Physics of Granular Media

The Physics of Granular Media Edited by Haye Hinrichsen and Dietrich E. Wolf The Physics of Granular Media Edited by ...

Author: Haye Hinrichsen | Dietrich E. Wolf

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The Physics of Granular Media Edited by Haye Hinrichsen and Dietrich E. Wolf



Editors Haye Hinrichsen Universität Würzburg, Germany e-mail: [email protected] Dietrich E. Wolf Gerhard Mercator Universität Duisburg, Germany e-mail: [email protected]

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library

Cover Picture: A. Kudrolli: Magnetic, granular particles arranged in chains

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available on the internet at .

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Composition Uwe Krieg, Berlin Printing Strauss GmbH, Mörlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim ISBN 3-527-40373-6

Contents

Preface

XI

List of Contributors

I

XV

Static Properties

1 Stress in Dense Granular Materials (I. Goldhirsch and C. Goldenberg) 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.2 Continuum Mechanics: A Brief Review . . . . . . 1.3 Constitutive Relations for Dense Granular Materials 1.3.1 Engineering Approaches . . . . . . . . . . 1.3.2 Recent Approaches . . . . . . . . . . . . . 1.3.3 Experiments and Possible Reconciliation . . 1.4 A Microscopic Approach . . . . . . . . . . . . . . 1.4.1 Displacement and Strain . . . . . . . . . . 1.4.2 Microscopic Derivation of Elasticity . . . . 1.5 Forces, Stress and Response Functions . . . . . . . 1.5.1 Force Models . . . . . . . . . . . . . . . . 1.5.2 Force Chains, Stress, Elasticity and Friction 1.5.3 Force Statistics . . . . . . . . . . . . . . . 1.6 Concluding Remarks . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Response Functions in Isostatic Packings (C. F. Moukarzel) 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rigidity Considerations for Contact Networks . . . . . . . . . . . . . . . . 2.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Network Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Isostaticity in the Limit of Large Stiffness to Load Ratio . . . . . . . 2.3 Consequences of Isostaticity . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Green Functions and the Virtual Work Principle . . . . . . . . . . . 2.3.2 Anomalous Fluctuations: Multiplicative Noise in Isostatic Networks

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Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Topologically and Positionally Regular Isostatic Networks . . . . 2.4.2 Topologically Regular Isostatic Networks with Positional Disorder 2.4.3 Topologically Disordered Positionally Regular Isostatic Networks 2.4.4 Topologically and Positionally Disordered Isostatic Networks . . . 2.4.5 Non-sequential Isostatic Networks . . . . . . . . . . . . . . . . . 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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32 33 33 35 36 37 39 41

Statistical Mechanics of Jammed Matter (H. A. Makse, J. Brujić, and S. F. Edwards) 3.1 Introduction to the Concept of Jamming . . . . . . . . . . 3.1.1 Jamming in Glassy Systems . . . . . . . . . . . . 3.1.2 Jamming in Particulate Systems . . . . . . . . . . 3.1.3 Unifying Concepts in Granular Matter and Glasses 3.2 New Statistical Mechanics for Granular Matter . . . . . . . 3.2.1 Classical Statistical Mechanics . . . . . . . . . . . 3.2.2 Statistical Mechanics for Jammed Matter . . . . . . 3.2.3 The Classical Boltzmann Equation . . . . . . . . . 3.2.4 “Boltzmann Approach” to Granular Matter . . . . . 3.3 Jamming with the Confocal . . . . . . . . . . . . . . . . . 3.3.1 From Micromechanics to Thermodynamics . . . . 3.3.2 Model System . . . . . . . . . . . . . . . . . . . . 3.4 Jamming in a Periodic Box . . . . . . . . . . . . . . . . . 3.4.1 Simulating Jamming . . . . . . . . . . . . . . . . 3.4.2 Testing the Thermodynamics . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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45 45 46 48 51 53 53 54 59 61 64 64 65 72 73 77 83

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Granular Gas The Inelastic Maxwell Model (E. Ben-Naim and P. Krapivsky) 4.1 Introduction . . . . . . . . . . . . . 4.2 Uniform Gases: One Dimension . . 4.2.1 The Freely Cooling Case . . 4.2.2 The Forced Case . . . . . . 4.3 Uniform Gases: Arbitrary Dimension 4.3.1 The Freely Cooling Case . . 4.3.2 The Forced Case . . . . . . 4.3.3 Velocity Correlations . . . . 4.4 Impurities . . . . . . . . . . . . . . 4.4.1 Model A . . . . . . . . . . . 4.4.2 Model B . . . . . . . . . . . 4.4.3 Velocity Autocorrelations . .

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89 89 90 90 94 96 96 100 101 102 103 106 108

Contents

4.5 Mixtures . . . 4.6 Lattice Gases . 4.7 Conclusions . References . . . . . .

VII

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108 109 111 113

5 Cluster Formation in Compartmentalized Granular Gases (K. van der Weele, R. Mikkelsen, D. van der Meer, and D. Lohse) 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Vertically Vibrated Experiment . . . . . . . . . . . . 5.3 Eggers’ Flux Model . . . . . . . . . . . . . . . . . . . . 5.4 Extension to More than two Compartments . . . . . . . . 5.5 Urn Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Horizontally Vibrated System . . . . . . . . . . . . . . . 5.7 Double Well Model . . . . . . . . . . . . . . . . . . . . 5.8 Further Directions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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117 117 119 121 124 127 132 134 135 136

III Dense Granular Flow

141

6 Continuum Modeling of Granular Flow and Structure Formation (I. S. Aranson and L. S. Tsimring) 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Order Parameter Description of Partially Fluidized Granular Flows . . . . . 6.3 Avalanches on an Inclined Plane . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Stability of Simple Solution . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Avalanches in a Single-mode Approximation . . . . . . . . . . . . 6.3.3 Comparison with Experiment . . . . . . . . . . . . . . . . . . . . 6.4 Fitting the Theory with Molecular Dynamics Simulations . . . . . . . . . . 6.4.1 Order Parameter for Granular Fluidization: Static Contacts vs. Fluid Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Couette Flow in a Thin Granular Layer . . . . . . . . . . . . . . . 6.4.4 Fitting the Constitutive Relation . . . . . . . . . . . . . . . . . . . 6.5 Surface-driven Shear Granular Flow Under Gravity . . . . . . . . . . . . . 6.6 Stick-Slips and Granular Friction . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Contact Dynamics Study of 2D Granular Media: Relevant Internal Variables (F. Radjaï and S. Roux) 7.1 A Geometry–Mechanics Dialogue . . . . . 7.2 A Granular Model . . . . . . . . . . . . . . 7.2.1 Contact Dynamics . . . . . . . . . . 7.2.2 Driving Modes . . . . . . . . . . .

143 143 144 147 148 149 150 152 152 153 153 154 155 158 162 163

Critical States and

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165 165 165 166 167

VIII

Contents

7.3

Macroscopic Continuum Description . . . . . . . . . . 7.3.1 Constitutive Framework . . . . . . . . . . . . . 7.3.2 Relation Between Micro- and Macro-descriptors 7.3.3 Internal Variables . . . . . . . . . . . . . . . . 7.4 Numerical Results . . . . . . . . . . . . . . . . . . . . 7.4.1 Critical States . . . . . . . . . . . . . . . . . . 7.4.2 Stress–Strain Relation . . . . . . . . . . . . . . 7.4.3 Dilatancy . . . . . . . . . . . . . . . . . . . . 7.4.4 Internal Variables . . . . . . . . . . . . . . . . 7.4.5 Evolution of Internal Variables . . . . . . . . . 7.4.6 Frictional/Collisional Dissipation . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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168 168 169 170 171 171 174 176 179 181 184 185 186

Collision of Adhesive Viscoelastic Particles (N. V. Brilliantov and T. Pöschel) 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Forces Between Granular Particles . . . . . . . . . . . . 8.2.1 Elastic Forces . . . . . . . . . . . . . . . . . . . 8.2.2 Viscous Forces . . . . . . . . . . . . . . . . . . 8.2.3 Adhesion of Contacting Particles . . . . . . . . . 8.3 Collision of Granular Particles . . . . . . . . . . . . . . 8.3.1 Coefficient of Restitution . . . . . . . . . . . . . 8.3.2 Dimensional Analysis . . . . . . . . . . . . . . . 8.3.3 Coefficient of Restitution for Spheres . . . . . . . 8.3.4 Coefficient of Restitution for Adhesive Collisions 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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189 189 190 190 193 196 199 199 200 202 205 207 208

IV Hydrodynamic Interactions 9

Fluidized Beds: From Waves to Bubbles (É. Guazzelli) 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . 9.2 Flow Regimes and Instabilities . . . . . . . . . . . 9.3 Instability Mechanism . . . . . . . . . . . . . . . . 9.4 Governing Equations . . . . . . . . . . . . . . . . 9.5 Primary Instability . . . . . . . . . . . . . . . . . . 9.6 Rheology of the Particle Phase . . . . . . . . . . . 9.7 Secondary Instability and the Formation of Bubbles 9.8 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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213 213 214 216 218 219 222 223 228 229

Contents

10 Wind-blown Sand (H. J. Herrmann) 10.1 Introduction . . . . . . 10.2 The Wind Field . . . . 10.3 Aeolian Sand Transport 10.4 Dunes . . . . . . . . . 10.5 Conclusion . . . . . . . References . . . . . . . . . . .

IX

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V Charged and Magnetic Granular Matter

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253

11 Electrostatically Charged Granular Matter (S. M. Dammer, J. Werth, and H. Hinrichsen) 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Charged Granular Matter in Vacuum . . . . . . . . . . . . . . . 11.3 Charged Granular Matter in Suspension . . . . . . . . . . . . . . 11.4 Agglomeration of Monopolar Charged Suspensions . . . . . . . 11.4.1 Mean Field Rate Equation . . . . . . . . . . . . . . . . 11.4.2 Self-focussing Size Distribution . . . . . . . . . . . . . 11.4.3 Brownian Dynamics Simulations . . . . . . . . . . . . . 11.5 Coating Particles in Bipolarly Charged Suspensions . . . . . . . 11.5.1 Coulomb Interaction vs. Translational Brownian Motion 11.5.2 Coulomb Interaction vs. Rotational Brownian Motion . . 11.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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255 255 256 260 262 263 265 267 271 273 276 277 278

12 Magnetized Granular Materials (D. L. Blair and A. Kudrolli) 12.1 Introduction . . . . . . . . . . . . 12.2 Background: Dipolar Hard Spheres 12.3 Experimental Technique . . . . . . 12.4 The Phase Diagram . . . . . . . . 12.5 The Non-equipartition of Energy . 12.6 Cluster Growth Rates . . . . . . . 12.7 Compactness of the Cluster . . . . 12.8 Migration of Clusters . . . . . . . 12.9 Summary . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .

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VI

Contents

Computational Aspects

13 Molecular Dynamics Simulations of Granular Materials (S. Luding) 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . 13.2 The Soft-particle Molecular Dynamics Method . . . 13.2.1 Discrete-particle Model . . . . . . . . . . . 13.2.2 Equations of Motion . . . . . . . . . . . . 13.2.3 Contact Force Laws . . . . . . . . . . . . . 13.3 Hard-sphere Molecular Dynamics . . . . . . . . . . 13.3.1 Smooth Hard-sphere Collision Model . . . 13.3.2 Event-driven Algorithm . . . . . . . . . . . 13.4 The Link between ED and MD via the TC Model . 13.5 The Stress in Particle Simulations . . . . . . . . . . 13.5.1 Dynamic Stress . . . . . . . . . . . . . . . 13.5.2 Static Stress from Virtual Displacements . . 13.5.3 Stress for Soft and Hard Spheres . . . . . . 13.6 2D Simulation Results . . . . . . . . . . . . . . . . 13.6.1 The Equation of State from ED . . . . . . . 13.6.2 Quasi-static MD Simulations . . . . . . . . 13.7 Large-scale Computational Examples . . . . . . . . 13.7.1 Cluster Growth (ED) . . . . . . . . . . . . 13.7.2 3D Ring-shear Cell Simulation . . . . . . . 13.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

297

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299 299 300 300 300 301 305 305 306 307 309 309 310 310 311 311 312 316 316 318 321 322

14 Contact Dynamics for Beginners (L. Brendel, T. Unger, and D. E. Wolf ) 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Discrete Dynamical Equations . . . . . . . . . . . . . . . . . . . . . 14.3 Volume Exclusion in a One-dimensional Example . . . . . . . . . . . 14.4 The Three-dimensional Single Contact Case Without Cohesion . . . . 14.5 Iterative Determination of Constraint Forces in a Multi-contact System 14.6 Computational Effort: Comparison Between CD and MD . . . . . . . 14.7 Rolling and Torsion Friction . . . . . . . . . . . . . . . . . . . . . . . 14.8 Attractive Contact Forces . . . . . . . . . . . . . . . . . . . . . . . . 14.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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325 325 326 327 329 333 336 337 339 340 341

Index

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345

Preface

The macroscopic dynamics of granular media like sand cannot adequately be described by natural laws known for solids, fluids or gases. The well known reasons are that grains, the elementary constituents of a granular medium, themselves contain so many degrees of freedom, that their interactions are in general irreversible and moreover highly nonlinear. It is the task of physics to extract the essence of the relationship between grain scale dynamics and the collective behavior of a granular medium. What this essence might be, is still for many aspects controversial, as reflected by several contributions in this book, a clear indication that this is a thriving research field. In the past the research on granular matter has been primarily a domain of engineering science. Based mostly on empirical methods a wealth of knowledge has been accumulated how to efficiently and safely handle granular matter, ranging from ultrafine powders to granular bulk goods. This is of utmost technical and economical importance in a large variety of industrial applications. Hence, granular media became an exemplarily interdisciplinary research field, where physicists work fruitfully together with mechanical and process engineers. For example, the recent workshop “Cooperative Grains: From Granular Matter to Nano Materials” which took place in 2003 at the Lorentz Center in Leiden (NL) reflected this joint research activity of engineers and physicists. This book on the physics of granular media summarizes the state of art in this research field from the point of view of physicists in form of a collection of review articles. It also includes a CD-ROM containing color figures and additional movies. The book is intended to serve beginners, who want to enter the field, as well as experts in physics and engineering as a reference, covering experiments, theoretical approaches, as well as advanced simulation techniques. In the context of modern physics granular media provide examples of complex systems far from equilibrium. Compared to other fields in physics, theory, experiment and computer simulation are in remarkably close interaction, often professionally done within the same research group. The community of physicists working on granular media is still small, consisting of a few hundred researchers worldwide. Nevertheless the systematic study of granular media from the physics viewpoint has led to amazing new insights. In order to reduce the enormous complexity, any quantitative modeling of a granular system is based on certain simplifying assumptions. Because of the diversity of experimental situations these assumptions are not as straight forward as in other fields in physics, instead they strongly depend on personal intuition about the essential underlying physical processes. As a result there is no “standard model” of granular matter, instead a large variety of possible

XII

Preface

approaches is discussed, each of them with certain advantages and disadvantages. In fact, research on the physics of granular matter is a young and quickly evolving field. The book is organized as follows. The first chapter deals with static properties of dense granular matter. The article by Goldhirsch and Goldenberg focuses on the interplay of a microscopic formulation in terms of forces and displacements and a macroscopic continuum description in terms of stress and strain. Moukarzel, on the other hand, discusses minimally rigid packings, concentrating on those aspects where a continuum description fails. In the third article, Makse, Brujić, and Edwards give a comprehensive introduction to the physics of jammed granular matter, suggesting a new Boltzmann-type statistical mechanics approach in order to describe such systems. Chapter 2 deals with the physics of granular gases. In a theoretical paper, Ben-Naim and Krapivsky consider models of totally inelastic granular gases, showing that the velocity distribution during cooling approaches a scaling form. They also consider binary mixtures and inelastic granular gases with external energy input in a steady state. In an experimental paper, van der Weele, Mikkelsen, van der Meer, and Lohse investigate the cooling of granular gases in compartmentalized driven systems, so-called Maxwell demon experiments. Reducing external driving the granular gas clusters reversibly in one of the compartments in a way that can be explained theoretically. In the last article, Brilliantov and Pöschel study the collision of adhesive viscoelastic particles. In particular, they derive the general solution for the contact problem of convex grains in the limit of sufficiently small impact velocities and a small viscosity relaxation time. Chapter 3 is devoted to phenomena occurring in dense granular flow. For such partially fluidized systems Aranson and Tsimring present a phenomenological continuum theory which describes phenomena such as avalanches in thin granular layers and surface shear flows in deep granular beds. On the other hand, Radjai and Roux employ the microscopic contact dynamics approach in order to describe plastic properties of two-dimensional packings. They focus in particular on the so-called critical state of granular packings, which may emerge after repeated plastic deformation. Topic of Chapter 4 is the hydrodynamic aspect of granular matter. The article by Guazzelli reviews the physics of fluidized beds and the transition from uniform flow via emerging regular waves to irregular flow and the formation of bubbles which can be explained in terms of certain instabilities. The second article by Herrmann studies the motion of sand dunes. The remarkable theoretical progress of the last few years in understanding this intriguing pattern formation relies on the careful combination of the physics of turbulent boundary layers and granular media. Chapter 5 deals with electrostatic and magnetic interactions in granular matter. The article by Dammer, Werth, and one of us reviews some important aspects of electrically charged granular gases and discusses the agglomeration of charged particles in a suspension and possible applications. As an entirely new problem, Kudrolli discusses the properties of magnetic granular media (see cover picture). Studying the clustering properties in a driven system experimentally, a rich phase structure is observed. Due to the anisotropic nature of magnetic interactions, the emerging clusters exhibit both simple and complex shapes. The last Chapter addresses recent development concerning advanced simulation techniques. A review by Luding discusses molecular dynamics simulations of soft particles as well as event-driven simulations of hard particles together with a concept to combine both ap-

Preface

XIII

proaches in the dense limit. Moreover, as an example for a micro-macro transition, the stress tensor is computed and compared using both methods. Finally, the last article by Brendel, Unger and one of us gives a tutorial on contact dynamics, a simulation method particularly suited for dense systems of rigid particles. We would like to thank the publisher VCH-Wiley, in particular Mrs. Wanka and Mr. Krieg, for excellent support.


• Igor S. Aranson, Ch. 6 Materials Science Division Argonne National Laboratory 9700 South Cass Avenue Argonne, IL 60439 USA

• Eli Ben-Naim,

Ch. 4

Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory Los Alamos, NM 87545 USA

• Daniel L. Blair, Ch. 12 Department of Physics Clark University Worcester, MA 01610 USA

Institut für Physik Universität Duisburg-Essen 47048 Duisburg Germany e-mail: [email protected]

Department of Physics Moscow State University 119899, Vorobievy Gory Moscow Russia

Nikolai V. Brilliantov Institute of Physics University of Potsdam Am Neuen Palais 10 14469 Potsdam Germany

• Jasna Brujic,

Ch. 3

Polymers and Colloids Group Cavendish Laboratory University of Cambridge Madingley Road Cambridge CB3 OEH UK

• Stephan M. Dammer, Ch. 11

• Lothar Brendel, Ch. 14

• Nikolai V. Brilliantov,

Current address:

Ch. 8

Institut für Physik Universität Duisburg-Essen 47058 Duisburg Germany

• Sam F. Edwards, Ch. 3 Polymers and Colloids Group Cavendish Laboratory University of Cambridge Madingley Road Cambridge CB3 OEH UK

XVI

• Chay Goldenberg, Ch. 1 School of Physics and Astronomy Faculty of Exact Sciences Tel-Aviv University Ramat-Aviv Tel-Aviv 69978 Israel e-mail: [email protected]

• Isaac Goldhirsch, Ch. 1 Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering Tel-Aviv University Ramat-Aviv Tel-Aviv 69978 Israel e-mail: [email protected]

• Élisabeth Guazzelli, Ch. 9 IUSTI-CNRS UMR 6595 Polytechnique Marseille Technopôle de Château-Gombert 13453 Marseille, cedex 13 France

• Hans J. Herrmann, Ch. 10 Institute for Computer Applications 1 University of Stuttgart Pfaffenwaldring 27 70569 Stuttgart Germany

• Haye Hinrichsen, Ch. 11 Fakultät für Physik und Astronomie Universität Würzburg 97074 Würzburg Germany

• Paul L. Krapivsky, Ch. 4 Center for Polymer Studies and Department of Physics, Boston University Boston, MA 02215 USA


• Arshad Kudrolli, Ch. 12 Department of Physics Clark University Worcester, MA 01610 USA

• Detlef Lohse,

Ch. 5

Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics, University of Twente P.O. Box 217 7500 AE Enschede The Netherlands

• Stefan Luding,

Ch. 13

Particle Technology, DelftChemTech TU Delft Julianalaan 136 2628 BL Delft The Netherlands e-mail: [email protected] Homepage: www.ical.uni-stuttgart.de/-lui

• Hernan A. Makse,

Ch. 3

Levich Institute and Physics department City College of New York New York, NY 10031 USA

• Devaraj van der Meer, Ch. 5 Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics, University of Twente P.O. Box 217 7500 AE Enschede The Netherlands

• Rene Mikkelsen,

Ch. 5

Department of Applied Physics and J.M. Burgers Centre for Fluid Dynamics, University of Twente P.O. Box 217 7500 AE Enschede The Netherlands


XVII

• Cristian F. Moukarzel, Ch. 2 Departamento de Física Aplicada Cinvestav Mérida 97310 Merida, Yucatán Mexico

• Lev S. Tsimring, Ch. 6 Institute for Nonlinear Science University of California at San Diego La Jolla, CA 92093-0402 USA

• Tamas Unger, Ch. 14 • Thorsten Pöschel, Ch. 8 Institut für Biochemie Charité, Monbijoustrasse 2 10117 Berlin Germany

• Farhang Radjaï,

Ch. 7

Laboratoire de Mécanique et Génie Civil CNRS/Université Montpellier II Place Eugène Bataillon F-34095, Montpellier cedex France

• Stéphane Roux,

Ch. 7

Laboratoire Surface du Verre et Interfaces CNRS/Saint-Gobain 39 Quai Lucien Lefranc F-93303 Aubervilliers cedex France


• Ko van der Weele, Ch. 5 Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente P.O. Box 217 7500 AE Enschede The Netherlands

• Jochen Werth, Ch. 11 Institut für Physik Universität Duisburg-Essen 47058 Duisburg Germany

• Dietrich E. Wolf,

Ch. 14


Part I

Static Properties

1 Stress in Dense Granular Materials Isaac Goldhirsch and Chay Goldenberg

1.1 Introduction

In the year 1776, Coulomb presented a study of granular materials to the “Academie Royale” in Paris. In the 19th century, granular materials were studied by Faraday, Reynolds, Hagen and others. Due to the numerous engineering applications involving granular matter, research in this field has been continually performed by engineers (cf. [1] for a recent summary of engineering approaches to this subject). The subject has found renewed interest in the physics community (as well as other communities) in the past two decades [2, 3]. Two main subfields have developed, that of granular gases (fluidized granular matter, in which the interactions are nearly instantaneous collisions), and statics or quasi-static dense flows. Granular gases have been successfully studied by using extensions of the kinetic theory of gases [4]. The behavior of dense granular matter, which is dominated by prolonged interparticle contacts, has proven more difficult for modeling. Quasi-static properties of granular materials are commonly modeled by engineers, using elasto-plastic models [5]. In recent years, a new class of models has been proposed for describing the statics of granular materials. These are based on the notion of “force propagation”, suggested by the observation of force chains, i.e., chains of contacts which experience “larger than average” forces, in experiments on granular materials [6], as well as simulations [7, 8]. These models (see e.g., [9, 10]) typically yield hyperbolic partial differential equations for the stress field, in contrast with the elliptic, non-propagating nature of the classical equations of static elasticity. It has been claimed that, at large scales, the hyperbolic models become elastic-like [11–13]. A certain degree of controversy has developed concerning the “correct” description of static granular matter. One of the goals of this chapter is to touch upon this controversy, and propose a way to reconcile the two approaches. This requires a critical study of the importance of different scales in granular systems and a cognitive differentiation between the notions of force and stress. Another goal is to briefly present the fundamentals of the various approaches, and add some recent results. As many models of granular systems ignore the effects of friction, it is a further goal of this chapter to present results and comments on these effects. Continuum mechanics [14, 15]) provides the basic terminology and description of many of the aspects of many-body systems; being based on the most fundamental conservation laws, it is not a subject of controversy. The basics of continuum mechanics are presented next.

4

1 Stress in Dense Granular Materials

1.2 Continuum Mechanics: A Brief Review Although at sufficiently small scales, materials are composed of discrete constituents (atoms or molecules for “regular” materials, macroscopic particles in granular materials), the continuum hypothesis is taken to apply at sufficiently large scales such that an “infinitesimal volume element” (or “material particle”) still contains a sufficiently large number of microscopic particles for the macroscopic fields to be continuous. The equations of continuum mechanics are based on the conservation of mass, momentum and energy. These are commonly expressed as differential equations for the (mass) density field, ρ(r, t), the momentum density field, p(r, t), and the energy density field , e(r, t). These equations (in the absence of body forces and sources or sinks of mass and energy) are: ρ˙ p˙ α e˙

= −div(ρV) ∂ = − [ρVα Vβ − σαβ ] ∂rβ ∂ = − [Vβ e − Vα σαβ + cβ ] ∂rβ

(1.1) (1.2) (1.3)

where a dot denotes a partial time derivative, Greek indices denote Cartesian coordinates (the Einstein summation convention is used), and the explicit dependence of the fields on r and t has been omitted. The velocity field, V, is defined by p ≡ ρV, σ is the stress tensor and c denotes the heat flux. As Eqs. (1.1–1.3) are obtained from fundamental conservation laws, their validity is very general. In order to close the set of continuum equations, σ and c must be expressed in terms of the basic fields, i.e., constitutive relations must be established. The constitutive relations are material-class specific, e.g., they are different for fluids and solids. In some cases, the above set of fields and corresponding constitutive relations need to be extended, e.g., to account for rotational degrees of freedom or frictional dissipation. Constitutive relations are often postulated on the basis of phenomenological arguments and empirical information, rather than on derivations from the underlying microscopic dynamics. This is the way fluid mechanics had originally been developed, though later the same result was substantiated by employing kinetic theory and, more generally, response and projection operator theories. The standard continuum description of solids requires the consideration of the notion of deformation (often, with respect to a reference state), expressed in terms of the displacement and strain fields [14, 15]. Let the Lagrangian coordinate, R, denote the coordinates of a material particle at time t = 0, and let r(R, t) denote its position at time t. The displacement field is given by: U(R, t) ≡ r(R, t) − R, and the velocity field is: V(R, t) = ∂U(R,t) . The Eulerian description employs r and t, instead of R and ∂t t as the independent variables. The constitutiverelation used in classical linear elasticity is ∂U ∂Uα is the linear strain tensor, and Cαβγδ is the σαβ = Cαβγδ γδ , where αβ = 12 ∂Rβα + ∂R β ∂Uα α tensor of elastic constants. Note the ∂R = ∂U ∂rβ to linear order in the gradients. When thermal β effects are neglected, the elastic energy density is given by e = 12 Cαβγδ αβ γδ . The equations of elasticity have been derived from a microscopic description in the case of (nearly) homogeneously strained lattice configurations, within the harmonic approximation of interparticle interactions [16,17]; the extension of these derivations to disordered systems is not trivial [18].

