Rheometry of Pastes, Suspensions, and Granular Materials: Applications in Industry and Environment

RHEOMETRY OF PASTES, SUSPENSIONS, AND GRANULAR MATERIALS

RHEOMETRY OF PASTES, SUSPENSIONS, AND GRANULAR MATERIALS Applications in Industry and Environment

Philippe Coussot

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright  2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Coussot, Philippe. Rheometry of pastes, suspensions, and granular materials : applications in industry and environment / Philippe Coussot. p. cm. Includes index. ISBN 13 978-0-471-65369-1 ISBN 10 0-471-65369-1 1. Amorphous substances. 2. Rheology. I. Title. TA418.9.A58C68 2005 2004021908 620.1
1—dc22 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

CONTENTS

PREFACE

xi

NOTATION

xv

INTRODUCTION

1

1 MATERIAL MECHANICS

4

1.1 Introduction 1.2 Continuum Mechanics 1.2.1 Definition of a Material 1.2.2 Continuum Assumption 1.2.3 Main Variables 1.2.4 Conservation Laws 1.3 Constitutive Equation 1.3.1 Physical Origin 1.3.2 General Characteristics 1.3.3 Effect of Change in Frame of Observation 1.3.4 Solids and Fluids 1.3.5 Simple Shear and Viscometric Flows 1.4 Viscometric Flows 1.4.1 Free Surface Flow over a Plane 1.4.2 Flow between Parallel Disks 1.4.3 Flow between a Cone and a Plate 1.4.4 Flow between Two Coaxial Cylinders

4 5 5 6 8 13 17 17 18 18 20 23 27 28 31 33 35 v

vi

CONTENTS

1.4.5 Flow in a Cylindrical Conduit (Poiseuille Flow) References 2 RHEOPHYSICS OF PASTES AND GRANULAR MATERIALS

38 40 41

2.1 Interactions between Material Elements 2.1.1 Hydrodynamic Interactions 2.1.2 Colloidal Interactions 2.1.3 Interactions between Bubbles or Droplets 2.1.4 Interactions between Two Solid Particles 2.1.5 Classification of Forces 2.2 Rheology of Soft Jammed Systems (Pastes) 2.2.1 Solid Regime: Viscoelasticity 2.2.2 Solid–Liquid Transition: Yielding 2.2.3 Liquid Regime: Flow 2.2.4 Time Effects: Thixotropy 2.2.5 Synthesis 2.3 Rheology of Granular Materials 2.3.1 Frictional Regime 2.3.2 Collisional Regime 2.3.3 Frictional–Collisional Regime Transition 2.4 Rheology of Granular Pastes 2.4.1 Frictional Regime 2.4.2 Lubricational Regime 2.4.3 Frictional–Lubricational Regime Transition References

42 44 49 51 51 55 57 59 60 62 64 67 67 69 72 73 73 74 77 77 78

3 EXPERIMENTAL PROCEDURES AND PROBLEMS IN PASTE VISCOMETRY

81

3.1 Experimental Procedures 3.1.1 Setup of the Material 3.1.2 Viscoelasticity in the Solid Regime 3.1.3 Yielding: Solid–Liquid Transition 3.1.4 Flow Curve 3.1.5 Thixotropy 3.1.6 Effect of Heterogeneity in Shear Rate 3.2 Wall Slip 3.2.1 Physical Origin

82 82 83 88 91 93 95 97 97

CONTENTS

3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.3 Shear 3.3.1 3.3.2 3.3.3 3.3.4

3.4

3.5

3.6

3.7 3.8 3.9

General Mechanical Characteristics Couette Flow Parallel Disks Cone–Plate Flow Capillary Flows Wall Slip, Yielding, and Fracture How to Avoid Wall Slip Localization Plateau in Flow Curve and Viscosity Bifurcation Modeling Shear Banding Implications for Paste Rheometry in the Continuum Regime 3.3.5 Paste Rheometry in the Discrete Regime Surface Tension Effects 3.4.1 Surface Tension in Simple Liquids 3.4.2 Surface Tension for Non-Newtonian Liquids Drying 3.5.1 Evaporation 3.5.2 Drying Regimes 3.5.3 Drying Rates during Rheometrical Tests 3.5.4 Effect of Drying on Rheometry 3.5.5 Countermeasures Phase Separation 3.6.1 Sedimentation 3.6.2 Migration 3.6.3 Segregation 3.6.4 Consequences in Rheometry Cracking Temperature Effects Inertia Effects and Turbulence 3.9.1 “Macroscopic” Inertia Effects 3.9.2 Turbulence References

