering: New Developments" (M. Ristic, ed.), pp. 285-292. Elsevier, Amsterdam, 1979. 122. Kosmac, T., Kolar, D., Komac, M., Trontelj, M., and Brloznik, M., Sci. Sintering 11, 97-104 (1979). 123. Aksay, I., and Pask, J. A., Science 183, 69 (1974). 124. Savitskii, A. P., and Martsunova, L. S., Sov. Powder Metall. Met. Ceram. (Engl. Transl.) 16, 333-337 (1977). 125. Savitskii, A. P., and Burtsev, N. N., Soy. Powder Metall. Met. Ceram. (Engl. Transl.) 18, 96-102 (1979). 126. Savitskii, A. P., Kim, E. S., and Martsunova, L. S., Sov. Powder Metall. Met. Ceram. (Engl. Transl.) 19, 593-596 (1980). 127. Takajo, S., Kaysser, W. A., and Petzow, G., Acta Metall. 32, 107-113 (1984). 128. Kingery, W. D., J. Appl. Phys. 30, 301-306 (1959). 129. Carter, R. E., J. Chem. Phys. 34, 2010 (1961). 130. Carter, R. E., J. Chem. Phys. 35, 1137 (1961). 131. Munir, Z. A., Am. Ceram. Soc. Bull. 67 (2), 342-349 (1988). 132. Parr, N. L., Res. Appl. Ind. 13, 261-269 (1960). 133. Knudsen, M., "The Kinetic Theory of Gases." Methuen, London, 1934. 134. Levenspiel, O., "Chemical Reaction Engineering," 2nd ed., p. 470. Wiley, New York, 1972. 135. Greskovich, C., J. Am. Ceram. Soc. 64(12), 725-730 (1981). 136. Priest, H. F., Priest, G. L., and Gazza, G. E., J. Am. Ceram. Soc. 60, 81 (1977). 137. Anderson, H. U., J. Am. Ceram. Soc. 57(9), 34-37 (1974). 138. Popper, P., in "Special Ceramics" (P. Popper, ed.), pp. 209-219. Academic Press, New York, 1960. 139. Spriggs, R. M., and Dutta, S. K., Sci. Sintering 6, 1 (1974). 140. Larker, H. H., Adlerborn, J., and Bohman, H., SAE Tech. Pap. Ser. 770335 (1977). 141. McColm, I. J., and Clark, N. J., "The Forming, Shaping and Working of High Performance Ceramics," p. 23. Blackie, London, 1988.

17

Finishing

17.1 O B J E C T I V E S This chapter is a very brief and qualitative review of the activities associated with finishing the ceramic after it has been sintered. The finishing processes discussed in this chapter include (1) grinding and polishing to give final dimensional tolerances and surface finish; (2) nondestructive evaluation of the final piece; (3) coating, glazing, and sealing the ceramic; and (4) final property measurements including density, electrical properties, and mechanical properties. These final property measurements are quality control inspections or proof tests. Such quality control inspections are part of a total quality assurance program for the production process, which also contains raw material and process specifications. The nature of the quality assurance program is also discussed.

17.2 I N T R O D U C T I O N After sintering, we have a dense ceramic piece. It consists ofpolycrystalline array of grains with mostly random orientation. At this stage, 875

876

Chapter 17 Finishing

ceramic powder processing is over for the most part. Because there is usually some degree of warping during sintering, the ceramic must be machined to the final dimensions desired within specified tolerances. This machining can cause surface flaws which will weaken the ceramic and effect other properties. To counteract these surface flaws, coatings and glazes are applied. These glazes also hermetically seal the ceramic. Glazes have been used through the ages as decoration, but healing surface flaws and sealing the ceramic are their most important properties. This final ceramic shape has certain properties which must be tested to verify that it will meet product specifications. Property measurements might include density, Vicker's hardness, wear resistance for abrasive materials, and various optical, magnetic, and electrical property measurements. Once tested, the ceramic is ready for shipping to the customer. This completes the treatment of ceramic powder processing. The final sections of this chapter is devoted to a short description of a quality assurance program for ceramic powder processing.

17.3 C E R A M I C M A C H I N I N G Sintered pieces can warp during densification. As a result they must be shaped to give the desired dimensional tolerances to the final ceramic piece. This shaping process is typically performed by grinding and polishing. Grinding uses an abrasive encrusted tool such as a grinding wheel. Polishing uses a free abrasive acting between a moving surface and the fixed ceramic surface. Two examples of polishing include (1) lapping, where an abrasive suspension is used between a rotating aluminum metal plate and a fixed ceramic piece, and (2) "sand blasting," where abrasive grit is directed at a ceramic surface by a high speed air jet. Because sintered ceramics are very hard, diamond encrusted tools and diamond grit are often used. These grinding and polishing tools are expensive, and for that reason, machining to near net shape at the green stage is desirable so that a large amount of the machining is not necessary after the ceramic has been sintered. Grinding wheels come in a number of configurations and compositions. Coarse abrasives are used for rough machining, where rapid stock removal is desired. Fine abrasives are used for final machining, where close tolerances and smooth surface finishes are required. Typically, a series of abrasive steps are used, decreasing the size of the abrasive as one approaches the final tolerance of the piece. This procedure removes the surface flaws created by the large-sized abrasives and improves the strength of the ceramic, as will be discussed later. Other types of machining include 1. Chemical machining, such as the hydrofluoric acid, HF, attack of silica glass, and the molten sodium borate, Na2B4OT, attack of alumina,

17.3 Ceramic Machining

8 77

FIGURE 17.1 Schematicof material deformation caused by grinding with a single

abrasive particle.

2. Photoetching [1], 3. Electrical discharge machining [2,3], 4. Laser machining [4-6].

17.3.1 Effect of Machining on Ceramic Strength To understand the effects of machining on the strength of a ceramic material, we must examine the interactions that occur at the toolwookpiece interface. The abrasive particles can plow a series of grooves into the surface of the ceramic. The ceramic in the region of the grooves is open to very high stress, temperature, and is often broken and deformed. Material adjacent to the abrasive particle is placed in compression and deforms plastically. After the abrasive particle passes, the material rebounds and either cracks or spawls off due to the resulting tensile stress. Thus, the size of the machining groove is larger than the size of the abrasive particle (or tool) used but within the same order of magnitude. Figure 17.1 is a schematic of the type of subsurface plastic deformation and cracks that form during grinding and machining. Median cracks are parallel to the direction of abrasion and perpendicular to the surface of the ceramic piece. These result from the high stress at the bottom of the tool groove. Because they are parallel to the direction of the groove, they have also been called l o n g i t u d i n a l cracks. These cracks are usually the deepest and produce the greatest strength reduction in ceramics. Cracks lateral and parallel to the surface extend away from the plastic zone. They result from high tensile stress at the

878

Chapter 17 Finishing

groove edge as the abrasive particle passes. Lateral cracks tend to curve toward the surface and often result in a chip being spawled off. This spawling accounts for a substantial portion of the stock removal during grinding. Lateral cracks are parallel to the surface, and they do not lead to stress concentration during subsequent mechanical loading. As a result, they do not significantly reduce the strength of the material. Radial cracks normally result from single particle impact or indentation and extend radially from the point of impact (as in the Vicker's hardness testing). They are perpendicular to the surface but they are usually shallow and do not degrade the strength of the ceramic as much as the median cracks. The cracks are shown in Figure 17.1 which are perpendicular to the abrasive groove are analogous to radial cracks. There are many of these cracks along the length of the abrasion groove. They had been referred to as transverse, chatter, or crescent cracks but their mechanisms of formations are generally not discussed. They are probably initiated by the high tensile stress that arises at the trailing edge of the contact of the abrasive particle and the workpiece. Such a biaxial stress mechanism with high friction is discussed in Richardson's book [7]. The effects of machining on other properties has been discussed by Stokes [8]. He found that the shape of the hysteresis loop in magnetic ferrites is altered by near-surface stresses resulting from machining. Polishing, annealing, and etching are routinely used to relieve nearsurface stress and obtain reproducible hysteresis loops. Ferro-electric, dielectric, and pyro-electric materials are also affected by machining and surface condition.

17.3.2 Effect of Grinding Direction on Ceramic Strength Most ceramics are machined with tools containing abrasive particles rather than just a single one. However, it is likely that the resulting surface flaws are the same as discussed previously with median and radial cracks concentrating stress during loading of the ceramic, thus controlling the strength. The surface flaw that will control the strength depends on the relative orientation of the grinding grooves to the direction in which the stress will be applied. This is shown schematically in Figure 17.2 for a specimen loaded in a bending mode. As a load is applied this specimen begins to bend. Stress concentration will occur at the tips of the cracks perpendicular to the stress axis but not at the cracks parallel to this stress axis. So, for specimens ground in a longitudinal direction, stress concentration will occur at the radial transverse cracks, leading to weakness. For specimens ground in the transverse direction, stress concentration will occur at the median or

17.3 Ceramic Machining

879

FIGURE 17.2

Grinding direction and the effect on ceramic strength. Refer to Figure 17.1 for crack nomenclature and orientation.

longitude cracks, leading to weakness. Because the median cracks are usually the largest ones caused by grinding, it is expected that the strength is the lowest for the transversely ground samples where the grooves and the median cracks are perpendicular to the tensile stress axis, as seen in Table 17.1. From this table, we find that there is a substantial reduction in the ceramic strength due to the direction used in grinding of its surface. This anisotropy of strength is an important consideration for an engineer designing a component that must withstand high stress in operation.

17.3.3 Effect o f Ceramic Microstructure on Strength The ceramic microstructure has a pronounced effect on the rate of machining and on the residual strength after machining. Rice [9] has shown that fine-grained ceramics require high grinding force and longer time to cut or machine. The sawing rate or the rate of wear (cm2/sec)

880

Chapter 17 Finishing

TABLE 17.1

Tensile Strength versus Grinding Direction a

Material

Longitudinal grinding (MPa )

Transverse grinding (MPa )

Hot pressed silicon nitride b Soda lime glass Mullite

669 97 319

428 68 259

MgF2 B4C

87 374

53 154

a Rice, R. W., and Mecholsky, J. J., NBS Spec. Publ. (U.S.) 562, 351-378 (1979). b Richerson, D. W., Schuldies, J. J., Yonushonis, T. M., and Johansen, K. M., in "Ceramics for High Performance Applications, II" (J. J. Buirke, E. N. Lenoe, and R. N. Katz, eds.), pp. 625-650. Brook Hill, Chestnut Hill, MA, 1978.

was found to be proportional to R~ 1/2, where Rg is the grain size of the ceramic. Porosity, distributed uniformly, increases the rate of machining but also decreases the smoothness of the surface finish that can be achieved. The strength reduction resulting from machining depends on a comparison of the size of the flaws initially present in the ceramic to those produced by the machining. Machining has very little effect on the strength of either high-porosity ceramics or large-grained ceramics because the flaws induced during machining are no longer than the microstructural flaws already present within the ceramic. On the contrary, the strength of fine-grain ceramics can be reduced significantly by surface flaws.

17.3.4 Grinding and Machining Parameters This section draws heavily on Chapter 12 of the book Modern Ceramic Engineering by Richardson [7]. The parameters such as sawing rate and force, selected for machining a ceramic, have a large effect on the rate of machining and the rate of tool wear, as well as the resulting properties of strength of the ceramic. The general trends associated with machining parameters are the sawing rate and the grain size and the static brittleness index of the ceramic, BI. We already have seen that the sawing rate or the rate of wear (cm2/sec) is proportional to R~ 1/2, where Rg is the grain size of the ceramic. The proportionality constant is a function of the static brittle index, which is given by the Vicker's hardness, Hv(GPa) divided by the value of the toughness, Klc (MPA m 1/2) [i.e., BI(m -1/2) = HvIKlc] [10]. The toughness is an integral of the stress curve for a ceramic under increasing amounts of strain from 0 to its point of rupture. Vicker's hardness testing is done by indentation using a diamond point. Both the hardness and the tough-

17.3 Ceramic Machining

881

ness can be determined by the size of the indent and the length of the cracks caused by various forces applied to the diamond point. Returning to the brittleness index, the larger is the value of BI, the lower is the energy required to grind. Typical values of BI are BI(A11203) [10] 3000 m -1/2, BI(MgO) [10] ~ 8000 m -1/2, BI(tetragonal - ZrO2)* ~ 2400 m

-1/2

.

As important as the absolute values of K~cand Hv are in determining the erosive wear of a ceramic surface, the parameter obtained by squaring the reciprocal of the brittleness index, B1-2 = (K~c/Hv)2(m) = CID, is more important because it is the critical indentation depth (CID) associated with wear, given a certain force and cutting speed. There are two types of wear mechanisms: plastic deformation wear and fracture wear. These two mechanisms are delineated by wear depths greater or less t h a n the CID. Wear depths greater t h a n the CID are associated with fracture wear, and those less t h a n the CID are associated with plastic deformation wear. When the abrasive particle size is small and applied loads are small, plastic deformation wear is favored over fracture wear. For ceramic surfaces, fracture wear removes material much faster by at least an order of magnitude t h a n plastic wear. Particle sizes and grinding force (and abrasive jet pressures) are chosen from a knowledge of the ceramic, that is, its K~c, Hv values and the CID, so that ablation goes beyond the critical indentation depth. This speeds surface erosion, making the operation much more economical. The erosive wear of a surface is also used to test the adherence of a coatings and films to the surface. Large abrasive particles result in greater depth of grinding damage and are used for rough operations for that reason. Finer abrasives are used to remove the subsurface damage produced by the larger abrasives during rough grinding and to achieve the final tolerances and strength associated with the surface finish and strengths desired. Different surface finishes are then obtained by using finer and finer diamond grits in an abrasive free machining, also called lapping. This results in less damage to the ceramic surface. Table 17.2 shows the various strengths that can be achieved with different surface finishes on single-crystal A1203. In addition to an increase in strength due to polishing by centerless grinding, further and drastic increases in strength can be obtained by annealing in 02, chemical polishing, and flame polishing. Hard ceramic materials are more difficult to cut and grind t h a n soft materials. Hard ceramics require a higher cutting or grinding force [12]. This causes increased grinding wheel wear, higher surface temperatures, and greater damage to the ceramic.

* Values of H e a n d Klc t a k e n from K i n g e r y et al. [11, p. 777].

882 TABLE 17.2

Chapter 17 Finishing Surface Preparation vs. Tensile Strength for Single-Crystal A1203

Surface preparation

Tensile strength (MPa)

As machined Polished by centerless grinding Annealed in oxygen after polishing by centerless grinding Chemically polished with molten borax Flame polished Pristine whiskers

440 590 1040 6860 7350 15900

Data taken from Stokes [8].

17.4 C O A T I N G A N D G L A Z I N G The damaged surface of a ceramic after finishing is sometimes healed by coating. Coating methods include (1) sol-gel, (2) laser glazing, (3) chemical vapor deposition (CVD), (4) sputtering, (5) plasma spraying, (6) flame spraying, and (7) glazing. In sol-gel methods, a sol is prepared (see Chapter 8, Sol-Gel Ceramic Synthesis). The sol is used to coat the ceramic surface. During the drying of the sol, a gel is formed. This gel is further dried and sintering into a film. In laser glazing, the laser is used to melt the ceramic surface. After the laser beam passes, the surface cools and solidifies. A modification of this technique sprays another ceramic powder into the laser beam in the proximity of the surface, melting it along with the ceramic surface, resulting in a solidified surface which is a mixture of the original ceramic and that of the added powder. CVD and sputtering are physical deposition methods. In CVD, metal-organic vapors are caused to thermally decompose at the surface of the heated ceramic. Controlling the partial pressure of various metal-organics and oxygen as well as the ceramic substrate temperature, a film with a particular stociometry can be produced. In sputtering, ions (typically argon) are caused to bombard a target made of the coating material. The argon ions cause the ejection of atoms of the target material, which are collected by ceramic surface placed in close proximity to the target. The ejected ions are neutralized and condense on the ceramic surface, forming a film. With both CVD and sputtering, the thickness growth rate is usually limited to several atomic layers per minute. With plasma and flame spraying, a powder is heated by one of these two heat sources and caused to attach to the surface of the ceramic piece, giving coating rates on the order of microns per minute. At the ceramic surface, the liquidified powder cools and solidifies, forming a coating. With glazing, a ceramic powder suspension or slip is used to coat the ceramic surface. The suspension is dried, the binder burned out and then sintered, often using either liquid phase

17.5 Quality Assurance Testing

883

sintering or viscous sintering of glassy phases to form a coating. Glazing is used to either hermetically seal the ceramic after manufacture to prevent its degradation (either strength or environmental attack) during use or to seal in decorations painted with various metal oxides, which give different colors on the surface of the ceramic piece prior to glazing. This is called underglaze decoration. Several glaze formulations with different metal oxides, which give different colors, can be used to paint decorations onto the surface of the ceramic piece. This technique is used for the multicolored dinnerware that has been commercially available since the 1700s. After coating the ceramic is ready for market. In some cases, customers require certification testing done before shipment. Certification testing is one way of assuring quality.

17.5 Q U A LI T Y A S S U R A N C E TESTING Quality assurance is a multistepped process required at each stage of ceramic processing (for an overview, see Richardson [7, Chapter 13]). The degree of quality assurance is determined by the criticality of the application. This might be the utmost of in-process quality assurance for an electrical component or a low-level quality assurance for bricks or floor tiles. Most applications require meeting specifications, which are written manufacturing procedures and, in addition, one or more certification tests to assure that the manufacturing procedure has been followed and the specifications has been met. More critical and demanding applications may require destructive proof testing and nondestructive inspection, NDI, to be discussed later. In-process quality assurance has two major objectives: (1) to maintain the process to be under control with the aim of achieving maximum yield and quality and (2) to reject unacceptable components and accept good components. The rejection criteria result from experience that is held jointly between the manufacturer and the end user. Therefore, the end user is usually involved in writing the specifications associated with the certification and fabrication quality assurance of a particular ceramic. The fundamental step in quality assurance is the preparation of a formal, written manufacturing procedure. All of the critical aspects of a ceramic manufacturing procedure including certification tests are described in this document. In addition, it also specifies the paperwork that must accompany the part through the process, such as, lot number for raw materials and processing conditions and the date(s) of production, list of signatures verifying that operations have been followed as specified. This paperwork is kept on file throughout the service life of that part to serve as a record of its fabrication.

884

Chapter 17 Finishing

The quality assurance process starts with the procurement of raw materials. It may consist of simply checking the chemical analyses, specific surface area, and particle size distribution submitted by the supplier to ensure that they are within specification or it may involve additional analyses within the plant to check the supplier's analyses and further test the raw materials. These extra analyses are performed when it has been found that a particular raw material property is critical to the quality of the ceramic processing. An example of extra analyses often performed on ceramic powders is zeta potential analyses to test the presence of surface contaminants not easily detected by bulk chemical analysis. To achieve the desired characteristics of the final product, for example, the optical transparency of an infrared radome for a missile, a great deal of care goes into the raw material testing for the chemical purity and particle size aspects which are extremely important to the optical transparency. Once within the ceramic process quality assurance again consists of chemical analyses of various other raw materials that go into the process; for example, the dopant chemicals (either in soluble form or as secondary powders), the polymers used as dispersants and binders, and the solvents. In addition, all the process conditions should be followed to their specifications; for example, mixing time and mixing temperature; molding time and temperature; drying conditions; the atmosphere, time, and temperature for binder burnout, and the atmosphere, time, and temperature for sintering. The nature of process quality assurance changes with the shape forming and densification steps of the manufacturing process. Much less interest is now centered on the bulk chemistry and much more interest on the dimensions of the piece and its electrical properties, as well as rejection-causing defects emerging during the manufacture. These defects~cracks, pores, inclusions, laminations, and knit l i n e s ~ can sometimes be detected by visual inspection but more frequently require using an optical microscope for surface defects or nondestructive investigation for interior defects.

17.5.1 Proof Testing The idea behind proof testing is to locate the ceramic pieces that have a value of a critical property, such as strength, dielectric strength, or pyro-electric figure of merit, below specification. This is done by testing all of the ceramic pieces for values of this property. The testing gives rise to a distribution of property values as shown in Figure 17.3. The figure shows the typical distribution of flaws within the ceramic which gives rise to a broad distribution of properties. Parts are rejected which have values of the property below a certain value--the test value.

17.5 Quality Assurance Testing

885

FIGURE 17.3 Proof testing of a ceramic to be used in an application for which its properties are critical. The initial distribution of the measured property is given by the curve. After proof testing at a value of the property slightly above the value expected during use, the acceptable pieces correspond to those in the truncated distribution.

The test value is typically larger than required of the ceramic in typical service. Those parts which are accepted correspond to a truncated fraction of the initial distribution of ceramic parts. Many different types of tests can be used for proof testing: density, Vicker's hardness, wear resistance for abrasive materials, and various optical, magnetic, and electrical property measurements. Density is one of the most frequent tests performed because it is performed nondestructively. Comparing the density of the piece of its theoretical density gives the volume fraction occupied by the solid in the ceramic piece. If there are any sufficiently large flows or residual pores within the ceramic, then the solid volume fraction will be less than 1.0. Vicker's hardness testing results in a damage at least to the surface of the ceramic piece. Sometimes the ceramic piece can be salvaged if the damaged surface is ground off before it is sold. Mechanical properties of ceramics are not usually very easy to qualify by proof testing. The final product is usually not of a configuration readily tested by mechanical testing machines. The options are to process appropriate mechanical property test shapes at the same time as the product or to cut test bars out of random samples of the product. Proof testing is then conducted on those bars. The proof testing of mechanical properties is again somewhat difficult because a ceramic piece after testing may contain cracks which have been induced by the testing procedure, and these cracks may not be significant enough to cause the piece to fail at low temperatures but sufficient to cause

886

Chapter 17 Finishing

failure at high temperatures in service. As a result, the piece will fail immediately, once stressed at these high temperatures. For this reason, mechanical property testing is a delicate and somewhat difficult process to develop for ceramic pieces.