1.3

Constitutive Relations for Dense Granular Materials

5

1.3 Constitutive Relations for Dense Granular Materials 1.3.1 Engineering Approaches Commonly used constitutive relations for dense granular materials are based on elasto-plastic models: the material is assumed to behave elastically under small stress and to yield (flow) plastically when a certain function of the stress components exceeds a threshold. In many approaches the elastic deformation is neglected and rigid-plastic models are used. The yield function Y (σαβ ) is a function of the stress components σαβ , chosen to vanish at yielding (it can be envisioned as defining a surface in the space of principal stresses; the material yields when the state of stress reaches the surface). Elasto-plastic models for granular materials have been developed mainly in the context of soil mechanics [5, 19, 20]. Most of these are based on extensions of metal plasticity [14], with a pressure-dependent yield function. The simplest yield function suggested for granular materials is based on the Coulomb yield criterion, according to which the shear stress at failure equals the product of the coefficient of friction, µ, and the normal stress, similar to the description of a block sliding down an inclined plane (µ is often characterized by the angle of friction: φ = tan−1 µ). The corresponding yield function is Y = |τ | + p tan φ, where τ is the shear stress and p is the normal stress (pressure). In three dimensions, the corresponding yield surface is in the shape of a cone (extended von Mises yield function) or hexagonal pyramid (extended Tresca yield function). According to this idealized (rigid-plastic) model, the granular material behaves rigidly until the shear stress exceeds a certain value, proportional to the normal stress, whereupon the material fails by forming two blocks sliding with respect to each other. Upon yielding, the ∂Y . strain rate, ˙αβ , is usually assumed to satisfy an “associated flow rule”: ˙αβ = λ ∂σ αβ Extensions of this model include: (i) A non-associated flow rule (replacing φ by the angle of dilatancy ν), which describes the experimentally observed increase in volume of the material under shear (known as Reynolds dilatancy) [5]. (ii) A curved yield surface (which provides an improved description of the energy dissipation and the asymptotic density obtained at large deformation) [19], and (iii) Strain hardening models (with yield functions which depend on the strain history, usually through the density). An example of the latter is the “critical state theory” of Schofield and Wroth [5, 19] with a curved yield surface whose volume in the principal stress space depends on the density. Other suggestions for constitutive models for granular materials include: (iv) Micropolar or Cosserat-type models (see e.g., [21]), which treat the angular momentum of the particles as a kinematic field that couples to translation by friction. In these models the stress tensor is asymmetric, and an additional couple-stress field is introduced [15]. One motivation for using such models is the description of localized deformations [21] – the deformation of granular materials is often concentrated in a thin layer (referred to as a shear band) with a typical width of about 10 particle diameters [5]. These models include an intrinsic length scale which imposes a finite thickness for these shear bands (in the usual plastic models they are infinitesimally thin); (v) Gradient-dependent theories which extend plastic models by considering the dependence of the yield function or the flow rule on higher-order strain gradients; and (vi) Hypoplastic models (extensions of hypoelasticity [15]), with no natural state for the material to return to when all stresses are removed. (vii) Suggestions for combining the collisional stress, which is important in rapid flows [4]

6


with the contact stress (the only kind of stress accounted for in quasistatic deformations) in order to obtain a theory of more general validity, have been proposed [22] as well.

1.3.2 Recent Approaches An alternative approach for the description of stress in static granular materials, which is very different from the standard elastic or elasto-plastic descriptions [9], is based on the assumption that the continuum equations can be closed by postulating relations among the stress components, without considering the deformation of the material. This class of models has been assumed to be relevant in the idealized limit of infinitely stiff grains. The resulting equations for the stress inside the material are hyperbolic, unlike the elliptic equations obtained from linear elasticity. As a microscopic justification, it has been suggested that random packings of infinitely rigid frictionless particles are isostatic [10,23]. This means that the forces among the particles can be determined uniquely from the configuration, geometry and boundary forces, using only Newton’s first and third laws [9, 10, 24–26]. It is easy to show that isostaticity implies that in d dimensions, the mean coordination number should be exactly z = 2d for frictionless spheres and z = d + 1 for frictional ones, i.e., the system should be rather tenuous. Recent numerical studies [27, 28] suggest that packings of frictionless (but not frictional) spheres do approach isostaticity in the limit of infinite rigidity. An embodiment of these ideas, inspired by experiments, is the force propagation model, according to which contact forces propagate along force chains [11–13]. This model has been claimed to yield elastic-like equations at large scales (though a physical interpretation of the fields satisfying these equations is lacking) [11–13]. It has also been proposed that granular materials may experience a crossover from a hyperbolic to an elliptic behavior as the degree of disorder increases [29]. As real granular materials have finite rigidity (and are frictional), one may question the relevance of the isostatic limit. It may be relevant for small confining forces, and perhaps one can develop a theory for near-isostatic systems around the singular [23] isostatic limit. It is important to note that a hyperbolic description for static systems is qualitatively very different from an elliptic description. In particular, a hyperbolic description implies that stress propagates along characteristic directions, so that boundary conditions cannot be specified arbitrarily on the entire boundary of the system, or, equivalently, that the stress distribution in the system is independent of (some of) the boundary conditions, e.g., the properties of the floor on which a granular pile is resting. Another model for stress propagation, known as the “q-model” [30], is of parabolic nature. This model assumes a random, diffusive transmission of forces which, when applied to the response to a localized force, yield a parabolic description [31].

1.3.3 Experiments and Possible Reconciliation One would have thought that experiment should decide whether (at least) rather hard granular materials should conform to one description or the other. However, this is not the case. Different experiments can be taken to support either one of these descriptions, or both. In [29, 31, 32], the interparticle force distribution has been measured in two-dimensional (2D) granular slabs, with a localized force applied at the top layer. In [33], the particle displacements for similar 2D systems have been measured. Prominent force chains have been

1.4

A Microscopic Approach

7

observed in ordered 2D systems; the force chains fade out with increasing disorder. For pentagonal particles in 2D (random) arrangements the measured force distribution is single peaked, and the width of the peak is linearly related to the vertical distance, in conformity with elasticity. The results for cuboidal particles obtained in [31] appear to suggest a parabolic behavior, consistent with a diffusive model, but the systems studied were quite small. In [33], the width of the measured distribution of displacements, as a function of the vertical distance from the particle which is directly displaced, follows a square root dependence (as expected from a diffusive model) for small distances of a few particle diameters, crossing over to a linear dependence at larger distances (consistent with an elastic description). In experiments on similarly forced three-dimensional (3D) ordered slabs [34–36], the vertical force distribution on the floor exhibits multiple force peaks for shallow systems [36] and less structure for deeper ones. These results have been interpreted in terms of a (hyperbolic) force-propagation model. In experiments on somewhat larger (in terms of number of particles), disordered 3D systems [34, 35], a single peak in the stress distribution measured at the floor was observed, whose width is proportional to the depth of the system. We propose that these seemingly contradictory experimental results (and theoretical explanations) are not necessarily at odds with each other. The basic observation is that even elastic systems are not described by the (macroscopic) equations of elasticity on small scales. Therefore, even when the interactions are, e.g., harmonic, one should expect significant corrections to the continuum elastic equations on small scales, which should be prominent mostly in the (relatively small) systems (whose size is typically a few dozen particle diameters) studied by physicists, but not in the large systems considered by engineers. At small scales local anisotropies and randomness play a major role and render the force networks strongly anisotropic, similar to the predictions of a hyperbolic theory. Indeed, simulations [18,37] reveal the existence of a crossover from microscopic to macroscopic behavior of granular assemblies (as well as other systems [38]) as a function of system size or resolution. We believe that such a crossover is observed in some of the experiments mentioned above. Furthermore, even classical elasticity is nearly hyperbolic for strongly anisotropic systems. These arguments leave open the question of the possible relevance of isostatic systems, interesting as they may be, as a singular limit of real systems of finite rigidity (and frictional interactions). An important point to notice is the distinction between force and stress. Whereas interparticle forces can exhibit force chains which look like they contradict elasticity, the latter is not (directly) concerned with interparticle forces at all, but rather with the stress field. This field involves an averaging over the forces (whose result is resolution dependent) and leads to less pronounced structure than the underlying force field, as shown explicitly below. Therefore the existence of force chains cannot be taken as an argument against elasticity or elasto-plasticity, or in favor of it. This issue is further discussed below.

1.4 A Microscopic Approach A coarse graining approach [18, 39] leading to expressions for the stress and other fields in terms of microscopic information is presented next. Unlike in [39], here we present only spatial (not temporal) coarse graining, since the granular systems of interest here cannot “sample phase space”, being usually stuck in metastable states. Although experimental results pre-

8


sented in the literature are averaged over several configuration to decrease the fluctuations, it is unclear what kind of averaging (if any) is appropriate for granular systems. Therefore we believe it is important to define the stress in such a way that it applies to single realizations. Consider a system of particles whose masses, center of mass positions and velocities at mass density time t are given by {mi ; ri (t); vi (t)}. Following [39] define the coarse-grained ρ(r, t) ≡ i mi φ[r − ri (t)], and the momentum density p(r, t) ≡ i mi vi (t)φ[r − ri (t)], where φ(R) is a non-negative coarse-graining function (with a single maximum at R = 0) of width w, the coarse-graining scale, and φ(R)dR = 1. The energy density is given by e(r, t) ≡

1 1 mi vi2 (t)φ[r − ri (t)] + Φ (rij (t)) φ[r − ri (t)], 2 i 2 i,j;i=j

where it is assumed that the particles interact by a pairwise potential Φ (rij ) (this assumption is not necessary for obtaining the continuum equations, but it does simplify the derivation). The conservation equations [Eqs. (1.1–1.3)] are obtained upon taking the time derivative of the macroscopic fields, ρ, p and e, and performing straightforward algebraic manipulations [39]. In addition to obtaining the standard equations of continuum mechanics from microscopic considerations, this procedure provides expressions for the stress tensor, σαβ , and the heat flux, c: mi viα (r, t) viβ (r, t)φ(r − ri (t)) σαβ = − i

1 1 − fijα rijβ dsφ[r − ri (t) + srij (t)], 2 0

(1.4)

ij;i=j

cβ

  1  = Φ (rij (t)) viβ φ[r − ri (t)] mi vi2 + 2 i j;j=i 1 1 + viα + vjα fijα (t)rijβ (t) dsφ[r − ri (t) + srij (t)] 4 0

(1.5)

ijα;j=i

vi (r, t)

≡ vi (t) − v(r, t) is the fluctuating velocity, fij = −∇i Φ (rij ) is the force where exerted on particle i by particle j, and rij ≡ ri − rj . The first line in Eq. (1.4) defines the kinetic stress (which vanishes in quasi-static deformations); the second is the contact stress.

1.4.1 Displacement and Strain In uniformly strained lattice configuration, the dependence of the relative microscopic particle displacements (denoted by uij ≡ ui −uj ) on the positions of the particles in the reference state are simply related to the macroscopic strain field αβ : uijα (r, t) = αβ rijβ (t = 0), i.e., the deformation is affine. This relation is often postulated for random configurations of particles, and for non-uniform strains, as a “mean field” approximation (see e.g., [40]); an improved version thereof minimizes the RMS difference between the actual relative displacements and the above mean field expression in a given volume [41]. However, an affine deformation

1.4

A Microscopic Approach

9

is inconsistent with local force equilibrium in generic (random) configurations, as has been explicitly shown [42]. Furthermore, the assumption that one can deduce microscopic information (single particle displacements) from macroscopic (hence, averaged) information (the strain field), is flawed. It is therefore not surprising that the time derivative of the linear strain field obtained from mean field approaches does not equal the symmetrized velocity gradient (the rate of strain tensor). This fact also leads to problems in the determination of the elastic moduli of granular solids [43]. An alternative expression [18] for the strain field, which is based on continuum mechanics, can be obtained by exploiting the relation between the displacement and velocity fields (in t the Lagrangian representation): V(R, t) = ∂U (R, t)/∂t, hence: U(R, t) = 0 V(R, t )dt . Substituting the microscopic expressions for the momentum and density fields, and using p ≡ ρV, one obtains: t i mi vi (t )φ[r(R, t ) − ri (t )] dt . U(R, t) ≡ (1.6) 0 j mj φ[r(R, t ) − rj (t )] The macroscopic displacement field, U, clearly depends on the entire trajectory of each particle. However, noting that u˙ i = vi , where ui ≡ ri (t) − ri (0) is the displacement of particle i, and invoking integration by parts in Eq. (1.6), one obtains [18] that the displacement, up to terms which are non-linear in the strain, is given (in the Eulerian representation) by: lin i mi ui (t)φ[r − ri (t)] (1.7) U (r, t) ≡ j mj φ[r − rj (t)] Unlike the full displacement field, Ulin depends only on the total displacement of each particle, ui (t). Using Eq. (1.7), the linear strain is given by:

1 ∂ ∂ lin αβ = 2 φ[r − ri ] . mi mj φ[r − rj ] uijα + uijβ (1.8) 2ρ ij ∂rβ ∂rα

1.4.2 Microscopic Derivation of Elasticity It is quite surprising that the microscopic derivations of elasticity produced heretofore are limited to lattice configurations [16]. The coarse graining method described in Section 1.4 and 1.4.1 can be used to obtain a derivation of elasticity for the case of a disordered system. The derivation below specializes to harmonic interactions, and as such it does not directly apply to (non-prestressed) non-cohesive granular systems, but it provides useful insights. For harmonicinteractions, the linearized force exerted by particle j on particle i is given by: fij −Kij ˆr0ij · uij ˆr0ij ,, where the superscript ‘0’ denotes the reference, zero force, configuration. Using Eq. (1.4) the linearized contact stress field can be written as: 1 1 lin 0 0 0 σαβ (r, t) = Kij rîjγ uijγ (t)ˆ rijα rijβ dsφ[r − r0i + sr0ij ]. (1.9) 2 ij 0 This expression is not manifestly proportional to the linear strain field [Eq. (1.8)]. The stress and strain are two different averages of the microscopic displacements, and they need not, a priori, be proportional to each other.

10


To see how linear elasticity still comes about [18], consider a volume Ω, whose linear dimension, W , is much larger than the coarse-graining scale, w, and let r be an interior point of Ω which is ‘far’ from its boundary. Let upper case Latin indices denote the particles in the exterior of Ω which interact with particles inside Ω. Since the considered system is linear, there exists a Green’s function G such that uiα= Jβ GiαJβ uJβ for i ∈ Ω. Define LijαJβ ≡ GiαJβ − GjαJβ . It follows that uijα = Jβ LijαJβ uJβ . Since a rigid translation (all {uJ } equal) results in zero relative particle displacements (uij = 0), one can write: uijα = Jβ LijαJβ [uJβ − Uβ (r)]. It follows that uijα =

{LijαJβ [Uβ (rJ ) − Uβ (r)] + LijαJβ [uJβ − Uβ (rJ )]} ,

(1.10)

Jβ

where uJβ − Uβ (rJ ) is a fluctuating displacement. The second term is subdominant when W sufficiently exceeds w. To leading order in a gradient expansion: Uβ (rJ ) − Uβ (r) ∂Uβ (r) γ ∂rγ (rJγ − rγ ) . Substituting this result in Eq. (1.9), we obtain (invoking the rotational lin (r) µν Cαβµν (r)µν (r), where symmetry of the microscopic Green’s function): σαβ Cαβµν (r) =

1 0 0 0 0 1 Kij LijγJµ rJν − rν rîjα rijβ rîjγ dsφ[r − r0i + sr0ij ]. 2 0 ijJγ

Thus linear elasticity is valid for small strains and small strain gradients. More importantly, elasticity is only valid on sufficiently large scales (which may be required to be larger as disorder increases [18]) , and even then the stress and strain field are generically inhomogeneous for disordered systems or inhomogeneous straining [18] (see also [38,44]). Further insight can be gained by considering the energy density e(r, t). It turns out [18] that there is a correction to the standard relation e = 12 σ : of linear elasticity, whose form is the divergence of a flux (representing the work of the fluctuating forces). This correction term becomes negligible on sufficiently large scales, i.e., classical elasticity is regained.

1.5 Forces, Stress and Response Functions 1.5.1 Force Models Perhaps the simplest model granular system is a collection of frictionless spherical particles. The interactions of the grains are usually taken to depend (assuming they are sufficiently stiff) on the imaginary overlaps of the spheres [7, 45, 46]), defined as the difference of the distance between their centers and the sum of the radii of the spheres. For two frictionless elastic spheres, a classical result by Hertz (see e.g., [47]) is that the force is proportional to the overlap to power 32 , whereas for cylinders it is linear in the overlap. Non-cohesive particles experience only repulsive forces. Even for frictionless particles, internal dissipation as in, e.g., viscoelastic particles, gives rise to a dependence of the force on the relative velocity as well (for some examples of force schemes commonly used in simulations, see e.g., [48–50]). The description of static and kinetic friction requires the use of more complicated force models, which depend on the particle orientations and their relative tangential velocities, and possibly

1.5

Forces, Stress and Response Functions

11

on the history of contact deformation (see e.g., [48–51]). Spherical particles are often used in carefully prepared experiments, but most natural or industrial granular materials are composed of non-spherical particles. Such particles are more difficult to treat theoretically, but several models for simulating their behavior have been suggested (e.g., [52]). In isostatic systems the forces can be determined from the equations of equilibrium alone, and are therefore independent of the force-displacement law; however, the particles’ actual displacements do depend on this law (consider, e.g., the infinite stiffness limit for systems of identical configuration but with different stiffness distributions, keeping the stiffness ratios fixed in each of the systems). Photoelastic particles are often used to measure the stress in granular systems [6, 29, 31, 32, 53]. These measurements probe the stress inside each particle, i.e., a “sub-microscopic” field, as the interparticle forces are the microscopic entities of the system.

1.5.2 Force Chains, Stress, Elasticity and Friction As mentioned above, experiments [6, 32, 53] and simulations [7, 8] reveal that granular materials possess inhomogeneous networks of ‘larger-than-average’ contact forces, referred to as “force chains”. These chains are visually apparent in plots in which forces are indicated by lines whose thickness is proportional to the force magnitude. In homogeneous systems one can easily define force chains as pertaining to forces above a certain cutoff (such as the average interparticle force). It is less obvious how to define force chains in inhomogeneous systems, e.g., in systems subject to gravity, in which the mean force increases with depth. Although the typical lengths of straight segments in force chains are of the order of a few particle diameters, it has been suggested [9, 54, 55]) that their existence invalidates elasticity and requires a very different description from that of “regular” materials. While we do not presume to know the ‘correct’ description of granular materials, it is worth noticing that force chains appear in systems of particles whose interactions are harmonic. Figure 1.1a depicts force chains in an assembly of 2D particles, initially positioned on a 2D triangular lattice with lattice constant d (with square-shaped boundaries), whose x and y coordinates have been randomly displaced from their respective lattice positions by ±0.04d. Points whose distance is less than 1.02d are connected by linear springs (whose equilibrium lengths equal the respective distances) with equal spring constants (this results in an average dilution of about 12% of the springs compared to the perfect lattice). A uniform isotropic compression of 1% is applied to the boundary particles. Similar force chains have been observed in a polydisperse Lenard– Jones system [38]. These results indicate that force chains are not specific to granular systems and their existence does not contradict large-scale elasticity. As a matter of fact, it is quite certain that if one could observe the individual interparticle ‘forces’ in atomic systems one would also observe force chains. Another notable observation is that a significant portion of the stress (even in a homogeneously strained system) in granular systems is carried by forces which do not belong to the force chains. An example is provided by a system of frictionless polydisperse disks (with radii uniformly distributed within 50% of the maximal radius) which are initially placed on a triangular lattice with spacing equal to the maximal radius, with walls that are then isotropically compressed, such that the final density is 97.5% that of a close-packed lattice of uniform disks. The interparticle forces are taken to be linear in the overlaps. Figure 1.2a shows the forces in

12


the system. Figure 1.2b shows the fraction of the vertical force carried by the forces whose magnitude is greater than the mean, for forces in horizontal “slices” of the system, as a function of the vertical coordinate, z. As seen in Figure 1.2b, only about 70% of the applied force is carried by the force chains. Furthermore, the force carried by the chains fluctuates with depth, so that the forces in the chains do not obey the conditions of force equilibrium (as assumed in models based on force-chain splitting [11, 13]). It is therefore questionable whether models which describe the stress exclusively in terms of the force chains are justified. A further argument showing that the presence of force chains does not necessarily imply that elasticity is invalid, can be obtained by considering the stress field in systems which exhibit force chains. Consider a two-dimensional system of uniform disks (arranged on a triangular lattice) subject to a vertical external force at the center of the top layer [37]. Experiments on such systems are described in [29, 32]. Consider first an unrealistic model for the grain interactions, i.e., the case of nearest neighbor harmonic forces. Figure 1.3 presents the forces in the system. Force chains are evident. A contour plot of the “vertical stress component” σzz [computed 1 −(|r|/w)2 , using Eq. (1.4)] for the same system is shown in Figure 1.3b (with φ(r) = πw 2e and w = d, the particle diameter, i.e., a fine resolution). The force chains are no longer evident. The model described above corresponds, in the continuum (long-wavelength) limit, to an isotropic 2D elastic medium. The observed force chains, which break isotropy, can be attributed to the fact that the local environment of a particle in contact with a finite number of other particles cannot be isotropic. Such anisotropy is inherent in any discrete system. When a localized force is applied to the system, as in the discussed case, the degree of anisotropy and inhomogeneity is increased, in particular in the neighborhood of the point of application. As the large-scale description of the discussed system corresponds to linear elasticity, one obtains that for sufficiently large systems, the distribution of forces on the floor corresponds closely to the stress calculated using linear elasticity [37], as expected. −1

10

−2

10

P −3

10

−4

10

−5

10

0

0.5

1

1.5

2

2.5

3

3.5

F

a)

b)

Figure 1.1: a) The forces in a bond-diluted distorted triangular network of linear springs. Forces with magnitude larger than the mean are indicated by solid lines whose width is proportional to the magnitude, and smaller forces by thin dotted lines. Compressive forces are indicated by gray lines; tensile forces by black lines. b) The distribution of force magnitudes in an ensemble of such networks.

1.5


13

Deviations from the elastic solutions for small systems may be important in some of the experiments mentioned in Section 1.3.3: Figure 1.4a compares the vertical stress at the floor of 2D slabs composed of a different number of discrete layers with isotropic elastic solutions for a finite slab (with rough or frictionless support) and a half plane (Boussinesq’s solution). Note that boundary conditions strongly affect the solution: the solution of the equations of elasticity for a finite slab with appropriate boundary conditions resembles the experimental result [35, 37], whereas the (infinite half plane) Boussinesq solution does not. The convergence to the appropriate (rough support) solution for a sufficient number of layers is evident. In order to examine this crossover in a 3D discrete system, whose continuum limit corre1

0.8

0.6

Φ 0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

z/h

a)

b)

Figure 1.2: a) The forces in a system of polydisperse frictionless disks under uniform compression. Top: all forces, bottom: only forces whose magnitude is larger than the mean. b) The fraction of the applied vertical force carried the forces whose magnitude is larger than the mean (i.e., those belonging to force chains), vs. the vertical coordinate, z, scaled by the system height, h.

a)

b)

Figure 1.3: a) Force chains in a 2D triangular lattice. A vertical force is applied at the center of the top layer. Line widths are proportional to the force magnitudes. Only the central part of the system is shown. b) Contour plot of hσzz in the same system (h is the slab height); reproduced from [37].

14


sponds to an isotropic medium, we used a simple cubic lattice, with equal springs coupling nearest neighbors and next-nearest neighbors [17]. The convergence to the elastic solution (Figure 1.4b) is apparent in this case as well, although a larger number of layers is required for this 3D system. A similar crossover is also observed in disordered systems [18], where the expected elastic behavior is recovered on sufficiently large scales (which seems to increase with the disorder, in accordance with the derivation presented in Section 1.4.2). In a disordered system, the deformation is inhomogeneous (on the considered scale) even under boundary conditions which would induce a homogeneous deformation in an ordered system (uniform compression). 2D triangular lattice, 40 layers 2D triangular lattice, 10 layers 2D triangular lattice, 5 layers Elastic − rough rigid support Elastic − frictionless rigid support Elastic − infinite half plane

0.9 0.7

0.7

0.5

2

0.5

h σzz

hσzz 0.3

0.3

0.1

0.1

−0.1 0

3D cubic lattice, 60 layers 3D cubic lattice, 10 layers 3D cubic lattice, 5 layers Elastic − rough rigid support Elastic − frictionless rigid support Elastic − infinite half space

0.9

0.5

1

x/h

a)

1.5

2

−0.1 0

0.5

1

x/h

1.5

2

b)

Figure 1.4: a) hσzz at the bottom of the 2D triangular lattice, compared to continuum elastic solutions. The applied force is unity. b) h2 σzz along the x-axis at the bottom of a cubic lattice, compared to continuum elastic solutions. The results are scaled by h; reproduced from [37].

A more realistic force model consists of ‘one-sided’ springs, i.e., springs that snap when in tension. Figure 1.6a presents the forces obtained for the same system as presented in Figure 1.3, but with ‘one-sided’ springs. Compared to the system of regular springs, the application of the localized force at the top of the packing leads to rearrangements in the contact network: some horizontal springs in the region close to the point of application of the force are disconnected (as also observed in [56] for a pile geometry) but the force chains in both systems are qualitatively similar. The force distribution vs. the horizontal coordinate at different depths (Figure 1.5) is in good agreement [37] with experiment [29, 32]. For slightly disordered systems [37], the force chains are qualitatively similar, though somewhat shorter. The corresponding vertical stress field σzz is shown in Figure 1.6b. The stress field in this case is clearly different from that obtained using ‘two-sided’, harmonic springs: the response for ‘one-sided’ springs is double-peaked. This is obviously related to the disconnected springs below the point of application of the external force. The effects of this change in the contact network on the macroscopic response may be understood in terms of elastic anisotropy. In [37], it has been shown that a 2D triangular lattice of harmonic springs in which the spring constant for the horizontal springs, K1 , is different from that of the oblique ones, K2 , corresponds (in the continuum limit) to an anisotropic elastic system. For sufficiently large K2 /K1 , the response of such an elastic system has two peaks [37] (see also [13] for a detailed analysis of the case of an infinite half-plane; the results presented here and in [37]

1.5


15

are for a finite slab on a rigid floor). The absence of horizontal springs corresponds to an isostatic system (in the extreme anisotropic limit K2 /K1 → ∞). The double peaked stress distributions are similar to those obtained from hyperbolic models. It follows that hyperboliclike behavior can be obtained using an anisotropic elastic model (whose large anisotropy limit is ‘hyperbolic’; see also [13, 57]). Figure 1.7 depicts the stress distribution (on the floor) for different values of K2 /K1 . The distribution is either single peaked (narrower than the isotropic one for K2 /K1 < 1, wider for K2 /K1 > 1) or double peaked for sufficiently large K2 /K1 . A similar double peaked distribution is obtained for the case of ‘one-sided’ springs, where some horizontal contacts are severed (Figure 1.6b). Weak anisotropy (K2 /K1 1) may explain some of the deviations from isotropic elasticity observed in experiments [35]. Interestingly, friction restores the single peak response, indicating that the harmonic force law may be more relevant to the description of granular response than one may have a-priori guessed. This effect is observed in discrete element simulations with normal and tangential linear spring-dashpot forces among the particles (see e.g., [7, 45]), possibly the simplest

a)

b)

c)

Figure 1.5: The norms of the interparticle forces vs. the horizontal position, x, scaled by the system height, h, at several depths. a) “Two-sided” springs (harmonic). b) “One-sided” springs (the legend indicates the depth measured from the top, in layer numbers; reproduced from [37]). c) Experiment (average over configurations; reproduced from [29]).

a)

b)

Figure 1.6: a) Force chains in a 2D triangular lattice of ‘one-sided’ springs. A gravitational force has been applied in order to stabilize the system (the applied force is 150 times the particle weight). b) Contour plot of hσzz in the same system (h is the slab height); reproduced from [37].

16


model for frictional disks. The simulation parameters were chosen to correspond to those of the experiments performed in the Duke group [29, 32, 53, 58]. The force chains, and the corresponding vertical stress field (σzz ) for a system of equal frictional disks with a vertical applied force are shown in Figure 1.8 (compare to Figure 1.6). The difference between the frictionless and frictional case is quite pronounced, as also observed in the bottom force profile (Figure 1.10a). Similar results (presented in Figure 1.10b) are obtained in a “pile geometry”: particles arranged in a triangular lattice with a slope of 30◦ (this is an unrealistic pile, since the angle of repose, even for frictional disks, is smaller than 30◦ , and thus the pile has to be supported by side walls). A similar geometry was used in simulations of frictionless disks in [56]). Although the contact networks are quite similar for the frictional and frictionless systems, in particular in the region below the point of application of the force, one observes that less contacts are disconnected in the frictional system. Note, however, that even a contact network with no horizontal contacts, which corresponds to the extreme anisotropic limit for a frictionless system, results in much smaller anisotropy for the frictional system (recall the transition K2/K1=1 K2/K1=0.5 K2/K1=2 K2/K1=10

0.9 0.7

Isotropic elasticity rough rigid support

0.5

hσzz 0.3 0.1 −0.1 0

0.5

1

x/h

1.5

2

Figure 1.7: hσzz at the floor of anisotropic triangular lattices composed of 40 layers of particles; reproduced from [37].

a)

b)

Figure 1.8: a) Force chains in a 2D triangular lattice of frictional disks. A vertical force is applied at the center of the top layer. Line widths and lengths are proportional to the force magnitudes. Only the central part of the system is shown. b) Contour plot of hσzz in the same system (h is the slab height).