4 LOCAL RHEOMETRY

4.1 Techniques for Measuring the Velocity Field in Fluids 4.1.1 Principles of NMR

vii

100 102 103 103 104 105 107 108 108 111 116 120 127 130 130 132 133 133 135 138 139 139 140 140 141 142 143 144 145 147 147 148 149 153

154 154

viii

CONTENTS

4.1.2 Principles of MRI 4.1.3 Principles of MRI Velocimetry 4.1.4 Resolution and Difficulties 4.2 Rheological Interpretation of Velocity Profiles 4.2.1 Aspects of Velocity Profiles in Couette and Capillary Flow for Typical Rheological Behavior 4.2.2 Rheological Analysis from One Velocity Profile 4.2.3 Rheological Analysis from a Set of Velocity Profiles 4.3 Velocity Profile Reconstruction From Rheometry 4.3.1 Local Flow Properties as a Function of Stress History 4.3.2 Reconstruction of the Profiles of Rotation Velocity and Rotation Angle 4.3.3 Application to Simple Shear Stress Histories 4.3.4 Specific Behavior Types 4.3.5 Wall Slip Effect 4.3.6 Inclined Plane Flow References 5 NONVISCOMETRIC FLOWS OF YIELD STRESS FLUIDS

5.1 Displacement of an Object Through a Yield Stress Fluid 5.1.1 Drag and Buoyancy Forces 5.1.2 Displacement of a Long Object along Its Axis 5.1.3 Displacement of a Compact Object of Any Shape 5.2 Squeeze or Stretch Flows 5.2.1 Lubricational Regime 5.2.2 Elongational Regime 5.2.3 Intermediate Cases 5.2.4 Squeeze or Stretch Flow between Two Planes Forming a Dihedral 5.2.5 Squeeze or Stretch Flow of a Long Band between Two Parallel Planes 5.2.6 Flow Instabilities 5.3 Spreading or Coating Flows 5.3.1 Gravity Flow of a Yield Stress Fluid over an Inclined Plane 5.3.2 Spreading Over a Rotating Surface 5.3.3 Coating by the Relative Displacement of Two Almost Parallel Planes 5.3.4 Flow Instability

156 157 159 160 161 168 170 173 174 176 177 180 181 182 183 185

186 186 187 189 193 194 198 199 200 201 202 206 207 212 213 215

CONTENTS

5.3.5 Role of Surface Tension in (Stable) Spreading Flows References 6 GRANULAR FLOWS IN FRICTIONAL REGIME

6.1 Viscometric Flows 6.1.1 Couette Flow 6.1.2 Annular Shear Cell 6.2 Free Surface Flows 6.2.1 Granular Heaps over Horizontal Surfaces 6.2.2 Granular Heaps over Inclined Surfaces 6.2.3 Granular Flow 6.3 Conduit Flows 6.3.1 Stress Distribution in a Straight Conduit 6.3.2 Flow Rate through a Conical Aperture 6.4 Compression Flows 6.5 Displacement of an Object 6.6 Free Surface Flow of a Granular Paste 6.6.1 Granular Paste Flowing over an Inclined Plane 6.6.2 Liquefaction of a Granular Paste at Rest over a Horizontal Plane References 7 PRACTICAL RHEOMETRICAL TECHNIQUES