17.6 N O N D E S T R U C T I V E

TESTING

In addition to the strength optical, magnetic, and electrical properties of the ceramic piece, the density can also be used to detect large pores and inclusions present after sintering (for an overview, see Richardson [7, Chapter 13]). For this reason, the density is probably the single most important factor used in testing a sintered ceramic piece. The density does not, however, give a complete picture. Small cracks and other defects like knit lines can be missed by density measurements, and for this reason, nondestructive investigations are used. One of the classical nondestructive techniques is radiology, where the sample is examined by an X-ray beam and a photograph of the interior of the ceramic piece taken. Such a procedure is shown in Figure 17.4, where the radiograph detects two types of flaws: one a high-density inclusion and the other a pore. The size of the defect that can be detected by X-ray depends on a combination of factors: the thickness of the part and its X-ray adsorption, the size of the flaw compared to the thickness of the part, the difference in the X-ray absorption between the flaw and the part, and the orientation of the flaw. Flaws present at levels of about 1-2% of the thickness of the ceramic piece can be detected. Because flaws can be missed based on their small dimension in one orientation, typically several pictures of the piece are taken at different angles, thus allowing cracks in one dimension to be seen where they are too small to be seen if photographed in another direction. In addition to these nondestructive investigation techniques, the properties of the ceramic piece can be determined by direct measurement. These properties include strength, electrical, magnetic, and optical properties. Sometimes this is done by random sampling of pieces, and in other cases it is done by proof testing of each piece. Proof testing of each piece is expensive but necessary if the piece is a critical component or used in a critical application. There are other methods of NDI other than X-ray radiography. One of them is ultrasonic NDI. In this technique, ultrasonic waves are used to investigate the interior of a ceramic piece. An ultrasonic transducer placed on the surface of the ceramic piece sends the ultrasonic waves received by a sensor found either within the ultrasonic transducer or at another location on the surface of the ceramic piece. Ultrasonic waves are scattered and reflected back from the surfaces and defects within

17.6 Nondestructive Testing

887

b) Top view of film after development Radiology schematic (a) and the resulting image (b) of a sintered sample with both low and high density defects. Figure 17.4(b) reprinted from Richardson [7, p. 627] by Marcal Dekker, Inc..

F I G U R E 17.4

the piece. These scattered ultrasonic waves are detected by an array of detectors and analyzed for the size, shape, and location of defects within the ceramic piece, using time of flight analysis for various characteristics of the scattered waves very much the same way the principles of sonar and radar are applied. This procedure is identical to the technique developed to study the internal organs of the h u m a n body using ultrasonic imaging. This ultrasonic testing method can be used on macroscopically large pieces as well as on microscopically small ceramics, such as ceramic thin films, using a new technique called ultrasonic microscopy, where defects micrometers in size can be detected. In addition to ultrasonics, high-frequency ultrasonics and microwaves can be used as well as laser and acoustic holographic techniques. With holographic techniques, the ceramic piece is imaged by coherent light or sound waves before and after it is stressed slightly. The holo-

888

Chapter 17 Finishing

graphic picture superimposes these two images. A interferometric fringe pattern is obtained which shows where the ceramic piece deforms very slightly, typically on the order of the wavelength of the light (or sound) used for illumination. These holographic techniques can point out regions where strength is not uniform, suggesting a defect in that location. The nonuniform deformation results from the concentration of stress at the flaw. The size of a subsurface flaw can be estimated by the size of the nonuniform fringe pattern. Microwaves are particularly useful for NDI because many ceramics are transparent to microwaves, allowing flaws to be detected. Metallic Si flaws 25 t~m in diameter within a reaction bonded Si3N4 piece can be detected with 91-98 GHz microwaves using a cross-polarized transmission techniques [13]. In addition, a very simple and low cost method can be used to detect surface flaws. This is the use of penetrants that are typically fluorescent dyes. Usually, a three-step procedure is used. The ceramic part is first soaked in a fluorescent dye. Then the part is dried and cleaned in a very controlled manner to remove the dye from smooth surfaces but not from the surface defects. When the part is examined under ultraviolet light, the surface defects such as cracks and porosity retain the dye and show up brilliantly. This method is used widely for surface inspection of ceramics and is frequently included as part of a quality assurance certification. Penetrants are effective for nonporous ceramics. With open porosity, the penetrant will enter all the pores of the ceramic, giving fluorescence to the whole ceramic piece, thus preventing detection of surface flaws. Not all penetrants are fluorescent dyes. Radioactive krypton can be used as a penetrant. It is retained in cracks or other defects and can be detected by either a Geiger counter or by carefully wrapping the ceramic piece in photographic film. After development of the film, the location of cracks and pores can be detected.

17. 7 S U M M A R Y This concluding chapter gives information on finishing, quality assurance, proof testing, and nondestructive investigations. Finishing a ceramic piece to the final size and surface roughness tolerances desired for a particular application have been discussed. The surface finish is critical to strength because grinding and machining create surface flaws which degrade its strength. Similar problems associated with surface defects degrade other material properties. Surface coatings are used to evade the strength degradation of machining induced surface flaws.

References

889

Quality assurance is practiced at all steps of ceramic manufacture. All raw materials and process conditions are subject to scrutiny. Certification tests are also performed on the product. Proof testing assures that the ceramic part will give properties above those desired for a particular application. Proof testing is performed only when the ceramic part is used for a critical application. Finally, nondestructive investigations of ceramic materials were discussed in very general terms. With this chapter the reader is ready to apply his or her knowledge of all ceramic processing steps discussed in previous chapters to develop the quality assurance specifications necessary for a ceramic part and its fabrication process. Only with care for each raw material's quality and consistency of processing can high-quality ceramics be produced that will meet product specifications on a routine basis.

References 1. Stookey, S. D., Ind. Eng. Chem. 45, 115-118 (1953). 2. Lee, D. W., and Geick, G., NBS Spec. Publ. (U.S.) 348, 197-211 (1972). 3. Petrofes, N. F., and Gadalla, A. M., Am. Ceram. Soc. Bull. 67(6), 1048-1052 (1988). 4. Lumley, R. M., Am. Ceram. Soc. Bull. 48(9), 850-854 (1969). 5. Brody, C. J., and Molines, J. L., in "Ceramics Applications in Manufacturing" (D. W. Richardson, ed.), pp. 142-150. Soc. Manuf. Eng., Dearborn, MI, 1988. 6. Copley, S. M., Bass, M., and Wallace, R. G., NBS Spec. Pub[. (U.S.) 562, 283-292 (1979). 7. Richardson, D. W., "Modern Ceramic Engineering," Chapter 8. Dekker, New York, 1992. 8. Stokes, R. J., NBS Spec. Publ. (U.S.) 348, 348-352 (1972). 9. Rice, R. W., in "Ceramics for High Performance Applications" (J. J. Burke, A. E. Gorum, and R. N. Katz, eds.), pp. 287-343. Brook Hill, Chestnut Hill, MA, 1974. 10. McColm, I. J., and Clark, N. J., "Forming, Shaping and Working of High Performance Ceramics," p. 105. Blackie, London, 1988. 11. Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., "Introduction to Ceramics," 2nd ed. Wiley (Interscience), New York, 1976. 12. Subramanian, K., Ramanath, S., and Matsuda, Y., Ceram. Ind. (Chicago) 30-32 (1990). 13. Bahr, A. J., in "Proceedings of the DARPA/AFML Review of Progress in Quantitative NDE, AFML-TR-78-25," pp. 236-241. 1979.

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Appendix A Ceramic Properties

891

0o

P r o p e r t i e s of S e l e c t e d C e r a m i c M a t e r i a l s

Material A l u m i n a (Al203) Magnesia (MgO) Spinel (MgAl204) Zirconia c (ZrO2) Zirconiaf (ZrO2) Fused silica (SiO2) Soda lime glass Borosilicate glass Silicon carbide (SIC) Silicon nitride (Si3N4) T i t a n i u m carbide (TIC) Cordierite (2MgO 92Al203 9 5SIO2) Mullite (3Al203 92SIO2) SiMon (SiAION) Porcelain A l u m i n u m Nitride (AlN) Boron Nitride (BN) Beryllia (BeO)

Density (g/cm 3)

Modulus of elasticity (GPA )

3.97 3.58 3.55 5.56 5.91 2.2 2.5 2.23 3.22 3.44 4.92 2.50

393 207 284 152 205 75 69 62 414 304 462 118

3.26 3.0 2.45 3.26 2.27 3.01

144 231 94 331 83 350

Poisson 's ratio

Hardness Klc (Knoop ) (MN/m sly) (kg/mm 2)

0.27 0.36

4.5

0.32

1.1 8.4

0.16 0.23 0.20 0.19 0.24

4.4 5.6

Modulus of rupture (MPa )

Electrical resistivity (12- m)

Thermal conductivity (W/m - K)

Heat capacity Cp (Cal/gmPC)

30 48 15.0" 2.0 2.0 1.3 1.7 1.4 9O 16-33" 17.2 4.18

0.2 0.25 0.25 0.18 0.12 0.2

4.18

0.15

1.67 117 55 251

0.28

2100 370 1600 1200

275-550 105 b 83-220 138-240

>1012 >1012

500 550

110 69 69 450-520 414-580 275-450

>1018 >1010 --1013 --10-2 >1012 --10-6

2500 2200 2600

68 2.2 1.6 1200 205 1200

Data are for fully dense m a t e r i a l s and at room temperature unless noted otherwise. Mean value t a k e n over the t e m p e r a t u r e range 0-1000~ b Sintered and containing approximately 5% porosity. c Stabilized with CaO. d Mean value t a k e n over the t e m p e r a t u r e range 0-3000~ Sublimes. f Stabilized with Y203.

100 276 230

--1012 ---1011 ~1012

0.23 0.20 0.17 0.17

Coefficient of thermal expansion [(~ -1 • 106]

Melting temperature or range

8.8 a 13.5 a 7.6 a 10.0" 10.5 0.5 a 9.0 d 3.3 d 4.7 3.6 a 7.4 2.0

2050 2850 2135 2500-2600 2600 1610

5.13 3.0 6.5-7.4 4.1 1.2 8.1

[~

2300-2500 e -1900 e 3160 1450 1810 1100 2000 e 3000 e 2530

Appendix B G a m m a Function

G a m m a Function: Values of F(n)

= f ~ e - Z x n-1

dx; F(n + 1)

= nF(n) a

n

F(n)

n

F(n)

n

F(n)

n

F(n)

1.00 1.01 1.02 1.03 1.04

1.00000 0.99433 0.98884 0.983655 0.97844

1.25 1.26 1.27 1.28 1.29

0.90640 0.90440 0.90250 0.90072 0.89904

1.50 1.51 1.52 1.53 1.54

0.88623 0.88659 0.88704 0.88757 0.88818

1.75 1.76 1.77 1.78 1.79

0.91906 0.92137 0.92376 0.92623 0.92877

1.05 1.06 1.07 1.08 1.09

0.97350 0.96874 0.96415 0.95973 0.95546

1.30 1.31 1.32 1.33 1.34

0.89747 0.89600 0.89464 0.89338 0.89222

1.55 1.56 1.57 1.58 1.59

0.88887 0.88964 0.89049 0.89142 0.89243

1.80 1.81 1.82 1.83 1.84

0.93138 0.93408 0.93685 0.93969 0.94261

1.10 1.11 1.12 1.13 1.14

0.95135 0.94739 0.94359 0.93993 0.93642

1.35 1.36 1.37 1.38 1.39

0.89115 0.89018 0.88931 0.88854 0.88785

1.60 1.61 1.62 1.63 1.64

0.89352 0.89468 0.89592 0.89724 0.89864

1.85 1.86 1.87 1.88 1.89

0.94561 0.94869 0.95184 0.95507 0.95838

1.15 1.16 1.17 1.18 1.19

0.93304 0.92980 0.92670 0.92373 0.92088

1.40 1.41 1.42 1.43 1.44

0.88726 0.88676 0.88636 0.88604 0.88580

1.65 1.66 1.67 1.68 1.69

0.90012 0.90167 0.90330 0.90500 0.90678

1.90 1.91 1.92 1.93 1.94

0.96177 0.96523 0.96878 0.97240 0.97610

1.20 1.21 1.22 1.23 1.24

0.91817 0.91558 0.91311 0.91075 0.90852

1.45 1.46 1.47 1.48 1.49

0.88565 0.88560 0.88563 0.88575 0.88595

1.70 1.71 1.72 1.73 1.74

0.90864 0.91057 0.91258 0.91466 0.91683

1.95 1.96 1.97 1.98 1.99 2.0O

0.97988 0.98374 0.98768 0.99171 0.99581 1.00000

a For large positive values of x, F(x) a p p r o x i m a t e s the asymptotic series

1 xZe -z

1+ ~

1 + 288x 2

] 51,840x2 - 2,488,320x 4 + ..- .

893

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Appendix C Normal Probability Function

Normal Probability Function f (x ) = - ~

1

exp [ - lx~ ]

F(x)=

=-~exp

-

s 2 ds

F ( - x ) = 1 - F(x)

f l - x ) = f(x) x

f(x)

F(x)

1 - F(x)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22

0.3989 0.3989 0.3989 0.3988 0.3986 0.3984 0.3982 0.3980 0.3977 0.3973 0.3970 0.3965 0.3961 0.3956 0.3951 0.3945 0.3939 0.3932 0.3925 0.3918 0.3910 0.3902 0.3894

0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5754 0.5793 0.5832 0.5871

0.5000 0.4960 0.4920 0.4880 0.4841 0.4801 0.4761 0.4721 0.4681 0.4641 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.4207 0.4168 0.4129

x

0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45

f(x)

0.3885 0.3876 0.3867 0.3857 0.3847 0.3836 0.3825 0.3814 0.3802 0.3790 0.3778 0.3765 0.3752 0.3739 0.3726 0.3712 0.3697 0.3683 0.3668 0.3653 0.3637 0.3621 0.3605

F(x)

0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736

1 - F(x)

0.4091 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264

895

896

Appendix C

f(x) 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95

0.3589 0.3572 0.3555 0.3538 0.3521 0.3503 0.3485 0.3467 0.3448 0.3429 0.3411 0.3391 0.3372 0.3352 0.3332 0.3312 0.3292 0.3271 0.3251 0.3230 0.3209 0.3187 0.3166 0.3144 0.3123 0.3101 0.3079 0.3056 0.3034 0.3011 0.2989 0.2966 0.2943 0.2920 0.2897 0.2874 0.2850 0.2827 0.2803 0.2780 0.2756 0.2732 0.2709 0.2685 0.2661 0.2637 0.2613 0.2589 0.2565 0.2541

F(x)

0.6772 0.6808 0.6844 0.6879 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.7258 0.7291 0.7324 0..7357 0.7389 0.7422 0.7454 0.7486 0.7518 0.7549 0.7580 0.7612 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.7881 0.7910 0.7939 0.7967 0.7996 0.8023 0.8051 0.8079 0.8106 0.8133 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289

1 - F(x)

0.3228 0.3192 0.3156 0.3121 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 0.2743 0.2709 0.2676 0.2644 0.2611 0.2579 0.2546 0.2514 0.2483 0.2451 0.2420 0.2389 0.2358 0.2327 0.2297 0.2266 0.2236 0.2207 0.2177 0.2148 0.2119 0.2090 0.2061 0.2033 0.2005 0.1997 0.1949 0.1922 0.1894 0.1867 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711

x

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 t:3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45

f(x)

0.2516 0.2492 0.2468 0.2444 0.2420 0.2396 0.2371 0.2347 0.2323 0.2299 0.2275 0.2251 0.2227 0.2203 0.2179 0.2155 0.2131 0.2107 0.2083 0.2059 0.2036 0.2012 0.1989 0.1965 0.1942 0.1919 0.1895 0.1872 0.1849 0.1827 0.1804 0.1781 0.1759 0.1736 0.1714 0.1692 0.1669 0.1647 0.1626 0.1604 0.1582 0.1561 0.1540 0.1518 0.1497 0.1476 0.1456 0.1435 0.1415 0.1394

F(x)

0.8315 0.8340 0.8365 0.8389 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265

1 - F(x)

0.1685 0.1660 0.1635 0.1611 0.1587 0.1563 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 0.1151 0.1131 0.1112 0.1094 0.1075 0.1057 0.1038 0.1020 0.1003 0.0985 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0..0869 0.0853 0.0838 0.0823 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735

Appendix C

1.46 1.47 1.48 1.49 1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.9 1.91 1.92 1.93 1.94 1.95

f(x)

F(x)

1 - F(x)

0.1374 0.1354 0.1334 0.1315 0.1295 0.1276 0.1257 0.1238 0.1219 0.1200 0.1182 0.1163 0.1145 0.1127 0.1109 0.1092 0.1074 0.1057 0.1040 0.1023 0.1006 0.0989 0.0973 0.0957 0.0940 0.0925 0.0909 0.0893 0.0878 0.0863 0.0848 0.0833 0.0818 0.0804 0.0790 0.0775 0.0761 0.0748 0.0734 0.0721 0.0707 0.0694 0.0681 0.0669 0.0656 0.0644 0.0632 0.0620 0.0608 0.0596

0.9279 0.9292 0.9306 0.9319 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9430 0.9441 0.9452 0.9463 0.9474 0.9485 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9700 0.9706 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744

0.0722 0.0708 0.0694 0.0681 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 0.0359 0.0352 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256

x

1.96 1.97 1.98 1.99 2 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.1 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.2 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.3 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.4 2.41 2.42 2.43 2.44 2.45

f(x)

0.0584 0.0573 0.0562 0.0551 0.0540 0.0529 0.0519 0.0508 0.0498 0.0488 0.0478 0.0468 0.0459 0.0449 0.0440 0.0431 0.0422 0.0413 0.0404 0.0396 0.0387 0.0379 0.0371 0.0363 0.0355 0.0347 0.0339 0.0332 0.0325 0.0317 0.0310 0.0303 0.0297 0.0290 0.0283 0.0277 0.0270 0.0264 0.0258 0.0252 0.0246 0.0241 0.0235 0.0229 0.0224 0.0219 0.0213 0.0208 0.0203 0.0198

F(x)

0.9750 0.9756 0.9762 0.9767 0.9773 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 0.9861 0.9865 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929

897

1 - F(x)

0.0250 0.0244 0.0239 0.0233 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 0.0139 0.0136 0.0132 0.0129 0.0126 0.0122 0.0119 0.0116 0.0113 0.0110 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071

898

2.46 2.47 2.48 2.49 2.5 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.6 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.7 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.8 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.9 2.91 2.92 2.93 2.94 2.95

Appendix C

f(x)

F(x)

1 - F(x)

0.0194 0.0189 0.0184 0.0180 0.0175 0.0171 0.0167 0.0163 0.0158 0.0154 0.0151 0.0147 0.0143 0.0139 0.0136 0.0132 0.0129 0.0126 0.0122 0.0119 0.0116 0.0113 0.0110 0.0107 0.0104 0.0101 0.0099 0.0096 0.0093 0.0091 0.0088 0.0086 0.0084 0.0081 0.0079 0.0077 0.0075 0.0073 0.0071 0.0069 0.0067 0.0065 0.0063 0.0061 0.0060 0.0058 0.0056 0.0055 0.0053 0.0051

0.9931 0.9932 0.9934 0.9936 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9980 0.9980 0.9981 0.9981 0.9982 0.9983 0.9983 0.9984 0.9984

0.0070 0.0068 0.0066 0.0064 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016

x

f(x)

F(x)

2.96 2.97 2.98 2.99 3 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.1 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.2 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.3 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.4 3.41 3.42 3.43 3.44 3.45

0.0050 0.0048 0.0047 0.0046 0.0044 0.0043 0.0042 0.0040 0.0039 0.0038 0.0037 0.0036 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 0.0025 0.0025 0.0024 0.0023 0.0022 0.0022 0.0021 0.0020 0.0020 0.0019 0.0018 0.0018 0.0017 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 0.0013 0.0013 0.0012 0.0012 0.0012 0.0011 0.0011 0.0010

0.9985 0.9985 0.9986 0.9986 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997

1

-

F(x)

0.0015 0.0015 0.0014 0.0014 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003

Appendix C

3.46 3.47 3.48 3.49 3.5 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.6 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.7 3.71 3.72 3.73

f(x)

F(x)

1 - F(x)

x

f(x)

F(x)

0.0010 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004

0.9997 0.9997 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

3.74 3.75 3.76 3.77 3.78 3.79 3.8 3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89 3.9 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 4

0.0004 0.0004 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001

0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

899 1 - F(x)

0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

This Page Intentionally Left Blank

Appendix D t Test w

m

Table for t Test of Significance b e t w e e n Two Sample Means (Xx a n d x2)

-t

0

+t

0.8

0.7

0.6

0.5

0.4

Degrees

of freedom

P = 0.9

1 2 3 4 5 6 7 8 9 10

0.158 0.142 0.137 0.134 0.132 0.131 0.130 0.130 0.129 0.129

0.325 0.289 0.277 0.271 0.267 0.265 0.263 0.262 0.261 0.260

0.510 0.445 0.424 0.414 0.408 0.404 0.402 0.399 0.398 0.397

0.727 0.617 0.584 0.569 0.559 0.553 0.549 0.546 0.543 0.542

1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700

1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879

11 12 13 14 15 16 17 18 19 20

0.129 0.128 0.128 0.128 0.128 0.128 0.128 0.127 0.127 0.127

0.260 0.259 0.259 0.258 0.258 0.258 0.257 0.257 0.257 0.257

0.396 0.395 0.394 0.393 0.393 0.392 0.392 0.392 0.391 0.391

0.540 0.539 0.538 0.537 0.536 0.535 0.534 0.534 0.533 0.533

0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687

0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860

21 22 23 24 25 26 27 28 29 30

0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127 0.127

0.257 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.256 0.256

0.391 0.390 0.390 0.390 0.390 0.390 0.389 0.389 0.389 0.389

0.532 0.532 0.532 0.531 0.531 0.531 0.531 0.530 0.530 0.530

0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683

0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854

0.12566

0.25335

0.38532

0.52440

0.67449

0.84162

901

902

Appendix D

Degrees

of 0.3

0.2

0.1

0.05

0.02

0.01

1 2 3 4 5 6 7 8 9 10

1.963 1.386 1.250 1.190 1.156 1.134 1.119 1.108 1.100 1.093

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.626 2.228

31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764

63.657 9.925 5.841 4.604 5.032 3.707 3.499 3.355 3.250 3.169

11 12 13 14 15 16 17 18 19 20

1.088 1.083 1.079 1.076 1.074 1.071 1.069 1.067 1.066 1.064

1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325

1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725

2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086

2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528

3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845

21 22 23 24 25 26 27 28 29 3O

1.063 1.061 1.060 1.059 1.058 1.058 1.057 1.056 1.055 1.055

1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310

1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1..697

2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042

2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457

2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750

1.03643

1.28155

1.64485

1.95996

2.35996

2.57582

freedom

Appendix E Reduction Potentials

R e d u c t i o n P o t e n t i a l s of E l e c t r o c h e m i c a l R e a c t i o n s a t 25~ a