1.5


17

from one to two peaks as the anisotropy is increased (Figure 1.7; see also [13]). This can be explained by the presence of the tangential springs, which modify the elastic behavior of the system by adding “directions” in which the forces act. These results suggest that frictional forces may have a significant effect on the quasi-static response of granular systems, rendering it, in particular, “more elastic” and “more isotropic”. One important characteristic of classical linear elasticity is superposition. We tested superposition in a model granular system (with small polydispersity, corresponding to that present in the experiments on nominally ordered systems performed in the Duke group [58]), as follows. Two vertical forces (of magnitude F = 150mg, ¯ at 90◦ ) were applied to two particles in the top layer of the system, both separately and together. The external forces were applied to systems relaxed under gravity, i.e., they are pre-stressed. Therefore, in order to test the superposition, the stress due to gravity (without additional forces) should be subtracted first. The results are shown in Figure 1.9. Although the system is not strictly elastic, due to the 10

Left Right Both Left + Right

Fy mg 5

0 0

100

50

150

x/r

¯ Figure 1.9: The forces on the floor for applied forces separated by 16R.

changes in the contact network discussed earlier, it appears that at sufficiently large distances from where the forces are applied, superposition is obeyed quite closely. Again, as explained above, the effects of rearrangements in the presence of friction seem to be smaller than without it. Furthermore, the particle displacements (at least for an elastic system) typically decay algebraically with the distance from the applied force, so that at large enough distances, the contact network would not be significantly modified (if at all). In this case, the long-range response is essentially elastic, and not sensitive to the detailed force distribution near the boundary (as suggested by Saint-Venant’s principle). An interesting example in which the effects of friction are evident on the microscopic scale is the experiment performed on regular 2D packings of photoelastic disks [32] with an external force applied at oblique angles. The directions and “strengths” of the observed force chains (averaged over realizations) appear to depend quite strongly on the angle of the applied force, with respect to the horizontal direction. A particularly interesting effect is that for some angles, force chains appear not only in the lattice directions (0, ±60◦ , ±120◦ , 180◦ for a triangular lattice), but also in new directions which can be identified as ±30◦ (in fact, in individual realizations, rather than their average as reported in the experiments of [32], force chains appear also at ±90◦ , i.e., the vertical direction [58]). These directions correspond to nextnearest neighbor directions in the triangular lattice. The fact that the forces themselves, and not just the contact points, appear to be aligned with these ±30◦ directions, suggests that frictional

18


forces among the particles (tangential to the contact normals, which result in interparticle torques) are necessary for obtaining forces (and chains) at angles different from the lattice directions. For an applied force at ±90◦ , it appears that the frictional forces are small enough such that the results obtained in this case [29, 32] are described quite well by a model of frictionless particles with linear force-displacement laws [37]. Simulations with an applied torque (in addition to the applied force) show that this torque does influence the observed force chains [59]. The results (for slightly polydisperse systems) shown in Figure 1.11 are qualitatively similar to those observed in the experiment [32]. Note that these results are obtained for a single configuration, while the results presented in [32] are for an average over 8 6

Fz mg

4 2 0

No Friction wall Friction: µ=0.94, µ =0.35

-2 20

40

60

x/d

a)

b)

Figure 1.10: a) The vertical forces on the bottom of a triangular lattice of frictionless (solid line) and frictional (dashed line) disks (with interparticle friction coefficient µ = 0.94 and particle-wall friction coefficient µwall = 0.35), for the center half of the system. The forces in the corresponding system with gravity alone have been subtracted. b) Contour plot of σzz in a pile composed of equal disks, supported by side walls; top: frictionless disks, bottom: frictional disks.

90◦

75◦

45◦

60◦

30◦

Figure 1.11: Force chains in 2D packings of slightly polydisperse frictional particles. A force F with magnitude 150 times the mean particle weight is applied to the particle at the center of the top layer. The angle of the force with respect to the horizontal is indicated below each picture. The same realization of the packing was used in all cases. The region shown is the central third of the upper half of the system.

1.6

Concluding Remarks

19

configurations. The results obtained in simulations for different realizations of the disorder are qualitatively similar [59]. The agreement of the results obtained using a relatively simple force model with the experiments is encouraging.

1.5.3 Force Statistics The distribution of force magnitudes in static granular packings has been extensively studied in experiments [60, 61] and simulations [8]. An exponential behavior of the distribution at large forces appears to be quite universal in experiments on granular systems, independent of the degree of disorder [61], the friction coefficient [61], or the rigidity of the particles [62], and has also been observed in simulations of granular systems with different models for the interparticle forces (e.g., [63, 64]). The universality of the force distribution appears to extend to other systems such as foams, glasses, colloids etc. (see [65] and references therein). The exponential tail of the distribution is reproduced in simple models such as the q-model [30]. The distribution for smaller forces appears to be less universal, and it has been suggested that the appearance of a peak in the force distribution near the mean force may signal the onset of jamming or a glass transition [66]. Interestingly, a qualitatively similar force distribution is obtained in purely harmonic disordered networks (of the type shown in Figure 1.1a). Figure 1.1b shows the force distribution obtained for an ensemble of 100 such networks, consisting of 1085 particles each. The force was normalized by the mean force in the ensemble. The tail of the logarithm of the distribution is fitted quite well by a line of slope −3.8, similar to that obtained in experiments on highly compressed disordered packings of soft rubber spheres [62] (similar distributions were obtained for a harmonic network of unequal springs in [63]). For the case of networks with no force dilution (the same connectivity as in the perfect lattice), the force distribution is Gaussian with a half-width of a few percent of the mean, i.e., a much narrower distribution. This indicates that a random connectivity should be consequential for the force distribution. A similar effect has been observed in simulations of granular systems under different applied pressures [63]. It has been suggested that the crossover from a Gaussian to an exponential distribution is related to the appearance of isolated force chains [27]. The (near-)universality of the force distribution implies that one cannot use it to disqualify or substantiate any specific force model. As mentioned, the same applies to the observation of force chains. A more sensitive and direct test is rendered by the response of a granular system to inhomogeneous external forcing, such as that provided by localized forces, which, as described in Section 1.5.2, seems to be consistent with elasticity.

1.6 Concluding Remarks There are several important issues which must be considered when studying the stress in dense granular (or other) systems. First, it is important to distinguish inherently microscopic quantities, such as the forces and the particle displacements, from the related but macroscopic fields, such as stress and strain. Scales are important in determining when a macroscopic continuum theory should be valid and when corrections due to the underlying discreteness are consequential. Another important issue is that of averaging. Static granular systems are

20


typically in metastable states due to geometric constraints or static friction. Therefore, the justification of any kind of ensemble averaging should be critically questioned. The fact that some experimental features obtained by averaging can be described using models of ordered packings is encouraging. When considering the merits of elastic or other models, one must remember that isotropy should not be taken for granted. As shown, even macroscopically isotropic systems may exhibit anisotropy at small scales or for small systems, due to large deformation gradients and/or disorder. Furthermore, a macroscopic behavior which is not usually associated with isotropic elasticity (such as a two-peaked response to a localized force, or a stress dip under a pile) may be perfectly consistent with anisotropic elasticity. Indeed, the limit of very large elastic anisotropy corresponds to a hyperbolic type of behavior, but is obtained as a limit of an elliptic theory, and does not suffer from the same conceptual problems (e.g., indeterminacy, or the constraints imposed on the boundary conditions). Boundary conditions play an important role: the solution of the equations of elasticity for a finite slab with appropriate boundary conditions resembles the experimental result, whereas the Boussinesq solution does not (hence, this is not a case against elasticity, rather against its misuse). Somewhat surprisingly, many aspects of the response of granular systems can be understood using models of frictionless particles. However, some effects do require the introduction of friction, e.g., the response to oblique applied forces. The rotations and torques present in this case suggest that an elastic Cosserat model [15] should be appropriate below yield. Many of the issues discussed above are not unique to granular systems, but are relevant to many other non-equilibrium systems such as glasses, foams, suspensions, nanoscale systems and micro or nanofluids. The unique opportunity provided by granular materials, in which the microscopic scales are relatively easy to access experimentally, as well as the similarities (and differences) between granular materials and “regular” materials, should prove useful in obtaining a better understanding of the latter through the study of the former.

Acknowledgments We appreciate helpful interactions with R. P. Behringer, J. Geng, E. Clément, D. Serero, H. Jaeger, T. A. Witten, N. Mueggenburg and J.-N. Roux. Support from the U.S.-Israel Binational Science Foundation, INTAS and the Israel Science Foundation is gratefully acknowledged.

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C.-L. Liao et al., Int. J. Solids and Structures 34, 4087 (1997). S. Alexander, Phys. Reports 296, 65 (1998). H. Makse et al., Phys. Rev. Lett. 83, 5070 (1999). J. P. Wittmer et al., Europhys. Lett. 57, 423 (2002). H. J. Herrmann and S. Luding, Cont. Mech. and Thermodynamics 10, 189 (1998). D. E. Wolf, in Computational Physics, edited by K. H. Hoffmann and M. Schreiber, pp. 64–94 (Springer, 1996). L. Landau and E. Lifshitz, Theory of Elasticity, 3rd Edition (Pergamon, 1986). J. Schäfer, S. Dippel, and D. E. Wolf, J. de Phys. I 6, 5 (1996). M. H. Sadd, Q. Tai, and A. Shukla, Int. J. Non-Linear Mech. 28, 251 (1993). O. R. Walton, in Mobile Particulate Systems, edited by E. Guazzelli and L. Oger, pp. 367–380 (Kluwer, 1995). L. Vu-Quoc and X. Zhang, Mech. Mat. 31, 235 (1999). O. R. Walton and R. L. Braun, in Joint DOE/NSF Workshop on Flow of Particulates and Fluids, 1993; H. G. Matuttis, S. Luding, and H. J. Herrmann, Powder Tech. 109, 278 (1999); A. Schinner, Granul. Matter 3, 35 (2001). D. W. Howell and R. P. Behringer, Chaos 9, 559 (1999). J. P. Wittmer et al., Nature 382, 336 (1996). J.-P. Bouchaud, M. E. Cates, and P. Claudin, J. de Physique I 5, 639 (1995). S. Luding, Phys. Rev. E 55, 4720 (1997). M. E. Cates et al., Chaos 9, 511 (1999). J. Geng and R. P. Behringer, private communication . C. Goldenberg and I. Goldhirsch, In preparation. D. M. Mueth, H. M. Jaeger, and S. R. Nagel, Phys. Rev. E 57, 3164 (1998). D. L. Blair et al., Phys. Rev. E 63, 041304 (2001). J. M. Erikson et al., Phys. Rev. E 66, 040301 (2002). M. L. Nguyen and S. N. Coppersmith, Phys. Rev. E 62, 5248 (2000). L. E. Silbert, G. S. Grest, and J. W. Landry, Phys. Rev. E 66, 061303 (2002). C. S. O’Hern et al., Phys. Rev. Lett. 86, 111 (2000). C. S. O’Hern et al., Phys. Rev. Lett. 88, 075507 (2002).

2 Response Functions in Isostatic Packings Cristian F. Moukarzel

2.1 Introduction Stress propagation in static granular packings demonstrates unexpected features [1–3] not easily explainable within the framework of elasticity theory. The observation of a pressure dip below conical piles, force chains, sudden macroscopic changes in stress patterns under slight perturbations, and exponential (instead of Gaussian) stress distributions, among other phenomena, have triggered intense theoretical and experimental work. Several stress propagation models have been put forward [4–15] in an attempt to explain some of these features. The q-model [4, 5] assumes diffusive behavior for the vertical stress component considered as a scalar quantity, and gives rise to an exponential distribution of stresses. Other scalar models in turn predict Gaussian [16] or power-law distributed [17] stresses. By postulating a linear relation between stress components [6], a wave-like equation [7] is derived for stress propagation, the so called OSL model [18]. This model reproduces the pressure dip [7], and is consistent with stresses in silos [19]. A memory formalism [8, 9] contains, as special limits, the wave-like and diffusive behaviors. Furthermore, a recent description in terms of scattering force-chains [10] gives rise to wave-like propagation on small scales, crossing over to something similar to classical elasticity on larger scales. Several recent approaches predict elastic behavior at large scales [10–15]. Linear elasticity describes the propagation of stresses in terms of differential equations of the elliptic type, wave-like propagation corresponds to the hyperbolic case, while diffusive behavior is the borderline, or parabolic case. These three descriptions give rise to very different responses [3, 20] when a small force is applied to a localized region on the upper surface of a packing. Linear elasticity predicts a bell-shaped response, having a width proportional to depth. A diffusive behavior, on the other hand, implies that the width of the response scales as the square root of depth. Finally, a wave-like propagation would be evidenced by a response that is maximum on a diffuse annulus of linearly growing radius (the “light-cone”) in three dimensions, and on two diverging peaks in two dimensions. In this article we do not discuss any specific stress propagation models. Instead, we concentrate on describing the surprising properties that isostatic contact networks have, specifically exploring stress propagation. Our discussion has relevance to stress propagation on granular packings, because disordered packings of undeformable frictionless spheres, have, in any dimension d, an isostatic contact network [21–23]. Isostatic networks have singular stress transmission properties [21–25] and some of their infinitesimal response properties do not have a well defined continuum limit [24,25]. Isostaticity has only been proven for frictionless packings [21, 26]. It is known that friction gives rise to indeterminacies [27, 28], and this

24

2 Response Functions in Isostatic Packings

is not compatible with isostaticity. In real packings, friction is important, and it is at present not clear whether isostaticity holds in some restricted sense, or not at all, in this case. Recent molecular dynamics results [29] on frictional deformable spheres are not compatible with the packing being isostatic, although previous similar studies [30] reached different conclusions. Although it is at present not clear to what extent isostaticity is important in real (frictional and deformable) packings, it is reasonable to expect that some of its relevance for idealized packings should carry over to this case as well. A detailed study of friction effects is beyond the scope of this work. Isostaticity is a topological property that amounts to minimal rigidity: an isostatic contact network is rigid but would cease to be so if any of its contacts where removed. The system of equilibrium equations for interparticle stresses is exactly determined, and therefore stresses do not depend on the rheology of the material. They are determined by the applied load and the network geometry [21]. A trivial implication of isostaticity for frictionless spherepackings is that the number of unknowns (contact stresses) is exactly equal to the number dN of equilibrium equations. This explains [21, 23] why the mean coordination number of stiff frictionless sphere-packings is six (four for disks). A second important though straightforward property of isostatic networks is that they enjoy an exact symmetry between force–force and displacement–displacement response functions: The excess stress induced on contact b by a vertical overload applied on sphere i (which is given by the stress–stress response) is exactly equal to the vertical displacement ∆ib that sphere i suffers when the interparticle contact b is stretched (defining the displacement response). This property has been recently used in experiments [50]. In a disordered packing, response (or Green) functions are random variables because of disorder. It than emerges that, even more important than the above simple consequences of isostaticity, is the following fact: On generic disordered isostatic networks, Green functions are the result of a random multiplicative process [31], i.e., stresses and displacements propagate in a way that the noise due to randomness is felt multiplicatively [24, 25]. This endows response functions on such networks with singular properties. If a packing of deformable spheres is compressed, extra contacts appear and isostaticity is eliminated. Such networks are hyperstatic (or overconstrained), and their elastic properties can be described by classical elasticity theory. In the following sections we will argue that the isostatic case is singular and that elasticity does not apply to it.

2.2 Rigidity Considerations for Contact Networks 2.2.1 Formulation We will consider d-dimensional packings of N spheres with radii Ri , interacting in pairs via an arbitrary central-force Hamiltonian Eij = kij U (ij) (dij ), where dij is the distance between particle centers, and kij > 0 are generalized stiffness constants. We will take the elastic potential U (ij) (d) to be constant for d ≥ Ri + Rj . Therefore elastic forces are non-zero only if particles i and j are “in contact” (dij < Ri + Rj ). We will also allow for an arbitrary set of external forces Fi assumed to act on the sphere centers, e.g., gravity or confining pressure, and represent it by means of a vector of dN components (“the load”) F = {F1 , F2 , . . . , FN }.

2.2

Rigidity Considerations for Contact Networks

25

We assume that the system is in static equilibrium and that the vector of equilibrium particle is known. positions X Let fij be the elastic force acting on particle i due to its contact with particle j. We write it as fij = λij µ ij , where µij is the unit vector pointing from j to i, and λij = λji = −kij ∂U (ij) (dij )/∂dij .

(2.1)

Using this convention, the “stress” λ is positive for compressive forces. The stiffness constants kij are assumed to be large enough with respect to the load so that versors µij can be taken to be constant (load-independent). This is the so-called linear approximation of elasticity. Notice that we are not requiring linear elasticity, i.e., that the potentials U (dij ) be quadratic in dij .

2.2.2 Network Rigidity The structure obtained by associating a site to each (hyper)sphere center i and a bond to each pair of spheres (ij) in contact is called a contact network. Let B be the total number of contacts, and let λ be the vector of B bond stresses. The equations for force equilibrium at each site take the form λij µij = 0 i = 1, . . . , N (2.2) Fi + j(i)

where the sum goes over sites j in contact with site i. We take λij to be our unknowns, thus obtaining a set of dN equations in B unknowns. These equations can be written in matrix form as ˜ λ = F . M

(2.3)

˜ , which has B columns and dN Because of our assumption of large kij ’s, the rigidity matrix M rows, does not depend on F . From the point of view of structural rigidity [32–34] networks can be classified into rigid and flexible. A network is rigid if a solution λ(F ) of Eq. (2.3) exists for any load F , and flexible (or deformable, or floppy) otherwise. It follows from the basic theory of linear equations that the rigidity of the network is determined solely by the ˜. rank K of M (a) If K < dN , the network is flexible (or hypostatic). In this case h = dN − K is its degree of hypostaticity (or number of floppy modes), which is the dimension of the subspace of non-resolvable loads, or equivalently the dimension of the subspace of infinitesimal deformations that use no energy. (b) If K = dN (it cannot be larger) the network is rigid. Because the rank K of an N × M matrix satisfies K ≤ min(N, M ) we have that K ≤ B. Thus B ≥ dN , with B the number of interparticle contacts, is a necessary (but not sufficient) condition for rigidity.

26


If the network is rigid, a solution λ(F ) of Eq. (2.3) exists for any F . The linearity of the force-equilibrium equations (for any elastic interaction law) implies then that λ(F ) can in general be written as λ = λ(load) + λ(self) ,

(2.4)

where λ(self) is a solution of ˜ λ(self) = 0, M

(2.5)

and thus does not depend on the external load F . Let us notice here that λ(self) depends on the stiffnesses kij but not on the load, while λ(load) depends on the load, but not on the stiffnesses. More specifically if all stiffnesses are rescaled by γk and all loads by γF , then λ → γF λ(load) + γk λ(self) .

(2.6)

It is useful to define the “stiffness to load” ratio Isl = γk /γF . This ratio measures the relative importance of self-stresses in comparison with load-induced stresses. Rigid networks, i.e., those whose rigidity matrix has rank K = dN , in turn admit a subclassification into two classes: hyperstatic and isostatic. (b1) If a rigid network has B > K = dN , in other words, if it has more contacts than necessary for rigidity, it is said to be hyperstatic. For hyperstatic networks, the problem of finding the stresses from equilibrium alone is underdetermined. In this case the null˜ ) of M ˜ has non-zero dimensionality H = B − K (H is the degree subspace N (M of hyperstaticity of the network), and thus Eq. (2.5) admits non-trivial solutions λ(self) . Only hyperstatic networks admit self-stresses. (b2) If the network is rigid and minimally so, i.e., if B = K = dN exactly, then there is a unique solution λ(F ) for each load F . Equivalently, λ(self) = 0 is the only solution of ˜ is Eq. (2.5). In this case the network is said to be isostatic, or minimally rigid, and M invertible. For an isostatic network one can write Eq. (2.3) as λ = F M ˜ −1

(2.7)

This last expression makes it explicit that, on isostatic networks, stresses are determined solely ˜ ), and do not depend on the rheology of the contacts. Let us by the load (F ) and geometry (M (ab) define the load-stress Green function Gi,α to be the excess stress induced in bond (ab) by a unit load acting on site i in the direction of axis α (α = 1, 2, . . . , d in d dimensions). Taking α as a vector index we can write all d Green functions for a given site and bond as a vector (ab) . Load-induced stress on bond (ab) can thus be written as G i (load) (ab) (2.8) Fi · G λab = i i

˜ −1 is a matrix of Green functions. Comparison between Eq. (2.7) and Eq. (2.8) shows that M Figure 2.1 shows a small two-dimensional pile, together with some possible contact networks

2.2

Rigidity Considerations for Contact Networks

a)

27

b) A

+1

+1 -1

+1

B

-1

-1

c)

+1

-2

-2 -1

+1

+1

-1

d) F2 F1

e)

f) F2 F1

F1

λ

g)

F2

λ

λ

h)

Figure 2.1: Isostatic and Hyperstatic networks. a) A two-dimensional pile of disks of radius ≈ R with extremely small polydispersity. b) A possible isostatic contact network. c) The addition of one redundant contact AB turns the network hyperstatic. Now stresses are no longer uniquely determined by the load. Thick lines indicate the overconstrained subgraph, i.e. bonds that may sustain self-stresses. d) A possible set of self-stresses. Any multiple of these is of course also a solution of Eq. (2.5). This is the origin of the indeterminacy in stresses. The overconstrained subgraph in c) is 1-redundant or minimally overconstrained. The removal of any of its bonds makes the network again isostatic (e)). 1 Illustration of the Virtual Work Principle: f) An isostatic system in equilibrium under external forces F 2 remains in equilibrium if one of its bonds is removed (g)) and replaced by a couple of forces and F λ that this bond provided. h) The system is now a mechanism, and can be deformed. Under small 2 and λ) adds up to zero. 1 , F displacements, the work done by all intervening forces (i.e. F

representing sets of active contacts. All networks are rigid, but b) and e) are isostatic while c) is hyperstatic. The hyperstaticity of the network c) is due to the existence of an additional contact AB. Network c) is said to have one redundant contact. On this network, Eq. (2.5) has non-trivial solutions, called self-stresses. The thick lines in Figure 2.1c indicate the subset of bonds which carry self stresses in this particular network, and a possible set of self-stresses is shown in Figure 2.1d. For later use we remark here that any set of self-equilibrated stresses must have at least one tensile stress [21, 22], in any dimension, if the overconstrained graph is bounded. This is important for the isostaticity proof in the next section.

2.2.3 Isostaticity in the Limit of Large Stiffness to Load Ratio We now show that, under general conditions, in the limit in which the sphere stiffness is very large, or the applied pressure small, there can be no overconstrained graphs in a polydisperse packing. Assume that all interactions are purely repulsive, that is, the system can only sustain

28


compressive forces between particles. Our packing is assumed to be externally compressed. Let us furthermore assume that the sphere radii have some non-zero amount of polydispersity. This amounts to the statement that all overconstrained subgraphs, if they exist, will suffer non-zero self-stresses. It can be then shown [21, 22] that, if the stiffnesses kij of spheres are large enough compared to the mean compressive forces, the contact network has to become isostatic. The reasoning goes as follows: under large enough pressure, sphere packings are hyperstatic, they have overconstrained subgraphs. Any overconstrained subgraph must have self-stresses because of polydispersity. Some bonds are then subject to tensile self-stress, as noticed in the previous section. The negativity of a self-stress is not in contradiction with the requirement of compressive total stress if the applied pressure is large, because the superposition of both contributions in Eq. (2.6) is still compressive. But when the stiffness-to-load ratio Isl is increased, either by reducing the external pressure (γF decreases) or increasing the stiffness (γk increases), self-stresses become dominant in Eq. (2.6), and therefore if any overconstrained subgraph existed, some of its bonds would have a tensile total stress, which is not allowed by assumption. Therefore, when the external pressure is lessened, overconstrained graphs must cease to exist. This comes about by the “opening” of those contacts that would otherwise develop a tensile total stress. Therefore on reducing the external pressure, overconstrained subgraphs progressively dissappear by elimination of redundant contacts, and the system finally becomes isostatic. One sees then that granular packings become isostatic in the limit of low pressure because tensile stresses are not possible. Roux [26] recently provided a formal discussion of isostaticity in granular packings or arbitrary shapes.

2.3 Consequences of Isostaticity Because of our considerations in Section 2.2.2, isostaticity trivially means that the total number of contacts B = dN exactly, for a packing of N spheres in d dimensions [21, 22, 35]. The polydispersity requirement is important [21] as, for example, an FCC packing is not isostatic if all spheres have equal radius. In this case each sphere has coordination 12, so there are six contacts per particle. However, as soon as a slight polydispersion is introduced, the mean coordination drops to six (a total of three contacts per particle) exactly. Let us now discuss some further consequences of isostaticity.

2.3.1 Green Functions and the Virtual Work Principle The virtual work principle (VWP) is well known in Classical Mechanics [36]. In its simplest form it establishes that, if an equilibrated system suffers a virtual displacement δxi consistent with all constraints, the total work done by the intervening forces adds up to zero. (2.9) Fi δxi = 0 i

Assume now that we are dealing with a rigid packing (it can resolve any load) in equilibrium under a given external load. Clearly a rigid system cannot be displaced while satisfying all constraints, so that the simple form of the VWP above would seem in principle useless. But if the packing is isostatic (rigid and minimally so) there is a simple trick that allows us to use this

2.3

Consequences of Isostaticity

29

form of the VWP with profit. The trick consists in removing one bond mn from the original equilibrated system, and replacing it by its stress λmn acting on sites m and n, as shown in Figure 2.1(f,g,h). After doing this, we are now left with a structure which is no longer rigid, but is still in equilibrium. This system is a mechanism (a structure with one degree of freedom) and a deformation is possible without violation of any constraint (i.e., maintaining all other bond lengths constant). Consider now changing the distance dmn between sites m and n by an infinitesimal amount δdmn . We can now apply Eq. (2.9), which for this particular situation reads (2.10) Fi δxi − λmn δdmn = 0 i (mn)

= δxi /δdmn one has that and now calling ∆ i (mn) = λmn Fi ∆ i

(2.11)

i

Comparison with Eq. (2.8) then allows us to conclude that (mn) = ∆ (mn) G i i

(2.12)

Thus on an isostatic system, the stress λmn induced in a contact (mn) by a unit force acting on site i in the direction eα , is equal to the component in the direction eα of the displacement induced in site i by a unit stretching of bond mn. This is the “symmetry” in response functions mentioned in the introduction. The force–force and displacement–displacement response functions are equal on isostatic systems (but notice that they were defined here in such a way that propagation of forces and displacements occurs in opposite directions). This symmetry only exists if the network is isostatic, because only in this case is the system transformed into a mechanism by the removal of a single bond (mn).

2.3.2 Anomalous Fluctuations: Multiplicative Noise in Isostatic Networks We now discuss the fact that, generically, disordered isostatic networks have singular distributions of response functions. This comes about because perturbations, i.e., slight overloads or displacements, propagate in such a way that the noise (due to randomness) acts multiplicatively on them. Let us first analyze a one-dimensional case that, despite its simplicity, already displays a complex behavior. Consider a packing consisting of two layers of polydisperse disks whose radii have a small dispersion, arranged in such a way that its contact network is as illustrated in Figure 2.2a. The lower layer (dark disks) is regarded as fixed, while the top layer (containing N white disks) rests on the lower one (say, under gravity) but is free to move laterally. Concentrating on the top layer, consider applying an infinitesimal horizontal force F0 to one of its ends. We may then pose the following question: What is FN at the other end such that the arrangement is equilibrated, and, how does the ratio between FN and F0 behave for large N ? In other words, we are interested in the properties of the stress–stress response function. If

30

2 Response Functions in Isostatic Packings C

fk

F

ek

F0

N

µk a)

f k-1

µk-1

b)

Figure 2.2: Simple example of multiplicative propagation. a) Two layers of a packing made of slightly polydisperse disks whose centers form a distorted square lattice. We want to calculate the horizontal force Fn needed to equilibrate a perturbation F0 applied at the right end of the second (white disks) layer. All disks on the lower layer are fixed, only the white disks can move due to the force. b) The contact network of this packing can be decomposed as a sequence of distorted squares. Relation (2.14) between fk and fk−1 results from the equilibrium equations (see text).

all disks had the same radius, the contact network would be a a strip from a regular square lattice and clearly FN = F0 . In the polydisperse case, contact forces between the upper and lower layers are not strictly vertical but have a small horizontal component, which adds to the horizontal-force equilibrium equation. Naively regarding all these small contributions as independent variables, one could expect FN to√be something like F0 plus a random term with Gaussian distribution, typically behaving as N for N large. The underlying assumption behind this picture would be that the stochastic equation describing the propagation of forces has additive noise, i.e., Fk = Fk−1 + ηk ,

(2.13)

where ηk are random variables. In fact this is not a good description, as there is multiplicative noise [24, 25] in this system, rather than additive noise, as we now show. Consider equilibrium of forces for a subsystem composed of two adjacent disks k and k − 1, as depicted in Figure 2.2b. We let µk be the versor which is normal to the “near vertical” contact below disk k, and ek the versor in the direction of the contact between disks k and k−1. Let us for convenience decompose the force Fk acting on disk k, due to its contact with disk k +1 (disk k +1 is not drawn), into two components: fk and vk , respectively parallel and normal to µk . The normal component vk is in the direction of the “near vertical” contact and will be assumed to be equilibrated by it. So we just have to consider equilibrium of the fk ’s. Equilibrium of forces takes a particularly simple form in terms of the fk ’s. Zero torque with respect to the point C where the projections of the vertical contacts intersect (Figure 2.2b) implies, after some trivial vector algebra, that µk · ek ) = fk ( µk−1 · ek ), fk−1 (

(2.14)

where · denotes scalar product. Therefore, the fk “propagate” according to fk = fk−1

( µk · ek ) . ( µk−1 · ek )

(2.15)

2.3

Consequences of Isostaticity

31

Since µ k, µk−1 and ek are almost parallel (assuming that the polydispersity is small), Eq. (2.15) can be written (expanding the cosines to second order in polydispersity and regrouping) as fk = fk−1 (1 + ηk ) ,

(2.16)

with ηk 1 random variables (not independent, though). Thus already for this simple onedimensional example, the noise term appears multiplicatively [24, 25] in the stochastic equation that determines force propagation. This has important consequences on the distribution and magnitude of the fluctuations, as multiplicative processes give rise to much larger deviations than additive processes [31].