7.1 Tests 7.1.1 7.1.2 7.1.3 7.2 Tests 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.3 Tests 7.3.1 7.3.2

Involving Displacement of an Object through Fluid Penetrometer Displacement of an Object Immersed in Fluid Fall of an Object under Gravity through a Vibrating Fluid Involving Squeeze Flows Simple Squeeze Tests Imperfect Squeeze Test Band Squeezing Slump Test Chopin Alveograph Involving Spreading or Coating Flows Consistometer Inclined Plane Test

ix

218 220 223

224 224 226 227 227 229 229 234 235 236 239 242 243 243 244 244 246

247 247 251 253 253 254 255 256 257 261 262 263 265

x

CONTENTS

7.4 7.5

7.6 7.7

INDEX

7.3.3 Ericksen Gauge 7.3.4 Atterberg Limits Tests Involving Conduit Flows Practical Tests for Granular Materials 7.5.1 Friction Coefficient from Quasistatic Flow Regime 7.5.2 Flows in Frictional Regime Study of Thixotropy using Practical Rheometrical Tests Applications to Industrial and Environmental Materials 7.7.1 Foodstuffs 7.7.2 Cosmetic and Pharmaceutical Materials 7.7.3 Environmental Materials 7.7.4 Civil Engineering Materials References

266 267 268 270 271 274 274 278 278 279 279 282 284 287

PREFACE

Although the behavior of clays and paints initially prompted Bingham [1] to introduce rheology as a new scientific field in the early 1920s, the main developments in that field since then have concerned polymers. The internal structure of polymers can now be easily characterized using various techniques and shows obvious physical similarities between materials. As a consequence, a general descriptive framework for these materials is available now, allowing us to determine fairly accurately the physicochemical origins of the mechanical behavior of polymers [2,3]. Besides polymers, industry and nature provide us with a vast range of materials composed of complex polymeric elements in suspension in a liquid: emulsions, foams, and suspensions of solid particles. When the concentration of suspended elements is weak, the behavior of such materials is qualitatively similar to that of the interstitial liquid, which is generally Newtonian. At sufficiently high concentrations, the suspended elements develop specific mutual interactions and hence we usually deal with a “pasty” material, incapable of flow when the force exerted onto it is below a critical value. We currently encounter this type of material in our daily lives in products such as shaving foam, Chantilly, solar cream, cosmetic cream, mayonnaise, puree, paint, modeling paste, peanut butter, marmalade, and hair gel, or in natural phenomena such as mining slurries, mudflows, debris flows, lavas, snow, and lahars (water–ash mixtures on volcanic hill slopes). Civil engineering, food, and cosmetic industries also use pasty or granular materials for transporting or storing solid matter or for product forming, with agents such as drilling fluids, cement paste, concrete, mortar glues, ceramic slip, foodstuff pastes, sewage sludges, sand, grains, and powders.

xi

xii

PREFACE

In contrast to polymers, all these materials have extremely different internal structures, ranging from the packing of submicrometric, platelet, clay particles for muds, to the crowding of soft bubbles (with diameters in the high micrometer range) for foams. More critical is the fact that most industrial materials contain a wide range of elements of various sizes and interaction patterns (polymers in liquid suspensions or adsorbed by solid particles, colloidal particles interacting at a distance via electrostatic forces, formation of viscoelastic droplets or bubbles, grains interacting via friction or collision through the liquid, etc.). In this context one often focuses on a specific material and notes that it exhibits unique characteristics, which certainly is true from a physicochemical perspective. Such individual approaches unfortunately do not promote the development of a unifying approach to define the relationship between internal structure and mechanical properties for this wide class of materials. A new concept has emerged in physics, known as “jamming” [4], in which all these materials are perceived as “jammed” systems. Although this concept is not yet precisely defined, it concerns the “pasty” materials as defined above since this internal jamming implies that these materials cannot flow unless some outside force is applied to “unjam” this structure. In this field physicists essentially seek generic thermodynamic properties at the origin of the link between this jamming and the evolutions of the internal structure of these materials. As I was studying various materials, I increasingly concluded that, considering the variety of internal structures of jammed materials, generic laws, if they exist, should be sought in the mechanical behavior of these materials. Indeed, in mechanics the specificities of the internal structure of each material may be put aside in favor of the qualitative characteristics of this internal structure, which induces different mechanical trends. The primary objective of this book is to propose a synthetic and general approach to define the mechanical behavior of pastes and granular materials. In practice, it is critical to apply reliable rheometrical techniques for relevant characterization of materials, specifically, to measure some physical parameters related to the effective rheological behavior of the material. For pasty and granular materials this approach still constitutes a challenge for several reasons; for instance, these materials may exhibit a strongly nonlinear behavior [they behave either as solids or liquids depending on flow conditions], several experimental problems may occur (wall slip, fracture, drying, etc.) with these materials, and for some of these substances (granular materials) there may exist no constitutive equation intrinsic to the material (i.e., independent of boundary conditions). Finally, probably because of these difficulties, a variety of practical techniques for characterization of these materials in a rapid and robust way (slump test, Marsh cone, penetrometric measurement, etc.) involve “nonviscometric flows,” specifically, flows that are more complex and controlled than the so-called viscometric flows used in rheometry. Often, each industrial field developed its own techniques that in fact involved the same basic flow types found in other fields. In this context the second major objective of this book was to review the experimental problems encountered with such materials and to examine the techniques