Reaction

Potential

Reaction

Potential

Ag +e-Ag A g 2+ + e = A g + (4f H C I O 4 )

+ 0.7996 +1.987

Cs + + e = Cs C u 2+ + e = C u +

-2.923 +0.158

A g 2 0 + H 2 0 + 2e = 2 A g + 2OH-

+0.342

Cu + + e = Cu C u 2+ + 2e = C u

+0.522 +0.3402

2 A g O + H 2 0 + 2e = A g 2 0 + 2OHA1a+ + 3e = A1 ( 0 . 1 f N a O H )

C u 2+ + 2e = C u ( H g )

+0.345

+0.599 -1.706

F e 3+ + e = F e + + F e 3+ + e = F e ++ ( l f H C 1 ) F e 3+ + e = F e ++ ( l f HC1Oa)

+0.770 +0.700 +0.747

F e 3+ + e = F e ++ ( l f H3PO4) F e 3+ + e = F e ++ ( 0 . 5 f H 3 S O 4) F e 2+ + 2e = F e

+0.438 +0.679 -0.409

AsO43- + 2 H 2 0 + 2e = AsO2 + 4OH- (lf NaOH) A u 3+ + 3e = A u

-0.08 + 1.42

B a 2+ + 2e = B a

-2.90

C a 2+ + 2e = C a

-2.76 -0.4026

2 H + + 2e = H 2

C d 2+ + 2e = C d C d 2+ + 2e = C d ( H g )

-0.3521

2 H 2 0 + 2e = H2 + 2 O H 2 H g e+ + 2e = Hg~ +

-0.761 -2.335 -1.4373

2Hg~ + + 2e = 2 H g H g 2+ + 2e = H g

+0.905 +0.7961 +0.851

Hg2(AcO)2 + 2e = 2 H g + 2AcO-

+0.5113

C d ( O H ) 2 + 2e = C d ( H g ) + 20HC d 3+ + 3e = Ce C e 3+ + 3e = C e ( H g ) Ce 3+ + e = C e +++ C e O H 3+ + H + + e C e 3+ Co 2+ + Co 3+ + C r 3+ +

=

+ H20 2e = Co e = Co 2+ ( 3 f H N O 2) e = C r 2+

C r 2+ + 2e = C r

+ 1.4430

HgO+ +1.7134 -0.28 +1.842 -0.41 -0.557

0.0000 -0.8277

2H20+2e=Hg+

2OHI n 3+ + e = I n ++ I n 2+ + e = I n + I n 3+ + 3e = I n

+0.097 -0.49 -0.40 -0.338

903

904

Appendix E

Reaction

Potential

Reaction

Potential

K§ + e = K

-2.924

Ru(IV) + e = R u ( I I I ) ( 2 f

Li t + e = Li

-3.045

HC1) Ru(IV) + e = R u ( I I I ) ( 0 . 1 f

+0.858

M g et + 2e = M g M n 2t + 2e = M n

-2.375 -1.029

HC104) Ru(III) + e = Ru(II)(0.1f

+0.49

M n O e + 4 H + 2e = M n 2§ + 2 H 2 0

+ 1.208

HClO4) Ru(III) + e = Ru(II)(1-6

M n O 4 + e = M n O 2-

+0.564

M n O 4 + 4 H § + 3e = MnOe + 2HeO

+1.679

Sb203 + 6 H t + 6e = 2 S b +

MnO4 + 2 H 2 0 + 3e = MnO2 + 4OH-

3H20 SbO+ + 2H+ + 3e = Sb +

+0.1445

+0.588

M n O 4 + 8 H § + 5e = M n 2t + 4 H 2 0

+1.491

2H20 Sn2+ + 2e = S n ( I V ) + 2e Sr(IV) + 2e S r 2t + 2e =

+0.212 -0.1364 -0.139 -2.89 -1.793

-0.11 f

HC1)

-0.084

Sn = S n 2t ( l f H C 1 ) = S r 2t Sr(Hg)

Na t + e = Na Nb(V) + 2e = N b ( I I I ) ( 2 f HC1) N d 3t + 3e = N d N i 2t + 2e = N i N i O 2 + 4 H § + 2e = N i 2§ +

-2.7109

2H20 NiO2 + 2H2 O§ + 2e =

+1.93

4H20 TeO2 + 4 H t + 4e = Te +

+0.472

Ni(OH) 2 + 2OHN p 3+ + 3e = Np

+0.76 -1.9

2H20 Ti 3t + e = Ti 2t T i ( O H ) 3t + H t + e = Ti 3+ +

+0.593 -2.00

0 2 + 4 H + + 4e = 2 H 2 0 02 + 2H2 O§ + 4e = 4 O H -

+1.229 +0.401

H20 T13t + 2e = T1 t

+0.06 +1.247

P b 3+ + 2e = P b P b 2t + 2e = P b ( H g ) P b O 2 + 4 H t + 2e = P b 2t +

-0.1263 -0.1205

T13t + 2e = T1 t (lfHC1) T1 t + e = T1 T1 t + e = T l ( H g )

+0.783 -0.3363 -0.3338

2H20

+0.344 -2.246 -0.23

TcO4 + 4 H t + 3e = TcOe(c) + 2H20 TcO4 + 8 H t + 7e = T c +

+1.46

UO~ + 4 H t + e = U 4t +

P b O + H 2 0 + 2e = P b + 2OH-

-0.576

2H20 UO22t + 4 H t + 2e = U 4t +

PbO2 2e P d 2t P d 2§

+1.685 +0.987 +0.623

+ S O ]- + 4 H § + = PbSO4 + 2H20 + 2e = P d ( 4 f H C 1 0 4 ) + 2e = P d ( l f HC1)

Rb § + e

=Rb

2H20

-2.925

R e O ~ + 8 H t + 7e = Re + 4H20

+0.367

ReO~ + 2 H + + e = ReO3(c) + H 2 0 RuO4(c) + e = R u O 4 R u O 4 + e = R u O 2-

+0.768 + 1.00 +0.59

V(OH)~ + 2 H t + e = VO 2t + 3 H 2 0 V(V) + e = V ( I V ) ( l f N a O H ) VO 2t + 2 H § + e = V 3+ + H2 O

+0.738

+0.62 +0.334

+1.000 -0.74 +0.337

V 3t + e = V 2t

-0.255

Z n 2t + 2e = Z n Z n 2t + 2e = Z n ( H g )

-0.7628 -0.7628

a T h e r e a c t i o n s a r e a l p h a b e t i c a l l y a r r a n g e d a c c o r d i n g to t h e e l e m e n t b e i n g r e d u c e d , a n d t h e s i g n s of t h e e l e c t r o d e p o t e n t i a l s a r e g i v e n a c c o r d i n g to t h e S t o c k h o l m C o n v e n t i o n ( I U P A C , 1935). T h e y c a n a l s o be c a l l e d r e d u c t i o n p o t e n t i a l s .

Appendix F

Thermodynamic Data

Thermodynamic Data for 298.15 K (25~ Substance and state

~'I~

cop

kcal tool -1

hG~ kcal mo1-1

S~ cal deg -1 mol -I

cal deg -1 mo1-1

Oxygen O(g) O2(g) O3(g)

59.553 0 34.1

55.389 0 39.0

38.467 49.003 57.08

5.237 7.016 9.37

Hydrogen H(g) H+(g) H+(aq) H2(g) OH-(aq) H20(li q) H20(g) H202(li q) H202(g)

52.095 367.161 0 0 -54.970 -68.315 -57.796 - 44.88 -32.58

48.581 362.6 0 0 -37.594 -56.687 -54.634 - 28.78 -25.24

27.391 26.01 0 31.208 -2.57 16.71 45.104 26.2 55.6

4.9679 4.968 0 6.889 -35.5 17.995 8.025 21.3 10.3

Helium He(g)

0

0

30.1244

4.9679

Neon Ne(g)

0

0

34.9471

4.9679

Argon Ar(g)

0

0

36.9822

4.9679

Fluorine F(g) F-(aq) F2(g) HF(g) HF(aq, undissoc.)

18.88 -79.50 0 -64.8 -76.50

14.80 -66.64 0 -65.3 -70.95

37.917 -3.3 48.44 41.508 21.2

5.436 -25.5 7.48 6.963

Chlorine Cl(g) Cl-(aq) C12(g) HCl(g)

29.082 -39.952 0 -22.062

25.262 -31.372 0 -22.777

39.457 13.5 53.288 44.646

5.220 -32.6 8.104 6.96

905

906

Appendix F

Substance and state

AH~ kcal mo1-1

AG~ kcal mol -~

S~ cal deg -1 mo1-1

cal deg -1 mo1-1

Bromine Br(g) Br-(aq) Br2(liq) Br2(g) HBr(g)

26.741 -29.05 0 7.387 -8.70

19.701 -24.85 0 0.751 - 12.77

41.805 19.7 36.384 58.641 47.463

4.968 -33.9 18.090 8.61 6.965

Iodine I(g) I-(aq) I2(c)

25.535 -13.19 0

16.798 -12.33 0

43.184 26.6 27.757

4.968 -34.0 13.011

-12.3 6.33

-12.3 0.41

57.2 49.351

6.969

0 0.08 66.636 24.45 -70.944 -77.194 -94.58 -151.9 -217.32 -4.93 -9.5 -149.67 -212.08 -145.51 - 194.548

0

7.60

5.41

56.951 11.87 -71.748 -71.871 -88.69 -116.3 -177.97 -8.02 -6.66 -126.15 -180.69 -128.56 - 164.938

40.084 102.98 59.30 38.7 61.34 -7. 4.8 49.16 29. 33.4 31.5 55.5 37.501

5.658 37.39 9.53

4.968 6.961 7.133 8.89 -20.7 9.19 18.47 20.2 8.38

I~(aq) HI(g) Sulfur S(c, rhombic) S(c, monoclinic) S(g) Sg(g) SO2(g) SO2(aq, undissoc.) SO3(g) SO2-(aq) SO2-(aq) H2S(g) H2S(aq) HSO~(aq) HSO~(aq) H2SO3(aq, undissoc.) H2SO4(liq) Nitrogen N(g) N2(g) NO(g) NO2(g) NO~ (aq) N20(g) N204(g) N205(g) NH3(g) NH3(a q, undissoc.) NH~(aq) N2H4(g) HN3(g) HN3(a q) HNO3(li q) HNO3(g) NH2OH(a q) NH2OH~(a q) NH4OH(aq, undissoc.) NH4NO3(c) NH4F(c) NH4Cl(c) NH4Br(c) NH4I(c) (NH4)2SO4(c)

112.979 0 21.57 7.93 -49.56 19.61 2.19 2.7 - 11.02 - 19.19 -31.67 22.80 70.3 62.16 -41.61 -32.28 -23.5 -32.8 -87.505 -87.37 -110.89 - 75.15 -64.73 -48.14 -282.23

108.886 0 20.69 12.26 -26.61 24.90 23.38 27.5 - 3.94 - 6.35 -18.97 38.07 78.4 76.9 - 19.31 -17.87

36.613 45.77 50.347 57.35 35.0 52.52 72.70 85.0 45.97 26.6 27.1 56.97 57.09 34.9 37.19 63.64

-63.04 -43.98 -83.36 - 48.51 -41.9 -26.9 -215.56

43.3 36.11 17.20 22.6 27. 28. 52.6

Phosphorus P(c, white) P(c, red, triclinic) P(g)

0 -4.2 75.20

0 -2.9 66.51

9.82 5.45 38.978

12.11 -70. 8.18

-20. 33.20

19.1 11.85 10.44 26.26 12.75

33.3 15.60 20.1 23. 44.81

5.698 5.07 4.968

Appendix F

Substance and state

PO~(aq) po3-(aq) P204-(aq)

AH~f kcal mo1-1

AG~ kcal mo1-1

S~ cal deg -1 mo1-1

-243.5 -458.7

-53. -28.

-644.8

54.70

-260.34

-8.0

-270.17

21.6

-267.5 -273.10

26.41 37.8

-485.7 -122.60

64. 77.76

Cp cal deg -1 tool -~

HPO3(c) HPO2-(aq) HPO42-(aq) H2PO~(aq) H2PO~(aq) H3PO3(c) H3PO4(c) H3PO4(a q, undissoc.) H4P207(c) H4P207(aq, undissoc.) POC13(g)

-233.5 -305.3 -542.8 -392.0 - 713.2 -226.7 -231.6 -308.83 -231.7 309.82 -230.5 -305.7 -307.92 -535.6 -542.2 -133.48

Arsenic As(c, gray) As(g) AsO43-(aq) As2Os(c) As406(c, octahedral) As4C6(g) AsH3(g) HAsO42-(aq, undissoc.) H2AsO~(aq, undissoc.) H3AsO4(c) H3AsO4(aq, undissoc.)

0 72.3 -121.27 -221.05 -314.04 -289.0 15.88 -216.62 -217.39 -216.6 -215.7

0 62.4 -155.00 -187.0 -275.46 -262.4 16.47 -170.82 -180.04

8.4 41.61 -38.9 25.2 51.2 91. 53.22 -0.4 -28.

-183.1

44.

0 62.7

0 53.1 -42.33 -81.32 -198.2 -97.4

10.92 43.06

-77.37 -72.0

44.0 80.71

25.8 18.33

0 49.5 -137.16 -170.0

0 40.2 -118.0

13.56 44.669 36.2

6.10 4.968 27.13

0 0.4533 171.291 -26.416 -94.051 -98.90 -161.84 17.88 -101.71 -165.39 -101.51 -90.48 -101.68 -167.22

0 0.6930 160.442 -32.780 -94.254 -92.26 -126.17 - 12.13 -83.9 -140.26 -86.38

1.372 0.568 37.7597 47.219 51.06 28.1 -13.6 44.492 22. 21.8 30.82

2.038 1.4615 4.9805 6.959 8.87

-89.0 -184.94

39. 44.8

P406(c) P4010(c)

50.60

25.35

20.30 5.89 4.968 27.85 45.72 9.10

Antimony

Sb(c) Sb(g) SbO+(aq) SbO~(aq)

Sb205(c) HSbO2(aq. undissoc.) H3SbO4(aq) SbC13(c) SbC13(g) SbOCl(g) Bismuth Bi(c) Bi(g) Bi203(c) Bi(OH)3(c) Carbon C(c, graphite) C(c, diamond) C(g) CO(g) CO2(g) CO2(aq, undissoc.) CO32-(aq) CH4(g) HCOO-(aq) HCO~(aq) HCOOH(Iiq)

HCOOH(g) HCOOH(aq, un-ionized) H2CO3(aq)

-232.3 -116.6 -216.8 -91.34 -75.0 -25.5

-

6.03 4.97

29.9 11.1

8.439 -21.0 23.67

907

908

Appendix F

Substance and state

CH3OH(liq) CH3OH(g) CH3OH(aq) CC14(liq) CS2(liq)

CS2(g) CN-(aq) HCN(g) HCN(aq, un-ionized) NH4HCO3(c) CO(NH2)2(c) CO(NH2)2(aq) C202-(aq) HC20~(aq) (COOH)2(c) CH3COOH-(a q) CH3COOH(Iiq) CH3COOH(g) CH3COOH(aq, un-ionized) C2HsOH(Iiq) C2HsOH(g) C2H5OH(aq) (CN)2(g) (C2Hs)20(liq)

Silicon Si(c) Si(g) Si02(c, quartz) SiH4(g) H2SiO3(c) H2SiO3(aq, undissoc.) H4SiO4(c) H4SiO4(aq. undissoc.) SiCl4(g) SiC(c, cubic) Germanium Ge(c) Ge(g) GeH4(g)

Tin Sn(c, white) Sn(c, gray) Sn(g) Sn2+(aq) Sn4+(aq) SnO(c)

SnO2(c) Sn(OH)2(c) Sn(OH)4(c) SnCl2(c) SnCl4(liq) SnCl4(g)

Lead Pb(c) Pb(g) pb2+(aq) PbO(c, yellow) PbO(c, red) PbO2(c)

AH~ kcal mol -~

hG~ kcal tool -1

S~ cal deg -1 mo1-1

C~ cal deg -1 mol -I

-57.04 -47.96 -58.779 -32.37 21.44 28.05 36.0 32.3 25.6 -203.0 -79.56 -75.954 -197.2 -195.6 -197.7 -116.16 -115.8 -103.31 -116.10 -66.37 -56.19 -68.9 73.84 -66.82

-39.76 -38.72 -41.92 -15.60 15.60 16.05 41.2 29.8 28.6 -159.2 -47.04

30.3 57.29 31.8 51.72 36.17 56.82 22.5 48.20 29.8 28.9 25.00

19.5 10.49

-161.1 -166.93

10.9 35.7

-88.29 -93.2 -89.4 -94.78 -41.80 -40.29 -43.44 71.07

20.7 38.2 67.5 42.7 38.4 67.54 35.5 57.79

-1.5 29.7 15.9

0 108.9 -217.72 8.2 -284.1 -282.7 -354.0 -351.0 -157.03 -15.6

0 98.3 -204.75 13.6 -261.1 -258.0 -318.6 -314.7 -147.47 -15.0

4.50 40.12 10.00 48.88 32. 26. 46. 43. 79.02 3.97

4.78 5.318 10.62 10.24

0 90.0 21.7

0 80.3 27.1

7.43 40.103 51.87

5.580 7.345 10.76

0 -0.50 72.2 -2.1 7.3 -68.3 -138.8 -134.1 -265.3 -77.7 -122.2 -112.7

0 0.03 63.9 -6.5 0.6 -61.4 -124.2 -117.5

12.32 10.55 40.243 -4. -28. 13.5 12.5 37.

6.45 6.16 5.081

-105.2 -103.3

61.8 87.4

39.5 23.5

0 46.6 -0.4 -51.94 -52.34 -66.3

0 38.7 -5.83 -44.91 -45.16 -51.95

15.49 41.889 2.5 16.42 15.9 16.4

6.32 4.968

31.49 18.1 10.85 8.57

22.26

26.64 15.64 13.58

21.57 6.42

10.59 12.57

10.94 10.95 15.45

Appendix F

Substance and state

Pb304(c) HPbO~-(aq) Pb(OH)2(c) PbC12(c) PbSO4(c)

AH~ kcal mo1-1

S~ cal deg -1 mo1-1

C~ cal deg -1 mo1-1

-143.7 -80.90

50.5

35.1

-123.3 -85.90 -219.87

-75.08 -194.36

32.5 35.51

24.667

0 134.5 -304.20 -261.55 -256.29

0 124.0 -285.30 -231.60 -231.56

1.40 36.65 12.90 21.23 38.8

2.65 4.971 15.04 19.45

Aluminum Al(c) Al(g) A13+(aq) AIO~-(aq) Al203(c, corundum) A1C13(c)

0 78.0 -127. -219.6 -400.5 -168.3

0 68.3 -116. -196.8 -378.2 -150.3

6.77 39.30 -76.9 -5. 12.17 26.45

5.82 5.11

Thallium Tl(c) Tl(g) Tl+(aq) T13+(aq) T1OH(c) TIC13(c)

0 43.55 1.28 47.0 -57.1 -75.3

0 35.24 -7.74 51.3 -46.8

15.34 43.225 30.0 -46. 21.

6.29 4.968

Zinc Zn(c) Zn(g) Zn2+(aq) ZnO(c) Zn(OH)2(c) ZnC12(c) ZnS(c, sphalerite) ZnSO4(c)

0 31.245 -36.78 -83.24 -153.74 -99.20 -49.23 -234.9

0 22.748 35.14 -76.08 -132.68 -88.296 -48.11 -209.0

9.95 38.450 -26.8 10.43 19.5 26.64 13.8 28.6

6.07 4.968 11. 9.62 17.3 17.05 11.0

Cadmium Cd(c) Cd(g) Cd2+(aq) CdO(c) Cd(OH)2(c) Cd(OH)2(aq) CdC12(c) CdSO4(c)

0 26.77 -18.14 -61.7 -134.0 -128.08 -93.57 -223.06

0 18.51 -18.542 -54.6 -113.2 -93.73 -82.21 -196.65

12.37 40.066 -17.5 13.1 23. -22.6 27.55 29.407

6.21 4.968

Mercury Hg(liq) Hg(g) Hg+l(aq) Hg2+(aq) HgO(c, red) HgCl2(c) Hg2C12(c) HgS(c, red) HgSO4(c) Hg2SO4(c)

0 14.655 40.9 41.2 -21.7 -53.6 -63.39 -13.9 -169.1 -177.61

0 7.613 39.30 36.70 -13.995 -42.7 -50.377 -12.1

18.17 41.79 -7.7 20.2 16.80 34.9 46.0 19.7

6.688 4.968

-149.589

47.96

31.54

Copper Cu(c) Cu+(aq)

0 17.13

0 11.95

7.923 9.7

5.840

Boron B(c)

B(g) B203(c) H3BO3(c) H3BOa(aq, un-ionized)

-171.7

AG~f kcal mo1-1

18.89 21.95

10.38

17.85 23.80

10.53

11.57

909

910

Appendix F

Substance and state

hH~ kcal mo1-1

AG~ kcal mo1-1

S~ cal deg -1 mo1-1

C~ cal deg -1 mo1-1

15.48 -37.6 -40.3 -32.8 -52.6 -12.7 -184.36

15.66 -31.0 -34.9 -28.65 -42.0 -12.8 -158.2

-23.8 10.19 22.26 20.6 25.83 15.9 26.