2.3.2.1 An Approximate Description of Propagation in Two-dimensional Packings Using the insight obtained in previous paragraphs, let us now discuss the slightly more general, though still idealized, situation of a disordered two-dimensional packing. This example will serve to illustrate the fact that multiplicative transmission is a generic property of disordered isostatic packings. Consider an arbitrary disordered packing (Figure 2.3) of non-cohesive, undeformable, frictionless disks under gravitational load, resting on a layer of fixed disks (shaded), from whose positions all other disks’ positions can be determined. Because of isostaticity we can use the symmetry between stress response and displacement response. The excess stress induced on one of the bottom disks by an infinitesimal overload acting on a disk at the top, is exactly the same as the displacement of this upper disk when the bottom disk is (infinitesimally) pushed upwards. Because the contact network is isostatic, an infinitesimal displacement of one of its boundary particles produces a deformation of some part of the system, without changing the distances between pairs of particles in contact. This suffices to demonstrate that particles in contact will remain so, at least for infinitesimal perturbations. Consider how this displacement is propagated along an arbitrary path 1-2-3-4-5 (full line in Figure 2.3) of disks in contact with each other. If we, for simplicity, assume for a moment that disks a, b, c, d do not move as a result of the perturbation (this may be true for the specific example in the figure, but not in general), displacements propagate as shown in Figure 2.3b, where black dots (“sites”) indicate disk centers, lines represent contacts between disks, and the shaded regions are fixed. Each site on the path is connected to three other sites: two of them (numbered) lay on the path and suffer a displacement, and the third is not on the path (indicated with letters, and assumed not to move). For i = 1, 2, . . ., let µi be the unit vector normal to the dashed bond that connects to site i, and let ei be the unit vector from i to i + 1. Because we assumed that sites a, b, . . . do not move, and because distances between particles in contact are not changed, the motion of site i has to be normal to the dashed bond, so let it be written i . Equivalently, let the displacement that site i + 1 suffers be δi+1 µ i+1 . The condition as δi µ i − δi+1 µi+1 ) · ei = 0, and that the distance between sites i and i + 1 be unchanged reads (δi µ from it we get for the scalar amplitudes δ: δi+1 = δi

i · ei µ . µi+1 · ei

(2.17)

32


5 d

4 3 c

b

1

δ2

b

2 1

δ1

a

a)

b)

a

3

2 c

δ3

5 δ5

4 δ4

d

Figure 2.3: Next example of multiplicative propagation. This time we consider the propagation of displacements. a) The propagation of displacements in a random packing of polydisperse disks like the one in the figure can be approximately analyzed by selecting an arbitrary path along disk centers in contact, and assuming other disks, not along this path, to be fixed. The approximation involved consists in ignoring the motions of these other disks. b) Under this approximation, the relevant contact network, along which the displacement propagates, is one-dimensional as in Figure 2.2. The basic result describing propagation along this path is given by Eq. (2.18).

Therefore for a chain of k particles, δk = Gk δ0 with Gk =

k−1 i=0

µ i · ei µi+1 · ei

(2.18)

A pure multiplicative process is of course only a zeroth-order approximation, since disks a, b, . . . do move in general. Still within an approximate treatment, the contribution of the motions of the supporting sites a, b, . . . can be (roughly) thought of as “additive noise” on top of a multiplicative process, i.e., δi+1 = δi

µ i · ei + ηi . µi+1 · ei

(2.19)

We have thus argued that multiplicative processes, possibly with additive noise, constitute an approximate description for the propagation of stress and displacement on disordered isostatic packings. It is known that random multiplicative processes with additive noise give rise to truncated power-law distributions [37–41] and this has been observed numerically for disordered isostatic contact networks [21–25]. In Section 2.4 we review several examples of contact networks and show numerically that their distributions of response functions are of the power-law type.

2.4 Specific Examples We now turn to a consideration of numerical studies for several kinds of disordered isostatic contact networks. We will argue that, because of the existence of multiplicative noise, the

2.4

Specific Examples

33

distribution of response functions is generically of the power-law type. All studies in this section make use of the symmetry between displacement- and stress-responses on isostatic lattices. This allows us to measure stress response functions G by slightly pushing up one site on the bottom layer of a packing, and finding the way in which this perturbation propagates upwards. The existence of random multiplicative processes in the propagation of perturbations (i.e., slight overloads or displacements) has the consequence that the distribution P (G) of response functions tends to a power-law at large distances. The first moment of this distribution is usually finite because of normalization requirements, but its higher moments diverge exponentially fast with distance. Disordered isostatic networks display anomalous sensitivity to perturbation.

2.4.1 Topologically and Positionally Regular Isostatic Networks Let us first consider the simplest regular (non-disordered) two-dimensional isostatic network, namely a square lattice. Consider then displacing one of the bottom sites upwards by an infinitesimal amount as shown in Figure 2.4a. Because our system is originally isostatic (minimally rigid), this displacement, or perturbation, produces a deformation without any change in the bond lengths (no elastic energy must be expended in order to produce this deformation). Only those sites along the diagonals (characteristics) stemming from the perturbation point will be displaced. Consequently, G takes some constant value (one-half in this case) along the characteristics, and is zero everywhere else. In other words, forces and displacements propagate independently along the diagonals. Let us next see how the introduction of disorder modifies this simple picture. We will separately consider the effects of positional (site locations) and connectivity (topological) disorder.

2.4.2 Topologically Regular Isostatic Networks with Positional Disorder We first introduce positional disorder by randomly displacing the square lattice sites by a small amount around their regular positions, as depicted in Figure 2.4b. Because bond angles are now random, induced displacements are non-zero everywhere inside the cone defined by the two characteristics stemming from the perturbation point. Recall that the induced displacement at point (x, y) measures the stress response, i.e., the excess stress on the bottom, produced by a unit overload at (x, y) (see Section 2.3.1). Two notable features of these disordered networks are as follows: First, some sites turn out to move downwards, meaning that an overweight there would reduce the compressive stress on the bottom site (the response function G is negative). Second, the upwards propagation of displacements is described [24,25] by a random multiplicative process. One of the observable consequences of the existence of multiplicative noise is that the distribution Ph (G) of response functions h layers above the perturbation, develops a power-law tail for large G, when h is large. This tail extends up to a cutoff value Gmax (h) that grows exponentially with h. These features of Ph (G) can be seen in Figure 2.4c. The average response function must obviously satisfy < G(x, y) > dx = 1. Therefore, very large positive fluctuations of G on a given layer must be canceled by equally large negative ones, on the same layer. Because of this, Ph (G) becomes almost symmetric around zero for large h. Notice that Figure 2.4c only shows the G > 0 part of P (G).

34

a)


00000000000000000 11111111111111111 11111111111111111 00000000000000000 00000000000000000 11111111111111111 00000000000000000 11111111111111111

10

000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111

b)

00000000000000000 11111111111111111 00000000000000000 11111111111111111 00000000000000000 11111111111111111 11111111111111111 00000000000000000

000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 111111111111111111111 000000000000000000000

1

Gy

P(δ)

10

10-3

10

c)

-1

-5

100 Layers 200 Layers 300 Layers 400 Layers

10-7 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 Displacement δ

Height

y

d)

Figure 2.4: a) The response function of a regular square lattice is non-zero only along the diagonals stemming from the perturbation point. b) If positional disorder is introduced, displacements propagate multiplicatively along the diagonals, and diffuse inside the cone. c) Green function distributions P (G) for position-disordered square lattices. The perturbation is applied on the first (lowermost) layer, and propagates upwards. Two hundred layers above, it is already evident that P (G) develops a power-law tail for large values of G. The dotted line is a power-law ∼ δ −0.78 . d) Average response G(x, y) on disordered square lattices. Fluctuations grow exponentially fast with height in the central region above the perturbation, but after averaging over 105 samples, the wave-like characteristics of the average response become visible. The number of samples needed to average fluctuations out, however, grows exponentially with height because of the multiplicative nature of noise.

Figure 2.4d shows the average response calculated from 105 piles 80 layers high, having periodic boundaries across the horizontal direction. Although fluctuations are strong (they grow exponentially fast with the height h of the pile), after taking averages over a large enough (of order eαh ) number of samples [25], the wave-like nature of the average response is uncovered. A recent theoretical analysis [6, 42] of stress propagation in granular packings assumes a linear relationship between the main stress components, to conclude that stresses propagate in a wave-like manner on these systems. Later analysis [43] suggested that it is precisely because of isostaticity that a linear relation between stress components may be justified. Our results [25] on isostatic lattices are entirely consistent with these predictions. However, it is worth stressing that a consistent treatment of fluctuations is lacking in [6, 42], where these were taken to be small. We notice that, since noise is multiplicative in these networks, fluctuations grow without bound, and therefore the physical relevance of expectation values becomes doubtful [31]. On systems with multiplicative noise, typical values differ from expectation values by an amount that grows exponentially with system size [31].

2.4

Specific Examples

35

2.4.3 Topologically Disordered Positionally Regular Isostatic Networks Let us next consider topological disorder on isostatic networks. For the purpose of numerical analysis we use networks whose sites are located on a regular triangular lattice. Isostaticity is ensured by letting each site be supported from below by two out of the possible three neighbors, see Figure 2.5a. The choice of supporting neighbors is made at random but in a way that ensures that no tensile stresses are generated. The distribution of stresses P (f ) turns out to decay exponentially for large f [21–25] on these networks, in accordance with recent experimental findings [4].

a) 0

10

-5

10

P(G)

10-10 10-15 -21

10

10-25 10-30 0 10

b)

1000 layers 750 layers 500 layers 250 layers 5

10

15

10 10 10 Response function G

20

10

c)

Figure 2.5: a) Compressive stress isostatic networks, whose sites form a regular triangular lattice. Each site is supported by two neighbors from below, such that all stresses are compressive. b) Distribution P (G) of stress–stress Green functions G at constant depth h. Even when stresses are well behaved (see [21–25]), stress responses become power-law distributed for large h. c) Average response function < G(x, y) > shows wave-like behavior after averaging over 4 × 107 samples.

Looking at the upwards propagation of displacements (which determine the stress response G), we again find that these can be described by a random multiplicative process. The multiplication of displacements in this case is due to the existence of particular configurations of contacts (called “pantographs” [21, 22]), which multiply displacements by an integer number. The distribution P (G) of response functions becomes power-law for large distances h, as shown in Figure 2.5b. Again in the case of topological disorder, as for distorted square lattices, (Section 2.4.1), we find that, despite the existence of strong fluctuations, the average response < G(x, y) > shows the distinctive wave-like shape [24,25] displayed in Figure 2.5c. This behavior is consistent with recent theories [6, 42] that suggest wave-like propagation of stresses on granular packings.

36


The results in this section show that it is possible to construct a model with wave-like response functions and exponential distribution of stresses [24, 25]. Thus clearly the q-model [4, 5] (which necessarily has diffusive response functions [3]) is not the only possibility in order to explain the exponential distribution of stresses observed experimentally.

2.4.4 Topologically and Positionally Disordered Isostatic Networks In the previous sections we have analyzed two types of disordered isostatic networks, and shown that the average response < G(x, y) > has a wave-like shape (Figures 2.4d and 2.5c). However, one could argue that the existence of “preferred directions” for propagation is a consequence of the underlying lattice structure, which is not completely absent in these models. Therefore it is important to see what happens on isostatic networks without any regular structure. For this purpose we now discuss numerical results on sequential packings of polydisperse disks [50, 51], and show that totally equivalent conclusions can be reached for this case.

a) 1e+05

P(D)

Gy

340 a.u. 300 a.u. 260 a.u. 220 a.u. 180 a.u. 140 a.u. 100 a.u. 60 a.u. 20 a.u.

1

1e-05

1e-10

1e-15

Height

(b) 1e-20

0.0001

1

10000

1e+08

1e+12

D

b)

c)

Figure 2.6: a) Sequentially deposited packing of polydisperse disks. b) Distribution P (D) of response functions D obtained by displacing one of the lowermost disks and measuring the upwards displacement on a disk in the bulk. The depth of the layer over which P (D) is measured is indicated in the labels in units of the average disk diameter. The total height of the packing is 400 diameters in this case. c) After disorder averages over 107 samples, it can be seen that the average response has a wave-like character. Thus, the existence of a preferred direction for propagation is not a consequence of an underlying lattice structure, but intrinsic to isostatic networks.

2.4

Specific Examples

37

The contact networks from which the results discussed in this section are obtained were built by pouring disks, one by one, into a rectangular die. Disks are deposited following a steepest-descent algorithm [44, 51] until a stable position is found. This packing procedure originates a sequentially deposited isostatic structure, as each disk rests on two lower neighbors, as illustrated by Figure 2.6a. However, notice that tensile stresses cannot be avoided with this simple algorithm. Once the assembly is ready, the central particle, that rests directly on the bottom, is displaced upwards. Upon perturbing the system, contacts remain unchanged, as appropriate for an idealized infinitesimal displacement. Again for this kind of disordered packing, we find that the distribution of Green functions is similar to a power-law (Figure 2.6b), i.e., the system presents strong fluctuations, due to multiplicative effects discussed in Section 2.3.2. After taking averages over 107 samples we obtain the results shown in Figure 2.6c for the average response < G(x, y) >. Comparable results have been reported for < G > using a more elaborate but very timeconsuming adaptive algorithm [45, 48] which avoids tensile stresses. The conclusion is then that the existence of preferred directions for propagation of stresses is clearly not dependent on the existence of an underlying regular lattice.

2.4.5 Non-sequential Isostatic Networks All isostatic networks analyzed in the previous sections are sequentially deposited: disks in a given layer rest on exactly two previously deposited disks. The sequential structure of these networks, in addition to isostaticity, is what makes it possible for stresses or displacements to be calculated by propagation, starting from a boundary (respectively top or bottom). Although one expects most disks to rest on two lower neighbors, it is in principle possible, and it does happen, that some disks rest on a different number of lower neighbors. On such a structure, stresses cannot be calculated by propagation from top to bottom, and the whole linear system of equilibrium equations (which is exactly determined because of isostaticity) has to be solved instead. Given that the propagative nature of solutions depends on an approximate assumption (sequential structure), it might be argued that the strong growth of fluctuations observed numerically in the previous examples could also be a consequence of this approximation, and not a generic property of isostatic packings. In this section we present numerical results for networks which are isostatic but not sequential. The numerical analysis of non-sequential isostatic networks poses two main problems. In the first place, calculation of stresses and displacements by propagation is no longer possible, so we solve the elastic equations iteratively by means of a conjugate gradient algorithm. The second complication arises when one wants to construct an isostatic contact network without using sequential deposition. Several algorithms exist for the generation of contact networks for frictionless disk packings [45, 46], that could be used for this purpose, and of course they give rise to isostatic networks [46, 47]. However, these algorithms are iterative and relatively time consuming. Here we only want to test whether the sequential deposition procedure is responsible for the power-law character of the distribution P (G) of response functions, so a simpler procedure is chosen. We have chosen to generate non-sequential isostatic networks by means of a bond-moving algorithm, starting from a sequential network. The algorithm works for packings of slightly polydisperse disks, whose centers are approximately located on the sites of a triangular lattice. Each disk may in principle be in contact

38


a) 2

10

0 moves 10 moves 1023 moves 10 moves

100 P(G) 10-2 -4

10

Response G b)

10-4

10-2

100

102

Figure 2.7: Non-sequential isostatic structures may be constructed by bond-swapping, starting from a slightly distorted square network (Figure 2.4a,b). a) The resulting network after 103 random bond-moves that preserve isostaticity. The two lowermost layers are not modified. b) The distribution of response functions G at a distance of 20 layers from the perturbation point, after respectively 0, 10, 102 and 103 bond moves. It is clear that the power-law character of P (G) is not a consequence of the sequential deposition structure of the lattice. On the contrary, when one starts from a sequentially deposited square lattice (0 moves) the distribution P (G) becomes more pronouncedly power-law for a larger number of moves, i.e., as the lattice becomes non-sequential.

with six other disks, and the connectivity is defined in such a way that isostaticity is respected at all times. First, a sequentially deposited isostatic network is built. For the starting network, each disk is supported by exactly two neighbors from below. Once the starting isostatic network is defined, the bond-moving procedure starts. At each step, a disk is chosen at random and a new supporting contact is added if there is room for it. Upon doing this, the network becomes overconstrained, i.e., can now sustain self-stresses, see Section 2.2.2. In order to restore isostaticity, it suffices to remove an arbitrary contact, taken at random from the subset that suffers self-stresses. This subset is identified by means of a matching algorithm [52, 53]. This bond-moving is done a predetermined number, B, of times. For B sufficiently large the network approaches a stationary state with no memory of the starting structure. We next discuss the distribution of responses P (G) at constant height, obtained by starting from a slightly distorted “square” lattice. An example of the structure obtained after 103 bond

2.5

Discussion

39

swaps is shown in Figure 2.7a. The distribution of response functions at a height h = 20 layers is shown in Figure 2.7b, for different numbers of bond moves. For zero moves, one has networks with the topology of a square lattice, and P (G) does not display a power-law (plusses) because h = 20 is still small. In Figure 2.4b one can see that clear power-law behavior only appears on these systems for h = 100 and larger. But as soon as ten bond swaps are performed (crosses), P (G) becomes similar to a power-law, and a tail of large G appears. This tendency is kept for larger numbers of bond moves. Our results thus show that the phenomenon of multiplicative propagation of perturbations is present in non-sequential isostatic lattices as well. In other words, disordered isostatic networks posses this property generically and not as a consequence of an imposed sequential structure.

2.5 Discussion We have reviewed some of the unusual displacement- and stress-transmission properties of networks that are isostatic, or minimally rigid. The contact network of disordered frictionless packings of spheres is isostatic in the limit of low pressure (or large stiffness) limit [21–23]. Some of the particularities of isostatic systems are as follows: A continuum elastic medium, whose stress transmission properties are described by elliptic equations, when discretized, gives rise to a hyperstatic system. The equilibrium equations for stresses are underdetermined, therefore the rheology of the system must be known in order to find the stresses that a given load produces. An isostatic system, on the contrary, has rheology-independent stresses once the geometry is fixed, since stress-equilibrium equations are exactly determined. A medium described by classical elasticity has little sensitivity to localized perturbations. Imagine drilling a small hole or somehow weakening a small region of size a in an elastic sheet under fixed-load conditions. Changes in the stress and displacement fields induced by this perturbation are negligible at distances from the perturbation point much larger than a. An isostatic system behaves in a fundamentally different way. The weakening of a single element may produce changes in the displacement field on a global scale. This is related to strong dependence on boundary conditions [43], which is normally not present in continuum elasticity. It has been argued recently [43, 45] that, in systems that satisfy isostaticity, principal stresses are linearly related. This in turn can be shown [6, 42] to imply that stress propagation is determined by wave-like, or hyperbolic equations. Stresses propagate, on average, along preferred directions, which amounts to the prediction that the average response < G > must show two peaks (in two dimensions) that separate linearly with distance. These claims are trivially true in ordered isostatic networks like, for example, a square lattice (Section 2.4.2), but it is not obvious whether the existence of preferred directions survives the introduction of disorder. It has been argued that disorder may produce a crossover from a two-peak response on small scales to a single-peak response [10] at large scales. Our own numerical results [50] (Section 2.4.4) show that, even in the presence of strong polydispersity, two peaks are evident in the average response (Figure 2.6c). Our results have been obtained with a simple sequential packing algorithm that does not avoid unphysical tensile stresses. However, comparable results have been reported for < G > using a more elaborate, though time-consuming, adaptive algorithm [45, 48] which avoids tensile stresses. The conclusion is then that preferred direc-

40


tions exist for propagation of stresses on disordered isostatic packings, i.e. stresses propagate in a wave-like manner on them. Notice, however, that the existence of friction might modify this picture substantially, since frictional packings are not isostatic [27, 29]. Most often, theoretical treatments of disordered systems deal with average properties, even when microscopic details are expected to depend on disorder. The assumption that justifies the neglect of fluctuations is that physically relevant observables are usually self-averaging. This means that their probability distribution is sufficiently narrow in a certain limit. In the particular case of stresses in disordered systems, one assumes that a coarse-grained stress field can be defined such that, for large enough coarse-graining scales, fluctuations are negligible. This is a reasonable assumption in general and allows one to write down continuum equations for stresses or displacements, for example. However, notice that the self-averageness assumption fails for isostatic networks. Response functions on disordered isostatic networks display very strong fluctuations because of the fact that noise (i.e., disorder) appears multiplicatively in the equations ruling their behavior [24, 25]. The effect of a perturbation on a disordered isostatic network may grow multiplicatively with distance [21–25], instead of decaying, as happens on a hyperstatic network. Notice that this discussion only applies for infinitesimal perturbations. These considerations do not hold for response functions under finite perturbations, in which case rearrangements of the contact network are unavoidable [26, 48, 49, 54]. For isostatic networks under infinitesimal perturbations, the existence of multiplicative noise [31] produces wildly fluctuating responses [21–25]. The scale of these fluctuations grows exponentially with distance, and the distribution of responses is of the power-law type. The appearance of power-laws has been reported recently in several systems that are simultaneously under the effect of multiplicative and additive noise [37–41]. The simplest example of a disordered isostatic lattice (Section 2.4.2), the randomly distorted square lattice, has power-law distributed responses (Figure 2.4b), and suffers stresses whose scale grows exponentially with system size [24, 25]. This is a remarkable property of these seemingly harmless networks, and amounts to the inexistence of a continuum limit for stresses. Even packings that by construction (Section 2.4.3) only have compressive (and consequently bounded), stresses, still have wildly fluctuating stress responses (Figure 2.5b), whose scale grows exponentially with distance. Since stresses are linear superpositions of responses, it may appear surprising that the former may remain bounded. However, this is just a consequence of correlations among response functions at the same point. These correlations are introduced by construction, because contact forces are required to be non-tensile. If this constraint is lifted (allowing for stresses of any sign) stresses grow exponentially with depth. Before closing we must once again remark that the unusual growth of fluctuations discussed in this work only refer to isostatic networks. Strict isostaticity is seldom found in natural structures, and this is not surprising given that they have such singular properties. An idealized packing of infinitely rigid frictionless polydisperse spheres, however, has a perfectly isostatic contact network in any dimension. But perhaps this is not a good model for granular materials, after all. An, as yet, unsettled point is to which extent these findings are applicable for real granular packings, which are made of deformable particles that usually have large friction coefficients. Loss of isostaticity because of particle deformations only comes about when the compressive stress is large enough to close interparticle gaps, establishing redundant contacts. For granular systems with large disorder and interparticle gaps of the order of the

References

41

mean particle size (like, e.g., gravel) it is reasonable to expect that redundant contacts will seldomly occur. However, friction invalidates isostaticity to some extent [27, 29]. It is not, however, clear whether overconstrained subgraphs remain localized, making for a system that is still effectively isostatic on a large scale, or percolate across the whole system. A clear understanding of these and similar issues in relation to friction and its effect on isostatic systems is still needed. However, we feel that, even when isostaticity may not hold strictly for real packings, it is quite possible that their properties will display strong remnants of the singular behavior found on model networks that have this property.

Acknowledgments I wish to thank H. Herrmann, J. Socolar, J. Goddard, J. P. Bouchaud, P. Claudin and A. Tkachenko for useful discussions and correspondence about the topics here discussed. I would also like to acknowledge the collaboration of A. M. Vidales and C. Ruiz-Suárez in some of the studies described in this work. The investigations here reported have been partially supported by CONACYT, México, through research project 36256-E.

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3 Statistical Mechanics of Jammed Matter Hernán A. Makse, Jasna Brujić, and Sam F. Edwards

Abstract A thermodynamic formulation of jammed matter is reviewed. Experiments and simulations of compressed emulsions and granular materials are then used to provide a foundation for the thermodynamics.

3.1 Introduction to the Concept of Jamming The act of jamming or the condition of being jammed stems from the following dictionary description: “a crowd or congestion of people or things in a limited space, e.g., a traffic jam”. The scientific translation defines “jamming” as a state which emerges when a many-body system is blocked in a configuration far from equilibrium, from which it takes too long a time to relax for the timescale to be a measurable quantity. Jamming is emerging as a fundamental feature of many diverse systems [1], such as • Granular materials: sand, sugar, marbles, dry powders. • Emulsions: mayonnaise, custard, milk. • Colloidal suspensions: paints, muds. • Structural glasses: polymer melts, silica glass. • Spin glasses. These distinct disordered systems are but a few examples of equilibrium systems, which are united by their behavior at the point of structural arrest. Whereas one can think of liquids or suspensions as consisting of particles which move very slowly compared to gases, a state may occur where all particles are in close contact with one another and therefore experience jamming. The process of jamming is specific to the system in question due to their different microscopic properties. The following examples illustrate this. While it suffices to pour a granular material into a closed container and shake it to jam up the particles, the emulsion droplets require a large “squeezing” force usually implemented by centrifugation. On the other hand, the interaction between colloidal particles can be tuned such that the interparticle attraction induces a jammed configuration even at low densities of the material. Furthermore,

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3 Statistical Mechanics of Jammed Matter

glassy materials can be cooled down to very low temperatures at which the molecules can no longer diffuse, thus trapping the system into jammed configurations. Hence, through very different jamming mechanisms, we arrive at the jammed state for a variety of systems. All these systems belong to a class of materials known as “soft” matter, referring to their complex mechanical properties which are neither fluid nor solid-like. This behavior is directly linked to the material’s capability of supporting a mechanical disturbance once it has reached a jammed state. While the concepts of crowding and the subsequent mechanical response unite these materials, the details of their constitutive particles introduce important differences. For instance, in a polymer melt, it is the physical chemistry of the individual strands which will govern the ensemble, whereas it is the interactions between the colloidal particles, rather than their constituent molecules, which will determine the system behavior in suspensions. Moreover, particles of sizes up to 1 µm are governed by the laws of statistical mechanics since their dynamics is due to thermal (Brownian) motion. Above that threshold size (e.g., grains), the gravitational energy exceeds kB T , where T is the room temperature and kB the Boltzmann constant, thus prohibiting motion. The colloidal regime is therefore defined for sizes between 1 nm and 1 µm, such that thermal averaging is present. It applies to glasses, colloids, surfactants and microemulsions, or in other words, to ‘complex fluids’. Most of the fundamental physics research has been performed on thermal systems until present, and many of the unifying concepts have arisen through the comparison of systems within this category. In the next two sections we first describe the structural arrest in thermal systems, where the classical statistical mechanics tools are applicable, and then proceed to athermal systems, such as granular materials and compressed emulsions, in which new situations suitable for a statistical analysis are introduced.

3.1.1 Jamming in Glassy Systems In a fluid at thermal equilibrium the particle dynamics is too fast to capture the detail of the underlying potential energy landscape, thus it appears flat. Decreasing the temperature slows down the Brownian dynamics, implying a limiting temperature below which the system can no longer be equilibrated in this way. Hence, the thermal system falls out of equilibrium on the timescale of the experiment and thus undergoes a glass transition [2]. The motion of each particle is no longer thermally activated and only the vibration inside the cage formed by its surrounding neighbors persists. However, even below the glass transition temperature the particles continue to relax, but the nature of the relaxation is very different to that in equilibrium. This phenomenon of a structural evolution beyond the glassy state is known as “aging”. The dynamics becomes dominated by the multidimensional potential energy surface which the system can explore as a function of the degrees of freedom of the particles, depicted in Figure 3.1. In order to describe this landscape Stillinger and co-workers [4], based on ideas introduced by Goldstein [3], developed the concept of inherent structures which are defined as the potential energy minima. The trajectory of a system aging at temperature T can be mapped onto the successive potential basins that the system explores. Computational methods are the only available technique for investigating this behavior, in which the inherent structures are found via steepest-descent quenching of the system configurations to the basins of the wells. The entropy of the system can be shown to be separable into contributions from the available configurations and the vibrational modes around each minimum. There have

3.1

Introduction to the Concept of Jamming

47

been many studies which have embarked on an investigation of aging through the exploration of the configurational space [5–7].