PREFACE

xiii

used in different fields through the basic nonviscometric flows they involved, in order to propose theoretical analysis that would enable one to extract relevant rheological parameters from such tests. This book is merely a sketch of what might be written on the topic of rheology. In the near future the development of new techniques for internal exploration of materials will probably give us a deeper insight into the behavior of pastes or granular materials, which will make rheometrical analysis even more accurate. However, the tools described here may provide an at least initially comprehensive approach to this field. This work results from the research I have carried out since 1989. At first my motivation for the study of pasty materials was to provide protection against mudflows in mountain streams, initiated by M. Meunier in Cemagref and supervised by J. M. Piau in the Laboratory of Rheology in Grenoble. I undoubtedly enjoyed a perfect launching site and hopefully I was worthy of it since these individuals provided me with much of my scientific formation in different ways. Another key stage was my meeting with O. Coussy, the then head of LMSGC. Not content with teaching me the details of research management, he also boosted rheology research by creating a mainline around rheophysics in our laboratory. During these years I had the opportunity to work with researchers whose ideas are reflected at various points in this book: C. Ancey (EPFL, Lausanne), who, among other things, introduced me to the specificities of granular materials; D. Bonn (LPS, ENS Paris), with whom we developed the very important concept of viscosity bifurcation; J. C. Baudez (Cemagref), who had a remarkable approach to the reconstruction of a velocity profile; and X. Chateau (LMSGC), with whom I am still daily discovering the subtleties of mechanics. Several individuals also kindly agreed to review parts of this book, expressed comments, and provided enlightening advice; I thank them warmly here, and hope that this work will be of some personal benefit to them as well: N. Alderman (Aspen Technology), G. Ovarlez (LMSGC), X. Chateau (LMSGC), B. Herzhaft (IFP), F. Chevoir (LMSGC), and S. Rodts (LMSGC). I would also like to thank John Wiley & Sons, Inc. for trusting me. Eventually, last but not least, many thanks to my wife, who managed once again to put up with seeing me somewhat tense, focused on a new objective for about 2 years.

REFERENCES 1. E. C. Bingham, Fluidity and Plasticity, McGraw-Hill, New York, 1922. 2. J. D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1970. 3. P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, NY, 1979. 4. A. J. Liu and S. R. Nagel, Jamming and Rheology, Taylor & Francis, New York, 2001.

NOTATION

Roman Symbols a acceleration vector b separation distance between the centers of two elements; half-height of a cylinder of material b volume force Ba Bagnold number (see Section 2.3) B magnetic field; buoyancy force d strain rate in elongational flow; characteristic element length (equivalent average diameter) ds surface element dv volume element D fluid width DI , DII , DIII invariants of the strain rate tensor D strain rate tensor E elastic modulus of a solid particle f friction coefficient between two solid surfaces Fc critical force for incipient motion or stoppage F0 imposed force in squeeze flow Fr Froude number [equation (5.57)] FD drag force Ft relative configuration gradient [equation (1.2)] g gravity G elastic modulus for Maxwell model G
, G

elastic and viscous moduli G magnetic field gradient xv

xvi

NOTATION

h hc hstop h0 H Hϑ