10.11 15.21 11.6 13.82 11.43 23.9

0 68.01 25.234 -7.42 -30.370 -23.99 -14.78 -7.79 -171.10 -29.73 34.9

0 58.72 18.433 -2.68 -26.244 -23.16 -15.82 -9.72 -147.82 -8.00 37.5

10.17 41.321 17.37 29.0 23.0 25.6 27.6 34.42 47.9 33.68 25.62

6.059 4.9679 5.2 15.74 12.14 12.52 13.58 18.29 31.40 22.24 15.95

0 87.5 -101.5 -8.3 -28.1

0 78.0 -75.77

11.33 43.115 45.3

6.075 4.968

Nickel Ni(c) Ni(g) Ni + +(aq) NiO(c) Ni203(c) Ni(OH)2(c) NiCl2(c)

0 102.7 -12.9 -57.3 -117.0 126.6 -72.976

0 91.9 -10.9 -50.6

7.14 43.519 -30.8 9.08

6.23 5.583

- 106.9 -61.918

21. 23.34

Cobalt Co(c) Co2+(aq) Co3+(aq) CoO(c) C0304 CoC12(c)

0 -13.9 22. -56.87 -213. -74.7

0 -13.0 32. -51.20 -185. -64.5

7.18 -27. -73. 12.66 24.5 26.09

5.93

Iron Fe(c) Fe2+(aq) Fe3+(aq) FeO(c) Fe203(c) Fe304(c) Fe(OH)2(c) Fe(OH)3(c) FeC12(c) FeC13(c) FeS(c) FeSO4(c) Fe2(SO4)3(c) Fe3C(c)

0 -21.3 -11.6 -65.0 -197.0 -267.3 -136.0 -196.7 -81.69 -95.48 -23.9 -221.9 -617.0 6.0

0 -18.85 -1.1

6.52 -32.9 -75.5

6.0O

-177.4 -242.7 -116.3 -166.5 -72.26 -79.84 -24.0 -196.2

20.89 35.0 21. 25.5 28.19 34.0 14.41 25.7

24.82 34.28

18.32 23.10 12.08 24.04

4.8

25.0

25.3

0 135.1

0 124.4

9.95 45.960

6.18 6.102

Cu2+(aq) CuO(c) Cu20(c) CuCl(c) CuCl2(c) CuS(c)

CuS04(c)

Silver Ag(c) Ag(g) Ag+(aq) Ag20(c)

AgCl(c)

AgBr(c)

AgI(c) Ag2S(c, orthorhombic) Ag2SO4(c) AgNO3(c)

AgCN(c)

Gold Au(c) Au(g)

Au(OH)3(c) AuCl(c) AuC13(c)

-

10.59

17.13

13.20 29.5 18.76

Platinum

Pt(c) Pt(g)

Appendix F

Substance and state

AH~f kcal mo1-1

AG~ kcal mo1-1

S~ cal deg -1 mo1-1

-84.1 -26.5 -120.3 -161.

-88.1 -117.

40. 52.6

0 -52.76 -92.07 -124.29 -129.4 -156. - 166.2 -115.03

0 -54.5 -86.74 -111.18 -106.9 -119.7 - 147.0 -105.29

7.65 -17.6 14.27 12.68 45.7 14. 23.7 28.26

6.29 12. 10.86 12.94

0 -34.3 -143. -140.9 -210.60 -272.4 -356.2 -209.9

0

5.68

5.58

-173.96 252.9 -311.0 -182.8

12.00 19.4 62.6 44.0

Tungsten W(c) W(g) W02(c) W03(c) WO2-(aq) WC(c) (*at 627'K) WC2(c)

0 203.0 -140.94 -201.45 -257.1 -3.143" +2.0*

0 192.9 -127.61 -182.62

7.80 41.549 12.08 18.14

Titanium Ti(c) Ti(g) TiO2(c , rutile) TiC14(g)

0 112.3 -225.8 -182.4

0 101.6 -212.6 -173.7

7.32 43.066 12.03 84.8

5.98 5.839 13.15 22.8

Beryllium Be(c) Be2+(aq) BeO(c) BeO22-(aq)

0 -91.5 -145.7 -189.0

0 -90.75 -138.7 -153.0

2.27 -31.0 3.38 -38.

3.93

0 35.30 -111.58 -143.81 -220.97 -268.5 -153.28 -307.1 -261.9

0 27.04 -108.7 -136.10 -199.23 -255.8 -141.45 -279.8 -241.9

7.81 35.502 -33.0 6.44 15.10 13.68 21.42 21.9 15.7

5.95 4.968

0 -129.74 -151.79 -235.68 -291.5 - 190.2

0 -132.30 -144.37 -214.76 -279.0 - 178.8

9.90 -12.7 9.50 19.93 16.46 25.0

6.05

Pt(OH)2(c) PtCl2(c) PtC12-(aq) PtCl~-(aq) Manganese Mn(c) Mn2+(aq) MnO(c) MnO2(c) MnO~(aq) MnO2-(aq) Mn(OH)2(amorph) MnC12(c)

C~ cal deg -1 mo1-1

17.43

Chromium

Cr(c) Cr2+(aq)

CrO2(c) CrO3(c) CrO42-(aq) Cr203(c) Cr202-(aq) HCrO~(aq)

28.38

5.80 5.093 13.41 17.63

19.233" 43.0*

6.10

Magnesium

Mg(c) Mg(g) Mg2+(aq) MgO(c) Mg(OH)2(c) MgF2(c) MgCl2(c) MgS04(c) MgCO3(c)

Calcium Ca(c) Ca2+(aq) CaO(c)

Ca(OH)2(c) CaF2(c) CaC12(c)

8.88 18.41 14.72 17.06 23.06 18.05

10.23 20.91 16.02 17.35

911

912

Appendix F

Substance and state

Cal2(c) CaSO4(c, anhydrite) CaC2(c) CaCO3(c, calcite) Strontium St(c) Sr(g) Sr2+(aq) SrO(c)

Sr(OH)2(c) SrC12(c)

SrSO4(c) SrSO3(c)

Barium Ba(c) Ba2+(aq) BaO(c) Ba(OH)2(c) BaCl2(c) BaSO4(c) BaCO3(c)

Lithium Li(c) Li(g) Li+(aq) Li20(c) LiOH(c) LiCl(c)

Sodium Na(c) Na(g) Na+(aq) Na20(c) NaOH(c) Na202(c) NaF(c) NaCl(c) NaBr(c) Na2SO3(c) Na2SO4(c) NaNO3(c) Na2NO3(c) NaHCO3(c)

Potassium K(c) K(g) K+(aq) K20(c) KOH(c) KF(c) KCI(c) KCIO3(c) KCIO4(c) KBr(c) Kl(c)

K2SO4(c) KNO3(c) KMn04(c) K2CO3(c)

hH~ kcal mol -I

AG~ kcal mo1-1

S~ cal deg -1 mo1-1

C~ cal deg -1 tool -1

-127.5 -342.76 -14.3 -288.46

-126.4 -315.93 -15.5 -269.80

34. 25.5 16.72 22.2

0 39.3 -130.45 -141.5 -229.2 - 198.1 -347.3 -291.6

0 31.3 -133.71 -134.3

12.5 39.32 -7.8 13.0

6.3 4.968

- 186.7 -320.5 -272.5

27.45 28. 23.2

18.07

0 -128.50 - 132.3 -225.8 -205.2 -352.1 -290.7

0 134.02 - 125.5

15.0 2.3 16.83

6.71

-193.7 -325.6 -271.9

29.56 31.6 26.8

17.96 24.32 20.40

0 38.09 -66.55 -143.10 -116.48 -97.69

0 30.6 -70.22 -134.35 -104.92

6.96 33.14 2.70 9.06 10.23

5.89 4.968

0 25.60 -57.433 -99.90 -101.766 - 122.66 -137.52 -98.279 -86.38 -260.6 -331.55 -111.54 -270.26 -226.5

0 18.475 -62.589 -90.61 -90.77 - 107.47 -130.28 -91.79 -83.48 -239.5 -303.38 -87.45 -250.50 -203.6

12.26 36.714 13.96 17.94 15.40 22.66 12.24 17.24 20.75 34.9 35.76 27.85 33.17 24.4

6.73 4.968

0 21.33 -60.271 -86.80 -101.51 -135.90 -104.33 -93.50 -102.80 -94.12 -78.37 -342.66 -117.76 - 194.4 -274.90

0 14.50 -67.466 -76.99 -90.57 -128.81 -97.70 -69.29 -71.79 -90.92 -77.20 -314.62 -93.96 - 170.6 -254.44

15.46 38.297 24.15 22.50 18.86 15.90 19.74 34.2 36.10 22.93 25.43 42.0 31.81 41.0 37.17

23.82 14.99 19.57

10.76

19.46

11.42

12.93 11.85

16.52 14.23 21.34 11.20 11.98 12.28 28.71 30.55 22.24 26.53 20.94

7.05 4.968 20.00 15.51 11.71 12.26 23.96 26.86 12.52 12.61 31.08 23.01 28.10 27.35

Appendix

Substance and state

F

AH~f kcal mo1-1

AG~ kcal mo1-1

S~ cal deg -1 tool -1

C~ cal deg -1 mo1-1

Rubidium Rb(c) Rb(g) Rb+(aq) RbCl(c)

0 19.33 -60.018 -103.99

0 13.35 -67.45

18.35 40.3 28.79 22.90

7.36 4.968

Cesium Cs(c) Cs(g) Cs+(aq) CsCl(c)

0 18.18 -61.673 -105.82

0 11.88 -67.41

20.33 41.942 31.75 24.18

7.70 4.968

913

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Appendix G Summary

of

Differential Operations Involving the V-Operator in Rectangular Coordinates (x, y, z) ( v - v) = aVx + a ~ _~ aVz ax ay az 028 028 ~2s (V2s) = ~ + ~ay2 + ~az2

(oq (O~xoq \ az / rxY \ ay ax l ( av~ + aVz~ + Zzz (O~z+ O~x~ + ryz az ay / \ ax az /

(r" vv) = ZXXk ax l + zYYk ay /

as

[VS]x

[Vs]y = [Vs]x

(A) (B)

(C)

ax

(D)

c]s

(E)

m

as az

(F)

aVz

O~,y az

(G)

OVx

OVz ax

(H)

[V x v]~ = a~,y Ox

a~'x

(I)

[vX,]x-Ty

[V x V]y = az

oy

915

916

~4ppendix G

[ V - r ] x = Orxx+ Orxy ~ Orxz

Oy

Ox . . [V . T]y

(j)

Oz

(~Txy Oryy C~Tyz (~X + Oy + Oz

. .

(K)

[V. Z]z = OZxz + C~TYz~ C~Zzz Ox Oy Oz

(L)

[V2p]x -- c32px c32px O2px Ox 2 + ~Oy2 ~ r 2 [V2p]y

02py

02py

(M)

02py

- 0x 2 + ~Oy2 ~ OZ2

02Pz

(N)

D2Pz C32Vz

[V2P]z = 0--~ + ~ 3 y2 -~ C3Z2

(O)

8Vx OVx 8v_._~x [v. Vv] x = Vx -~x + Vy -~y + Vz Oz

(P)

C~Py [~. W]y - ~x ~ OVz

C~Py

+ ~y ~ OVz

OPy

+ ~z oz Ov__zz

[~" V~lz = ~X-~x + ~y~y + ~z oz

(Q) (R)

Note: Operations involving the tensor z are given for symmetrical z only. Data from R. B. Bird, W. E. Stewart, and E. N. Lightfoot, "Transport Phenomena," p. 738. Wiley, New York, 1960.

Appendix H Summary of Differential Operations Involving the V-Operator in Cylindrical Coordinates

(r, O, z)

(V. v ) = 1 O lOvo OVz r ~ (rvr) + -r O0 + ~Oz

(A)

( V 2 s ) = l O ( r ~ r-~raS) +-a-z-za~ r"102s ~

(B)

('r:Vp)

Or / + %o

= Trr \

[

r -~r

nt- TrO

(1

+ ~O2s az 2

O0

+ -r aO J + r~

\ Oz / aO + az ]

(C)

(O~z O~r~ Or + Oz ]

-+" Trz \

8s [VS]r = --

Or

(D)

i as

[Vs]e - r aO

(E)

as [VS]z- Oz

(F) 1 aVz

[ V X " ] r -- ~

r O0

[V x v] o [V

x

V]z

OVr az

8Vo Oz

(G)

(H)

OVz Or

1 a = r ~r (rv~

i aP r r O0

(I)

917

918

Appendix H

[V " r] r

1 a 1 a = r - ~ r (rgrr) + r -~ TrO -

-

-

-

1 Orrz r 7.00 H Oz

-

(J)

[V 9r]o -

1 0~'oo + OTrO _+_2 OTo~ r O0 Or 7 7"rO-~ OZ

(K)

[V'r]z

1 0

(L)

l Oroz Orzz r O0 -4 Oz

= -r-~r (r'rrz) + -

O(lO )lO r2O oO r O(lO )

[V2p]r = ~r

~ r (r12r)

[V2v]~ = ~

7r (rye)

[V2v]z

10 (Or) = r-~r r-~r

-~

r 2 0 02

r 2 00 f- 0z 2

1 02120 f. 2 012r 02120 r 2 002 - ~ - ~ q Oz 2

102Vz 02Vz + - ~ - - ~ -~ Oz 2

(~Vr _t 120 012r [P" VP]r = Pr Or r O0

122 t- Vz Ovr r Oz

OVO @ _VO _4_ OPO _~_ 12rPO + Vz O v o [12 V12]o = 12r Or r O0 r Oz

OVz + _vo OVz + v z 012z r O0 Oz

[12 ~12]z-- Pr Or

(M) (N) (O) (P) (Q) (R)

Note: Operations involving the tensor r are given for s y m m e t r i c a l 9 only. Data from R. B. Bird, W. E. Stewart, and E. N. Lightfoot, "Transport Phenomena," p. 738. Wiley, New York, 1960.

Appendix I Summary of Differential Operations Involving the V-Operator in Spherical Coordinates (r, O, 6)

1 0 ( V - p ) = ~ ~rr

(r2pr) -t r sin1 0 O00 (Vo sin O) + r sin 0 04)

lO( )

(V2s) = ~-~~rr r 2

1 0 (sinoOS) 1 02s + r 2 sin 0 0--0 0-0 + r 2 sin 2 0 04)2

(T" VP) = Trr \ Or / + %o

(B)

O0

1 Oveo Pr v 0 cot O) r r s i n O 04) e - r- +

+%~

(OPO lOPr ;o)(c~P6 Or 4 +

nt- TrO \

+ T~

(A)

1

r O0

fro \ Or

(lOv~ 1 Ore O0 + r sin 0 04)

cotO r

) 11,4,

OP r

r sin 0 04)

(C)

[V S] r -- '''L~

(D)

[Vs]o

10s

(E)

1 Os r sin 0 06

(F)

Or

=r~

~_,rVsl~-

919

920

Appendix I

[I~7 X P]r --

1 0 (v 6 sin O) r sin 0 O0

[VXV]o-

1 OVr rsinOOrb

1 OVo r sin 0 06

(G)

1 0 (rye) r Or

1 0 [V X v]~ = r ~ r r (rv~

(H)

10Vr r O0

1 0 [ V ' T ] r = ~ ~ r r (r2Trr) -~

(I)

1 Orr6 r sin 0 06

1 0 (TrO sin O) r sin 0 O0

Too + T66

(J) 1 0 [V" 7"10 = r--2~rr (r2Tr0) q

t "grO r

cot 0 m

r

1 Oro6 1 0 (too sin O) + r sin 0 O0 r sin 0 06 ,r6rb

(K)

1 O 1 Oro6 1 0%6 2 cot 0 [ V . T] 6 = ~ ~rr (r2rr6) + --~ Jr- Tr~ + ~ r 0 6 r O0 r s i n O 04) r 2v r

[V2V]r = V2Vr

re

20v o r e O0

2v o cot 0 2 Ov6 re - r e sin 00~b

[Vgv]o = Vgvo _~ 2 0 P r PO __ 2 COS 0 0 V 6 r 200 r 2 s i n 20 r 2 s i n eO04~ 2v~ 2 OPr t 2 cos 0 8v o [V2P]6 = V 2 P 6 2 -~r sin e 0 r 2 sin 00~b r 2 sin20 04~ OP_..__r_jr_ PO OPr 4- Pd) OPr [P" VP]r = Pr Or 0---0 00 r sin 0 04) OVo + v---~ o OVo +

[v" Vv]o = Vr Or

r O0

Ov~ v o Ov 6 [P VP] 6 = Pr Or + r O0

p2_~_ p~

r

(M) (N) (O) (P)

r

v6 Or~ 4 VrV~ v~ cot O r sin 0 04) r r v 6 0 % + v6v r + v6v 6 c o t O r s i n 0 &b

(L)

r

(Q) (R)

Operations involving the tensor r are given for symmetrical r only. Data from R. B. Bird, W. E. Stewart, and E. N. Lightfoot, "Transport Phenomena," p. 738. Wiley, New York, 1960. Note:

Appendix J Liquid Surface Tensions

Surface Tension of Various Liquids

Substance name Acetaldehyde Acetaldoxime Acetamide Acetanilide Acetic acid Acetone

Acetonitrile Acetophenone Allyl alcohol Aniline

Azoxybenzene Benzaldehyde Benzene

Benzylamine d-sec-Butyl alcohol n-Butyl alcohol

Formula

In contact with

Temperature (C ~)

Surface tension (dynes/cm)

C2H40 C2H5NO C2H5NO C2HsNO C2H402 C3H60 C3H60 C3H60 C2H3N CsHsO C3H6O C6HvN C6H7N C6HvN CleHloN20 C7H60 C6H 6 C6H 6 C6H6 C6H6 C7H9N C4HloO C4HloO C4H100 C4H100

vapor vapor vapor vapor vapor air or vapor air or vapor air or vapor vapor vapor air or vapor air vapor air vapor air air air saturated w/vapor air vapor vapor air or vapor air or vapor air or vapor

20 35 85 120 10 0 20 40 2O 20 20 10 20 5O 51 20 10 20 20 30 2O 10 0 2O 5O

21.2 30.1 39.3 35.6 28.8 26.21 23.70 21.16 29.30 39.8 25.8 44.10 42.9 39.4 43.34 40.04 30.22 28.85 28.89 27.56 39.5 23.5 26.2 24.6 22.1

921

922

Appendix J

Substance name tert-Butyl alcohol Carbon tetrachloride Chlorobenzene Chloroform Dichloroethane Diethyl phthalate Dimethylamine Ethyl acetate

Ethyl acetoacetate Ethyl alcohol

Ethylamine Ethylene oxide

Ethyl ether dl- Ethyl lactate Ethyl mercaptan Ethyl salicylate Formamide Formic acid Furfural Glycerol

Formula

C4HloO CC14 C4HsC1 CHC13 C2H4C12

C12H1404 C2HTN C2HTN C4HsO2 C4HsO2 C4HsO2 C6HloO3 C2H60 C2H60 C2H60 C2H60 C2H7N C2H40 C2H40 C2H40 C4HloO C4HloO C5HloO3 C2H6S C9HloO3 CH3NO

CH202 C5H402 C3H803 C~HsO3

C3HsO3 Glycol n-Hexane Hydrogen peroxide Isobutyl alcohol Isobutylamine Isobutyl chloride Isobutyric acid Isopentane Isopropyl alcohol Methyl acetate Methyl alcohol

Methylamine

n-Methylaniline

C2H602

C6H14 H202 C4HloO C4HllN C4H9C1

C4H802 C5H12 C3HsO C3H602 CH40 CH40 CH40 CH3NH2 CH3NH2 CH3NH2 C7H9N

In contact with air or vapor vapor vapor air air vapor nitrogen nitrogen air air air air or vapor air vapor vapor vapor nitrogen vapor vapor vapor vapor vapor air air or vapor vapor vapor air air or vapor air air air air or vapor air vapor vapor air air air or vapor air air or vapor air or vapor air air vapor nitrogen vapor nitrogen air or vapor

Temperature (C~)

Surface tension (dynes/cm)

20 20 20 20 35.0 20 0 5 0 20 5O 20 0 10 20 30 0 -20 0.0 20 20 5O 2O 2O 20.5 2O 2O 2O 2O 9O 150 2O 20 18.2 20 68 2O 2O 20 20 2O 0 2O 5O -12 -20 -70 2O

20.7 26.95 33.56 27.14 23.4 37.5 18.1 17.7 26.5 23.9 20.2 23.9 24.05 23.61 22.75 21.89 21.3 30.8 27.6 24.3 17.01 13.47 29.9 22.5 38.33 58.2 37.6 43.5 63.4 58.6 51.9 47.7 18.43 76.1 23.0 17.6 21.94 25.2 13.72 21.7 24.6 24.49 22.61 20.14 22.2 23.0 29.2 39.6

Appendix J

Substance name Methyl benzoate Methyl chloride Methyl ether Methylene chloride Methyl ethyl ketone Methyl formate Methyl propionate Methyl salicylate Naphthalene n-Octane n-Octyl alcohol Oleic acid Phenol Propionic acid n-Propyl acetate n-Propyl alcohol n-Propylamine n-Propyl bromide n-Propyl chloride n-Propyl formate Pyridine Quinoline Ricinoleic acid Styrene Tetrabromoethane 1,1,2,2Tetrachloroethane 1,1,2,2Tetrachloroethylene Toluene m-Toluidine o-Toludine p-Toluidine Trimethylamine Vinyl acetate m-Xylene o-Xylene p-Xylene

Formula C8H802 CH3C1 C2H60 CH2C12 C4H80 C2H402 C4H802 CsHsO~ CloH8 CsHls CsHlsO C18H3402 C6H60 C3H602 C5H1002 C3HsO C3HgN C3HTBr C3HTC1 C4HsO 2 CsHsN CgHTN C18H3403 CsHs C2H2Br4 C2H2C14 C2C14 C7H8 C7H9N CTHgN CTHgN C3HgN C4H602 C8H10 C8H10 C8H10

In contact with air or vapor air vapor air air or vapor vapor air or vapor nitrogen air or vapor vapor air air air or vapor vapor mr or vapor vapor air vapor vapor vapor air air air air air air vapor vapor vapor air or vapor air nitrogen vapor vapor air vapor

923

Surface Temperature tension (C~ (dynes/cm) 20 20 -10 20 20 20 20 94 127 20 20 20 20 20 20 20 20 71 20 20 20 20 16 19 20 22.5 20 10 20 20 50 -

4

20 20 20 20

37.6 16.2 16.4 26.52 24.6 25.08 24.9 31.9 28.8 21.80 27.53 32.50 40.9 26.7 24.3 23.78 22.4 19.65 18.2 24.5 38.O 45.0 35.81 32.14 49.67 36.03 31.74 27.7 36.9 40.0 34.6 17.3 23.95 28.9 30.10 28.37

Data taken from R. C. Weast, ed., "Handbook of Chemistry and Physics," 47th ed. CRC Press, Cleveland, OH, 1967.