Figure 3.1: The multidimensional energy landscape dominates the dynamics below the glass transition, as the system explores the inherent structures defined as the potential energy minima. A trajectory through the landscape is shown and the analogy between these inherent structures and the jammed states in granular materials is examined.

The importance of the inherent structure formalism is in enabling the comparison of jamming in particulate systems with glasses [8]. The entropy arising from the inherent configurations of the glass at very low temperatures and the exploration of these configurations due to the vibrational modes of the particles could be viewed as analogous to the configurational changes in particulate packings under slow tapping or shear. However, in granular materials there is an added effect of friction, which dissipates the analogous vibrations at once. Unlike granular materials, a thermal system is never permanently trapped in the bottom of a valley, but escapes in other accessible unstable directions through intrinsic thermal vibrations. At any finite temperature the system will not resemble the granular system in that it continuously evolves toward a maximum density state. Thus, the only true analogous situation between glasses and granular materials is valid at zero temperature. However, there are characteristic features of the glassy relaxation at a finite T which act as useful tools for the description of granular systems by exploiting the analogy between the relaxation of powders and aging in glassy systems [9]. For instance, theories developed during the late eighties and nineties in the field of spin glasses [10, 11] have led to a better understanding of glassy systems through the generalization of usual equilibrium relations, such as the fluctuation–dissipation relation, to situations far from equilibrium [12]. This approach developed by Cugliandolo, Kurchan and collaborators yielded macroscopic observable properties, such as an “effective temperature” for the slow

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modes of relaxation, which could then be compared between various glassy systems. Furthermore, the existence of an effective temperature with a thermodynamic meaning in glasses at very low temperature suggests an ‘ergodicity’ for the long-time behavior of the system [13]. This ergodicity is closely related to the statistical ideas for granular systems [14–16] which we will introduce in the following sections. In support of this argument, the effective temperature in glasses is found to be an adequate concept for describing granular matter [17], as it will be shown in Section 3.4. From a theoretical point of view, these systems are still only understood in terms of predictions of a general nature and many open questions remain. There is still much debate on issues such as the precise mechanisms of “surfing” the energy landscape, the effects of memory in the system, the slowing down of the system with time, and the discrepancies between the behavior of different glassy systems, but they are beyond the scope of this work.

3.1.2 Jamming in Particulate Systems In a sense one would imagine there is no simpler physical system than a granular assembly. After all it is just a set of packed rigid objects with no interaction energy. It is the inability to describe the system on the continuum level in any other way except according to its geometry which has led to a lack of a well established granular theory until present. Mostly due to their industrial importance, there has been a vast literature describing phenomenological observations without an encompassing theory. In the words of de Gennes, the state of granular matter can be compared to solid state physics in the 30’s or critical phenomena in phase transitions before the renormalization group. In other words, there is a need to describe the universal features of the observed behavior within a theoretical framework devised for these and other jammed systems. In parallel with the extensive research on glasses, described earlier, a decade ago Edwards and collaborators postulated the existence of a statistical ensemble for granular matter, despite the lack of thermal motion and the absence of an equilibrium state [18–22]. The main postulate was based on jamming the granular particles at a fixed total volume such that all microscopic jammed states are equally probable and become accessible to one another (ergodic hypothesis) by the application of a type of external perturbation such as tapping or shear, just as thermal systems explore their energy landscape through Brownian motion. Hence, let us consider granular jamming in more detail. Pouring sugar into a cup is the simplest example of a fluid to solid transition which takes place solely because of a density increase. In terms of physics, in particulate materials such as emulsions and granular media, a jammed system results if particles are packed together so that all particles are touching their neighbors, which obviously requires a sufficiently high density. In these athermal systems there is no kinetic energy of consequence; the typical energy required to change the positions of the jammed particles is very large compared to the thermal energy at room temperature (∼ 1014 times, see Section 3.4). As a result, the material remains arrested in a static state and is able to withstand a sufficiently small applied stress. There is a subtle, but crucial difference, between a configuration in mechanical equilibrium and a jammed configuration, particularly in the context of this research. The mechanism of arriving at a static configuration by an increase in density, which is an intuitively obvious process, is not always sufficient to satisfy the jamming condition in our definition. This applies

3.1


49

especially to systems which bear knowledge of the process of their creation. For instance, pouring grains into a container results in a pile at a given angle of repose. This mechanical equilibrium configuration is not jammed because in response to an external perturbation, the constituent particles will irreversibly rearrange, approaching a truly jammed configuration. The statistical mechanics which we are aiming to test implies an ergodic hypothesis, which is not valid in such history-dependent samples1 . It turns out that, by allowing the system to explore its available configurational space through external mechanical perturbations, the system will rearrange such that all possible configurations (w.r.t. the perturbation) become accessible to one another. Continuing with the analogy in the real world, the gentle tapping on a table of the cup filled with sugar will initially change the unstable angle of repose of the sugar pile and flatten its top surface, and therefore its density, until it settles into a desired configuration which depends on the strength of the tap. We can only perform a statistical analysis on the resulting configurations which have no memory of their creation, i.e., the true jammed configurations. Thus we arrive at a jammed ensemble, suitable for the application of statistical mechanics, described in Section 3.2. Since the particles can jump across the energy landscape during the tap, but then stop at once due to frictional dissipation, there is an analogy to the inherent structure formalism in glassy systems. This new statistical mechanics is able to provide unifying concepts between previously unrelated media. 3.1.2.1 Applications of the Jamming Condition The statistical mechanics which we are aiming to develop implies an ergodic hypothesis, which is not valid in history-dependent samples. In fact, there are many experimental situations in which the statistical mechanics cannot be applied due to the lack of ergodicity. For instance, convection cycles have been observed in granular systems under vigorous tapping [24] – an effect which is closely associated with the segregation process of different granular species. These types of closed loops in phase space cannot be described within the thermodynamic framework. Rapid granular flows observed in pouring sand in a pile, or vigorously shaken granular systems at low density are out of the scope of the present approach since the systems are exploring configurations far from the jammed states [25]. Kinetic theories of inelastic gases are more appropriate for treating these situations [26]. The physics of the angle of repose [27] may not be understood under the thermodynamic framework due to the absence of the jamming condition of the pile, despite the fact that it is static. In many practical situations, heterogeneities appear which also preclude the application of a thermodynamic approach. For instance, when granular materials are sheared in a sufficiently large shear stage, shear bands appear where the strain is discontinuous [28]. Such local effects cannot be captured by the present thermodynamic approach. On the other hand, if the application of statistical physics to jammed phenomena were to prove productive, then one could anticipate a more profound insight into the characterization and understanding of the system as a whole. For instance, the thermodynamic hypothesis 1

Theories attempting to describe such systems have been developed by Bouchaud et al., proposing a model for the ‘fragile’ systems, i.e., systems which rearrange under infinitesimal stresses [23]. However, this situation will not be considered here.

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Figure 3.2: Compaction curve for a packing of glass beads under an oscillating pressure. Increasing the amplitude of oscillation initially increases the density by filling the loose voids, after which a reversible regime is achieved (from [32]).

would lead to the prediction of macroscopic quantities such as viscosity and complex shear moduli, which would in turn provide a complete rheological characterization of the system. As a matter of theoretical interest, a statistical ensemble for jammed matter could be one of the very few generalizations of the statistical mechanics of Gibbs and Boltzmann to systems out of equilibrium. 3.1.2.2 Achieving the Jammed State Experimentally, the conditions for a statistical ensemble of jammed states can be achieved by pre-treating the granular assembly by tapping or via slow shear-driving. Experiments at the University of Chicago involving the tapping of granular columns were the first to show the existence of a reversible regime in which the system configurations are independently sampled [29]. Starting with a loose packing of the grains, the tapping routine initially removes the unstable loose voids and thus eliminates the irreversible grain motion. Once all the grains are touching their neighbors, the density of the resulting configuration becomes dependent on the tapping amplitude and the number of taps; the larger the amplitude, the lower the density. The mechanism of the compaction process leading to a steady-state density is extremely slow, in fact, it is logarithmic in the number of taps. This dependence of the density of grains on the external perturbation of the system once the memory effects of the pile construction

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51

details have been removed, is known as the reversible branch of the ‘compaction curve’, see Figure 3.2. Despite the presence of friction between grains (implying memory effects) this curve is reversible, establishing a new type of equilibrium state. It is along this curve that the thermodynamics for granular matter can be applied. There have been several further experiments confirming these results for different system geometries, particle elasticities and compaction techniques. For example, the system can be mechanically tapped or oscillated, vibrated using a loudspeaker, slowly sheared in a couette geometry, or even allowed to relax under large pressures over long periods of time, all to the same effect [30–33]. Here we show a new compaction regime under an oscillating pressure where the same density dependence of a packing of glass and acrylic beads is noted for varying amplitudes of the pressure oscillation. These experiments have been performed at Schlumberger-Doll Research [32]. The resulting curve of the achieved volume fraction as a function of the amplitude of the pressure oscillation is shown in Figure 3.2. Moreover, experiments in the Cavendish laboratory [33] have shown how the conductivity of powdered graphite can also be a measure of the particle density as it is being vibrated, in which the direct link to the volume function is less obvious, but the qualitative results indicate the same trends. The methodology for achieving jammed configurations has also been established numerically for the purpose of rheological and thermodynamical studies and it will be described in Section 3.4. At this point, it is important to note that we have only considered infinitely rigid, rough grains in which an increase in the pressure of the system, for instance by placing a piston on top of the grains, causes no change in the shape of the grains and therefore no change in the packing density. On the other hand, real grains have a finite elastic modulus, thus the application of a sufficiently large external pressure will always result in grain deformation and therefore a density increase unrelated to the tapping. In soft particles, such as emulsions, the effect of pressure is more significant. The tapping experiment described above measured the resulting densities at atmospheric pressure, which is considered to be the zero reference pressure. The same experiment can be repeated at finite pressures giving rise to equivalent compaction curves, depicted in Figure 3.3. Whereas hard grains, such as glass beads, require extremely large pressures (∼1 MPa) to deform and the amount of deformation is limited by their yield stress, softer particles, such as rubber, are able to reach higher densities with relative ease. Droplets and bubbles, being the softest particles one can have, are capable of reaching the density of 1, corresponding to a biliquid foam and a foam, respectively, by an application of much smaller pressures (∼1 kPa). They have the advantage of the whole pressure range being accessible to them. Another distinction between granular materials and emulsions is the presence of friction in the former and the smoothness of the latter. Since friction plays an important role in inducing memory into the system, its absence leads to a much easier achievement of the jammed state, described above. For instance, in the case of emulsions, allowing the particles to cream under gravity will suffice in order to arrive at the reversible part of the compaction curve, bypassing the irreversible branch, as it will be shown in Section 3.3.

3.1.3 Unifying Concepts in Granular Matter and Glasses In the preceding paragraphs, it has been shown under which conditions both thermal and athermal systems explore the configurational energy landscape, which possibly results in common-


1

Volume fraction (

reversible branch at finite p ( deformable particles)

0.64

RCP

reversible branch at p=0

0.59

RLP

0.55

minRLP

Increasing pressure p

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Foams

Deformable grains

Rigid grains

Emulsions

irreversible branch Suspensions

Amplitude of tapping ()

Figure 3.3: Compaction curves of volume fraction φ versus amplitude of oscillation Γ for different external confining pressures, p. Increasing the amplitude of oscillation initially increases the volume fraction by filling the loose voids (irreversible branch), after which a reversible regime is achieved. For infinitely rigid grains (the “zero pressure” curve) the minimum volume fraction along the irreversible branch is the random loose packing. The reversible branch goes from the random loose packing fraction to the random close packing fraction. Below the minimum RLP only suspensions can exist.

alities in their behavior. At present, new unifying theoretical descriptions for jammed matter are being sought, as well as new experimental evidence to unify the predicted state for all varieties of jammed systems. The prediction of how different systems jam with respect to the applied stress, density and temperature has led to a speculative diagram proposed by Liu and Nagel in their article “Jamming is not just cool anymore” in Nature [34–36]. It links the behavior of glasses (thermal systems) and bubbles, grains, droplets (athermal systems) by the dynamics of their approach to jamming. Since the observable properties such as applied strain, temperature and density can be obtained by consideration of only the jammed configurations in a given system, the thermodynamics of jamming, discussed in the next section, is intimately related to the ideas put forward in the jamming phase diagram.

3.2

New Statistical Mechanics for Granular Matter

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3.2 New Statistical Mechanics for Granular Matter This section aims to justify the use of statistical mechanics tools in situations where the system is far from thermal equilibrium, but jammed. In what follows, we present the classical statistical mechanics theorems to an extent which facilitates an understanding of the important concepts for the development of an analogous granular theory, as well as the assumptions necessary for belief in such a parallel approach. Thereafter, we present a theoretical framework to fully describe the exact specificities of the granular packing, and the shaking scenario which leads to the derivation of the Boltzmann equation for a jammed granular system. This kind of an analysis paves the path to macroscopic quantities, such as the compactivity, characterizing the configurations from the microstructural information of the packing. It is according to this theory that the jammed configurations obtained from experiments and simulations are later characterized.

3.2.1 Classical Statistical Mechanics In the conventional statistical mechanics of thermal systems, the different possible configurations, or microstates, of the system are given by points in the phase space of all positions and momenta {p, q} of the constituent particles. The equilibrium probability density ρeqm must be a stationary state of Liouville’s equation which implies that ρeqm must be expressed only in terms of the total energy of the system, E [37]. The simplest form for a system with Hamiltonian H(p, q) is the microcanonical distribution: ρeqm (E) =

1 , Σeqm (E)

(3.1)

for the microstates within the ensemble, H(p, q) = E, and zero otherwise. Here, Σeqm (E) =

δ E − H(p, q) dp dq,

(3.2)

is the area of energy surface H(p, q) = E. Equation (3.1) states that all microstates are equally probable. Assuming that this is the true distribution of the system implies accepting the ergodic hypothesis, i.e., the trajectory of the closed system will pass arbitrarily close to any point in phase space. It was a remarkable step of Boltzmann to associate this statistical concept of the number of microstates with the thermodynamic notion of entropy through his famous formula Seqm (E) = kB ln Σeqm (E).

(3.3)

Thus, in classical statistical mechanics, the total energy of the system is sufficient to describe the probability density of states. Whereas the study of thermal systems has had the advantage of available statistical mechanics tools for the exploration of the phase space, an entirely new statistical method, unrelated to the temperature, had to be constructed for grains.

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3.2.2 Statistical Mechanics for Jammed Matter We now consider a jammed granular system composed of rigid grains (deformable particles will be considered in Section 3.4). Such a system is analogously described by a network of contacts between the constituent particles in a fixed volume, V , since there is no relevant energy E in the system. In the case of granular materials, the analogue of phase space, the space of microstates of the system, is the space of possible jammed configurations as a function of the degrees of freedom of the system {ζ}. It is argued that it is the volume of this system, rather than the energy, which is the key macroscopic quantity governing the behavior of granular matter [18–20]. If we have N grains of specified shape which are assumed to be infinitely rigid, the system’s statistics would be defined by a function W(ζ), a function which gives the volume of the system in terms of the specification of the grains. In this analogy one replaces the Hamiltonian H(p, q) of the system by the volume function, W(ζ). The average of W(ζ) over all the jammed configurations determines the volume V of the system in the same way as the average of the Hamiltonian determines the average energy E of the system. 3.2.2.1 Definition of the Volume Function, W One of the key questions in this analogy is to establish the ‘correct’ W function, the statistics of which is capable of fully describing the system as a whole. The idea is to partition the volume of the system into different subsystems α with volume W α , such that the total volume of a particular configuration is W α. (3.4) W(ζ) = α

It could be that considering the volume of the first coordination shell of particles around each grain is sufficient; thus, we may identify the partition α with each grain. However, particles further away may also play a role in the collective system response due to enduring contacts, in which case W should encompass further coordination shells. In reality, of course, the collective nature of the system induces contributions from grains which are indeed further away from the grain in question, but the consideration of only its nearest neighbors is a good starting point for solving the system, and is the way in which we proceed to describe the W function. The significance of the appropriate definition of W is best understood by the consideration of a response to an external perturbation to the system in terms of analogies with the Boltzmann equation which we will describe in Section 3.2.4. Perhaps the most straightforward definition of the function W α is given in terms of the Voronoi diagram which partitions the space into a set of regions, associating all grain centroids in each region to the closest grain centroid, depicted by line OP in the diagram in Figure 3.4a. The loop formed by the perpendicular bisectors (ab) of each of the lines joining the central grain to its neighbors is the Voronoi cell, depicted in red. Even though this construction successfully tiles the system, its drawback is that there is no analytical formula for the enclosed volume of each cell. Recently Ball and Blumenfeld [38, 39] have shown by an exact triangulation method that the volume defining each grain can be given in terms of the contact points

3.2


55 Cq

Cp

C

O

a

Cp

p

v

q

P

Cq

O P

P’

b

(a) Voronoi

(b) Ball and Blumenfeld

(c) Edwards

Figure 3.4: Different volume functions as discussed in the text: a) Voronoi, b) Ball and Blumenfeld and c) Edwards construction.

C using vectors constructed from them (see Figure 3.4b). The method consists in defining shortest loops of grains in contact with one another (p,q loops), thus defining the void space around the central grain. The difficulty arises in three dimensions since this construction requires the identification of void centers, v. This is not an obvious task, but is currently under consideration. The resulting volume (red) is the antisymmetric part of the fabric tensor, the significance of which is its appearance in the calculation of stress transmission through granular packings [38]. A cruder version for the volume per grain, yet with a strong physical meaning, has been given by Edwards. For a pair of grains in contact (assumed to be point contacts for rough, rigid grains) the grains are labeled α, β, and the vector from the center of α to that of β is denoted as R αβ and specifies the complete geometrical information of the packing. The first step is to construct a configurational tensor C α associated with each grain α, based on the structural information, αβ αβ α = Ri Rj . (3.5) Cij β

Then an approximation for the area in 2D or volume in 3D encompassing the first coordination shell of the grain in question is given as α. (3.6) W α = 2 Det Cij This volume function is depicted in the Figure 3.4c, with grain coordination number 3 in two dimensions, where Eq. (3.6) should give the area of the triangle (red) constructed by the centers of grains P which are in contact with the grain α. The above equation is exact if the area is considered as the determinant of the vector cross product matrix of the two sides of the triangle, but its validity for higher coordination numbers and in 3D has not been tested. Surprisingly, this approximation works well according to our experimental studies in Section 3.3. This is due to the partitioning of the obtained volumetric objects into triangles/pyramids, intrinsic to the method, and subsequently summing over them to obtain the resulting volume. This definition is clearly only an approximation of the space available to each grain since there is an overlap of W α for grains belonging to the same coordination shell. Thus, it over-

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α estimates the total volume of the system: W > V . However, it is the simplest approximation for the system based on a single coordination shell of a grain. 3.2.2.2 Entropy and Compactivity Now that we have explicitly defined W, it is possible to define the entropy of the granular packing. The number of microstates for a given volume V is measured by the area of the surface W(ζ) = V in the phase space of jammed configurations and it is given by: Σjammed (V ) = δ V − W(ζ) Θ(ζ) dζ, (3.7) where now dζ refers to an integral over all possible jammed configurations and δ(V − W(ζ)) formally imposes the constraint to the states in the sub-space W(ζ) = V . Θ(ζ) is a constraint that restricts the summation to only reversible jammed configurations as opposed to the merely static equilibrium configurations as previously discussed. This function will be discussed in detail in Section 3.4.2.2. The radical step is the assumption of equally probable microstates which leads to an analogous thermodynamic entropy associated with this statistical quantity: (3.8) S(V ) = λ ln Σjammed (V ) = λ ln δ V − W(ζ) Θ(ζ) dζ, which governs the macroscopic behavior of the system [19, 20]. Here λ plays the role of the Boltzmann constant. The corresponding analogue of temperature, named the “compactivity”, is defined as XV−1 =

∂S . ∂V

(3.9)

where the subscript V refers to the fact that it is the derivative of the entropy with respect to the volume. This is a bold statement, which perhaps requires further explanation in terms of the actual role of compactivity in describing granular systems. We can think of the compactivity as a measure of how much more compact the system could be, i.e., a large compactivity implies a loose configuration (e.g. random loose packing, RLP) while a reduced compactivity implies a more compact structure (e.g., random close packing, RCP, the densest possible random packing of monodisperse hard spheres). In terms of the reversible branch of the compaction curve, large amplitudes generate packings of high compactivities, while in the limit of the amplitude going to zero, a low compactivity is achieved. In terms of the entropy, many more configurations are available at high compactivity, thus the dependence of the entropy on the volume fraction can be qualitatively described as in Figure 3.5. In the figure, for monodisperse packings the RCP is identified at φ ≈ 0.64 [40], the RLP fraction is identified at φ ≈ 0.59 [41], while the crystalline packing, FCC, is at φ = 0.74, but cannot be reached by tapping. At any given tapping amplitude, there exists an equilibrium volume fraction toward which the system slowly evolves. For instance, a system may find itself at a lower entropy than the equilibrium curve by the application of an internal constraint at a given volume fraction. This situation can be achieved by creating small crystalline regions within a packing configuration

3.2


S()

57

High compactivity

Low compactivity RLPmax ~ 0.59

RCP ~ 0.64

FCC ~ 0.74

Volume fraction Figure 3.5: Interpretation of the compactivity and entropy in terms of different packings. See also Figure 3.2.

of a lower density, and looser regions compensating for the volume reduction such that the total volume of the system remains constant. This configuration, given that it is not jammed, will tend toward the equilibrium packing via the application of a small perturbation by increasing its entropy. Such an example will be made more explicit in the derivation of the Boltzmann equation for granular materials. At volume fractions beyond the RCP (and at atmospheric pressure) the system is not able to explore the configurations as they can only be achieved by the partial crystallization of the sample, where there are very few configurations available. It becomes clear from Eq. (3.9) that the compactivity is only applicable in equilibrated jammed states. As an analogue of temperature, it should also obey the zero-th law of thermodynamics. Hence, two different powders in physical contact with one another should equilibrate at the same compactivity, given a mechanism of momentum transfer between the two systems. Indeed, we may think of an appropriate laboratory experiment which would test this hypothesis under certain conditions necessary for creating the analogous situation to heat flow. Two powders, A and B, of different grain types are poured into a vertical couette cell as shown in Figure 3.6. The grains must experience an equivalent tapping or shearing regime, which is achieved by the rotation of the inner cylinder of the couette cell. The species are separated by a flexible diaphragm, such that momentum transfer between the two systems is ensured. The two powders must be well separated such that there is no mixing involved, but in contact nevertheless. The grains are kept at a constant pressure by a piston which is allowed to move freely to accommodate for the changes in volume experienced by the two types of grains. Gravity may play a role in the experiment, which is avoided by density matching the particles with a suspending fluid. The experiment consists in placing powders A and B together in the above cell and slowly shearing them at a given velocity. The powders should come to equilibrium volumes VA

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3 Statistical Mechanics of Jammed Matter Moving piston

w A = sand B = sugar

Soft membrane

C = rice

Couette cell

XA=XB

XB=XC

XA=XC

Figure 3.6: ABC experiment to test the zero-th law of granular thermodynamics.

and VB , with equivalent respective compactivities, XA = XB = X. While it is easy to measure the volumes of the two systems, the measurement of their compactivities employs more sophisticated methods, discussed in Section 3.3.2.3. In the absence of a compactivity scale, we use powder B as a ‘thermometer’ by placing it in contact with a third powder C. The volume B is kept at VB and the volume of C is allowed to fluctuate until it reaches the equilibrium state. Finally, powders A and C are put together to test if they will reach the same volumes as they did in previous runs in contact with B, thus proving the zero-th law. A form of the zero-th law of thermodynamics will be shown to be valid numerically in Section 3.4. 3.2.2.3 Remarks To summarize, the granular thermodynamics is based on two postulates: (1) While in the Gibbs construction one assumes that the physical quantities are obtained as an average over all possible configurations at a given energy, the granular ensemble consists of only the jammed configurations at the appropriate volume. (2) As in the microcanonical equilibrium ensemble, the strong ergodic hypothesis is that all jammed configurations of a given volume can be taken to have equal statistical probabilities. The ergodic hypothesis for granular matter was treated with skepticism, mainly because a real powder bears knowledge of its formation and the experiments are therefore history dependent. Thus, any problem in soil mechanics or even a controlled pouring of a sand pile does not satisfy the condition of all jammed states being accessible to one another as ergodicity has not been achieved, and the thermodynamic picture is therefore not valid. This point has been discussed in Section 3.1.2. The Chicago experiments of tapping columns [30] showed the existence of reversible situations. For instance, let the volume of the column be V (n, Γ) where n is the number of taps and Γ is the strength of the tap. If one first obtains a volume V (n1 , Γ1 ), and then repeats the experiment at a different tap intensity and obtains V (n2 , Γ2 ), when we return to tapping at (n1 , Γ1 ) one obtains a volume V which is V (n1 , Γ1 ) = V (n1 , Γ1 ). Moreover, in simulations of slowly sheared granular systems, the ergodic hypothesis was shown to work [17] as we will discuss in Section 3.4.

3.2


59

It is often noted in the literature that, although the simple concept of summing over all jammed states which occupy a volume V works, there is no first principle derivation of the probability distribution of the Edwards ensemble as it is provided by Liouville’s theorem for equilibrium statistical mechanics of liquids and gases. In granular thermodynamics there is no justification for the use of the W function to describe the system as Liouville’s theorem justifies the use of the energy in the microcanonical ensemble. In Section 3.2.4 we will provide an intuitive proof for the use of W in granular thermodynamics by the analogous proof of the Boltzmann equation. The comment was also made that there is no proof that the entropy Eq. (3.8) is a rigorous basis for granular statistical mechanics. In the next section we will develop a Boltzmann equation for jammed systems and show that this analysis can be used to produce a second law of thermodynamics, δS ≥ 0 for granular matter, and the equality only comes with Eq. (3.8) being achieved. Although everyone believes that the second law of thermodynamics is universally true in thermal systems, the only accessible proof comes in the Boltzmann equation, as the ergodic theory is a difficult branch of mathematics which will not be covered in the present discussion. By investigating the assumptions and key points which led to the derivation of the Boltzmann equation in thermal systems, it is possible to draw analogies for an equivalent derivation in jammed systems. It should be noted that there is an extensive literature on granular gases [25, 26], which are observed when particles are fluidized by vigorous shaking, thus inducing continuous particle collisions. There is a powerful literature on this topic, but it is not applicable to the problem of jamming.

3.2.3 The Classical Boltzmann Equation Entropy in thermal systems satisfies the second law, ∂S ≥ 0, ∂t

(3.10)

which states that there is a maximum entropy state which, according to the evolution in Eq. (3.10), any system evolves toward, and reaches at equilibrium. A “semi”-rigorous proof of the Second Law was provided by Boltzmann (the well-known “H-theorem”), by making use of the “classical Boltzmann equation”, as it is now known. In order to derive this equation, Boltzmann made a number of plausible assumptions concerning the interactions of particles, without proving them rigorously. The most important of these assumptions were: • The collision processes are dominated by two-body collisions (Figure 3.7a). This is a plausible assumption for a dilute gas, since the system is of very low density, and the probability of there being three or more particles colliding is infinitesimal. • Collision processes are uncorrelated, i.e., all memory of the collision is lost on completion and is not remembered in subsequent collisions: the famous Stosszahlansatz. This is also valid only for dilute gases, but the proof is more subtle.

60


v’

v

c0

K

K v1

v1’

c1’

c1 (b) Granular Boltzmann

(a) Classical Boltzmann

C1

C4 C0 C3

c0’

A)

C2

C’4

C’1 C’ C’3 0 C’2

Before

After

(c) First coordination cell rearrangements

Figure 3.7: a) Collision of two particles in a dilute gas. b) “Collision of two configurations” given in terms of two contact points in a jammed material. c) Rearrangements inside a pocket of mobile grains under the first coordination shell approximation for grain α = 0.

Thus, Boltzmann proves Eq. (3.10) for a dilute gas only, but this is a readily available situation. The remaining assumptions have to do with the kinematics of particle collisions, i.e., conservation of kinetic energy, conservation of momentum, and certain symmetry of the particle scattering cross-sections. Let f (v, r) denote the probability of a particle having a velocity v at position r. This probability changes in time by virtue of the collisions. The two-particle collision is visualized in Figure 3.7a where v and v1 are the velocities of the particles before the collision and v and v1 after the collision. On timescales larger than the collision time, momentum and kinetic energy conservation apply: 1 1 1 1 2 mv 2 + mv12 = mv 2 + mv 1 . (3.11) mv + mv1 = mv + mv1 , 2 2 2 2 Then, the distribution f (v, r) evolves with time according to ∂f ∂f +v + K(v, v ; v1 , v1 ) f (v)f (v1 ) − f (v )f (v1 ) d3 v1 d3 v d3 v1 = 0. (3.12) ∂t ∂r The kernel K is positive definite and contains δ-functions to satisfy the conditions (3.11), the flux of particles into the collision and the differential scattering cross-section. We consider the case of homogeneous systems, i.e., f = f (v), and define S = −kB f ln f. (3.13) Defining x = f f1 /f f1 we obtain ∂S = K ln x (1 − x)d3 v1 d3 v d3 v1 , ∂t

(1 − x) ln x ≥ 0,

Hence ∂S/∂t ≥ 0 (see standard text books on statistical mechanics).