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Appendix K Drago E a n d C Parameters

925

926

Appendix K

Drago E and C Parameters

Parameters Acid 1. Iodine 2. Iodine monochloride b 3. Iodine monobromide b 4. Thiophenol b,e

5. P-tert-Butylphenol 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

p-Methylphenol c Phenol p-Fluorophenol p-Chlorophenol m-Fluorophenol m-Trifluoromethylphenol tert-Butyl alcohol b Trifluoroethanol Hexafluoroisopropyl alcohol Pyrrole b (C4H4NH) Isocyanic acid b (HNCO) Isothiocyanic acid (HNCS) Boron trifluoride b'f'/ Boron trifluoride (g)b,f Boron trimethyl f Trimethylaluminum Triethylaluminum c Trimethylgallium Triethylgallium Trimethylndium Trimethyltin chloridef Sulfur dioxide b Bis (hexafluoroacetylacetonate) copper (II) b Antimony pentachloridef Chloroform 1-Hydroperfluoroheptane [CF5 (CF2)sH] b

CA (Kcal/mole)ll2

EA (Kcal/mole)ll2

1.00 0.830 1.56 0.198 0.387 0.404 0.442 0.446 0.478 0.506 0.530 0.300 0.434 0.509 0.295 0.258 0.227 3.08 1.62 1.70 1.43 2.04 0.881 0.593 0.654 0.0296 0.808 1.40 5.13 0.150 0.226

1.00 5.10 2.41 0.987 4.06 4.18 4.33 4.17 4.34 4.42 4.48 2.04 4.00 5.56 2.54 3.22 5.30 7.96 9.88 6.14 16.9 12.5 13.3 12.6 15.3 5.76 0.920 3.39 7.38 3.31 2.45

Number of heats used to determine the parameters for the specified acid. b Tentative value calculated from limited data or data limited to bases with similar C/E ratios. In latter cases, can be confidently used only with bases with C/E ratios less than 4.0. c Tentative value calculated from estimated enthalpies. d P a r a m e t e r is a standard. e The number of enthalpies estimated from infrared frequency shifts agree with these parameters. f Steric effects commonly encountered. g Accuracy of input data estimated to be at best 10%. h Marginal and conditional standard derivations. / Data from 1,2-dichloroethane displacement reactions. a

Appendix K

92~

Parameters

CB (Kcal/mole) 1/2

EB (Kcal/mole) 112

Base 1. Pyridine 2. Ammonia 3. Methylamine 4. Dimethylamine 5. Trimethylamined 6. Ethylamine b 7. Diethylamine b 8. Triethylamined 9. Acetonitrile 10. Chloroacetonitrile 11. Dimethylcyanamide 12. Dimethylformamide 13. Dimethylacetamide 14. Ethyl acetate 15. Methyl acetate 16. Acetone 17. Diethyl ether d 18. Isopropyl ether b'd 19. n-Butyl ether d 20. p-Dioxane [(CH2)402] 21. Tetrahydrofuran [(CH2)40] 22. Tetrahydropyran 23. Dimethyl sulfoxide 24. Tetramethylene sulfoxide [(CH2)4SO] 25. Dimethyl sulfide 26. Diethyl sulfide 27. Trimethylene sulfide [(CH2)3S] 28. Tetramethylene sulfide 29. Pentamethylene sulfide 30. Pyridine N-oxide 31. 4-Methypyridine N-oxide 32. 4-Methoxypyridine N-oxide ~ 33. Tetramethyluread 34. Trimethylphosphine ~ 35. Benzene 36. p-xylene 4 37. Methylene

6.40 3.46 5.88 8.73 11.54 6.02 8.83 11.09 1.34 0.530 1.81 2.48 2.58 1.74 1.61 2.33 3.25 3.19 3.30 2.38 4.27 3.91 2.85 3.16 7.46 7.40 6.84 7.90 7.40 4.52 4.99 5.77 3.10 6.55 0.707 1.78 2.19

1.17 1.36 1.30 1.09 0.808 1.37 0.866 0.991 0.886 0.940 1.10 1.23 1.32 0.975 0.903 0.987 0.963 1.11 1.06 1.09 0.978 0.949 1.34 1.38 0.343 0.339 0.352 0.341 0.375 1.34 1.36 1.37 1.20 0.838 0.486 0.416 0.574

This Page Intentionally Left Blank

Appendix L Hildebrand Solubility P a r a m e t e r a n d Hydrogen Bond Index

Alphabetical List of Solvents with Values of Hildebrand Solubility P a r a m e t e r 8 and Hydrogen Bond Index (1-10) ~/ Solvent

8

~/

Acetone Acetonitrile Acetyl acetone Acrylonitrile Amyl acetate (n) Amyl alcohol (n) Amyl chloride Amyl formate (n) Amyl formate (iso) Aniline

10.0 11.9 9.5 10.5 8.5 10.9 8.3 8.5 8.0 11.8

5.7 4.5 5.2 4.3 5.4 8.9 2.7 4.7 4.7 8.7

Benzaldehyde Benzene Benzonitrile Benzyl alcohol Butyl acetate (n) Butyl acetate (n), 90-92% Butyl alcohol (n) Butyl alcohol (iso) Butyl alcohol (sec) Butyl bromide (n) Butyl n-butyrate (n) Butyl (iso) butyrate (iso) Butyl "Carbitol ''2. Butyl "Carbitol" acetate 2 Butyl "Cellosolve ''2

9.4 9.2 8.4 12.1 8.5 8.5 11.4 10.5 10.8 8.7 8.1 8.0 8.9 8.5 8.9

5.2 2.2 5.0 8.9 a 5.4 5.4 a 8.9 a 8.9 a 8.9 2.7 5.4 5.4 6.9 a 5.6 a 6.9 a

929

930

Appendix L Solvent Butyl "Cellosolve" acetate 2 Butyl formate (n) Butyl lactate (n) Butyl propionate (n) Butyraldehyde (n) Butyrolactone (T) Butyronitrile

8.5 8.7 9.4 8.8 9.0 12.6 10.5

4.9 4.7 5.2 6.4 5.7 5.0

Capronitrile "Carbitol," low gravity 2 "Carbitol" solvent 2 "Carbitol" acetate 2 Carbon disulfide Carbon tetrachloride "Cellosolve" solvent 2 "Cellosolve" acetate 2 Chlorobenzene Chloroform Cyclohexane Cyclohexanol Cyclohexanone Cyclopentanone

9.4 9.6 10.9 8.5 10.0 8.6 9.9 8.7 9.5 9.3 8.2 11.4 9.7 10.4

5.0 6.9 a 7.6 a 5.6 a 2.2 2.2 6.9 a 5.6 a 2.7 2.2 2.2 8.9 a 6.4 5.2

6.6 9.2 6.1 9.8 9.0 8.2 9.9 8.8 14.2 8.9 8.5 9.6 8.5 8.5 9.6 7.4 10.6 8.8 8.6 15.4 7.8 6.9 8.0 10.8 9.7 9.9 12.1

2.2 6.9

Decane (n)

Diacetone alcohol Dichlorodifluoromethane Dichloroethyle ether (fl,fl') Dichloropropane (1,2) Dichloropropane (2,2) Diethylacetamide (N,N) Diethyl carbonate Diethylene glycol Diethylene glycol monobutyl ether Diethylene glycol monobutyl ether acetate Diethylene glycol monomethyl ether Diethylene glycol monoethyl ether acetate Diethylene glycol monoethyl ether acetate Diethylene glycol monomethyl ether Diethyl ether Diethylformamide (N,N) Diethyl ketone Diethyl oxalate Diformylpiperazine (N,N') Diisobutyl ketone Diisopropyl ether Diisopropyl ketone Dimethylacetamide (N,N) Dimethylaniline Dimethyl carbonate Dimethylformamide (N,N)

5.6 a

2.5 a

5.2 2.7 2.7 6.6 4.0 8.5 a

6.9 5.6 6.9 5.6 5.6 6.9 6.9 6.4 5.0 4.5 >9.4 a 5.2 6.6 5.2 6.6 7.3 4.0 6.4

Appendix L

931

Solvent Dimethyl sulfoxide Dioxane (1,4) Dipentene Dipiopyl sulfone

13.0 9.9 8.5 11.3

5.0 5.7 2.7 5.0

Ethyl acetate, 99% Ethyl acetate, 85-88% Ethyl alcohol, undenatured, abs. Ethyl alcohol, 2B? 95% Ethyl alcohol, C c, 95% Ethyl amylketone Ethylbenzene Ethyl benzoate Ethylbutyl (2) alcohol Ethyl cyanoacetate Ethylene carbonate Ethylene dichloride Ethylene glycol Ethylene glycol diacetate Ethylene (glycol monobutyl ether) Ethylene (glycol monobutyl ether) acetate Ethylene glycol monomethyl ether Ethylene(glycol monomethyl ether) acetate Ethylene oxide Ethylformamide (N) Ethyl formate Ethylhexyl (2) alcohol Ethyl lactate Ethyl silicate

9.1 9.1 12.8 13.6 13.1 8.2 8.8 8.2 10.5 11.0 14.7 9.8 14.6 10.0 8.9 9.9 10.8 9.2 11.1 13.9 9.4 9.5 10.0 6.9

5.2 5.2 a 8.9 a 8.9 a 8.9 a 5.0 2.7 4.5 8.9 a 4.0 4.0 2.7 9.6 5.2 6.9 6.9 6.9 5.6 5.8 6.5 5.2 8.9 a 4.7 2.5 a

Formamide FREON | 11 propellant (FREON MF solvent) 5 FREON 12 propellants 5 FREON TF solvent 5 FREON TMC solvent 5 Furfural

19.2 7.8 6.1 7.2 8.5 11.2

>16.2 a 2.5 a 2.5 a 2.5 a 2.6 a 4.7

Glycerol

16.5

8.5 a

Heptane (n) Hexane (n) Hexyl alcohol (n) Hexylene glycol

7.3 7.3 10.7 9.7

2.2 2.2 8.9 a 8~ a

Isoamyl Isobutyl Isobutyl Isobutyl Isobutyl Isodecyl

10.1 8.3 10.5 7.5 8.0 9.6

8.9 5.5 8.9 a 5.5 a 5.4 8.5 a

alcohol acetate alcohol heptyl ketone isobutyrate alcohol

932

Appendix L

Solvent Isooctyl alcohol "Isopar" E isoparaffinic solvent 6 "Isopar" G isoparaffinic solvent 6 "Isopar" H isoparaffinic solvent 6 "Isopar" K isoparaffinic solvent 6 "Isopar" L isoparaffinic solvent 6 "Isopar" M isoparaffinic solvent 6 Isophorone Isopropyl acetate Isopropyl alcohol, 99% Kerosene

10.0 7.1 7.2 7.1 7.2 7.3 7.4 9.4 8.4 11.5 7.2

8.5 a 2.2 a 2.2 a 2.2 a 2.2 a 2.2 a 2.2 a 7.0 a 5.3 8.9 a 2.2

Mesityl oxide Methyl acetate Methyl alcohol Methyl amyl acetate Methyl n-amyl ketone Methyl "Carbitol ''2 Methyl "Cellosolve''2 Methyl "Cellosolve" acetate 2 Methyl chloroform Methyl cyclohexane Methyl cyclohexanone Methylene chloride Methyl ethyl ketone Methyl formate Methyl isoamyl ketone Methyl isobutyl carbinol Methyl isobutyl ketone Methyl isopropyl ketone Methyl nonyl ketone Methyl propionate Methyl n-propyl ketone Mineral spirits (Sun spirits 7) Mineral spirits, low odor

9.0 9.6 14.5 8.0 8.5 9.6 10.8 9.2 8.3 7.8 9.3 9.7 9.3 10.2 8.4 10.0 8.4 8.5 7.8 8.9 8.9 7.6 6.9

5.7 5.2 8.9 a 5.4 5.0 6.9 a 6.9 a 5.6 a 2.2 a 2.2 5.4 2.7 5.0 5.5 4.9 8.8 4.9 5.1 5.0 5.2 5.1 2.2 a 2.2

Nitrobenzene Nitroethane Nitromethane Nitropropane (1) Nitropropane (2)

10.0 11.1 12.7 10.3 9.9

3.2 3.1 3.1 3.1 3.1

Octyl alcohol (n) Octyl alcohol (iso)

10.3 10.0

8.9 a 8.5 a

Pentane (n) Perchlorethylene Pine oil Piperidine

7.0 9.3 8.6 8.7

2.2 2.2 a 2.2 a 10.9

Appendix L

Solvent

6

933

~/

Propiolactone Propionitrile Propyl acetate (n) Propyl acetate (iso) Propyl alcohol (n) Propyl alcohol (iso), 99% Propylene 1,2-carbonate Propylene glycol Propylene oxide Propyl propionate (n) "Pentoxone" solvent 1 Pyridine

13.3 10.8 8.8 8.4 11.9 11.5 13.3 15.0 9.2 8.5 8.5 10.7

5.0 5.0 5.3 5.3 8.9 a 8.9 a 4.0 9.4 5.8 4.7 5.5 a 8.7

Quinoline

10.8

8.7

"Solvesso" 100 aromatic solvent 6 "Solvesso" 150 aromatic solvent 6 Styrene Sun Spirits 7

8.6 8.5 9.3 7.6

3.8 3.8 2.7 2.2 a

Tetrahydrofuran THF 4 "Texanor' solvent 3 Toulene Trichlorethane (1,1,2) Trichlorethane ( 1,1,2) Trichlorofluoromethane Trichlorotrifluoroethane Turpentine

9.1 8.2 8.9 9.6 9.6 7.8 7.2 8.1

5.3 5.5 a 3.8 2.7 2.7 2.5 a 2.5 a 3.8 a

VM & P n a p h t h a

7.6

2.2

Water Xylene

23.5 8.8

16.2 3.8

Estimated, no data on O D shift available, b Typical value for technical grade. c Denatured alcohol containing by volume 0.95% methyl isobutyl ketone, 4.05% ethyl acetate (99%) and S. D. A. 3A (95%) Data taken from Paint Technology Manuals, oil TD881:058 no. AP-103 and Oil and Colour Chemists' Association (Great Britain), Solvents, oils, resins, and driers. 2nd ed., London. Chapman & Hall, 1969. a

Key to suppliers: 1 Shell Chemical Co. 2 Union Carbide Corp. 3 Eastman Chemical Products, Inc. 4 Du Pont, Industrial Chemicals Dept. 5 Du Pont, Freon | Products Div. 6 Humble Oil & Refining Co. 7 Sun Oil Co.

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Appendix M Hydrated Cation Radii

H y d r a t e d Cation Radius for Various Metals for Use with E q u a t i o n 9.64

Metal in oxide

Valence

Hydrated cation radius (A )

Mg Fe Co Ni Pb Cd La Be Cu Zn Y A1 Th Pu Fe Hg Ce Zr Cr Ti U Sn Mn Si W

+2 +2 +2 +2 +2 +2 +3 +2 +2 +2 +3 +3 +4 +4 +3 +2 +4 +4 +3 +4 +4 +4 +4 +4 +4

3.67 3.44 3.06 2.97 2.97 3.93 4.02 2.50 3.06 2.50 3.75 3.75 5.00 4.78 3.67 2.39 5.00 4.68 3.00 4.68 4.89 3.24 3.28 3.06 2.57

Recalculation of d a t a from Parks, G. A., Chem Rev. 65, 177 (1965).

935

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Index

Accumulation term, 86 Acheson process, 38, 167 Acid-base, surface, 388 Actuators, 4 Adhesion, wetting, 364, 366, 369 Adiabatic flame temperature, 261,265 Adipic acid, precipitation, 218-219 Adsorption, 379-380, 382-383; see a l s o Adsorption isotherms BET, 65-66 binary solvent adsorption, 384-386 ceramic paste formation, 398-403 chemical adsorption, 64 crystal habit modification, 216-220 diffuse double layer, 390-394 gas adsorption, 52, 64 heat of adsorption, 409-410 ionic surfactants, 398-403 ions, 386-389 diffuse double layer, 390-394 Stern layer adsorption, 389-390 surface charge, 394-395 isoelectric point, 398, 412-413 Langmuir, 64-65, 217, 379, 382 multilayer, 65-66, 401-402 polymers, 403-410, 453 Stern layer adsorption, 387, 389-390 zeta potential, 387, 393-398, 470, 471 ceramic paste formation, 395-398 geothite, 393-394 measurement, 395-398 sedimentation, 503 titania, 443-444

Adsorption free energy, 389 Adsorption isotherms, 380-384, 392 excess adsorption, 384-385 Gibbs, 379, 380-381 Langmuir, 64-65, 217, 379, 382, 383, 386, 401 solid-liquid interface, 382-384, 387, 411 Aero-gels, 354-355 Aerosols, 496 Aggregate networks, 345 Aggregate number density, 378 Aggregates, 557 comminution, 374 fractal aggregates, 819-824 fractal dimensions, 243, 479-480 population balance, 476 shape, 214-216 Aggregation, 557 diffusion limited (DLA), 196-210, 228, 229, 231 doublet formation, 467-488 large aggregate clusters, 475-481 models, 214-215 particle size distribution, 229-233 batch reactor, 240-244 continuous stirred tank reactor, 233-240 population balance, 230, 249, 289-296, 476 quenching, 296-301 reaction limited (RLA), 481, 583 shear aggregation, 233, 486-488

937

938

Index

Aggregation (continued) simultaneous precipitation and coaggregation, 244, 246-248 sols and gels, 348 Aggregation rate, 230 Aggregation rate constant, 232 Air classifiers, 115-117 Alkoxide precursors, 343-344 Alumina colloidal stability, 469, 470 comminution, 108, 109 compaction, 574, 656, 657, 660 densification, 783 die pressing, 657, 659 electric arc melting, 355 gel, 338 grain boundary migration, 843 hot isostatic pressing, 866 microstructure of solution, 483, 485 properties, 892, 936 surface energy, 786 viscosity, 561-562 sintering, 783, 788, 808 spray dry, 312 synthesis, 35 plasma synthesis, 267 uses, 36 zeta potential, 470 Aluminum hydrated cation radius, 937 thermodynamic data, 909 Aluminum minerals, 34 Aluminum nitride, 39 plasma synthesis, 267 properties, 892, 936 Alusite, 34 Amblygonite, 34 Amino group, hydrolysis, 399, 401 Ammonium polyuranate, precipitation, 235 Angle of repose, 598-599 Anglesite, 35 Anisotropic particles dip coating, 641 rheology, 577-583 sedimentation casting, 636 suspensions, 551-554 Antimony, thermodynamic data, 907 Apatite, isoelectric point, 399 Apparent bypass, classifiers, 125-126 Archimedian growth spiral, 204-206 Argon, thermodynamic data, 905

Array formulation, comminution, 106-115 Arsenic, thermodynamic data, 907 Atomization, 312, 313-319 Attractive interaction energy, 422-424 Hamaker constant, 424-427, 428, 722 polymer-coated particles, 427 Average density, green body, 719

Baddeleyite, 38 Ball mill, 97, 100, 109 mechanical losses, 102-103 Barium, thermodynamic data, 912 Barium ferrite, 319 Barium minerals, 35 Barium sulphate, 184, 187 Barium titanate, 4 grain boundary migration, 843 Eh diagram, 415 polymer adsorption, 409 polymer oxidation catalyzed by, 748-749 sintering, 803, 810, 841 zeta potential, 474 Batch comminution (grinding), 95, 107-110 array formulation, 111 conservation of length, 89-91 Batch mill, array formulation, 111 Batch precipitation reactors, 226-229 aggregation, 240-244 population balance, 240-241 Bayer process, 35 Beer-Lambert law, 272 Bernoulli equation, 316 Beryllia grain boundary migration, 843 hot isostatic pressing, 866 properties, 892, 936 surface energy, 786 Beryllium, 911,937 BET adsorption, 65-66 BET equation, 65 Binary solvent adsorption, 384-386 Binder burnout, 681, 729-731, 771-772 heat transfer, 731-732 kinetics, 730, 752-755 carbon removal, 762-767 oxidative degradation, 749-750, 755-758 pyrolysis without oxygen, 761-762