K ≥ 0.

(3.14)

3.2


61

It is also straightforward to establish the equilibrium distribution where ∂S/∂t = 0 since it occurs when the kernel term vanishes, i.e., when the condition of detailed balance is achieved, x = 1: f (v)f (v1 ) = f (v )f (v1 ).

(3.15)

The solution of Eq. (3.15) subjected to the condition of kinetic energy conservation is given by the Boltzmann distribution f (v) =

βm 2π

3/2

1

2

e− 2 βmv ,

(3.16)

where β = 1/kB T . Equation (3.16) is a reduced distribution and valid only for a dilute gas. The Gibbs distribution represents the full distribution and is obtained by replacing the kinetic energy in (3.16) by the total energy of the state to obtain: P (E) ∝ e−βE .

(3.17)

The question is whether a similar form can be obtained in a granular system in which we expect P (W) ∝ e−W/λX ,

(3.18)

where X is the compactivity in analogy with T = ∂E/∂S. Such an analysis is shown in the next section in an approximate manner.

3.2.4 “Boltzmann Approach” to Granular Matter The analogous approach to granular materials consists in the following: the creation of an ergodic grain pile suitable for a statistical mechanics approach via a method for the exploration of the available configurations analogous to Brownian motion, the definition of the discrete elements tiling the granular system via the volume function W (the sum of which provides the analogous ‘Hamiltonian’ to the energy in thermal systems), and an equivalent argument for the energy conservation expressed in terms of the system volume necessary for the construction of the Boltzmann equation. We have already established the necessity of preparing a granular system adequate for real statistical mechanics so as to emulate ergodic conditions. The grain motion must be wellcontrolled, as the configurations available to the system will be dependent upon the amount of energy/power put into the system. This pretreatment is analogous to the averaging which takes place inherently in a thermal system and is governed by temperature. As explained, the granular system explores the configurational landscape by the external tapping introduced by the experimentalist. The tapping is characterized by a frequency and an amplitude (ω, Γ) which cause changes in the contact network, according to the strength of the tap. The magnitude of the forces between particles in mechanical equilibrium and their confinement determine whether each particle will move or not. The criterion of whether a particular grain in the pile will move in response to the perturbation will be the Mohr– Coulomb condition of a threshold force, above which sliding of contacts can occur and below

62


a

Immobile grains

b

c

a)

b)

Figure 3.8: a) Regions of mobile grains a, b, c in a matrix of immobile grains below the Coulomb threshold. b) Detail of pocket of mobile grains a surrounded by immobile grains which are shaded.

which there can be no changes. The determination of this threshold involves many parameters, but it suffices to say that a rearrangement will occur between those grains in the pile whose configuration and neighbors produce a force which is overcome by the external disturbance. The concept of a threshold force necessary to move the particles implies that there are regions in the sample in which the contact network changes and those which are unperturbed, shown in Figure 3.8. Of course, since this is a description of a collective motion behavior, the region which can move may expand or contract, but the picture at any moment in time will contain pockets of motion encircled by a static matrix. Each of these pockets has a perimeter, defined by the immobile grains. It is then possible to consider the configuration before and after the disturbance inside this well-defined geometry. The present derivation assumes the existence of these regions. It is equivalent to the assumption of a dilute gas in the classical Boltzmann equation, although the latter is readily achieved experimentally. The energy input must be on the level of noise, such that the grains largely remain in contact with one another, but are able to explore the energy landscape over a long period of time. In the case of external vibrations, the appropriate frequency and amplitude can be determined experimentally for different grain types, by investigating the motion of the individual grains or by monitoring the changes in the overall volume fraction over time. It is important that the amplitude does not exceed the gravitational force, or else the grains are free to fly up in the air, re-introducing the problem of initial creation just as they would if they were simply poured into another container. Within a we have a volume α∈a W α and after the disturbance a volume which a region is now α∈a W α as seen in Figure 3.7b and 3.7c. In Section 3.2.2.1 we have discussed how to define the volume function W α as a function of the contact network. Here the simplest “one grain” approximation is used as the “Hamiltonian” of the volume as defined by Eq. (3.6). In reality it is much more complicated, and although there is only one label α on the contribution of grain α to the volume, the characteristics of its neighbors may also appear.

3.2


63

Instead of energy being conserved, it is the total volume which is conserved while the internal rearrangements take place within the pockets described above. Hence Wα = W α . (3.19) α∈a

α∈a

We now construct a Boltzmann equation. Suppose z particles are in contact with grain α = 0, as seen in Figure 3.7c. For rough particles z = 4, while for smooth z = 6, at the isostatic limit (see Section 3.4). The probability distribution will be of the packing configurations which are represented by the tensor C α , Eq. (3.5), for each grain, where α ranges from 0 to 4 in this case. So the analogy of f (v) for the Boltzmann gas equation becomes f (C 0 ) for the granular system and represents the probability that the external disturbance causes a particular motion of the grain. We therefore wish to derive an equation ∂f (C 0 ) + K(C α , C α ) f0 f1 f2 f3 f4 − f0 f1 f2 f3 f4 dC 0 dC α dC α = 0. (3.20) ∂t α=0

K contains the condition that the volume is conserved (3.19), i.e., it must contain The term δ( W α − W α ). The cross-section is now the compatibility of the changes in the contacts, i.e. C α must be replaced in a rearrangement by C α , Figure 3.7b (unless these grains part and make new contacts in which case a more complex analysis is called for). We therefore argue that the simplest K will depend on the external disturbance Γ, ω and on C α and C α , i.e., z z ∂f (C 0 ) + δ W α− W α J (C α , C α ) fα − fα dC 0 dC α dC α ∂t α α α=0 α=0 α=0

= 0. where J is the cross-section and it is positive definite. The Boltzmann argument now follows. As before f0 f1 f2 f3 f4 S = −λ f ln f, x= , f0 f1 f2 f3 f4

(3.21)

(3.22)

and ∂S ≥ 0, ∂t the equality sign being achieved when x = 1 and

(3.23)

α

e−W /λX fα = , Z with the partition function α Z= e−W /λX Θα , α

and the analogue to the free energy being Y = −X ln Z, and X = ∂V /∂S.

(3.24)

(3.25)

64


The detailed description of the kernel K has not been derived as yet due to its complexity. Just as Boltzmann’s proof does not depend on the differential scattering cross-section, only on the conservation of energy, in the granular problem we consider the steady state excitation externally which conserves volume, leading to the granular distribution function, Eq. (3.24). It is interesting to note that there is a vast and successful literature of equilibrium statistical mechanics based on exp(−H/kB T ), but a meagre literature on dynamics based on attempts to generalize the Boltzmann equation or, indeed, even to solve the Boltzmann equation in situations remote from equilibrium where it is still completely valid. It means that any advancement in understanding of how it applies to analogous situations is a step forward.

3.3 Jamming with the Confocal 3.3.1 From Micromechanics to Thermodynamics The first step in realizing the idea of a general jamming theory is to understand in detail the characteristics of jammed configurations in particulate systems. Thus, the main aim of this section is to design an experiment to provide a microscopic foundation for the statistical mechanics of jammed systems. The understanding of the micromechanics on the scale of the particle, together with the respective statistical measures, pave the path towards an experimental proof of the existence of such an underlying thermodynamics. The problem with the characterization of the jammed state in terms of its microstructure is that the condition of jamming implies an optically impenetrable particulate packing. The fact that we cannot take a look inside the bulk to infer the structural features has confined all but one three-dimensional study of packings to numerical simulations and the walls of an assembly. In the old days Mason, a graduate student of Bernal, took on the laborious task of shaking glass balls in a sack and ‘freezing’ the resulting configuration by pouring wax over the whole system. He would then carefully take the packing apart, ball by ball, noting the positions of contacts (ring marks left by the wax) for each particle [40]. The statistical analysis of his hard-earned data led to the reconstruction of the contact network in real space, a measurement of the radial distribution function, g(r), and also the number of contacts of each particle satisfying mechanical equilibrium which gives z ≈ 6.4 for close contacts. This has been the only reference point for simulators and theoreticians to compare their results with those from the real world and it therefore deserves a particular mention. In search of an alternative method of experimentation, more in line with the automated nature of our times, we developed a model system suitable for optical observation. Moreover, our aim was to investigate the jammed state in a different jammed medium, to probe the universality of the configurational features. Finally, one needs to solve the system geometry as well as the stresses propagating through it in order to come up with a general theory. To probe the stress propagation through the medium, rather than its configuration alone, the particles must have well-defined elastic properties. The system which could satisfy all our requirements was found in a packing of emulsion droplets [42, 43].

3.3

Jamming with the Confocal

65

3.3.2 Model System Emulsions are a class of material which is both industrially important and exhibits very interesting physics [44]. They belong to a wider material class of colloids in that they consist of two immiscible phases one of which is dispersed into the other, the continuous phase. Both of the phases are liquids and their interface is stabilized by the presence of surface-active species. Emulsions are amenable to our study of jamming due to the following properties: 1. Transparency. An alternative way of “seeing through” the packing is to refractive index match the phases in the system – i.e., the particles and the continuous medium filling the voids. Since an emulsion is made up of two immiscible liquids, it is possible to raise the refractive index of the aqueous phase to match that of the dispersion of oil droplets. However, transparency is not the only requirement, since the particles are then dyed to allow for their optical detection. 2. Alternative Medium. The emulsion is made up of smooth, stable droplets in the 1–10 µm size range, as compared to rigid, rough particles in the above described granular system. Both systems are athermal, but the length scale and the properties of the constituent particles of the system are very different. 3. Elasticity. Emulsion droplets are deformable, stabilized by an elastic surfactant film, which allows for the measurement of the interdroplet forces from the amount of film deformation upon contact. Moreover, the elasticity facilitates the measurement of the dependence of the contact force network on the external pressure applied to the system. Our model system consists of a dense packing of emulsion oil droplets, with a sufficiently elastic surfactant stabilizing layer to mimic solid particle behavior, suspended in a continuous phase fluid. The refractive index matching of the two phases, necessary for 3D imaging, is not a trivial task since it involves unfavorable additions to the water phase, disturbing surfactant activity. The successful emulsion system, stable to coalescence and Ostwald ripening, consisted of Silicone oil droplets (η = 10cS) in a refractive index matching solution of water (wt = 51%) and glycerol (wt = 49%), stabilized by 20 mM sodium dodecylsulphate (SDS) upon emulsification and later diluted to below the critical micellar concentration (CMC= 13 mM) to ensure a repulsive interdroplet potential. The droplet phase is fluorescently dyed using Nile Red, prior to emulsification. The control of the particle size distribution, prior to imaging, is achieved by applying very high shear rates to the sample, inducing droplet break-up down to a radius mean size of 3.4 µm. This system is a modification of the emulsion reported by Mason et al. [45] to produce a transparent sample suitable for confocal microscopy. 3.3.2.1 Characterization of a Jammed State Having prepared a stable, transparent emulsion we use confocal microscopy for the imaging of the droplet packings at varying external pressures, i.e., volume fractions. The key feature of this optical microscopy technique is that only light from the focal plane is detected. Thus 3D images of translucent samples can be acquired by moving the sample through the focal plane of the objective and acquiring a sequence of 2D images.

66


Figure 3.9: 2D slices of emulsions under varying compression rates: a) 1g, b) 6000g, and c) 8000g.

Since the emulsion components have different densities, the droplets cream under gravity to form a random close packed structure. In addition, the absence of friction ensures that the system has no memory effects and reaches a true jammed state before measurement. If the particles are subjected to ultracentrifugation, configurations of a higher density are achieved as the osmotic pressure is increased. The random close packing fraction reached under gravity depends on the polydispersity of the emulsion, or in other words, the efficiency of the packing. The sequence of images in Figure 3.9 shows 2D slices from the middle of the sample volume after: a) creaming under gravity, b) centrifugation at 6000g for 20 minutes and c) centrifugation at 8000g for 20 minutes. The samples were left to equilibrate for several hours prior to measurements being taken. The volume fraction for our polydisperse system shown in Figure 3.9b is φ = 0.86, determined by image analysis. This high volume fraction obtained at a relatively small osmotic pressure of 125 Pa is achieved due to the polydispersity of the sample. The 3D reconstruction of the 2D slices is shown in Figure 3.10. We have developed a sophisticated image analysis algorithm which uses Fourier Filtering to determine the particle centers and radii with subvoxel accuracy of all the droplets in the sample [42]. This data was previously unavailable from true 3D experiments. Since the droplets are deformable and they exert forces on one another upon contact, the area of droplet deformation gives an approximation of the force. The deformed areas appear brighter than the rest of the image due to an enhanced fluorescence at the contact [42], allowing for an independent measurement of the forces between droplets as can clearly be seen in Figure 3.11. A system of random close packed particles is fully described by the geometry of the system configuration and the distribution of forces and stresses in the particulate medium. This means that if P is the probability distribution of configurations and of interparticle forces, it consists of two independent components, P = Pf (forces)Pc (configurations),

(3.26)

which give the full description of the particulate system. The above statement has been presented in a theory context, but must be supported by experiment. In the next two sections we present experimental results that test the basic granular theory and some of the assumptions within it by separately measuring the distribution of forces [43] and the distribution of configurations [46].

3.3


67

104.0 m

Z

X

Y 76

.8

m

Figure 3.10: Confocal image of the densely packed emulsion system.

3.3.2.2 Force Distributions in a Jammed Emulsion The micromechanics of jammed systems has been extensively studied in terms of the probability distribution of forces, P (f ). However, the experiments were previously confined to 2D granular packs [28, 47] or the measurement of the forces exerted at the walls of a 3D granular assembly [48–52] thus reducing the dimensionality of the problem. On the other hand, numerical simulations [52–55] and statistical modeling [56] have provided the P (f ) for a variety of jammed systems, from structural glasses to foams and compressible particles, in 3D. Our novel experimental technique can be compared with all previous studies in search of a common behavior. Apart from being the first study of P (f ) in the bulk, it is also the first study of jamming in an emulsion system. The result of the confocal images analysis, shown in Figure 3.12, shows the probability distribution of interdroplet forces, P (f ), for the sample shown in Figure 3.9b. We use a linear force model (see Section 3.4.1) to obtain the interdroplet forces from the contact area data extracted from the image analysis described above. The distribution data shown are extracted from 1234 forces arising from 450 droplets. The forces are calculated from the bright, fluorescent patches that highlight the contact areas between the droplets as seen in Figure 3.11. The droplets are only slightly deformed away from spherical at the low applied pressures. This indicates that the system is jammed near the RCP. The data shows an exponential distribution at large forces, consistent with results of many previous experimental and simulation data on granular matter, foams, and glasses. The quan-

68


Figure 3.11: A novel fluorescence mechanism using confocal microscopy is providing new insight into the microstructure and the mechanics of jammed matter. The suitable model system is an emulsion comprised of oil droplets of approximately 3.4 µm dispersed in a refractive index matching solution. The system forms a random close packed structure by creaming or centrifugation giving rise to a force network. Based on the bright contact areas, the interdroplet forces can be extracted. The resulting micromechanics is being used to develop statistical theories of jammed materials.

1 0.9

P(x)=3.7x exp(-1.9x) Patch Data

P(f/)

0.1

0.01

0.001 0

1

2

f/

3

4

5

Figure 3.12: Probability distribution of the contact forces for the compressed emulsion system shown in Figure 3.9b. We also show a fit to the theory developed in [43].

titative agreement between P (f ) of a variety of systems suggests a unifying microstructural behavior governed by the jammed state. The salient feature of P (f ) in jammed systems is the exponential decay above the mean contact force. The behavior in the low force regime indicates a small peak, although the power law decay tending towards zero is not well pronounced.

3.3


69

The best fit to the data gives a functional form of the distribution (see Figure 3.12): ¯

P (f ) ∝ f 0.9 e−1.9f /f ,

(3.27)

with f¯ is the mean force. This form is consistent with the theoretical q-model [56] and with the model proposed in [43] which predicts a general distribution of the form P (f ) ∝ ¯ f n e−(n+1)f /f , where the power law coefficient n is determined by the packing geometry of the system and the coordination number.

Figure 3.13: Force chains in emulsions: Plot of the interdroplet forces inside the packing of droplets obtained in the experiments. We plot only the forces larger than the average for better visualization. Each rod joining the centers of two droplets in contact represents a force. The thickness and the color of the rod is proportional to the magnitude of the force.

Our experimental data allows us to examine the spatial distribution of the forces in the compressed emulsion, shown in Figure 3.13. In this sample volume, the forces appear to be uniformly distributed in space and do not show evidence of localization of forces within the structure. Moreover, we find that the average stress is independent of direction, indicating isotropy. For comparison, we also show computer simulation results for isotropic packings of Hertz–Mindlin spherical particles in Figure 3.14 (see Section 3.4) where force chains are not prominent either. On the other hand 2D packings clearly show the existence of force chains under isotropic pressure, indicating that their existence may be related to the dimensionality of the problem. Furthermore, force chains can be obtained in 3D by uniaxially compressing an isotropic packing, as shown in Figure 3.14c. It should be noted that in the case of Figure 3.14c an algorithm which looks for force chains is applied by starting from a sphere at the top of the system, and following the path of maximum contact force at every grain. Only the paths which percolate are plotted, i.e., the stress paths spanning the sample from the top to the bottom.

70


Interestingly, the salient feature of all the particulate packings shown in Figure 3.13 and 3.14, irrespective of their spatial characteristics, is an exponential distribution of forces. This indicates that force chains are not necessary to obtain such a distribution. The rationalization of this observation has been exploited in the theory developed in [43] which is based on the assumption of uncorrelated force transmission through the packing.

a)

b)

c)

Figure 3.14: Force chains in granular matter. a) Frictionless isotropic granular system at p = 100 KPa in 3D from simulations. We plot only the forces larger than the average. Force chains are tenuous and not well defined. b) Clearly visible force chains in a 2D frictional system from simulations. c) Frictional system under uniaxial compression from simulations. Percolating force chains are seen in this case.

The mean coordination number of the system, Z, is another theoretically important parameter. It has been shown that the isostatic limit is achieved (in 3D) for Z = 4 for frictional systems and Z = 6 for smooth particles (see Section 3.4.1.1). This has not been tested in the real world until the present, except in the famous experiment of Bernal for smooth particles who obtained a coordination number for spheres in contact, of 6.4 for metallic balls of 1/8”

3.3


71

Figure 3.15: An example of a volume W as the polyhedron constructed from the 3D images. The center grain (red) has six grains in contact (black), the centers of which are joined to form the polyhedron.

diameter. Even there, all particles in contact were counted, whereas the theory predicts the coordination number assuming only those particles which are exerting a force. Our experiments at low confining pressures (up to 200 Pa) show that the mean coordination number is Z ≈ 6 which can be interpreted as the isostatic limit for frictionless spheres. This completes the study of the jammed structures in terms of the force distributions. There are more subtle ways in which the static structure of the configuration can be investigated as a statistical ensemble as we describe in the next section. 3.3.2.3 Experimental Measurement of W and X Having investigated the probability distribution of forces within the system, we now consider the distribution of the configurations of the packing. In Section 3.2 we have justified the application of statistical mechanics to jammed conditions, provided there is a mechanism for changing the configurations by tapping. The probability distribution of configurations is governed by Eq. (3.18). Using an extension to the same image analysis method used to measure the distribution of forces, the 3D images of a densely packed particulate model system also allow for the characterization of the volume function W. This is performed by the partitioning of the images into first coordination shells of each particle, described in Section 3.2.2.1. The polyhedron obtained by such a partitioning is shown in Figure 3.15 and its volume is calculated from Eq. (3.6). The ability to measure this function and therefore its fluctuations in a given particle ensemble, enables the calculations of the macroscopic variables. We calculate the probability distribution of the volume per particle in the whole image and find an exponential behavior: P (W) ∝ e−W/λX ,

(3.28)

The exponential probability distribution of W enables the measurement of the compactivity X according to Eq. (3.18). The value obtained in this way is X = 94 µm3 /λ, shown

72


P(W)

1

Data P(W) = (1/94) exp(-W/94)

0.1

0.01

0

100

200

300

400

500

3

W [µm ] Figure 3.16: Probability distribution of W fitted with a single exponential. The decay constant is the compactivity, X = 94 µm3 /λ.

in Figure 3.16. The conversion of this measurement of the compactivity into a measurement of the analogue of temperature requires a new temperature scale for granular matter. In other words, λ (the analogue of the Boltzmann constant in thermal systems) provides the link between volume fluctuations (energy) and compactivity (temperature) and needs to be determined for jammed matter. We have shown that we can arrive at the thermodynamic system properties from the knowledge of the microstructure. Many images, i.e., configurations, can be treated in this way to test whether system size influences the macroscopic observables. If the particles are subjected to ultracentrifugation resulting in configurations of a higher density, the influence of pressure on the macroscopic variables can also be tested. Such a characterization of the governing macroscopic variables, arising from the information of the microstructure, allows one to predict the system’s behavior through an equation of state. This is the first experimental study of such statistical concepts in particulate matter and opens new possibilities for testing the above described thermodynamic formulation. In principle, one can apply low amplitude vibrations to the system and observe the droplet configuration before and after the perturbation, thus testing the ideas proposed in the Boltzmann derivation.

3.4 Jamming in a Periodic Box The following section describes the potential for using computer simulations in testing the thermodynamic foundations raised in the previous sections. Rather than employing rough rigid grains for which most of the theoretical concepts have so far been devised, computer simulations are obliged to introduce some deformability into the constituent particles to facilitate the measurement of the particle interactions with respect to their positions. As a result,

3.4

Jamming in a Periodic Box

73

the entropic considerations which have been explained only in terms of the volume in the case of rigid grains in Section 3.2 will now be generalized to situations in which there is a finite energy of deformation in the system. The energy of the system will parametrise the compaction curves at varying confining pressures shown in Figure 3.3. The entropy, Eq. (3.8), can then be redefined as a function of both energy and volume, S(E, V ) = λ ln Σjammed (E, V ).

(3.29)

The introduction of energy into the system implies a corresponding compactivity, −1 XE =

∂S , ∂E

(3.30)

where the subscript E denotes that the compactivity is now the Lagrange multiplier controlling the energy of the jammed configuration, not the volume. Notice that XE differs from the temperature of an equilibrium system T = ∂E/∂S because the energy in Eq. (3.30) is the energy of the jammed configurations and not the thermal equilibrium energy. The assumptions of ergodicity and equally probable microstates for a given energy and volume are still valid here, just as they were for rigid grains in the previous sections. The canonical distribution in Eq. (3.18) is generalized to Pν =

e(−Eν /λXE −Wν /λXV ) Θν . Z

(3.31)

The term Θν assures that we are considering only the jammed configuration. Its significance will be discussed in Section 3.4.2.2.

3.4.1 Simulating Jamming In Section 3.1.2 we described the need for a true jammed configuration before any statistical measurement can be applied. While this process is achieved via external perturbations in a laboratory experiment, the equivalent procedure guaranteeing reproducible results using simulations, requires a particular ‘equilibration’ procedure. At each pressure, the grains are pretreated in the following way in order to ensure that all memory effects have been lost in the system. We perform Molecular Dynamics (MD) simulations of an assembly of spherical grains in a periodically repeated cubic cell. The system is composed of soft elasto-frictional spherical grains interacting via Hertz–Mindlin contact forces, Coulomb friction and viscous dissipative terms. Two model systems are investigated: granular materials and compressed emulsions. 1. Granular matter model. Particles are modelled as viscoelastic spheres with different coefficients of friction. Interparticle forces are computed using the principles of contact mechanics [57]. Full details are given in Refs. [17, 52, 58]. The normal force Fn has the typical 3/2 power law dependence on the overlap between two spheres in contact (Hertz force), while the transverse force Ft depends linearly on the shear displacement between the spheres, as well as on the value of the normal displacement (Mindlin tangential elastic force). As the shear displacement increases, the elastic tangential force Ft

74


reaches its limiting value given by Amonton’s law for no adhesion, Ft ≤ µf Fn , which is a special case of Coulomb’s law. Viscous dissipative forces, proportional to the relative normal and tangential velocities of the particles, are also included to allow the system to equilibrate. A granular system with tangential elastic forces is path-dependent since the work done in deforming the system depends upon the path taken and not just the final state. On the other hand, a system of spheres interacting only via normal forces is said to be pathindependent, and the work does not depend on the way the strain is applied. It turns out that this is a good model for a compressed emulsion system since they do not exhibit frictional forces. 2. Compressed emulsions model. A system of frictionless viscoelastic spherical particles could be thought of as a model of compressed emulsions [59,60], see also [42] for details. Even though they can be modelled in this way, an important difference arises in the interdroplet forces, which are not given in terms of the bulk elasticity, as they are in the Hertz theory. Instead, forces are given by the principles of interfacial mechanics [44]. For small deformations with respect to the droplet surface area, the energy of the applied stress is presumed to be stored in the deformation of the surface. The simplest approximation considers an energy of deformation which is quadratic in the area of deformation [44], analogous to a harmonic oscillator potential which describes a spring satisfying Hooke’s law.

a)

b)

Figure 3.17: Preparation protocol for granular materials and droplets. We start with a system of noninteracting spheres (a) and then we apply a compression protocol to reach the jammed state (b)

There have been several more detailed numerical simulations [59] to improve on this model and allow for anharmonicity in the droplet response by also taking into consideration the number of contacts by which the droplet is confined. Typically these improved models lead to a force law for small deformations of the form Fn ∝ Ab , where A is the area of deformation and b is a coordination number dependent exponent ranging from 1 (linear model) to 3/2 (Hertz model). For simplicity and for a better comparison with the physics of granular ma-

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75

terials, in the following we will show results only considering the non-linear 3/2 dependence of the normal force. A more realistic non-linear dependence is considered elsewhere in [61]. Thus we adjust the MD model of granular materials to describe the system of compressed emulsions by only excluding the transversal forces (tangential elasticity and Coulomb friction). The continuous liquid phase is modelled in its simplest form, as a viscous drag force acting on every droplet, proportional to its velocity. Preparation protocol. Our aim is to introduce a numerical protocol designed to mimic the experimental procedure used to achieve the jammed state, as explained in Section 3.1.2.2. The simulations begin with a gas of noninteracting grains distributed at random positions in a periodically repeated cubic cell, depicted in Figure 3.17, showing snapshots of our typical simulations with 10,000 particles of size R = 100 µm. To avoid issues of path-dependency introduced by the shear forces, the transverse force between the grains is excluded from the calculation (Ft = 0). Because there are no transverse forces, the grains slip without resistance and the system reaches the high volume fractions found experimentally, thus avoiding the irreversible branch of the compaction curve. This procedure essentially mimics the path to jammed states for the compressed emulsion system. The protocol is then repeated for grains with friction. Initially, a fast compression of the grains brings the system to the irreversible branch of the compaction curve. It is then necessary to apply a compression protocol in order to reach a target pressure. This pressure is maintained with a “servo” mechanism by the continuous application of an oscillatory strain until the system reaches the jammed state [62]. The servo mechanism is analogous to the application of a small tapping amplitude to reach the reversible branch of the compaction curve, Figure 3.2. In general, we find that by preparing the system with frictional and elastic tangential forces, the system reaches states of lower volume fractions. At the end of the preparation protocol (depicted by the dashed lines in Figure 3.18a) we obtain a set of jammed systems at different stresses (for granular materials) or osmotic pressures (for droplets) [52].

7

10

frictionless packs, 3D

elastic limit

frictional packs, 3D

6

Z=6

Z(p)

Z(p)

isostatic limit

5

5

(b)

(a) 0 -5 10

a)

10

-3

10

-1

p [MPa]

10

1

10

4 -2 10

3

b)

10

-1

10

0

10

1

10

2

p [MPa]

Figure 3.18: Coordination number versus pressure. a) Frictionless packs in 3D (µ = 0). The system is isostatic Zc = 2D = 6 as p → 0. b) Frictional packs in 3D (finite µ).

76


3.4.1.1 Isostatic Jamming Consider a static packing of grains under an external force f ext . The internal stresses obey the Cauchy equations: ∂i σij + fjext = 0.