Index

thermal degradation, 737-738 volatiles loss, 758-761 mass transfer, 732-733 polymer degradation oxidative, 738-739, 740-741 thermal, 733-738, 740-741 stresses induced, 767-768 thermal stress, 768-770 volatile flow, 770-771 volatiles, 730, 731, 734-735, 738, 750-752 flow, 770-771 kinetics, 758-761 Binders, 725-726, 730 Bingham plastic fluid, 548-549, 620-628 Birth function, 105-106 aggregation, 230, 232, 233 comminution and classification, 134 rapid flocculation theory, 294 Bismuth, thermodynamic data, 907 Bispherical coordinates, PoissonBoltzmann equation, 433, 434-437 Blending, powder samples, 75-77 Boehmite, 36 Boiling point elevation, 509, 510-511 Boltzmann distribution of ions, 391 Bond percolation, 482-483 Bond's law, 101, 102 Boron, thermodynamic data, 909 Boron carbide, 38, 39, 157-158 plasma synthesis, 267, 274 tensile strength, 880 Boron nitride, 39 plasma synthesis, 267 properties, 892, 936 Boron oxide, carboreduction, 161-162 Borosilicate glass, properties, 892, 936 Boundary layer, 155, 686 Box kiln, 777-778 BPS analysis, 199 Bragg equation, 526 Brittle index, 880 Bromine, thermodynamic data, 906 Bronze, history, 10 Brownian aggregation, 214 Brownian coagulation, 467-473 Brownian diffusion, 495, 504-509, 584 Brownian diffusion coefficient, 504 Brownian flocculation, 473-475 Brownian motion, 422, 504 Brunauer, Emmett, and Teller (BET) adsorption, 65-66

939

Bulk diffusion sintering a string of spheres, 798-800 vacancies, 791-795 Buoyant force, 497 Burton-Prim-Slichter (BPS) analysis, 199 Bypass, classifiers, 125-126 Cadmium, 909, 937 Cadmium sulphide, 39 Cake clogging, 617-618 Calcination reactions, thermodynamics, 141-142, 343 Calcite, 34-35, 142 Calcium, thermodynamic data, 911-912 Calcium carbonate compaction, 660 decomposition, 142, 158-160 Calcium oxide, surface energy, 786 Capacitors, 4 Capillary stress, green body drying, 716-718 Carbides, 38 Carbon, thermodynamic data, 907-908 black, 252 removal, kinetics, 762-767 Carbon removal, binder burnout, 762-767 Carbothermic reduction boron oxide, 161-162 silica, 167 tungsten oxide, 168 Cascading, continuous stirred tank reactors, 224-226 Caustic soda, synthesis, 37 Celadon green, 14 Centrifugal casting, 631-636 Centrifugal sedimentation, 503 Centrifuge, classification, 115 Ceramic film stress during drying, 718 synthesis metal organic decomposition, 339-340 sol-gel synthesis, 344-355 Ceramic green body, see Green bodies; Green body drying; Green body formation Ceramic machining, 673, 876-878, 880-882 Ceramic paste formation, 357-358, 359, 416-417

940

Index

Ceramic paste formation ( c o n t i n u e d ) adsorption, 379-380 binary solvent adsorption, 384-386 diffuse double layer, 390-394 ionic surfactants, 398-403 ions, 386-389 isoelectric point, 398 isotherms, 380-384 polymers, 403-410 Stern layer adsorption, 389-390 surfactant selection, 410-413 zeta potential, 395-398 chemical stability, 414-416 deagglomeration, 359 comminution, 374-375 ultrasonification, 375-378 wetting, 359, 360-374 heat of wetting, 370-373 internal wetting, 368-369 partial wetting, 368 rough solid surface, 368 solvent selection, 373-374 Ceramic pastes green body formation, 643-644 extrusion molding, 644-651 injection molding, 651-653 rheology, 585-590 visco-elastic fluid behavior, 667 Ceramic powder characterization, 43-44, 78 blending powder samples, 75-77 density, 63 morphology, 56-63 particle size comparison of samples, 73-75 cumulative distribution, 66-67 distribution, 66-67 log-normal distribution, 67, 69-72 mean particle size, 52-55 normal distribution, 68-69 Rosin-Rammler distribution, 72, 108 size distribution accuracy, 55-56 statistial diameters, 48-52 sampling, 44-47 surface area, 64-66 Ceramic powder processing, 1-2 adsorption, s e e Adsorption binder burnout, s e e Binder burnout characterization, s e e Characterization comminution, s e e Comminution; Grinding concepts, 4-6

deagglomeration, s e e Deagglomeration drying, s e e Drying finishing, s e e Finishing green bodies, s e e Green bodies; Green body drying; Green body formation heat treatment, s e e Binder burnout; Green body drying machining, s e e Machining paste formation, s e e Ceramic paste formation polymer burnout, s e e Binder burnout sintering, s e e Sintering wetting, s e e Wetting Ceramic powders, s e e a l s o Dry ceramic powders characterization, s e e Ceramic powder characterization classification, s e e Classification history, 8-27, 538 processing, s e e Ceramic powder processing properties chemical stability, 414-416 HLB values, 412 isoelectric point, 398, 399, 412-413 raw materials, s e e Raw materials suspensions, s e e Ceramic suspensions synthesis, s e e Ceramic powder synthesis Ceramic powder synthesis Acheson process, 38, 167 alumina, 35, 267 Bayer process, 35 caustic soda, 37 freeze drying, 341-344 gas-phase reactants, 259-264, 307 aggregation, 293-305 chemical vapor decomposition, 266 collisional growth theory, 279-281 dispersion model, 284-292 flame synthesis, 260-266 furnace decomposition, 260-263, 266, 271 homogeneous nucleation, 274-279 kinetics, 267-274 laser synthesis, 260-263, 266, 267, 268, 272, 273 particle shape, 305-307 plasma synthesis, 260-263, 266, 267, 273, 303-304 population balance, 282-284, 293-300 liquid-phase precipitation, 179-183, 249

Index

coprecipitation, 244-248 crystal shape, 210-220 growth kinetics, 183-210 nucleation kinetics, 193-192 particle size distribution effects, 229-244 magnesia, 36-37, 267 melt solidification, 355-357 metal borides, 39, 267, 268 metal carbides, 38, 267, 268, 274 metal halides, 265 metal nitrides, 39, 267, 268 metal oxides, 267 population balance, 85-93 precursor solution chemistry, 346-350 sialon, 266 silica, 260, 264, 267, 274, 306 silicon carbide, 38, 167, 266, 267, 273 silicon nitride, 38-39, 267 soda ash, 37 sol-gel synthesis, 344-355, 817-818 solid-phase reactants, 139-141 fluid-solid reactions, 141-166, 176-177 kinetics, 151-162 liquid-solid reactions, 151 nitridation reactions, 148 oxidation reactions, 144-147 reduction reactions, 147 solid-solid reactions, 166-177 thermodynamics, 141-151 Solvay process, 37 spray drying, 311-335 atomization, 312, 313-319 droplet drying, 312, 319-331 equipment design, 334-335 gas-droplet mixing, 331-333 spray roasting, 313, 335-339 titania, 37, 240-241, 260, 264, 267, 290-292, 306 titanium tetrachloride, 37 zinc oxide, 37, 144, 267 zirconia, 38, 267 Ceramic strength, machining, 877-879 Ceramic suspensions, 612 colligative properties, 497, 498, 509-516 osmotic pressure, thermodynamics, 517-526 sedimentation, 495, 497-504 colloid properties, 495-497, 532 Brownian diffusion, 495, 504-509 colligative properties, 495, 497, 509-516

941

ordered suspensions, 516-532 particle movement, 497 particle size, 496 particle structure, 495 sedimentation, 495, 497-504 colloid stability, 421-422, 488-489 alumina, 469, 470 ceramic systems, 448, 488-489 interaction energy, 422-467 kinetics of coagulation and flocculation, 467-488 silica, 469, 472-475 surfactants, 448 titania, 469-472 green body formation, 612-613 centrifugal casting, 631-636 dip coating, 638-643 electrodeposition, 636-638 filter pressing, 618-620 sedimentation casting, 629-636 slip casting, 613-618 tape casting, 620-629 mechanical properties, 541, 543 ordered suspensions, 516-532 defects, 528-530 ordered domain size, 530-532 structure, 526-527 rheology, 531, 537-539, 543, 550-551, 613 colloidally stable concentrated suspensions, 564-583 concentrated ceramic systems, 562-585 concentrated polymer solutions, 562-564 dilute suspension viscosity, 551-562 unstable concentrated suspensions, 583-585 Cerium oxide, 227 Cerium, hydrated cation radius, 937 Cerussite, 35 Cesium, thermodynamic data, 913 Chain scission, binder burnout, 731,734, 739 Characterization ceramic green bodies after drying, 718-726 after formation, 674-675 Charged plates, electrostatic repulsion, 429-431, 432 Charged spheres, electrostatic repulsion, 432-440 Chatter cracks, 878

942

I~dex

Chemical adsorption, surface area measure, 64 Chemical machining, 876 Chemical stability, powder in a solvent, 414-416 Chemical vapor decomposition, 266 Chemical vapor deposition (CVD), 892 China, ceramic history, 8-26 Chlorine, thermodynamic data, 905 Chromia, 36 grain boundary migration, 843 isoelectric point, 399 Chromic oxide, 36 Chromite, 36 Chromium hydrated cation radius, 937 thermodynamic data, 911 Chromium carbide, 38 Chromium ion, polymer oxidation catalyzed by, 742-743 Chronomal analysis, 227-228 Classification, 95-96, 115, 136 collision, 120-122 comminution and classification circuit, 129, 132-135 equipment, 115-117, 127-129 forces, 117-120 nonspherical particles, 122 Classification function, 133, 222-223 Classifiers, 117 air classifiers, 115-116 collision, 120-122 comminution and classification circuit, 129, 132-135 dry, 115-117 efficiency, 124-127 forces, 117-120 nonspherical particles, 122 population balance, 133-134 recovery, 123-124 size selectivity, 123, 133 wet, 115, 117, 127-129 Clausius-Clapeyron equation, 143, 322 Clays, 27-31 history, 8-9 Cluster-cluster aggregates, 215-216 CMC, s e e Critical micelle concentration Coaggregation, simultaneous precipitation and coaggregation, 244, 246-248 Coagulation, 466, 467, 557 Brownian coagulation, 467-473 half-life, 467-468 kinetics, 467-488

Coarsening, 784, 797 Coating, 881-882 supersaturation, 191-192 Cobalt, 910, 937 Cobalt oxide, grain boundary migration, 843 Coefficient of pressure at rest, 592-594 Coefficient of thermal expansion, table of values, 936 Colburn analogy, 152, 154 Colligative properties ceramic suspensions, 497, 498, 509-516 solutions, 497, 509 suspensions, 497, 509 Collision, classifiers, 120-122 Collisional growth theory, 279-281 Colloid properties ceramic suspensions, 495-497, 532 Brownian diffusion, 495, 504-509 colligative properties, 495, 497, 509-516 ordered suspensions, 516-532 osmotic pressure, 513-516 particle movement, 497 particle size, 496 particle structure, 495 sedimentation, 495, 497-504 Colloid stability alumina, 469, 470 ceramic suspensions, 421-422, 488-489 in ceramic systems, 448, 488-489 interaction energy, 422-467 kinetics of coagulation and flocculation, 467-488 silica, 469, 472-475 surfactants, 448 titania, 469-474 Colloid stability factor, 470 Colloid stability ratio, 468, 469, 481, 584 Combustion sintering, 861 Comminution, 95-96, 135-136 aggregates, 374 array formulation, 110-115 batch comminution, 107-110 array formulation, 111 conservation of length, 89-91 birth and death functions, 105-107 comminution and classification circuit, 129, 132-135 conservation of volume, 105-106 deagglomeration, 374-375 efficiency, 102-103

Index

energy required for size reduction, 101-102 equipment, 96-100 population balance models, 103-110 population balance, 103-110 Comminution efficiency, 102-103 Compact body, 777, 782 deformation, 590, 594 fractal, 819 nonfractal, 820 Compaction equation, 663 Compaction pressure, 663 Compressibility, 520, 521, 522, 565, 708 Computer modeling, aggregate shape, 214-215 Condensation rate, 275 Configurational osmotic pressure, 514 Conservation of length, population balance, 89-91 Conservation of volume, comminution, 105-106 Consolidated layer thickness, 619 Constant rate period drying, 706 drying rate, 693-695 drying shrinkage, 695-697 flow stress, 713-715 gel drying, 353 stresses, 706 Constitutive equations, 544 dry powders, 545 fluids, 545-550 Contact nucleation, 192 Continuity equation, 543-544 Continuous grinding, array formulation, 112-115 Continuous stirred tank reactor (CSTR), 220-223 aggregation, 233-240 cascaded, 224-226 population balance, 236, 239 with recycle, 223-224 Continuous tunnel kilns, 778 Cooldown, sintering, 779 Cooldown period, 706 Cooling, after sintering, 867-869 Cooper-Eaton equation, 660 Copolymers, 451 Copper hydrated cation radius, 937 oxylate, 240 polymer oxidation catalyzed by, 742 thermodynamic data, 909-910 Copper acetate meta arsenate, 245

943

Coprecipitation, 244-248 Cordierite, properties, 892, 936 Corundum, 34 Coulomb yield criterion, 543, 597-598, 602, 661 Coulter counter, 52 Cracks drying, 682, 692, 705-718, 870 machining, 877-878 Creep compliance, 587 ceramic paste, 588-589 sintered ceramic, 826 Crescent cracks, 878 Cristobalite, 30, 32 Critical flocculation temperature, 456 Critical indentation depth (CID), 881 Critical micelle concentration (CMC), 400-401 Critical overlap radius, 583 Critical point, gel drying, 353 Critical radius, 275 Critical Weber number, 315, 317 Cross equation, 558, 567, 623 Cross-linking, binder burnout, 742 Cross-viscosity, 550 Crushers, 96-100 Crystal growth, see a l s o Liquid-phase precipitation kinetics, 193-195; see a l s o Nucleation, kinetics aggregation, 229-246 diffusion controlled growth, 196-201 growth effectiveness factor, 201 growth rates, 207, 208-210 screw dislocation growth, 204-208 stages of growth, 196 surface nucleation of steps, 202-203 two-dimensional growth of surface nuclei, 203-204 mixing, 199 supersaturation, 199, 207, 208 Crystallizer, constant stirred tank crystallizer, 88-89 Crystals, see Crystal growth; Crystal shape Crystal shape, 210 aggregate shape, 214-216 equilibrium shape, 210-212 impurities, 216-220 kinetic shape, 212-214 diffusion shape, 213-214 growth spiral shape, 214 surface nucleation shape, 214

944

Index

CSTR, s e e Continuous stirred tank reactor Cumulative distribution function, 221 Cumulative mass basis, population balance, 92-93, 104-105 Cumulative particle size distribution, 575 Cumulative progeny matrix, 111 Cuprous oxide, polymer oxidation catalyzed by, 742 Cyclization, binder burnout, 742 Cyclone classifier, 115, 127-129 Cyclonic separators, 309 Cylindrical green body, drying, 702-703

Damkohler number, 201 Darcy's law, 614, 615, 686, 698, 737 DCCA, s e e Drying control chemical additives Deagglomeration, 359 comminution, 374-375 ultrasonification, 375-378 Death function aggregation, 230, 232, 233 comminution and classification, 134 rapid flocculation theory, 294 Death rate, grinding, 105 Debye-Huckel linearization, 430-431 Decreasing rate period, 706 flow stress, 716 Defects die pressing, 669 green body characterization, 674-675 ordered suspensions, 528-530 Deformation, 590-592 coefficient of pressure at rest, 592-594 compact body, 590, 594 plastic body deformation, 590-591, 595-596 visco-elastic solids and fluids, 667 yield criteria for packings, 596-597 Deformation tensor, 545 Densification, 781, 784, 859 Densification rate, 812 Density ceramics, table of values, 936 green body, 784 Density distribution function, 453, 454 Depletion flocculation, 465 Depletion interaction energy, 464-465 Depletion stabilization, 450, 465 Depletion zone, polymer segments, 465 Derjaquin approximation, 437, 461

Derj aquin-Landau-Verwey-Overbeek (DLVO) theory, 465, 471 Diatomite, 32 Die ejection, 667-670 Die pressing, 653, 656-670 Diffuse double layer, 387 adsorption, 390-394 electro-osmosis, 395, 396 Diffusion Brownian diffusion, 495, 504-509, 584 bulk diffusion, 791-795, 798-800 in ceramic solids, 344 equilibrium with sedimentation, 505-506 Fick's laws, 196-198, 284-285, 505, 690, 691 green body drying, 698 nonspherical particle diffusion, 504-505 rotational diffusion, 506-509, 552-554 Diffusion coefficient, 504 Knudsen diffusion coefficient, 152, 698, 703, 760 Diffusion coefficient ratio, 468 Diffusion controlled crystal growth, 196210, 229, 231 growth chronomals, 228 Diffusion limited aggregation (DLA), 481-482, 583 Diffusion shape, 213-214 Diffusive flux, 152, 759 Dilatent fluid, 548, 549 Dimensionless double layer thickness, 572 Dimethylchlorosilane, thermal decomposition, 266 Dip coating, 638-643 Disc atomizers, 318-319 Discrete mass basis, population balance, 91-93, 104, 107 Disengagement rate, 563 Dispersed plug flow, 284 Dispersion, 410; s e e a l s o Aggregation HLB system, 411 surfactant selection, 410-413 ultrasonic dispersion, 375-378 Dispersion model, gas synthesis reactors, 284-292 Dispersion number, 286 Dispersions, s e e a l s o Ceramic suspensions doublet formation, 467-475 large aggregate clusters, growth and structure, 475-471

Index

Dissociation reaction, surfactant, 399-400 Dissolution, liquid phase sintering, 845, 853, 857 DLA, s e e Diffusion limited aggregation DLVO theory, 465, 471 Dodecylamine, 400, 401, 403 Dolomite, 35 Domain size, 529-532 Dopants, 260 Double layer thickness, 391-392, 428440, 514 dimensionless, 572 Double metal hydroxides, precipitation, 245 Double pipe heat exchanger, quenching, 297, 298 Double salts, precipitation, 245 Doublet formation, 467-488 Drag coefficient, 119-120 Drag diameter, 50 Drag force, 498, 500 Drago E and C values, 370-373, 410 table of values, 926-927 Drain casting, 613-614 Driving force, sintering, 786-788, 795798, 866 Droplet drying, 312, 319-331 Droplet drying time, 329-331 Dry bag isostatic pressing, 671 Dry ceramic powders green body formation die pressing, 653, 656-670 dry pressing, 1, 2, 653-673, 730 isostatic pressing, 653, 654, 671-673, 782, 865 ramming, 653, 654, 671 tapped density, 654-655 mechanical properties, 541,543, 590-592 coefficient of pressure at rest, 592-594 compact body, 594 constitutive equation, 545 Coulomb yield criterion, 597-598, 602 plastic body, 595-596 yield behavior at low pressure, 599-602 yield criteria for packings, 596-597 Dry classification, 115-117 Drying droplet drying, 312, 319-331 gels, 353-354, 684-685

945

green body, s e e Green body drying shrinkage, s e e Drying shrinkage Drying control chemical additives (DCCA), 353 Drying rate constant rate period, 693-695 falling rate period, 698-702 Drying shrinkage, 684, 690-691, 694 constant rate period, 695-697 thermal stress, 708-713 Drying time, calculation, 700-701 Dry pressing, 1, 2, 653-673, 730 Dynamic shape factor, 58-59 Effective cavitation temperature, 375 Effective diffusion coefficient, 152, 328, 330 porous layer, 756, 862 Effective local stress, 707, 768 Effective pore diffusion coefficient, 717-718 Effective volume fraction, particle rotation, 554 Egypt, ceramic history, 20 Einstein equation, 556 Elastic solid, 588 Elbow classifier, 115 Electrical resistivity, table of values, 936 Electric arc melting, alumina, 355 Electrochemical reactions, reduction potentials, 903-904 Electrodeposition, 636-638 Electrolytes adsorption diffuse double layer, 390-394 reaction with ions, 386-389 osmotic pressure, 511-512 Electro-osmosis diffuse double layer, 395, 396 zeta potential measurement, 396-397 Electrophoresis, zeta potential measurement, 397-398 Electrostatic interaction energy, 431-432 Derjaquin approximation, 437 Electrostatic interaction potential, 566 Electrostatic repulsion, 428 charged plates, 429-431, 432 charged spheres, 432-440 Electrostatic stabilization, 519, 523 Electrosteric stabilization, 448-449, 465 Electro-viscous effect, 554-555 Electro-zone counting, 52 Embryo concentrations, homogeneous nucleation, 186-189, 270-271

946

Index

Emery, 34 Emulsions, 496 Enstatite, 31 Enthalpic stabilization, 446, 447 Entropic stabilization, 446, 447 Equations of motion, 543-550 Equilibrium shape, 210-212 Equivalent spherical diameter, 49 Expansion chamber, classification, 115 Extruder, flow in, 644-646 Extrusion die, flow in, 646 Extrusion molding, ceramic pastes, 644-651

Falling rate period, drying rate, 698-702 Feldspar, 31-32, 399 Ferett's diameter, 48, 51 Ferric ion, polymer oxidation catalyzed by, 742-743 Ferrites, hot isostatic pressing, 866 Fick's laws, 196-198, 280-281, 505, 690, 691 Filter pressing, 618-620 Final stage sintering, 784, 803-809 Finishing, 779-780, 875-876, 888-889 coating and glazing, 882-883 machining, 673, 876-882 First falling rate period, gel drying, 353 First layer adsorption, 64-65 Flame synthesis, 260-266 Flat plate green body, 121, 703 drying, 703-705 flow between, 647-649 Flocculation, 422, 557 Brownian flocculation, 473-475 depletion flocculation, 465 Gibbs free energy, 446 kinetics, 467-488 polymer exclusion zone, 464 Flory-Huggins parameter, 458, 488-489 Flory-Huggins theory, 453, 455 Flory point, 512-513 Flow annular space between two concentric cylinders, 649-651 drying, flow of liquid in pores, 689-690 in extruder, 644-646 in extrusion die, 646 into injection molding die, 651-653 between two flat plates, 647-649 Flow dispersion, gas synthesis reactors, 284-292 Flow stress, green body drying, 713-716

Fluid energy mill, 97, 100 Fluids, constitutive equations, 545-550 Fluid-solid reactions, 139-141, 176-177 kinetics, 151-162, 343-344 limitations, 336-337 reactors, 162-166 thermodynamics multiple reaction systems, 148-151 nitridation reactions, 148 oxidation reactions, 144-147 reduction reactions, 147 Fluid-solid reactors, 162-166 Fluorine, thermodynamic data, 905 Fluorine minerals, 34 Fluorite, 34 Fluorspar, 34 Fluxes, melt solidification, 355 Fluxing agent, feldspar, 31 Flux of heat, binder burnout, 731 Force balance, 497, 498 Forces, classification, 117-120 Fourier transforms, shape regeneration, 59-60 Fractal ceramic powders, 60-62, 214 Fractal dimensions, 584 aggregates, 243, 479-480 Fractal sintering model, 819-824 Fractional surface coverage, 389 Free energy adsorption, 389 Gibbs free energy, 148-151,446, 510, 518 oxidation reaction, 145-147 standard free energy, 142-414 Free-falling diameter, 50 Free radicals, binder burnout, 739 Freeze drying, ceramic powder synthesis, 341-344 Freezing point depression, 509, 510 Freezing time, 342-344 Friction factor, 499 Funicular condition, 686 Furnace decomposition, 260-263, 266, 271 Fusion, Gibbs free energy, 510

Galena, 35 Gamma function, 72, 481 table of values, 893 Gas adsorption, 52, 64 Gas classifier, 115, 118-119 Gas-droplet mixing, 331-333

Index

Gas mixing quench, 304-305 Gas-phase powder synthesis, 259-264, 307 aggregation population balance, 293-300 quenching, 300-305 chemical vapor decomposition, 266 collisional growth theory, 279-281 dispersion model, 284-292 multipoint nucleation, 289-290 single-point nucleation, 288-289 flame synthesis, 260-266, 271 furnace decomposition, 260-263, 266 homogeneous nucleation, 274-279 kinetics, 267-269 combination reactions, 269-271 complex reaction mechanisms, 273-274 laser reactions, 272-273 plasma reactions, 273 thermal decomposition, 273 laser synthesis, 260-263, 266-267, 268, 272-273 particle shape, 305-307 plasma synthesis, 260-263, 266, 267, 273, 303-304 population balance, 282-284 with aggregation, 293-300 rapid flocculation theory, 294-296 Gas-solid reactive sintering, 861-864 Gels, 344 aero-gels, 354-355 drying, 349-350, 684-685 film formation, 347-349 sintering, 350-351 sol-gel synthesis, 344-355, 817-818 sol-gel transition, 481-486 thermal decomposition, 355 Geothite, zeta potential, 393-394 Germanium, thermodynamic data, 908 Gibbs adsorption isotherm, 379, 380-381 liquid-vapor interface, 380-382 Gibbs free energy, 518 flocculation, 446 fusion, 510 multiple reaction systems, 148-151 Gibbsite, 35, 160 Girifalco-Good-Fowkes equation, 427 Glasses, 102, 892, 936 Glass spheres, strength, 103 Glazes enstatite, 31 history, 12-14, 14, 17, 19 wollastonite, 33

947

Glazing, 881-882 Global rate constant, 162 Gold, thermodynamic data, 910 Gouy layer, 387, 390-394 Grain boundary velocity, 836 Grain death rate, 835-840 Grain growth, 824-843, 869 abnormal, 840-843 kinetics, 829 liquid phase sintering, 859 normal, 827-840 population balance, 834-835 pressure sintering, 864-867 reactive sintering, 844-864 sintering, 815, 824-843 Grain structure, 781 Green bodies, 537 drying, s e e Green body drying formation, s e e Green body formation history, 19 microstructure, 4-5, 539, 674-675 uniformity, 719-720 sintering, s e e Sintering strength, 721-726 Green body drying, 681-686 characterization after, 718-726 cylindrical green body, 702-703 diffusion, 698 flat plate green body, 703-705 flow of liquid in the pores, 689-690 heat transfer, 686-687, 693 mass transfer, 687-683, 693 shrinkage, 684, 690-691, 694 spherical green body, 693-702 stresses induced, 691-692, 706-707 capillary stress, 716-718 effective local stesss, 707 flow stress, 713-716 thermal stress, 708-813 total stress, 707 warping and cracking, 682, 692, 705718, 876 Green body formation, 609-612, 675 ceramic pastes, 643-644 extrusion molding, 644-651 injection molding, 651-653 ceramic suspensions, 612-613 centrifugal casting, 631-636 dip coating, 638-643 electrodeposition, 636-638 filter pressing, 618-620 sedimentation casting, 629-636 slip casting, 613-618 tape casting, 620-629

948

Index

Green body formation ( c o n t i n u e d ) characterization after drying, 718-726 after formation, 674-675 die ejection, 667-670 dry ceramic powders die pressing, 653, 656-670 dry pressing, 1, 2, 653-673, 730 isostatic pressing, 653, 654, 671-673 ramming, 653, 654, 671 tapped density, 654-655 rheology, 644-656 Green body strength, 721-726 Green density, 719, 784 sintering kinetics and, 811-812 Green machining, 673, 876-878, 880-882 Griffith fracture theory, 4, 102, 723-725, 826 Grinding, 95, 876; s e e a l s o Comminution; Grinding mills comminution and classification circuit, 129, 132-135 comminution efficiency, 102-103 conservation of length, 89-91 continuous grinding, array formulation, 112-115 death rate, 105 energy required for size reduction, 101-102 inefficiencies, 102-103 parameters, 880-882 Grinding mills, 96-100 comminution and classification circuit, 129, 132-135 population balance models, 103-110 Growth effectiveness factor, 201 Growth kinetics, s e e Crystal growth, kinetics Growth spiral, 204-206, 214 Gyratory crusher, 97

Hafnium nitride, plasma synthesis, 267 Hafnium oxide, grain boundary migration, 843 Half-life aggregation, 584 coagulation, 467-468 Hamaker constant, 424-427, 428, 722 Hammer mill, 97, 100 Hardness, table of values, 936 Hard-paste porcelain, history, 25

Hard spheres high concentration, 566-569 interaction energy, 519-523, 566 stress-strain curve, 569 HBI, s e e Hydrogen bond index Heat of adsorption, polymers, 409-410 Heat capacity, table of values, 936 Heat conduction, green body drying, 698 Heat flux, 154, 328 drying, 686-687 freezing fluid, 341 Heat of reaction, calcination, 142 Heats of adsorption, 384 Heat transfer binder burnout, 731-732 boundary layer, 686 green body drying, 686-687, 693 quenching, 301,302 sintering, 779 liquid phase, 846 Heat transfer flux, 322, 323, 687 binder burnout, 731 steady state, 704 Heat of wetting, 370-373 Heaviside step function, 520 Hectorite, viscosity, 578 Helium, thermodynamic data, 905 Helmholtz free energy, 518 Helmholtz-Smoluchowski equation, 637 Hematite, isoelectric point, 399 Herring scaling law, 5-6, 812 Heterogeneous nucleation, 183, 189-192 Hildebrand solubility parameter, 403-404 table of values, 929-933 Hillert expression, 836 Hindered settling, 500-503 HIP, s e e Hot isostatic pressing History ceramics, 7-26 colloid, 496 HLB number, 411 Hollingsworth radius, 452 Holography, testing, 887 Homogeneous nucleation embryo concentrations, 186-189 gas-phase synthesis, kinetics, 270-275 liquid-phase precipitation, 183-189 Hot isostatic pressing (HIP), 865 Hot pressing, 864-865 HPC, s e e Hydroxypropyl cellulose Hydrated cation radius, table of values, 937

I~d~x Hydrocyclones, 115, 127-129 Hydrogen, thermodynamic data, 905 Hydrogen bond index (HBI), 373-374 table of values, 929-933 Hydrolysis reaction, surfactants, 399-400 Hydroperoxides, binder burnout, 739 Hydroxyapatite, 178, 182 Hydroxypropyl cellulose (HPC), 473-474 IEP, s e e Isoelectric point IHP, s e e Inner Helmholtz plane Illite, 31 Ilmenite, 37 Impurities crystal habit modification by, 216-220 sintering, 830-831 Individual packing fractions, 819 Induction period, binder burnout, 739 Initial stage sintering, 782, 783, 786 bulk diffusion, 791-795 kinetics, 788-800 sintering stress, 795-798 string of spheres, 798-800 temperature gradient, 795 vapor transport, 789-791 Injection molding, 651-653, 730 Inner Helmholtz plane (IHP), 387, 390 Interaction energy colloid stability attractive interaction energy, 422-428 electrostatic repulsion, 428-445 steric repulsion, 445-465 total interaction energy, 440-445, 466-467 van der Waals forces, 422-445 depletion interaction energy, 464-465 electrostatic interaction energy, 431432, 439, 465 hard sphere interaction energy, 519523, 566 nonadsorbing polymer, 464-465 soft sphere interaction energy, 523-526 sterically stabilized particles, 446, 448 total interaction energy, 440-445, 466-467 two plates, 459-461 two spheres, 461-464 Interdiffusion, solid-solid, 170-176 Intermediate stage sintering, 782, 784 kinetics, 800-803

949

Internal porosity, 62-63 Internal wetting, 368-369 Interparticle spacing, ordered array, 526-627 Interpenetrational domain, 450 Intramolecular expansion factor, 452 Iodine, thermodynamic data, 906 Ion adsorption, 386-389 diffuse double layer, 390-394 Stern layer adsorption, 389-390 surface charge, 394-395 Ionic surfactants adsorption, 398-403 osmotic pressure, 512 Iron, 910, 937 Iron oxide zeta potential, 393 Iron pyrite, roasting, 144 Irregularity, 366 Isoelectric point (IEP), 398, 412-413 metal oxides, 399 Isostatic pressing, 653, 654, 671-673, 782, 865 Isothermal sintering, titania, 821-822

Jasperware, history, 26 Jaw crusher, 96-97

Kaolin isoelectric point, 399 origin of word, 27 as raw material, 29-30 viscosity, 588 Kaolinite, 28-30 decomposition, 156 Kawakita compaction, equation, 660 Kayanite, 34 Kelvin tetrakaidecahedraon, 838 Kick's law, 101, 102 Kilns, 777-779 Kinetics binder burnout, 730, 752-755 carbon removal, 762-767 oxidative degradation, 749-750, 755-758 pyrolysis without oxygen, 761-762 thermal degradation, 737-738 volatiles loss, 758-761 chemical stability of powder in solvent, 414-416 coagulation, 467-488

950

Index

Kinetics (continued) flocculation, 467-488 fluid-solid reactions, 151-162, 343-344 limitations, 336-337 gas-phase synthesis, 267-269 combination reactions, 269-271 complex reaction mechanisms, 273-274 laser reactions, 272-273 plasma reactions, 273 thermal decomposition, 271 nucleation, 161-162 heterogeneous nucleation, 183, 189-192 homogeneous nucleation, 183-189 secondary nucleation, 183, 192 polymer adsorption, 410 sintering, 788 final stage, 803-809 fractal aggregate effect, 817-824 grain growth, 829 green density effect, 811-812 initial stage, 788-800 intermediate stage, 800-803 particle size distribution effect, 812-817 Kinetic shape, 212-214 Kinetic theory of gases, 280 Kink site, 196 Knudsen diffusion coefficient, 152, 698, 703, 760 Laminar flow, drag force, 500 Langmuir adsorption, 64-65, 217, 379, 382 Langmuir adsorption isotherm, 64-65, 217, 379, 382, 383, 401 ionic surfactant, 401 solvents, 386 Langmuir equation, 64, 382, 406 Langmuir term, 412 Lanthanum, hydrated cation radius, 937 Lapping, 881 Laser synthesis, 260-263, 266-267, 268 kinetics, 272-273 Lateral cracks, 878 Lead, 908-909, 937 Lead minerals, 35 Leather-hard point, 684 Lepidolite, 34 Lewis acid-base interactions, 370

Light diffraction, ordered domain size by, 530-531 Light scattering, particle size distribution, 52 Lindemann's rule, 516 Liquid-phase precipitation, 179-183, 249 coprecipitation, 244-248 crystal growth kinetics, 193-195 aggregation, 229-246 diffusion controlled growth, 196-201 growth effectiveness factor, 201 growth rates, 208-210 screw dislocation growth, 204-208 stages of growth, 196 surface nucleation of steps, 202-203 two-dimensional growth of surface nuclei, 203-204 crystal shape, 210 aggregate shape, 214-216 diffusion shape, 213-214 equilibrium shape, 210-212 growth spiral shape, 214 impurities, 216-220 kinetic shape, 212-214 surface nucleation shape, 214 nucleation kinetics, 183 heterogeneous nucleation, 183, 189-192 homogeneous nucleation, 183-189 secondary nucleation, 183, 192 simultaneous precipitation and coaggregation, 244, 246-248 Liquid-phase sintering, 844-860, 869 Liquid-solid interface adsorption HLB system, 411 ions, 387 isotherms, 382-384 Liquid-solid reactions, 151 Liquid-vapor interface, Gibbs adsorption isotherm, 380-382 Litharge, isoelectric point, 399 Lithium, thermodynamic data, 912 Lithium minerals, 34 Local effective stress, 716 Local electrostatic potential, 428 Local surface supersaturation ratio, 206 Loess, 11-12 Log-normal distribution, ceramic powder particle size, 67, 69-72 London retardation wavelength, 722-723 Longitudinal cracks, 877-878 Low temperature plasma, 257

Index Machining, 673, 876 ceramic strength, 877-878 parameters, 880-882 Macroscopic population balance, 87-88, 92, 93 aggregation, 230 discrete mass basis, 104 precipitators, 220 Magnesia compaction, 660 grain boundary migration, 807, 843 properties, 892, 936 isoelectric point, 399 surface energy, 786 synthesis, 36-37 plasma synthesis, 267 uses, 37 Magnesite, 34-35 Magnesium, 911,937 Magnesium fluoride, tensile strength, 880 Magnesium hydroxide, thermal decomposition, 160-161 Magnesium nitride, plasma synthesis, 267 Manganese hydrated cation radius, 937 oxidation, free energy, 145-147 thermodynamic data, 911 Marongoni effect, 697 Martin's diameter, 48, 51 Mass balance, 543 Mass basis, population balance, 91-93 Mass flux, 283, 322 binder burnout, 731 drying, 687, 688, 689 Mass fractal, 480-481,583 Mass transfer binder burnout, 732-733 green body drying, 687-683, 693 Mass transfer coefficient, 318, 693 Mass transfer flux, steady state, 704 Maximum drying time, 690 Maximum packing fraction, 574 Mean droplet diameter, 312, 314 Mean field theory, 458 Mean particle size, 52-55 Mechanical properties, 542 ceramic suspensions, 541, 543 dry ceramic powders, 541, 543, 590-592 coefficient of pressure at rest, 592-594

951

compact body, 543, 590, 594 Coulomb yield criterion, 597-598, 602, 661 plastic body, 595-596 yield behavior at low pressure, 599-602, 661 yield criteria for packings, 596-597 equations of motion, 543-545 constitutive equation, 545-550 continuity equation, 543-544 momentum balance, 544, 545 rheology, 550, 551 concentrated ceramic systems, 562-585 dilute suspension viscosity, 551-562 dry ceramic powders, 590-602 viscous fluids, 588 Media mills, 97, 100 Melting temperature, table of values, 936 Melt solidification, 355-357 Mercury, 909, 937 Metal alkoxide, 347-348 Metal borides, 39, 267, 268 Metal carbides, 38, 267, 268, 274 Metal carboxylates, 339 Metal halides flame synthesis, 265 oxidation reactions, 273 Metalization, 620 Metallurgy, history, 19 Metal matrix composites, 844 Metal nitrides, synthesis, 39, 267, 268 Metal oxidation, 145, 171 Metal oxides acid-base properties, 388 adsorption, 388 isoelectric points, 399 polymer oxidation catalyzed by, 742-743 synthesis, 267 Metal salts, decomposition, 332 Metals, polymer oxidation catalyzed by, 742-743 Metal silicides, 39 Metal sulphides, 39 Mica, 31,433, 642 Micellization, surfactants, 386 Microscopic population balance, 86-87, 92, 93 Microscopy particle size distribution, 52 ultrasonic microscopy, 887

952

Index

Microstructure alumina solution, 483, 485 ceramic strength, 880 green bodies, 4-5, 539, 674-675 uniformity, 719-720 Microwaves, ceramic testing, 888 M i s e - e n f o r m e , see Green body formation M i s e - e n p d t e , see Ceramic paste formation Mixed metal oxides, adsorption, 389 Mixed metal precipitates, 245 Mixed oxide ceramic powders, synthesis, 166-167, 350 Mixed suspension, mixed product removal crystallizer (MSMPR), 220 Mixing, 199, 461 Modulus of elasticity, table of values, 936 Modulus of rupture, table of values, 936 Mohr circle, 599 Molar Gibbs free energy, 181 Mold filling, 565, 592-594 Molybdenum boride, 39 Molybdenum carbide, 38 Molybdenum oxide, isoelectric point, 399 Molybdenum silicide, 39 Momentum balance, 543, 544, 638-643 Monolayer capacity, 64 Mononuclear surface, 228, 229 Monosurface nucleation, 203-204, 209 Montmorillonite, 31 Morphology, see Particle morphology MSMPR, see Mixed suspension, mixed product removal crystallizer Mullite, 30, 160, 855 properties, 399, 880, 892, 936 Multilayer adsorption, 65-66, 401-402 Multimetal carboxylates, 245 Multipoint nucleation, 289-290 Mumford and Moodie separator, 117, 118

Navier-Stokes equation, 544 Neck formation stage, 785, 788 Neon, thermodynamic data, 905 Net flux, 171 Net local stress, 707, 708 Nickel, 910, 937 Nickel aluminate, 168 Niobium nitride, plasma synthesis, 267 Niobium silicide, 39 Nitridation reactions ceramic powder synthesis, 148 Gibbs free energy, 150

Nitrogen, thermodynamic data, 906 Nondestructive testing, 886-888 Noninteractional domain, 450 Nonspherical particles casting, 636, 641-643, 652, 674 classification, 118 crystallization, 212-218 diffusion, 504-504 electrostatics, 428-432 settling, 500 suspension viscosity, 577-582 van der Waals, 422-426 Nonwetting, 364 Normal distribution, ceramic powder particle size, 68-69 Normalized population weight distribution, 333 Nucleation, see also Liquid-phase precipitation contact nucleation, 192 gas-phase synthesis homogeneous nucleation, 274-279 multipoint nucleation, 289-290 single-point nucleation, 288-289 kinetics, see also Crystal growth, kinetics heterogeneous nucleation, 183, 189-192 homogeneous nucleation, 183-189 secondary nucleation, 183, 192 primary nucleation, 191 seeding, 191 surface nucleation, 202-203, 214 total nucleation rate, 190 Nucleation rate, 183, 189, 190 Number density, 85-86

OHP, see Outer Helmholtz plane Ordered domain size, 530-532 Ordered suspensions, 516-532 defects, 528-530 ordered domain size, 530-532 structure, 526-527 Osmotic compressibility, 565 Osmotic pressure, 509 double layer, 513-516 electrolyte solutions, 511-512 polydisperse suspension, 514 polymer solution, 455, 497, 512-513 suspensions, 497 Ostwald ripening, 209 Outer Helmholtz plane (OHP), 387

Index

Oxidation reactions ceramic powder synthesis, 144-147 free energy, 145-147 gas phase, kinetics, 269-271 Gibbs free energy, 150 manganese, 145-147 metal halides, 271 polymer binder burnout, 731, 738-739, 741-752 silicon tetrachloride, 270 thermodynamics, 144-147 titanium tetrachloride, 269 zinc sulphide, 165-166 Oxygen, thermodynamic data, 905

PAA, see Poly(acrylic acid) Packing, 611-612 disorder, 527-531 filter cakes, 620 maximum packing fraction, 574 slip casting, 617 tape casting, 629 yield criteria, 596-597 Parachor numbers, 366 Paris green, 245 Partial pressure, plasticizers, 735-736 Partial wetting, 372 Particle arrays, 516-532 Particle diameters, 48-52 Particle growth, see a l s o Liquid-phase precipitation; Nucleation collisional growth theory, 274-279 mixing, 199 Particle morphology, 56-63 fractal shapes, 60-62 internal porosity, 62-63 shape analysis, 59-60 shape factors, 57-59 Particle networks, 345 Particle rotation, effective volume fraction, 554 Particle shape, gas-phase powder synthesis, 305-307 Particle size, 5 array formulation, 106-115 colloidal systems, 496 distribution, 66-67 accuracy, 55-56 aggregation, 229-244 blending powder samples, 75-77 broad size distributions, 573-577 comparison of two samples, 73-75

953

cumulative distribution, 66-67 cumulative particle size distribution, 575 gas-droplet mixing, 333 gas phase, 287 log-normal distribution, 67, 69-72 multipoint nucleation, 289-290 normal distribution, 68-69 Rosin-Rammler distribution, 72, 108 sedimentation casting, 632-636 single-point nucleation, 288-289 sintering kinetics, 812-817 mean particle size, 52-55 reduction, see Comminution; Grinding statistical diameters, 48-55 Particle strength, 102, 103 Particle structure, in suspension, 495 PDI, see Potential determining ions Pechini process, 350 Peclet number, 553, 567, 570 Penetration, wetting, 364, 366, 369 Percent greater/less than distribution, 67 Percolation limit, 482, 557, 559 Percolation probability, 559 Percolation threshold, 559 Percus-Yevick equation, 520 Perimeter diameter, 51 Permeability, particle size distribution, 52 Petalite, 34 Phosphorus, thermodynamic data, 906-907 Photon correlation spectroscopy, particle size distribution, 52 Physical adsorption, surface area measure, 64 Piezoelectricity, 4 Plasma, spray roasting, 331 Plasma synthesis, 260-263, 266, 267 kinetics, 273 quenching heat transfer, 303-304 Plastic body deformation, 590-591, 595-596 Plasticizers, 656, 730, 735 partial pressure, 735-736 Platinum, thermodynamic data, 910-911 Plug flow reactor, 284, 333, 338 Plutonium, hydrated cation radius, 937 P M A A , see Poly(methacrylic acid) PMMA, see Poly(methyl methacrylate) Point of zero charge (PZC), 388 Poisson-Boltzmann equation, 391,430 bispherical coordinates, 433, 434-437