(3.32)

Since there are three equations (in 3-D) for six independent stress components (the stress tensor is symmetric), then the system is indeterminate, and the Cauchy equations must be augmented by additional constitutive equations. The conventional elastic approach is then to consider the deformability of the packing which is described by the strain field. Linear constitutive relations are introduced to relate the strain to the stress via the elastic constants of the material (Hooke’s law). For an isotropic elastic body only two elastic constants (for instance, the shear modulus and the Poisson ratio) are sufficient to fully describe the stress transmission in an elastic packing [63]. In the limit of infinitely rigid grains the strain field is ill-defined and the validity of elasticity theory is open to debate. In this case, it has been argued that it is possible to solve the stress distribution, based on Newton’s equations alone without resorting to the existence of the strain, only when the system is at a particular minimal coordination number [23, 38, 64]. The granular indeterminacy is then solved by resorting to the configurational information alone. Such approaches are intimately related to the thermodynamics of jamming [39]. The minimal coordination number can be understood in terms of simple constraint arguments for a system of N rigid spherical grains in D dimensions [64–66]. In the case of frictionless grains, ZN/2 normal forces have to be determined with DN equations of force balance. The critical coordination number for which the equations of force balance are soluble is found to be Zc = 2D. Similar arguments lead to a minimal coordination of Zc = D + 1 for infinitely rough spherical grains, i.e., grains with finite tangential forces, Ft , but with an infinitely large friction coefficient (µ → ∞). Below Zc the system cannot be jammed and it exists only in suspensions. Above Zc the system is underconstrained and elasticity theory may give the correct approach to describe such a packing of deformable grains. At the minimal coordination number the system is in a state of marginal rigidity [38], otherwise known as the isostatic limit [65, 66]. In order to test the existence of the isostatic limit we study the coordination number dependence on pressure for the two cases: frictionless grains and those with friction. The preparation protocols explained above are performed to achieve different target pressures and we obtain the average coordination number Z(p) of the jammed states as a function of the pressure, as shown in Figure 3.18. In the case of frictionless grains we find that the coordination number of the pack approaches the minimal value Zc ≈ 6 as p → 0. At low pressures compared to the shear modulus of the beads (p 26 GPa) the system behaves most like a pack of rigid balls, thus approaching the isostatic limit. The same preparation protocol gives Zc ≈ 4 in 2D confirming the Zc = 2D relationship (results not shown here). The preparation protocol for grains with infinite friction in 3D gives rise to different packings with lower coordination numbers. Our results suggest that the isostatic limit Zc = 4 is achieved for infinitely rough grains as p → 0 [61]. Grains with finite friction also seem to tend to the minimal coordination number Zc ≈ 4 as seen in Figure 3.18b, but at much smaller pressures than those achieved in our

3.4


77

calculations. Therefore, it is difficult to draw conclusions as to whether the isostatic limit requires even lower confining pressures or not. In fact, recent computational studies have suggested that the isostatic limit may not exist in packs with finite friction [67]. However, this study uses a different compression protocol more akin to a fast quench, which may leave the system trapped in the irreversible branch of the compaction curve due to the system’s inability to explore all the available configurational space. The approach to the marginal rigidity state Z → Zc can be seen as a jamming transition between a solid-like state with a finite shear modulus and a liquid-like state with no resistance to shear, observed in suspensions. In fact we find that the stress σ and the shear modulus G of the packing vanish according to a power law as the system approaches a critical density, φc , corresponding to the jamming transition [52]: σ ∼ (φ − φc )α ,

G ∼ (φ − φc )β .

(3.33)

The exponents α, β can easily be calculated in terms of the microscopic law of interparticle interactions. For instance, Hertz theory predicts the values of α = 3/2 and β = 1/2, in agreement with with our simulation results. The critical density φc depends on the interaction potential between the grains. A value φc = 0.63 ≈ φRCP is achieved for frictionless grains and corresponds to the volume fraction at RCP [40]. On the other hand φc < φRCP are achieved for grains with friction. These states correspond to RLPs. The jamming transition can be thought of as a particular second-order phase transition because the exponents are not universal; they depend on the details of the microscopic interactions between the grains [68]. The coordination number also approaches the critical minimal value as a power law. Empirically, we find (Figure 3.18) 1/3 p . (3.34) Z(p) = Zc + 10 MPa with Zc = 6 and Zc = 4 for the infinitely smooth and infinitely rough grains, respectively. After having characterized the jammed state, the computational study proceeds to develop the thermodynamics using the states depicted in Figure 3.18 as the starting point.

3.4.2 Testing the Thermodynamics If it were true that a thermodynamic framework could describe the behavior of jammed systems, it stands to reason that the compactivity XE of the granular pack can be measured from a dynamical experiment involving the exploration of the energy landscape. We examine the validity of this statement with computer simulations in the following discussion. Following the equilibration procedure, the exploration of the energy landscape equivalently needs a driving mechanism such that all appropriate configurations are sampled. This is achieved via a slow shearing procedure which has for an aim to probe each static configuration by allowing the system to evolve at a very slow shear rate. We first introduce a procedure to obtain a dynamical measurement of the compactivity via a diffusion-mobility protocol. We call this quantity the effective temperature Teff and show that it satisfies a form of the zero-th law of thermodynamics and thus has a thermodynamic meaning. The next crucial test for this assumption is to show that the effective temperature obtained dynamically can also be obtained via a flat average over the jammed configurations. Such a test

78


has been performed in [17], where it was indeed shown that Teff is very close to the compactivity of the packing XE . This result will be shown explicitly in Section 3.4.2.2. We conclude that the jammed configurations explored during shear are sampled in an equiprobable way as required by the ergodic principle. Moreover the dynamical measurement of compactivity renders the thermodynamic approach amenable to experimental investigations. In the next sections we calculate the effective temperature of the packing dynamically and show its relation to the compactivity, calculated by employing a configurational average. 3.4.2.1 Exploring the Jammed Configurations Dynamically: Effective Temperature Teff Consider a ‘tracer’ body of arbitrary shape immersed in a liquid in thermal equilibrium. As a consequence of the irregular bombardment by the particles of the surrounding liquid, the tracer performs a diffusive, fluctuating ‘Brownian’ motion. The motion is unbiased, and for large times the average square of the displacement goes as |x(t) − x(0)|2 = 2Dt, where D is the diffusivity. On the other hand, if we pull gently on the tracer with a constant force F , the liquid responds with a viscous, dissipative force. The averaged displacement after a large time is [x(t) − x(0)] = F χt, where χ is the mobility. Although both D and χ strongly depend on the shape and size of the tracer, they turn out to always be related by the Einstein relation T = D/χ,

(3.35)

(a form of the Fluctuation–Dissipation Theorem, FDT), where T is the temperature of the liquid. The Einstein relation is strictly valid for equilibrium thermal systems. However, it has been shown that fluctuation–dissipation relations are relevant to describe the thermodynamics of out-of-equilibrium systems. Insight into the validity of fluctuation–dissipation relations to describe systems far from equilibrium first came from glass theory [10, 11]. Recent analytic schemes for glasses have shown that a generalized FDT gives rise to a well-defined effective temperature which is different from the bath temperature [12]. It governs the heat flow and the slow components of fluctuations and responses of all observables. Explicit verifications of this approach have been made until present within the mean-field/mode-coupling models of the glass transition [13]. More recent studies have supported the existence of effective temperatures in schematic finite-dimensional models of glassy systems [15, 16]. The existence of a similar effective temperature for out-of-equilibrium sheared granular systems is by no means obvious. Moreover, its existence may provide a profound link between the physics of granular materials and glasses. In the absence of a first principle derivation, one needs to ascertain its validity for every particular case experimentally, or numerically at least. In order to test the existence of an effective temperature with a thermodynamic meaning for dense slow-moving granular matter we perform a numerical study of a diffusion-mobility computation in conditions that can be reproduced in the laboratory [17]. We consider a system of large and small spherical grains in a periodic cell. The simulations involve the application of a gentle shear on the particles in the y-z plane, at constant volume (see Figure 3.19). The shearing mechanism moves the periodic images at the top and the bottom of the cell with velocities γL/2, ˙ where γ˙ is the shear rate (Lees–Edwards boundary

3.4


79

Figure 3.19: Simulations of grains of 100 microns interacting via Hertz–Mindlin contact forces. A slow shear flow, indicated by the arrows, is applied to the jammed system. We follow the tracer particle trajectories to obtain the diffusivity. An external force F is then applied to the tracers in response to which we measure the particle mobility. These dynamical measurements yield an “effective temperature” obtained from an Einstein relation which is indeed very close to the compactivity obtained by a flat average over the ensemble of jammed configurations.

conditions [69]). A linear, uniform velocity profile along the z-direction is obtained using a modified set of equations to suit our computations (see Chapter 8 of [69]). Periodic boundary conditions are enforced in the x-direction and the y-direction of the flow. We focus our study on the region of slow shear rates, where the system is always close to jamming. We avoid shear bands by imposing a linear velocity profile and avoid segregation which may occur at much longer timescales than those employed in our computations. The time resolved displacement of the particles are measured to obtain their random fluctuations as well as the response function of the tracers to allow the determination of the effective temperatures via the Einstein relation (3.35). We measure the spontaneous fluctuations |x(t) − x(0)|2 and force-induced displacements [x(t) − x(0)] /F , where F is a small external force in the x-direction, for two types of tracers with different sizes. We calculate Teff by using parametric plots of |x(t) − x(0)|2 versus [x(t) − x(0)] /F with t as parameter. The parametric procedure of obtaining the effective temperatures allows the study of Teff for different timescales [12, 16]. Hence we determine the temperature of the different modes of relaxation (see also [70, 71]). Our data is consistent with a granular Einstein relation (Figure 3.20): |x(t) − x(0)|2 = 2 Teff

[x(t) − x(0)]

, F

(3.36)

valid for all tracers with the same fluctuation–dissipation temperature Teff for widely separated timescales. If there is an underlying thermodynamics, it will impose that the effective temperature be independent of the tracer’s size: a strong condition required for the thermodynamic hypothesis. We then verify whether all tracers have “equilibrated” at the same effective temperature

80


large grains no tangential forces

2

Diffusion: /2

0.4

small and large grains with tangential forces

0.3

small grains no tangential forces

0.2

Teff

0.1

0.0

0

5000

10000

15000

Response: /f

Figure 3.20: Parametric plot of diffusion vs response function for small and large grains and for spheres interacting with tangential forces (grains) and without tangential forces (emulsions). The fitting at long timescales shows the existence of a well-defined temperature which is the same for small and large grains: Teff = 2.8 × 10−5 for grains without transverse forces and Teff = 1.2 × 10−4 for grains with Mindlin transverse forces and Coulomb friction. These effective temperatures (measured here in reduced units [17]) are in fact very large, i.e., ∼ 1014 times kB T at room temperature, as expected. We calculate the response function for several small external fields and find the same temperature indicating that we are in the linear response regime. Plotted are results for a system without transverse forces using: F = 1.7 × 10−5 (small grains , large grains ❍) and F = 2.6 × 10−5 (small grains ■, large grains ●). For a system with tangential forces and Coulomb friction we show the case F = 6 × 10−5 (small grains , large grains ×).

by calculating the effective temperature for small and large tracers and finding that indeed they are the same. Figure 3.20 shows the parametric plot of the diffusion versus response function for systems with friction and frictionless grains. In both cases we consider the temperature of the small and large grains and obtain the same temperature. The fact that the effective temperature is the same for both types of particles and that it is only valid for the long-time relaxation suggest that Teff can be considered to be the temperature of the slow modes. 3.4.2.2 Exploring the Jammed Configurations via a Flat Average. Test of Ergodicity: Teff = XE The systems under investigation have exponentially large (in the number of particles) number of stable states jammed at zero bath temperature. In the previous section we explored such an energy landscape via slow shear. Next, we develop an independent method to study the configurational space. It allows us to investigate the statistical properties of the jammed states available at a given energy and volume. In turn we investigate whether it is possible

3.4


81

to relate the dynamical temperature obtained above via a diffusion-mobility protocol to the configurational compactivity based on jammed states. In order to calculate XE and compare with the obtained Teff we need to sample the jammed configurations at a given energy and volume in a equiprobable way. In order to do this we sample the jammed configurations with the following probability distribution: ν /Taux ] Pν ∼ exp[−E ν /T ∗ − Ejammed

(3.37)

Here the deformation energy E corresponds to the Hertzian energy of deformation of the grains or the Princen energy for droplets. The extra term added in Eq. (3.37) allows us to perform the flat sampling of the jammed states and plays the role of the term Θν in Eq. (3.31). The jammed energy is such that it vanishes at the jammed configurations: Ejammed ∝ |Fa |2 , (3.38) a

where Fa is the total force exerted on particle a by its neighbors. We introduce two “bath” temperatures (these temperatures are ∼ 1014 times the room temperature) which will allow us to explore the configuration space and calculate the entropy of the packing assuming a flat average over the jammed configurations. We perform equilibrium MD simulations with two auxiliary “bath” temperatures (T ∗ , Taux ), corresponding to the partition function (3.37). Annealing Taux to zero selects the jammed configurations (Ejammed = 0), while T ∗ fixes the energy E. In practice we perform equilibrium MD simulations with a modified potential energy: U=

Taux E + Ejammed , T∗

(3.39)

and calculate the force on each particle from F = −∇U . Since we need to calculate the force from a potential energy, only conservative systems can be studied with this method. Thus we focus our calculations on the system of frictionless particles. The auxiliary temperature Taux is controlled by a thermostat which adjusts the velocities of the particles to a kinetic energy determined by Taux . We start by equilibrating the system at high temperatures (Taux and T ∗ ∼ ∞) and anneal slowly the value Taux to zero and tune T ∗ so as to reach the value of E that corresponds to the average deformation energy obtained during shear. The partition function is ν Z= exp[−E ν /T ∗ − Ejammed /Taux ], (3.40) ν

from which we obtain the compactivity as T∗ =

∂E ∂S

Taux → 0

−→

XE ,

(3.41)

Thus at the end of the annealing process (Taux → 0), T ∗ (E) = XE (E), since in this limit we are sampling the configurations with vanishing fraction of moving particles at a given E. At the end of the protocol the compactivity at a given deformation energy can be obtained as illustrated in Figure 3.21. The remarkable result is that the compactivity and the


Elastic Energy: E

82

A

Path 1 Path 2 Path 3 Energy distribution during shear −2

10

*

T >Teff

B

*

−3

10

T =Teff

*

T τ d d A (τ , τ ) = v(τ )v(τ ) = v (τ ) [dv(τ )/dτ ] dτ dτ (4.15) = −(1 − p) v (τ ) [v(τ ) − u(τ )] = −(1 − p) A (τ , τ ) . This derivation reflects averaging over all possible collisions between the tagged particle of velocity v and another particle of velocity u, with the collision rule, v → v − (1 − p)(v − u). Therefore, the autocorrelation, A(τ , τ ) = A(τ , τ ) exp [−(1 − p)(τ − τ )], decays exponentially with the collision number, for τ > τ [45]. In terms of the original time variable: 1/p−2

A (t , t) = A0 (1 + t /t0 )

(1 + t/t0 )−1/p ,

(4.16)

with A0 = T0 . In particular, A(t) ≡ A(0, t) ∝ t−1/p , so memory of the initial conditions decays algebraically with time. The autocorrelation function decays faster than the temperature, A(t) ≤ T (t) (the two functions coincide in the completely inelastic case, p = 1/2).

94

4 The Inelastic Maxwell Model

Generally, the autocorrelation is a function of the waiting time t and the observation time t, and not simply of their difference, t − t . This history dependence (“aging”) merely reflects the fact that the collision rate keeps changing with time.

The spread in the position of a tagged particle, ∆2 (t) ≡ |x(t) − x(0)|2 , is obtained t t from the autocorrelation function using ∆2 = 2 0 dt 0 dt A (t , t ). Substituting the autocorrelation function (4.16), we find that asymptotically, the spread grows logarithmically with time √ √ ∆ ∝ τ ∝ ln t. (4.17) This behavior reflects the t−1 decay of the overall velocity scale and is consistent with inelastic hard-sphere results [26, 60–62].

4.2.2 The Forced Case In experimental situations, granular ensembles are constantly supplied with energy, typically through the boundaries, to counter the dissipation occurring during collisions2 . Theoretically, it is convenient to consider uniformly heated systems, as discussed by Williams and MacKintosh [14] in the case of one-dimensional inelastic hard rods and then extended to higher dimensions (see [28–30] and references therein). We therefore study a system where, in addition dv to changes due to collisions, velocities also change due to an external forcing: dtj |heat = ξj . We assume standard uncorrelated white noise: ξj = 0 and ξi (t)ξj (t ) = 2Dδij δ (t − t ). Such white noise forcing amounts to diffusion with a ‘diffusion’ coefficient D in velocity space. Therefore, the Boltzmann equation (4.2) is augmented by a diffusion term ∂2 ∂ − D 2 P (v, t) = K du1 P (u1 , t) du2 P (u2 , t) (4.18) ∂t ∂v × {δ [v − u1 + (1 − p)g] − δ(v − u2 )} The temperature changes according to dT /dt + λT 3/2 = 2D, so the steady state temperature is T∞ = (2D/λ)2/3 . The relaxation toward the steady state is exponential, |T − T∞ | ∼ exp(−const. × t). At the steady state, the Fourier transform F∞ (k) ≡ F (k, ∞) satisfies (1 + Dk2 )F∞ (k) = F∞ (k − pk) F∞ (pk). (4.19) √ with D = D/ T∞ . Conservation of the total number of particles and the total momentum (0) = 0, respectively. The solution is found recursively to give impose F∞ (0) = 1 and F∞ the following infinite product [40] ∞ l

−(ml ) 2m 2(l−m) 2 F∞ (k) = 1 + p (1 − p) . Dk

(4.20)

l=0 m=0 2

When the mean-free path is comparable with the system size, the boundary effectively plays the role of a thermal heat bath.

4.2

Uniform Gases: One Dimension

95

Thus in one dimension the Fourier transform is determined analytically in the steady state. To extract the high-energy tail from (4.20) we note that F∞ (k) has an infinite series of poles √ −1/2 . The simple poles at ±i/ D closest to the origin located at ±i p2m (1 − p)2(l−m) D imply an exponential decay of the velocity distribution [49, 53], √ A(p) −|v|/v∗ P∞ (v) e , with v∗ = D, (4.21) v∗ when |v| → ∞. A (lengthy) re-summation similar to that used in [40] yields the residue to this pole, and in turn, the prefactor ∞ 1 1 − a2n (p) 1 exp , (4.22) A(p) = 2 n a2n (p) n=1 ∞ 1 p2n + (1 − p)2n 1 exp . = 2 n 1 − p2n − (1 − p)2n n=1

In the interesting quasi-elastic limit (p → 0) we expand the denominator. Keeping only the ∞ π2 1 dominant terms simplifies the sum to n=1 2pn 2 = 12p and to leading order, the prefactor is A(p) ∝ exp[π 2 /(12p)]. A detailed analysis of the next poles shows that, for sufficiently small velocities, v vc , the velocity distribution is essentially Maxwellian, exp(−v 2 ) [57]. For convenience, the velocities are normalized here by the rms velocity. Matching this with the exponential tail √ exp(−|v|/ p) yields the crossover velocity vc ∼ p−1/2 (this estimate holds up to a logarithmic correction [57]). Thus, although in general the asymptotic decay of the velocity distribution is exponential, in the quasi-elastic limit, the velocity distribution is Maxwellian over a growing velocity range. A similar interplay between a generic exp(−|v|3/2 ) tail and an exp(−|v|3 ) decay in the quasi-elastic limit is found for inelastic hards spheres as well [30]. The leading high-energy behavior can also be derived by using a useful heuristic argument [26,28,49]. For sufficiently large velocities, the gain term in the collision integral in Eq. (4.18) is negligible. The resulting equation for the steady state distribution d2 P∞ (v) = −P∞ (v) (4.23) dv 2 yields the exponential high-energy tail (4.21). This argument applies to arbitrary collision rates. For example, if K ∝ v δ , the right-hand side of (4.23) becomes −v δ P∞ implying that P∞ (v) ∝ exp(−|v|γ ) with γ = 1 + δ/2. For hard spheres (δ = 1) one finds γ = 3/2 [28], and curiously, the Gaussian tail arises only for the so-called very hard spheres (δ = 2) [35]. Finally, we notice that steady state properties in the heated case are intimately related to the relaxation properties in the cooling case. This can be seen via the cumulant expansion ∞ 2 n (4.24) ψn (−Dk ) . F∞ (k) = exp D

n=1

Replacing the term (1 + Dk2 ) with exp[− n=1 n−1 (−Dk2 )n ], and substituting the cumulant expansion yields ψn = [na2n (p)]−1 . The cumulants κn are defined via into Eq. (4.19) n ln F∞ (k) = n=1 κn (ik) /n!. Therefore, the steady-state cumulants are directly related to n the relaxation coefficients (4.12), κ2n = (2n)! n D /a2n (the odd cumulants vanish).

96


4.3 Uniform Gases: Arbitrary Dimension 4.3.1 The Freely Cooling Case In general dimension, the colliding particles exchange momentum only along the impact direction. The post-collision velocities v1,2 are given by a linear combination of the pre-collision velocities u1,2 , v1,2 = u1,2 ∓ (1 − p) (g · n) n.

(4.25)

Here g = u1 − u2 is the relative velocity and n the unit vector connecting the particles’ centers. The normal component of the relative velocity is reduced by the restitution coefficient 2 r = 1 − 2p, and the energy dissipation is given by ∆E = −p(1 − p) (g · n) . In random collision processes, both the impact direction and the identity of the colliding particles are chosen randomly. In such a process, the Boltzmann equation ∂P (v , t) = K dn du1 P (u1 , t) du2 P (u2 , t) (4.26) ∂t × δ [v − u1 + (1 − p) (g · n) n] − δ (v − u1 ) exactly describes the evolution of the velocity distribution function P (v, t). √ The overall collision rate is chosen to represent the typical relative velocity, K = T , with the granulartemperature now being the average velocity fluctuation per degree of freedom, T = d1 dv v 2 P (v , t) with v ≡ |v |. The evolution equation involves integration over all impact directions, and this angular integration should be normalized, dn = 1. We tacitly ignored the restriction g ·n > 0 on the angular integration range in Eq. (4.26) since the integrand obeys the reflection symmetry n → −n. Several temporal characteristics such as the temperature and the autocorrelation behave as in the one-dimensional case. For example, the temperature satisfies Eq. (4.3) with prefactor λ = 2p(1 − p)/d reduced3 by a factor d. The temperature √ therefore decays according to Haff’s law (4.4), with the timescale t0 = d/ p(1 − p) T0 set by the initial temperature. Similarly, the decay rate of the autocorrelation function is merely reduced by the spatial did mension, dτ A (τ , τ ) = − 1−p d A (τ , τ ). Consequently, the non-universal decay (4.16) and the logarithmic spread (4.17) hold in general. The Fourier transform, F k, t = dv eik·v P (v , t), satisfies ∂ F k, τ + F k, τ = dn F (k − p, τ ) F ( p, τ ) (4.27) ∂τ with p = (1 − p) k · n n reflecting the momentum transfer occurring during collisions.

This equation was obtained by multiplying Eq. (4.26) by eik·v and integrating over the veloc3

The reduction in rate by a factor d is intuitive because, of the d independent directions, only the impact direction R is relevant in collisions. Mathematically, the prefactor λ = 2p(1 − p) d n n21 is computed by using the identity R n21 + . . . + n2d = 1 that yields d n n21 = 1/d.

4.3

Uniform Gases: Arbitrary Dimension

97

ities. The power of the Fourier transform is even more remarkable in higher dimensions4 as it reduces the (3d − 1)−fold integral in Eq. (4.26) to the (d − 1)−fold integral in Eq. (4.27). Hereinafter, we consider only isotropic distributions. The Fourier transform de velocity pends only on k ≡ k , so we write F k, τ = F (y, τ ) with y = k2 . To perform the angular integration, we employ spherical coordinates with the polar axis parallel to k, so that kˆ · n = cos θ. The θ-dependent factor of the angular integration measure dn is proportional to (sin θ)d−2 dθ. Denoting angular integration with brackets, f ≡ dnf , and using µ = cos2 θ gives 1 d−3 1 µ− 2 (1 − µ) 2 1 d−1 f (µ),

f = (4.28) dµ B 2, 2 0 where B(a, b) is the beta function. This integration is properly normalized, 1 = 1. The governing equation (4.27) for the Fourier transform can now be rewritten in the compact from ∂ F (y, τ ) + F (y, τ ) = F (ξy, τ ) F (ηy, τ ) , ∂τ

(4.29)

with the shorthand notations ξ = 1−(1−p2 )µ and η = (1−p)2 µ. Unlike the one-dimensional case, explicit solutions of this non-linear and non-local rate equation are cumbersome and practically useless. Nevertheless, most of the physically relevant features of the velocity distributions including the large velocity statistics and the time-dependent behavior of the moments can be obtained analytically. We seek a scaling solution: P (v , t) = T −d/2 P(w) with w = vT −1/2 , or equivalently F (y, τ ) = Φ(x) with x = yT . The scaling function Φ(x) satisfies −λx Φ (x) + Φ(x) = Φ(ξx) Φ(ηx)

(4.30)

and the boundary condition Φ(x) = 1 − 12 x + · · · as x → 0. In the elastic case, the velocity distribution is purely Maxwellian, Φ(x) = e−x/2 . Indeed, λ = 0 and ξ + η = 1 in this case. A stochastic process of elastic collisions effectively randomizes the velocities and leads to a thermal distribution [31]. In practice, this collision algorithm is used in molecular dynamics simulations to thermalize velocities. From the one-dimensional case, we anticipate that the velocity distribution has an algebraic large velocity tail. Generally, the large velocity behavior can be determined from the small wave number behavior of the Fourier transform. For example, the small-x expansion of the one-dimensional solution (4.10) contains both regular and singular terms: Φ(x) = 1 − 12 x + 13 x3/2 + · · · , and the dominant singular x3/2 term reflects the w−4 tail of P(w). In general, if P(w) has an algebraic tail, P(w) ∼ w−σ as w → ∞, (4.31) √ ∞ then Φ(x) ∝ 0 dw wd−1 P(w) eiw x contains, apart from regular terms, the following dominant singular term: Φsing (x) ∼ x(σ−d)/2 as x → 0. The exponent σ can be now obtained 4

For elastic Maxwell molecules, the Fourier transform was first used in unpublished theses by Krupp [34], and then rediscovered and successfully utilized by Bobylev (see [35, 36] for a review).

98


10 d=2 d=3 Eq. (33)

8 6 σ/d

4 2 0

0

0.1

0.2

0.3

0.4

0.5

p Figure 4.1: The exponent σ versus the dissipation parameter p for d = 2, and 3. For comparison with the leading large-dimensional behavior (4.33), σ is rescaled by d.

by inserting Φ(x) = Φreg (x) + Φsing (x) into Eq. (4.30) and balancing the dominant singular terms. We find that σ is a root of the integral equation5 σ − d (σ−d)/2 = ξ (4.32) 1−λ + η (σ−d)/2 . 2 This equation can be recast as the eigenvalue problem λν = νλ1 using λν = 1 − ξ ν − η ν and ν = (σ − d)/2. It can also be expressed using the hypergeometric function 2 F1 (a, b; c; z) [63] and Euler’s gamma function: σ−d+1 d Γ σ−d d−σ 1 d 2 σ−d Γ σ2 1 2 . = 2 F1 , ; ; 1 − p + (1 − p) 1 − p(1 − p) d 2 2 2 Γ 2 Γ 2 Clearly, the exponent σ ≡ σ(d, p) depends in a non-trivial fashion on the spatial dimension d as well as the dissipation parameter p. There are two limiting cases where the velocity distribution approaches a Maxwellian, thereby implying a divergence of the exponent σ. In the quasi-elastic limit (p → 0) we have σ → d/p. Even a minute degree of dissipation strongly changes the character of the system, and hence, energy dissipation is a singular perturbation [24, 29]. As the dimension increases, the relative weight of the impact

diminishes, and so does the role of dissipation. For direction large dimensions, the integral η (σ−d)/2 in Eq. (4.32) vanishes exponentially and asymptotic analysis of the remaining integral shows that the exponent grows linearly with the dimension: 1/2 1 + 32 p − p3 − p1/2 1 + 54 p σ→d (4.33) p(1 − p2 ) 5

This result was derived independently in [44, 47].

4.3


99

when d → ∞ [44]. Equation (4.33) provides a decent approximation even at moderate dimensions (Figure 1). Overall, the exponent σ(d, p) increases monotonically with increasing d, and additionally, it increases monotonically with decreasing p. Both features are intuitive as they mirror the monotonic dependence of the energy dissipation rate λ = 2p(1 − p)/d on d and p. Remarkably, the exponent is very large. The completely inelastic case provides a lower bound for the exponent, σ(d, p) ≥ σ(d, 1/2) with σ(d, 1/2) = 6.28753, 8.32937, for d = 2, 3, respectively. For typical granular particles, σ(d = 3, p = 0.1) ∼ = 30, and such algebraic decays are impossible to measure in practice. The algebraic tail of the velocity distribution implies that moments of the scaling function Φ(x) with sufficiently large indices diverge. In the scaling regime, moments of the velocity distribution can be calculated by expanding the Fourier transform in powers of x, Φ(x) ∼ =

[ν]

φn (−x)n .