954

I~dex

Poisson's ratio, table of values, 936 Polishing, 876 Poly(acrylamide), critical flocculation temperature, 456 Poly(acrylic acid) (PAA) adsorption on barium titanate, 410 binder burnout, 734, 748, 752 critical flocculation temperature, 456 Poly(butyl methacrylate), binder burnout, 734 Poly(dimethyl siloxane), critical flocculation temperature, 456 Polyethylene, binder burnout, 744 Poly(ethyl methacrylate), binder burnout, 762 Poly(isobutylene), critical flocculation temperature, 456 Polymer brush assumption, 556 Polymer exclusion zone, 464 Polymeric surfactants, crystal habit, 219-220 Polymer mole fraction distribution, 459, 461 Polymers, 451 adsorption, 403-410, 453 binder burnout, 740-741 oxidative degradation, 731, 738-739, 741-752 thermal degradation, 733-738, 740-741 electrosteric stabilization, 448-449 flocculation, 455-458 heat of adsorption, 409-410 mixing, 461 particles coated with, 427 properties, 451-453 solubility, 403-404 steric stabilization, 449, 458 viscosity, 556-557 Polymer solutions, 343, 344 osmotic pressure, 455, 497, 512-513 rheology, 562-564 thermodynamics, 453-459 Poly(methacrylic acid) (PMAA), binder burnout, 744, 745 Poly(methyl methacrylate) (PMMA) binder burnout, 734, 744-748, 760761,762 interaction parameter, 455, 457 Poly(methyl phenylacrylate), binder burnout, 734 Poly(methyl styrene) binder burnout, 734, 762

critical flocculation temperature, 456 Polymorphism, 868 Polynuclear growth, curves, 229, 230 Polynuclear surface, growth chronomals, 228 Poly(oxyethylene), critical flocculation temperature, 456 Poly(oxymethylene), binder burnout, 762 Polyphosphate surface active agents, 413 Polystyrene adsorption on glass, 406, 407 aggregation, 477 binder burnout, 734, 771 critical flocculation temperature, 456 ordered arrays, 517, 526 viscosity, 564, 567, 573, 585 Polysurface nucleation, 203-204, 209 Poly(tetramethylene oxide), binder burnout, 762 Poly(vinyl alcohol), critical flocculation temperature, 456 binder, 659 Poly(vinyl butyral), binder burnout, 743744, 745 Population balance, 85-86, 93 aggregation, 230, 249, 293-300, 476 batch precipitation reactors, 226, 240-241 classification, 134 classifier, 133-134 comminution, 103-110, 134 conservation of length, 89-91 continuous stirred tank reactor, 236, 239 cumulative mass basis, 92-93, 104-105 discrete mass basis, 91-93, 107 gas-phase synthesis, 282-284 with aggregation, 293-300 rapid flocculation theory, 294-296 grain growth, 834-835 macroscopic, 87-88, 92, 93, 220, 230 mass basis, 91-93 microscopic, 86-87, 92, 93 physical constraint, 292-295 precipitators, 220 Population balance equation gas phase reactions, 283 solutions, 299-300 Population balance models, grinding mills, 103-110 Porcelain history, 21-22, 26 properties, 892, 936

Index

Pore diffusion, shrinking core model, 155 Pore flow, 689-690 Pores, sintering, 833-834 Porosity, 62-63 Position-dependent diffusion equation, 476 Potassium, thermodynamic data, 912 Potential determining ions (PDI), 387 Potential distribution, charged spheres, 434-436 Powder density, 63 Powder sampling, 44-47 Prandtl number, 694 Precipitation classification function, 222-223 droplet drying, 324-326 liquid phase, 179-183, 249; see a l s o Precipitators coprecipitation, 244-248 crystal shape, 210-220 growth kinetics, 193-210 nucleation kinetics, 183-192 particle size distribution effects, 220-244 Precipitators batch precipitation reactors, 226-229 continuous stirred tank reactors, 220-226 population balance, 220 Pressure filtration, 618-620 Pressure nozzle, 316-318 Pressure sintering, 864-867, 869 Primary nucleation, 191 Primary progeny, 105-106, 107 Product population, 134 Progeny function, 106-107, 110 Projected area diameter, 49, 50 Proof testing, 884-886 Propagation reactions, binder burnout, 739 Pseudo-isostatic pressing, 865 Pseudo-plastic fluid, 548, 549, 550 Pyrophyllite, 31 PZC, see Point of zero charge

Quality assurance testing, 883-886 Quartz, 32, 399, 402 Quenching aggregation, 296-301 gas mixing quench, 300-301 heat transfer quench, 294-300

955

Radial distribution function, 520 Radial springback, 669 Radius of gyration, 452 Ramming, 653, 654, 671 Rate of coagulation, 468 Rate of condensation, 789-790 Rate of deformation tensor, 545 Raw materials, 39; see a l s o Ceramic powder synthesis history, 23, 26 natural, 27-34 precursor solution chemistry, 343-347 selecting, 40-41 specialty chemicals, 35 synthetic, 34-39 Reaction bonded silicon nitride, 861 Reaction limited aggregation (RLA), 481, 583 Reactive sintering, 779, 844-864, 860861,869 gas-solid, 861-864 liquid phase, 844-860 Rebound, 668 Recovery, classifiers, 123-124 Recycle ratio, 133 Reduction potentials, electrochemical reactions, 903-904 Reduction reactions carbothermic boron oxide, 161-162 silica, 167 ceramic powder synthesis, 147 thermodynamics, 147 Rehbinder effect, 375 Rejection rate, sintering, 784-785 Reprecipitation, liquid phase sintering, 845, 853, 857-859 Residence time distribution, 114, 162163, 282 Retardation effects, 424, 425 Rheology anisotropic particles, 577-583 Bingham plastic fluid, 548-549, 620-628 ceramic pastes, 585-590 ceramic suspensions, 531,537-539, 543, 550-551, 613 colloidally stable concentrated suspensions, 564-583 concentrated ceramic systems, 562-585 concentrated polymer solutions, 562-564

956

Index

Rheology ( c o n t i n u e d ) dilute suspension viscosity, 551-562 unstable concentrated suspensions, 583-585 green body fabrication, 644-656 polymer solutions, 562-564 Rittenger's law, 101, 102 RLA, s e e Reaction limited aggregation Root-mean-square end-to-end distance, 451-452 Root-mean-square radius of gyration, 452 Rosin-Rammler distribution, 72, 108 Rotational diffusion, 506-509, 552-554 Rotational Peclet number, 553 Roughness, 193-195, 366, 668 Rubidium, thermodynamic data, 913

Saggars, 778 Sampling, s e e Powder sampling Saturation ratio, 274-275, 324 Scaling law theory, 458-459, 562 Schmidt number, 288 Screens, classification, 115 Screw dislocation crystal growth, 204208, 209 Secondary nucleation, 183, 192 Second falling rate period, gel drying, 353 Second virial coefficient, 512, 513, 520 Sectional representation, 300 Sedimentation centrifugal sedimentation, 503 ceramic suspensions, 495, 497-504 equilibrium with diffusion, 505-506 hindered settling, 500-503 nonspherical particle settling, 500 sedimentation equilibrium, 506 sedimentation potential, 503-504 terminal settling velocity, 499 zeta potential, 503 Sedimentation casting, 629-631 anisotropic particles, 636 particle size distribution, 632-636 Sedimentation classifiers, 115 Sedimentation potential, 503-504 Seeding, nucleation, 191 Segment density distribution function, 454 Segregation drying, 720 sintering, 634

Selectivity, size, 119 Self-consistency, 458 Self-consistent mean field theory, 458, 459 Settling, s e e Sedimentation Shape factors, 57-59 Sharpness index, 124-126 Shear aggregation, 233, 486-488 turbulent flow, 487-488 Shear modulus, compact body, 594 Shear rate, 548, 586 Shear stress Newtonian fluids, 622 viscous fluids, 586-587 Shear thickening, 570, 571 Shear thinning, 568, 569 Sherwood number, 322 Shrinkage drying, 684, 690-691,694 constant rate period, 695-697 sintering, 811-812 intermediate stage, 815 Shrinkage rate, 812 Shrinking core model, 155, 335, 337 Shrinking sphere model, 140, 157-158 Sialon plasma synthesis, 266 properties, 892, 936 Sieve diameter, 49, 51 Sieving, 49, 52 Sieving times, 49 Silane, thermal decomposition, 273, 276-279 Silica, 32-33 aggregate, 478 carbothermic reduction, 167 colloidal stability, 469, 472-475 compaction, 527-528, 660 drying, 685 polymorphic transformations, 868-869 polymorphs, 868 properties, 892, 936 gelation time, 483, 484 surface energy, 786 spreading water on, 364 synthesis, 260, 264, 267, 274, 306 viscosity, 589 Silicon compaction, 660 drying, 706 hydrated cation radius, 937 synthesis, 268-271 thermodynamic data, 908

Index

Silicon carbide deagglomeration, 377 properties, 892, 936 surface energy, 786 synthesis, 38, 163-164, 167 furnace decomposition, 266 laser synthesis, 273 plasma synthesis, 267 sedimentation, 378 ultrasonication, 378 uses, 38 wetting, 372-373 Silicon nitride hot isostatic pressing, 866 properties, 892, 936 tensile strength, 880 reactive sintering, 861-864 synthesis, 38-39 plasma synthesis, 267 uses, 39 Silicon tetrachloride, oxidation, 270 Sillimanite, 34 Silver, thermodynamic data, 910 Similarity function, 107 Similarity solution, comminution, 107, 108, 110 Simultaneous precipitation and coaggregation, 244, 246-248 Single fluid nozzle, 316 Single-point nucleation, 288-289 Singular surfaces, 195 Sintering, 5-6, 777, 781-785, 869 combustion sintering, 861 cooling after sintering, 867-869 driving force, 786-788 gels, 355-356 grain growth, 824-843 abnormal grain growth, 840-843 kinetics, 829 normal grain growth, 827-840 reactive sintering, 844-864 ressure sintering, 864-867 green density, 811-812 isothermal sintering, 821-822 kinetics, 788 final stage, 803-809 fractal aggregate effect, 817-824 grain growth, 820 green density effect, 811-812 initial stage, 788-800 intermediate stage, 800-803 particle size distribution effect, 812-817

957

presintering heat treatment, 681-682 pressure sintering, 864-867 reactive sintering, 779 segregation, 634 solid state mechanisms, 785-788 stages, 782-784 final, 784, 803-809 initial, 782-783, 786, 788-780 intermediate, 782, 784, 800-803 vacancies and domain size, 531-532 Sintering forces, 787 Sintering pressure, 787 Sintering stress, 787, 795-798, 866 Site percolation, 482-483 Size selectivity, classifiers, 123, 133 Slip casting, 613-618 Slurries, drying, 313, 319 Small's equation, 404 Small's molar attractors, 404, 405 Soda ash, synthesis, 37 Soda glass, lithium ions and, 102 Soda-lime glass, properties, 880, 892, 936 Sodium, thermodynamic data, 912 Soft-paste porcelain, history, 21-22, 26 Soft sphere interaction energy, 523-526 Soft spheres, high concentration, 572-573 Sol, 496 Sol-gel synthesis, 340-351,817-818 Sol-gel transition, 481-486 Solid casting, 613 Solid-liquid interface adsorption HLB system, 411 ions, 387 isotherms, 382-384 Solid-phase synthesis, 139-141 fluid-solid reactions kinetics, 151-162 reactors, 162-166 thermodynamics, 141-151 kinetics, 151-162, 336-337, 343-344 liquid-solid reactions, 151 solid-solid reactions, 139-141, 166176, 344 thermodynamics, 141-144 multiple reaction systems, 148-151 nitridation, 148 oxidation reactions, 144-147 reduction reactions, 147 Solid-solid interdiffusion, 170-176 Solid-solid reactions, 139-141, 166-167, 176-177, 344

958

Index

Solid-solid reactions ( c o n t i n u e d ) solid-solid interdiffusion, 170-176 vaporization of one solid reactant, 167-169 Solid state reactive sintering, 860-861 Sols, see Sol-gel synthesis; Sol-gel transition Solutions, colligative properties, 497, 509 Solvay process, 37 Solvents binary adsorption, 384-386 chemical stability of powder in solvent, 414-416 Drago E and C values, 370-373, 410, 926-927 green body drying, 683-686 Hildebrand solubility parameter, 929-933 hydrogen bond index, 929-933 selection, 373-374 surface tension, 921-923 Sonic velocity, 316 Specific breakage rate, 105 Specific cake resistance, 615 Spectroscopy, particle size distribution, 52 Spherical green body, drying, 693-702 Sphericity, 59 Spinel, properties, 892, 936 Spodumene, 34 Spray dryer, 334-335 Spray drying, 311-313 atomization, 312, 313-319 droplet drying, 312, 319-331 equipment design, 334-335 gas-droplet mixing, 331-333 Spray roasting, 313, 335-339 Spreading, wetting, 364, 366, 369 Spreading coefficient, 364 Stabilization depletion stabilization, 450, 465 electrostatic stabilization, 519, 523 electrosteric stabilization, 448-449, 465 enthalpic stabilization, 446, 447 entropic stabilization, 446, 447 polymers, 448-449, 458 steric stabilization, 445, 449, 458, 519 Standard free energy, 414 calcination, 142 Static brittleness index, 880 Statistical diameters, 48-55

Steric interaction energy two plates, 459-461 two spheres, 461-464 Steric repulsion, 445-465 Steric stabilization, 445, 519 polymers, 449, 458 Stern layer, 387, 389-390 Stokes diameter, 49, 50 Stokes-Einstein equation, 504 Stokes number, inertial collision, 121 Stokes' law, 498 Stoneware, history, 26 Streaming potential, zeta potential measurement, 397 Strength ceramic, grinding direction, 878-879 green bodies, 721-726 grinding direction, 878-879 machining, 877-878 Stresses binder burnout, 767-768 thermal stress, 768-770 volatile flow, 770-771 green body drying, 691-692 capillary stress, 716-718 ceramic film, 718 thermal stress, 708-713 shear stress, 586-587, 622 sintering stress, 787, 795-798, 866 Stress tensor, 545 Strontium, thermodynamic data, 912 Strontium titanate, doped, 827 Strontium zirconate, doped, 832 Sublimation, freeze drying, 339 Sulfur, thermodynamic data, 906 Supersaturation, 180 coating, 191-192 crystal growth, 199, 207, 208 Ostwald ripening, 209 surface nucleation, 202-203 vapor phase, 274 Surface area, see Powder surface area Surface charge, ion adsorption, 394-395 Surface diameter, 50 Surface energy, solids, 363 Surface flux, 206 Surface nucleation, 202-203, 214 Surface potentials, charged spheres, 434-436 Surface reaction, shrinking core model, 155 Surface reaction rates, 153

Index

Surface roughness, 193-195, 366, 668 Surface shape factor, 558 Surface temperature, 156 Surface tension solids, 363 solvents, table of values, 921-923 wetting, 360 Surface volume diameter, 50 Surfactants aqueous, 411-413 colloid stability, 448 crystal habit, 219-220 dissociation reaction, 399-400 hydrolysis reaction, 399-400 ionic adsorption, 398-403 osmotic pressure, 512 micellization, 386 selection, 410-413 viscosity and, 555-556 Suspensions, see also Ceramic suspensions anisotropic particles, 551-554, 618 anisotropic properties, 551-554 colligative properties, 497, 509 electro-viscous effect, 554-555 osmotic pressure, 497, 514 polydisperse, 514 viscosity, 551-562, 564-583, 583-585

Talc, 31 Tantalum carbide, 38, 267 Tantalum nitride, plasma synthesis, 267 Tape casting, 620-629, 730 Tape casting machine, 621 Tapped density, 654-655 Terminal settling velocity, 499, 630 turbulent flow, 499 Termination reactions, binder burnout, 742 Testing nondestructive testing, 886-888 proof testing, 884-886 quality assurance, 883-886 ultrasonic testing, 887 Thallium, thermodynamic data, 909 Thermal conductivity, table of values, 936 Thermal decomposition binder burnout, 733-738, 740-741

959

calcium carbonate, 158-160 dimethylchlorosilane, 266 gas phase, 271 gels, 351 kinetics, 152-154, 271 magnesium hydroxide, 160-161 metal organics, 335-336 silane, 273, 276-279 spray roasting, 313 Thermal plasma, 261 Thermal stress binder burnout, 768-770 during drying, 708-713 Thermodynamics calcination reactions, 141-142 data table, chemical elements, 905-913 fluid-solid reactions multiple reaction systems, 148-151 nitridation reactions, 148 oxidation reactions, 144-147 reduction reactions, 149 multiple reaction systems, 148-151 osmotic pressure, ceramic suspension, 517-526 polymer solutions, 453-459 Theta point, 457, 513 Theta temperature, 455 Thixotropy, 557, 564, 577 Thorium, hydrated cation radius, 937 Thorium oxide, grain boundary migration, 843 Tin, 908, 937 Tin-glazed ware, history, 26 Titania adsorption properties, 389, 391 colloidal stability, 469-474 isoelectric point, 399 isothermal sintering, 821-822 polymer oxidation catalyzed by, 742 synthesis, 37 flame synthesis, 260, 264, 290-292, 306 liquid-phase precipitation, 240-241, 244 particle arrays, 527 plasma synthesis, 267 sintering, 821 viscosity, 560-561, 570 zeta potential, 443-444, 471 Titanium, 911,937 Titanium boride, 39, 866

960

Index

Titanium carbide, 38, 165 plasma synthesis, 267 properties, 892, 936 Titanium nitride, 39 Titanium tetrachloride oxidation, 265 synthesis, 37 Total colloidal stability ratio, 476 Total interaction energy, 440-445, 466-467 Total nucleation rate, 190 Translational diffusion coefficient, ellipsoidal particles, 504 Translational Peclet number, 567, 570 Transverse cracks, 878 True coprecipitation, 244-246 t-statistic, 73 t-test, 73-74 table of values, 901-902 Tungsten, 911,937 Tungsten boride, 39 Tungsten carbide, 38, 165 liquid phase sintering, 844 plasma synthesis, 267 Tungsten oxide carbothermic reduction, 168 isoelectric point, 399 Tungsten silicide, 39 Tunnel kiln, 777, 778 Turbulent flow shear aggregation, 487-488 terminal settling velocity, 499 Two-dimensional crystal growth, surface nuclei, 203-204 Two-fluid nozzles, 317-318

Ultrasonic testing, 887 Ultrasonification, 375-378 Uniformity, 4-5, 719-720 Unrolled diameter, 48, 51 Uranium, hydrated cation radius, 937 Uranium oxide grain boundary migration, 843 hot isostatic pressing, 866

Vacancies, sintering and, 532 Vanadium carbide, 38 Vanadium nitride, 39 plasma synthesis, 267 Vanadium silicide, 39

van der Waals forces, 375 attractive interaction energy, 422-424 Hamaker constant, 424-427, 428 polymer-coated particles, 427 electrostatic repulsion, 428 charged plates, 428-431, 432 charged spheres, 432-444 green body strength, 721-723 total interaction energy, 440-445, 466-467 van't Hoff infinite dilution, 512 Vapor condensation, 259 Vaporization, solid-solid reactions, 167-169 Vapor pressure, 509, 510 Vickers hardness, 880 Visco-elastic models, 586-590, 667 Visco-elastic solid, 588 Viscosity, 548-550, 583-585 alumina, 561-562 dilute suspensions, 551-562 anisotropic particles, 551-554 electrostatically stabilized suspensions, 560-562 electro-viscous effect, 554-555 polymer effect, 556-557 slightly aggregated suspensions, 557-560 surfactant effect, 555-556 pH and, 560-562 polymers, 556-557 surfactants and, 555-556 titania, 560-561 Viscous drag force, 497 Viscous fluids mechanical behavior, 588 shear stress, 586-587 Volatiles binder burnout, 730, 731, 734-735, 738, 750-752 flow, 770-771 kinetics, 758-761 Volume diameter, 50 Volume fraction, 514, 583 Volume shape factor, 57 Volume to surface mean diameter, 72 Volumetric flux, 206-207 von Smoluchowski's growth kinetics, 481, 486

Walden's rule, 362 Wall friction, 669

Index

Warping, green body, 705-718, 876 Water, chemical stability of powder in solvent, 414-415 Weber number, 311-312 Wet bag isostatic pressing, 671 Wet classifiers, 115, 117, 127-129 Wet paste extrusion, 1, 2 Wetting, 359, 360-374 heat of wetting, 370-373 internal wetting, 368-369 partial wetting, 368 rough solid surface, 368 solvent selection, 373-374 surface tension, 360 White ware, 14 Witherite, 35 Wollastonite, 33-34

Yield, classifiers, 119 Yield behavior, powders at low pressure, 599-602 Yield criteria Coulomb yield criterion, 543, 597-598, 602, 661 for packings, 596-597 Yield locus, 599, 600 Young's equation, 363, 542 Young's modulus, compact body, 594 Yttrium, hydrated cation radius, 937 Yttrium iron garnet, grain boundary migration, 843 Yttrium oxide, grain boundary migration, 843

961

Yttrium barium, copper oxide, 248, 335-336

Zeta potential, 387, 395-398, 470, 471 ceramic paste formation, 395-398 geothite, 393-394 measurement, 395-398 sedimentation, 503 titania, 443-444 Zinc, 909, 937 Zinc aluminate, 172 Zinc oxide, 37 grain boundary migration, 843 isoelectric point, 399 polymer oxidation catalyzed by, 742 synthesis, 37 plasma synthesis, 267 thermodynamics, 144 Zinc sulphide, 144, 165-166 Zircon, 38 Zirconia, 38 grain boundary migration, 843 green body strength, 723, 724-725, 726 isoelectric point, 399 plasma synthesis, 267 polymorphic transformations, 868 properties, 892, 936 slip casting, 617 synthesis, 38, 240, 267 Zirconium, hydrated cation radius, 937 Zirconium carbide, 38 Zirconium nitride, plasma synthesis, 267

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