(4.34)

n=0

The order of terms in this expansion must be smaller than the order of the singular term, xν [47]. The coefficients φn , needed for calculating transport coefficients [54], yield the leading asymptotic behavior of the velocity moments, Mk (t) = dv v k P (v , t), via the relation (2n)! T n φn µn M2n . Inserting the moment expansion into (4.30) yields a closed hierarchy of equations (λn − nλ1 )φn =

n−1

λm,n−m φm φn−m ,

(4.35)

m=1

with λn = 1 − ξ n − η n and λm,l = ξ m η l . Calculation of these coefficients requires the following integrals [45] Γ d2 Γ n + 12 2(n − 1) + 1 1 3 n =

µ = 1 ··· . dd+2 2(n − 1) + d Γ 2 Γ n + d2 Of course, φ0 = 1 and φ1 = 1/2; further coefficients can be determined recursively from (4.35), e.g., 2

1−p 1 1 − 3 d+2 . φ2 = 8 1 − 3 1+p2 d+2

(4.36)

This coefficient is finite only when λ2 > 2λ1 or d < d2 = 1 + 3p2 . Generally φn is finite only when the left hand side of Eq. (4.35) is positive, λn > nλ1 , or equivalently, when the dimension is sufficiently small, d < dn . The crossover dimensions dn are determined from λn = nλ1 . For d > dn , moments with index smaller than 2n are characterized by the temperature, while higher moments exhibit multi-scaling asymptotic behavior. The time evolution of the moments can be studied using the expansion F (y, τ ) =

∞ n=0

fn (τ ) (−y)n .

(4.37)

100


The actual moments are related to the fn coefficients via (2n)!fn = µn M2n . Substituting the expansion (4.37) into (4.29) yields the evolution equations n−1 d fn + λn fn = λm,n−m fm fn−m . dτ m=1

(4.38)

We have f1 ∝ e−λ1 τ , this is just Haff’s law. From df2 /dτ + λ2 f2 = λ1,1 f12 we see that f2 is a linear combination of two exponentials, e−λ2 τ and e−2λ1 τ . The two decay coefficients are equal λ2 = 2λ1 at the crossover dimension d = d2 . As expected, when d < d2 the fourth moment is dictated by the second moment, f2 ∝ f12 . Otherwise, f2 ∝ exp(−λ2 τ ). In general, exp(−nλ1 τ ) d < dn ; (4.39) M2n ∝ exp(−λn τ ) d > dn . Fixing the dimension and the dissipation parameter, moments of sufficiently high order exhibit multi-scaling asymptotic behavior. In practice, the exponent σ is large, and multi-scaling occurs only for very high order moments.

4.3.2 The Forced Case We consider white noise forcing as in the one-dimensional case. The steady state Fourier transform, F∞ (k) ≡ F∞ (k2 ), satisfies

(4.40) (1 + Dk2 )F∞ (k2 ) = F∞ (ξk2 ) F∞ ηk2 . This equation is solved recursively by employing the cumulant expansion (4.24). Writing 1 + Dk2 = exp − n≥1 (−Dk2 )n /n , we recast Eq. (4.40) into 1=

∞ n ψn − n−1 −Dk2 exp −

,

(4.41)

n=1

with the auxiliary variables ψn = ψn (1 − ξ n − η n ). The coefficients ψn are obtained by evaluating recursively the angular integrals of the auxiliary variables, ψn , and then using the identities ψn = ψn /λn . The few first coefficients can be determined explicitly, e.g., ψ1 = 1/λ and ψ2 =

4(d + 2)(1 −

p2 )

3 d2 . − 12(1 − p)2 (1 + p2 )

(4.42)

The nonvanishing second cumulant shows the steady state distribution is not Maxwellian. √ Moreover, the poles of the Fourier transform located at k = ±i/ D indicate that the large velocity tail of the distribution is exponential as in the one-dimensional case (4.21). Exponential decay is also suggested by the heuristic argument detailed above.

4.3


101

1.5 cooling forced

1 U

0.5

0

0

0.1

0.2

0.3

0.4

0.5

p Figure 4.2: The correlation measure U for spatial dimension d = 3. Shown are the freely cooling case and the forced case.

4.3.3 Velocity Correlations Maxwellian velocity distributions were originally obtained for random elastic collision processes (see Ref. [32], p. 36). Maxwell’s seminal derivation involves two basic assumptions: (1) The velocity distribution is isotropic, and (2) Correlations between the velocity components are absent. In inelastic gases, the velocity distributions are non-Maxwellian – therefore, there must be correlations between the velocity components6 . The quantity U=

vx2 vy2 − vx2 vy2

vx2 vy2

(4.43)

provides a natural correlation measure. A non-vanishing U indicates that velocity correlations do exist, and the larger U the larger the correlation. To compute U , we apply the identities: 2

∂ 2 F vx = ∂kx2 k=0

and

2 2

∂ 2 ∂ 2 vx vy = F . ∂kx2 ∂ky2 k=0

(4.44)

The correlation measure is directly related to the fourth cumulant of the isotropic velocity distribution. In the freely cooling case, U = 8φ2 − 1; in the forced case, U = ψ2 /ψ12 . 6

Inelastic collisions discriminate the impact direction, thereby generating correlations among the velocity components.

102


Substituting the corresponding coefficients yields Ucooling

=

Uforced

=

6p2 , d − (1 + 3p2 ) 6p2 (1 − p) . (d + 2)(1 + p) − 3(1 − p)(1 + p2 )

(4.45) (4.46)

Generally, correlations increase monotonically with p. Correlations are much weaker in the presence of an energy source because the nature of the driving is random (Figure 2). The perfectly inelastic case provides an upper bound. For example, when d = 3 and p = 1/2 we have Ucooling = 6/5 and Uforced = 2/15. Correlations decay as U ∝ p2 and U ∝ d−1 in the respective limiting cases of p → 0 and d → ∞, where Maxwellian distributions are recovered.

4.4 Impurities The dynamics of an impurity immersed in a uniform granular fluid often defies intuition. The impurity does not generally behave as a tracer particle, and instead it may move either faster or slower than the fluid. Despite extensive studies, many theoretical and experimental questions regarding the dynamics of impurities remain open [64–72]. Impurities represent the simplest form of polydispersity, an important characteristic of granular media [73, 74]. Theoretically, the impurity problem is a natural first step in the study of mixtures as it involves fewer dissipation parameters. The fluid background is not affected by the presence of the impurity, and the impurity can be seen as “enslaved” to the fluid background. We study dynamics of a single impurity particle in a uniform background of identical inelastic particles. We set the fluid particle mass to unity and the impurity mass to m. Collisions between two fluid particles are characterized by the dissipation parameter p as in Eq. (4.25), while collisions between the impurity and any fluid particle are characterized by the dissipation parameter q. When an impurity particle of velocity u1 collides with a fluid particle of velocity u2 its post-collision velocity v1 is given by v1 = u1 − (1 − q) (g · n) n.

(4.47)

Two restitution coefficients, rp ≡ r = 1 − 2p and rq = m − (m + 1)q, characterize fluid–fluid and impurity–fluid collisions, respectively. The restitution coefficients obey 0 ≤ r ≤ 1, and m the dissipation parameters accordingly satisfy 0 ≤ p ≤ 1/2 and m−1 m+1 ≤ q ≤ m+1 . The energy 2 dissipated in an impurity–fluid collision is ∆E = −m(1 − q)[2 − (m + 1)(1 − q)] (g · n) . Let Q (v , t) be the normalized velocity distribution of the impurity. In a random collision process, the impurity velocity distribution evolves according to the linear Lorentz–Boltzmann equation ∂Q (v , t) = Kq dn du1 Q (u1 , t) du2 P (u2 , t) ∂t × {δ [v − u1 + (1 − q)(g · n)n] − δ (v − u1 )} , (4.48)

4.4

Impurities

103

while the fluid distribution obeys (4.26). Two rates, Kp ≡ K and Kq , characterize fluid–fluid and fluid–impurity collisions, respectively. These rates can be eliminated from the respective t equations (4.26) and (4.48) by introducing the collision counters, τp ≡ τ = 0 dt Kp (t ), t and τq = 0 dt Kq (t ). The equation is further simplified by using the Lorentz–Boltzmann Fourier transform, G k, t = dv eik·v Q(v , t) ∂ G k + G k = dn G(k − q ) F (q) , (4.49) ∂τq with q = (1 − q) k · n n. This equation supplements the fluid equation (4.27). We consider two versions of the Maxwell model: An idealized case with equal collision rates, Kp = Kq (model A); and a more physical case with collision rates proportional to appropriate average relative velocities (model B). In the forced case, one can obtain the impurity velocity distribution using the cumulant expansion, and generally, velocity statistics of the impurity are similar to the background. Below, we discuss the, more interesting, freely cooling case.

4.4.1 Model A When the two collision rates are equal, all pairs of particles are equally likely to collide with each other. For simplicity we set the overall rate to unity, Kp = Kq = 1; then both collision counters equal time, τp = τq = t. In the freely cooling case, the fluid temperature decays exponentially, T (t) = T0 exp [−2p(1 − p)t/d] .

(4.50)

The impurity temperature, Θ(t), defined by Θ(t) = temperature via a linear rate equation

1 d

dv v 2 Q (v , t), is coupled to the fluid

d 1 − q2 (1 − q)2 Θ=− Θ+ T. dt d d

(4.51)

The solution to this equation is a linear combination of two exponentials Θ(t) = (Θ0 − c T0 ) e−(1−q

2

)t/d

+ c T0 e−2p(1−p)t/d .

(4.52)

The constant c = (1 − q)2 /[1 − q 2 − 2p(1 − p)] is simply the ratio between the impurity temperature and the fluid temperature in the long time limit. This generalizes the elastic fluid result c = (1 − q)/(1 + q) [75, 76]. Generally, the impurity and the fluid have different energies, and this lack of equi-partition is typical of granular particles [6, 77] There are two different regimes of behavior. When 1 − q 2 > 2p(1 − p), the impurity temperature is proportional to the fluid temperature asymptotically, Θ(t) T (t) → c as t → ∞. In the complementary region 2p(1 − p) > 1 − q 2 the ratio of the fluid temperature to the impurity temperature vanishes. Since the system is governed by three parameters (m, rp , rq ), it is convenient to consider the restitution coefficients as fixed and to vary the impurity mass.

104


From the definition of the restitution coefficients the borderline case, q 2 = 1 − 2p(1 − p), defines a critical impurity mass ! rq + (1 + rp2 )/2 ! m∗ = . (4.53) 1 − (1 + rp2 )/2 This critical mass is always larger than unity. Asymptotically, the impurity to fluid temperature ratio exhibits two different behaviors c m < m∗ , Θ(t) → (4.54) T (t) ∞ m ≥ m∗ . We term these two regimes, the light impurity phase and the heavy impurity phase, respectively. In the light impurity phase, the initial impurity temperature becomes irrelevant asymptotically, and the impurity is governed by the fluid background. In the heavy impurity phase, the impurity is infinitely more energetic compared with the fluid and practically, it sees a static fluid. A qualitatively similar phase transition is observed for impurities in inelastic hard spheres [71, 72]. Interestingly, the dependence on the dimension is secondary as both the critical mass, m∗ , and the temperature ratio c are independent of d. Below we study velocity statistics including the velocity distribution and its moments, primarily in one dimension, where explicit solutions are possible. 4.4.1.1 The Light Impurity Phase In this phase, m < m∗ , the impurity is governed by the fluid background. Wetherefore seek scaling solutions of the same form as that of the fluid: Q(v, t) = T −1/2 Q vT −1/2 and G(k, t) = g |k|T 1/2 . From the Lorentz–Boltzmann equation (4.49), the latter scaling function satisfies the linear equation −p(1 − p)zg (z) + g(z) = g(qz) [1 + (1 − q)z] e−(1−q)z .

(4.55)

Since the fluid scaling function is a combination of z n e−z with n = 0 and n = 1, we try the series ansatz g(z) =

∞

gn z n e−z

(4.56)

n=0

for the impurity. The first few coefficients, g0 = g1 = 1 and g2 = (1 − c)/2, follow from the small-z behavior g(z) ∼ = 1 − 12 cz 2 . The rest are obtained recursively gn =

q n−1 (1 − q) − p(1 − p) gn−1 . 1 − q n − np(1 − p)

(4.57)

The Fourier transform can be inverted to obtain the impurity velocity distribution function κ n −z explicitly. The inverse Fourier transform of e−κz is π1 κ2 +w 2 ; the inverse transforms of z e

4.4

Impurities

105

can be obtained using successive differentiation with respect to κ. Therefore, the solution is a series of powers of Lorentzians Q(w) =

∞ 2 1 Qn π n=2 (1 + w2 )n

(4.58)

The coefficients Qn are linear combinations of the coefficients gk ’s with k ≤ n + 1, e.g., Q2 = 1 − 3g2 + 3g3 and Q3 = 4g2 − 24g3 + 60g4 . There are special values of q for which the infinite sum terminates at a finite order. Of course, when q = p, the impurity is identical to the fluid and Qn = 0 for all n > 2. When q 2 (1 − q) = p(1 − p), one has Qn = 0 for all n > 3, so the velocity distribution includes only the first two terms. Regardless of q, the first squared Lorentzian term dominates the tail of the velocity distribution Q(w) ∼ P(w) ∼ w−4 , as w → ∞. One can show that this behavior extends to higher dimensions, Q(w) ∼ P(w) ∼ w−σ . Therefore, the impurity has the same algebraic extremal velocity statistics as the fluid. While the scaling functions underlying the impurity and the fluid are similar, more subtle features may differ. Moments of the impurity velocity distribution, Ln (t) = dv v n Q(v, t), obey the recursive equations n−2 n d Ln + bn Ln = q m (1 − q)n−m Lm Mn−m , (4.59) dt m m=2 with bn (q) = 1 − q n . Asymptotically, the fluid moments decay exponentially according to Mn (t) ∝ e−an (p) t . Using this asymptotic behavior, we analyze the (even) impurity moments. The second moment, i.e., the impurity temperature, was shown to behave similar to the fluid temperature when a2 (p) < b2 (q). The fourth moment behaves similarly to the fourth moment of the fluid, L4 ∝ M4 , when a4 (p) < b4 (q). However, in the complementary case, the fourth moment behaves differently, Ln (t) ∝ e−b4 (q)t . In general, when an (p) < bn (q), the nth impurity moment is proportional to the nth fluid moment, Ln ∝ Mn . Otherwise, when an (p) > bn (q), the nth impurity moment is no longer governed by the fluid and Ln (t) ∝ e−(1−q

n

)t

.

(4.60)

This series of transitions, affecting moments of decreasing order, occurs at increasing impurity masses, m 1 > m2 > · · · > m ∞ .

(4.61)

When m ≥ mn , the ratio M2k /L2k diverges asymptotically for all k ≥ n. The transition masses 1 rq + 12 (1 − rp )2n + (1 + rp )2n 2n (4.62) mn = 1 1 − 12 [(1 − rp )2n + (1 + rp )2n ] 2n are found from q 2n = p2n + (1 − p)2n and the definitions of the restitution coefficients. All of the transition masses are larger than unity, so the impurity must be heavier than the fluid for any transition to occur. The largest transition mass is m1 ≡ m∗ , and the smallest transition mass is m∞ = limn→∞ mn = (1 + rp + 2rq )/(1 − rp ). Impurities lighter than the latter mass, m < m∞ , mimic the fluid completely.

106


4.4.1.2 The Heavy Impurity Phase In this phase, m > m∗ , the velocities of the fluid particles are asymptotically negligible compared with the velocity of the impurity. Fluid particles become stationary as viewed by the impurity, and effectively, u2 ≡ 0 in the collision rule (4.47): v = u − (1 − q) (u · n) n.

(4.63)

Mathematically, this process is reminiscent of a Lorentz gas [78]. Physically, the two processes are different. In the granular impurity system, a heavy particle scatters off a static background of lighter particles, while in the Lorentz gas the scatterers are infinitely massive. We first consider the one-dimensional case. Setting u2 ≡ 0 in the Lorentz–Boltzmann equation (4.48), integration over the fluid velocity u2 is trivial, du2 P (u2 , t) = 1, and integration over the impurity velocity u1 gives ∂ 1 v Q(v, t) + Q(v, t) = Q ,t . (4.64) ∂t q q This equation can be solved directly by considering the stochastic process which the impurity particle experiences. In a sequence of collisions, the impurity velocity changes according to v0 → qv0 → q 2 v0 → · · · with v0 the initial velocity. After n collisions the impurity velocity decreases exponentially, vn = q n v0 . Furthermore, the collision process is random, and therefore, the probability that the impurity undergoes exactly n collisions up to time t is Poissonian tn e−t /n!. Thus, the velocity distribution function reads Q(v, t) = e−t

∞ n t 1 v , Q 0 n n n! q q n=0

(4.65)

where Q0 (v) is the initial velocity distribution of the impurity. The impurity velocity distribution function is a time-dependent combination of “replicas” of the initial velocity distribution. Since the arguments are stretched, compact velocity distributions display an infinite set of singularities, a generic feature of the Maxwell model. In contrast to the velocity distribution, the moments Ln (t) exhibit a much simpler behavior. Indeed, from Eq. (4.64) one finds that every moment is coupled only to itself, d n dt Ln = −(1 − q )Ln . Solving this equation we recover Eq. (4.60); in the heavy-impurity phase, however, it holds for all n. Therefore the moments exhibit multi-scaling asymptotic behavior. The decay coefficients, characterizing the n-th moment, depend on n in a non-linear fashion.

4.4.2 Model B For hard-sphere particles, the collision rates are proportional to the relative velocity. The overall collision rates model represent the average relative velocity, and a " in the Maxwell # √ (v1 − v2 )2 ∝ (T1 + T2 )/2. Therefore, the rates Kp = T and natural choice is

4.4

Impurities

107

# Kq = (T + Θ)/2, should be used in the Boltzmann equation (4.26) and the Lorentz– Boltzmann equation (4.48), respectively. This modification suppresses the heavy impurity phase, although the secondary transitions corresponding to higher-order moments remain. The fluid and the impurity temperatures obey √ d 2p(1 − p) T = − T T , dt d " 1 − q2 (1 − q)2 d T +Θ Θ = − Θ+ T . dt 2 d d Consequently, the temperature ratio, S = Θ/T , evolves according to " 1 − q2 2p(1 − p) 1 d 1+S (1 − q)2 √ S= − S+ + S. 2 d d d T dt

(4.66)

The loss term, which grows as S 3/2 , eventually overtakes the gain term that grows only linearly with S. Therefore, the asymptotic temperature ratio remains finite, S → c, where c is the root of the cubic equation " 1−q 2p(1 − p) 1+c c− = c. (4.67) 2 1+q 1 − q2 Consequently, there is only one phase, the light impurity phase. Intuitively, as the impurity becomes more energetic, it collides more often with fluid particles. This mechanism limits the growth rate of the temperature ratio. As in model A, energy equipartition does not generally occur, and the behavior is largely independent of the spatial dimension. Qualitatively, results obtained for the light impurity phase in model A extend to model B. The impurity velocity distribution follows a scaling asymptotic behavior. The only difference # is that the collision terms are proportional to β −1 = (1 + c)/2, and Eq. (4.55) generalizes as follows −βp(1 − p)zg (z) + g(z) = (1 + αz) e−αz g(qz).

(4.68)

Seeking a series solution of the form (4.56) leads to the following recursion relations for the coefficients gn =

(1 − q)q n−1 − βp(1 − p) gn−1 . 1 − q n − nβp(1 − p)

(4.69)

Again, the velocity distribution is a combination of powers of Lorentzians as in Eq. (4.58). Moreover, the coefficients Qn are linear combinations of the coefficients gn ’s as in model A. Most importantly, the large-velocity tail is generic Q(w) ∼ w−4 . The fluid moments evolve according to (4.11) with τ ≡ τp , while the impurity moments evolve according to n−2 n d q m (1 − q)n−m Mm Lm−j , Ln + bn Ln = (4.70) dτq m m=2

108


Note that asymptotically τq → τp /β, so the fluid moments decay according to Mn ∝ e−βan (p)τq . Hence, when βan (p) < bn (q), the impurity moments are enslaved to the fluid moments, i.e., Ln ∝ Mn asymptotically. Otherwise, sufficiently large impurity moments behave differently from the fluid moments, viz. Mn ∝ e−bn τq . Although the primary transition affecting the second moment does not occur (m1 ≡ m∗ = ∞), secondary transitions affecting larger moments do occur at a series of masses, as in Eq. (4.61). The transition masses mn are found by solving βan (p) = bn (q) simultaneously with Eq. (4.67). For example, for completely inelastic collisions (rp = rq = 0) one finds m2 = 1.65. These transitions imply that some velocity statistics of the impurity, specifically large moments, are no longer governed by the fluid.

4.4.3 Velocity Autocorrelations The impurity autocorrelation function satisfies dA/dτq = −(1 − q)A. Therefore, its decay is similar to (4.15) A τq , τq ∼ A0 exp −(1 − q) τq − τq

(4.71)

with q replacing p and τq replacing τ ≡ τp . For model A where the collision counters equal time, the decay remains exponential. However, the impurity autocorrelation decays with a different rate from the fluid. For model B, one can show that the algebraic decay (4.16) holds asymptotically with p replaced by q. We conclude that, while one-point velocity statistics of the impurity are governed by the fluid, two-time statistics are different.

4.5 Mixtures Granular media typically consists of mixtures of granular particles of several types. In contrast with the impurity problem, the different components of a mixture are coupled to each other. Moreover, there are numerous parameters, making mixtures much harder to treat analytically [51]. Remarkably, the impurity solution can be generalized to arbitrary mixtures7 . We detail the freely cooling case in one dimension. Consider a binary mixture where particles of type 1 have mass m1 and concentration c1 , and similarly for particles of type 2. We set c1 + c2 = 1. Also, unit collision rates are considered for simplicity. Collisions between a particle of type i = 1, 2 and a particle of type j = 1, 2 are characterized by the dissipation parameter pij = (mi − mj rij )/(mi + mj ), with rij the restitution coefficient. We denote the normalized velocity distribution of component i by Pi (v, t) and its Fourier transform by Fi (k, t). The governing equations now couple the two distributions 2

∂ Fi (k) + Fi (k) = cj Fj (k − pij k) Fi (pij k) . ∂t j=1 7

Mixtures were treated analytically in the elastic Maxwell model [79, 80].

(4.72)

4.6

Lattice Gases

109

Let Ti = dv v 2 Pi (v, t) be the temperature of the ith component. Writing Fi (k, t) ∼ = 1 − 12 k2 Ti , the temperatures evolve according to: d T1 T1 λ12 λ = − 11 . (4.73) λ21 λ22 dt T2 T2 The diagonal matrix elements are λii = 2ci pii (1 − pii ) + cj (1 − p2ij ) with j = i, and the offdiagonal matrix elements are λij = −cj (1 − pij )2 . Therefore, the temperatures are sums of two exponential terms, exp(−λ± t). The decay coefficients are the two (positive) eigenvalues

# 1 λ11 + λ22 ± (λ11 − λ22 )2 + 4λ12 λ21 . (4.74) λ± = 2 Asymptotically, the term with the smaller decay rate λ ≡ λ− dominates Ti (t) Ci e−λt ,

with

Ci =

(λ+ − λii )Ti (0) − λij Tj (0) . λ+ − λ−

(4.75)

The ratio between the two temperatures approaches C2 /C1 = (λ − λ11 )/λ12 , and the two components have different temperatures. Furthermore, as long as the components are coupled, both temperatures are finite. For example, a vanishing C2 implies that one of the concentrations vanishes as λ = λ1,1 . As in the cases of homogeneous gases and impurities, we seek a scaling solution of the form Pi (v, t) = eλt/2 Pi (w) with w = v eλt/2 , or equivalently, Fi (k, t) = fi (z) with z = |k| e−λt/2 . The scaling functions fi (z) are coupled via the non-local differential equations 2

1 cj fj (z − pij z) fi (pij z). (4.76) − λzfi (z) + fi (z) = 2 j=1 n −z Substituting the series solution (4.56), fi (z) = ∞ , yields recursion relations n=0 Ai,n z e for the coefficients Ai,n 2 n nλ λ Ai,n + Ai,n−1 = 1− cj Aj,m Ai,n−m (1 − pij )m pn−m . (4.77) ij 2 2 j=1 m=0 The small z behavior fi (z) = 1 − 12 Ci z 2 implies the first three coefficients Ai,0 = Ai,1 = 1, and Ai,2 = (1−Ci )/2. For n ≥ 3, the coefficients (A1,n , A2,n ) are solved recursively in pairs. Each such pair satisfies two inhomogeneous linear equations. Therefore, as in the impurity case, the velocity distribution is an infinite series of powers of Lorentzians (4.58). Although the two velocity distributions are different, they have the same extremal behavior, Pi (w) ∼ w−4 . It is straightforward to generalize the above to mixtures with an arbitrary # number of components, and to incorporate different collision rates, in particular, Kij = (Ti + Tj )/2.

4.6 Lattice Gases Inelastic collisions generate spatial correlations and consequently, inelastic gases exhibit spatial structures such as shocks. Thus far, we have considered only mean-field collision processes where there is no underlying spatial structure. Random collision processes can be

110


naturally generalized by placing particles on lattice sites and allowing only nearest neighbors to collide. Consider a one-dimensional lattice where each site is occupied by a single particle. Let vj be the velocity of the particle at site j. The velocity of such a particle changes according to Eq. (4.1) due to interactions with either of its two neighbors. Time is conveniently characterized by the collision counter τ . In an infinitesimal time interval dτ , the velocity of a particle changes as follows   prob. 1 − 2dτ ; vj (τ ) (4.78) vj (τ + dτ ) = vj (τ ) − (1 − p) [vj (τ ) − vj−1 (τ )] prob. dτ ;   vj (τ ) − (1 − p) [vj (τ ) − vj+1 (τ )] prob. dτ. This process is stochastic and we are interested in averages over all possible realizations of the process, denoted by an overline. We consider random initial conditions where the average velocity vanishes and no correlations are present: vj (0) = 0 and vi (0)vj (0) = T0 δi,j . Spatial velocity correlations satisfy closed equations as in the Ising–Glauber spin model [81]. For example, from the dynamical rules (4.78), the average velocity Vj (τ ) = vj (τ ), obeys a discrete diffusion equation dVj = (1 − p)(Vj−1 − 2Vj + Vj+1 ). dτ

(4.79)

From the initial conditions, Vj (0) = 0, we obtain Vj (t) = 0. Consider the spatial correlation function vi vj . The initial state is translationally invariant, so this property persists. The correlation functions Rn = vj vj+n satisfy dRn = 2(1 − p)(Rn−1 − 2Rn + Rn+1 ), n ≥ 2; dτ dR1 = 2(1 − p) [pR0 − (1 + p)R1 + R2 ] ; (4.80) dτ dR0 = 4p(1 − p) [R1 − R0 ] . dτ The initial conditions are Rn (0) = T0 δn,0 . Since we are interested in the asymptotic behavior, we employ the continuum approximation. The correlation function satisfies the diffusion equation ∂R/∂τ = 2(1 − p)∂ 2 R/∂ 2 n, so the solution is the Gaussian T0 n2 . (4.81) exp − Rn (τ ) # 8(1 − p)τ 8(1 − p)πτ The temperature, T ≡ R0 decays as T (τ ) T0 [8(1 − p)πτ ]−1/2 in the long time limit8 . Although no correlations were present in the initial conditions, spatial correlations develop at later times. The corresponding correlation length ξ grows diffusively with the collision counter, ξ ∼ τ 1/2 . The system consists of a network of domains of typical size ξ. Inside a domain, velocities are strongly correlated, and momentum conservation yields a relation 8

An exact solution of the discrete equations (4.80) is possible. It yields an identical asymptotic expression for the temperature.

4.7

Conclusions

111

between the average velocity and the domain size, v∗ ∼ ξ −1/2 ∼ τ −1/4 . This scale is consistent with the temperature behavior above, T ∼ v∗2 . In arbitrary dimension, this scaling argument yields T ∼ τ −d/2 [22, 43]. At least for scalar velocities, the correlation functions obey closed equations in arbitrary dimension [82], and this behavior can be also obtained analytically. The actual time dependence is determined from the collision rate. We consider two choices. In model A, the rate is proportional to the typical velocity K 2 = T . In the physically more realistic model B, the rate is proportional to the average relative velocity K 2 = (vj+1 − vj )2 = 2(R0 − R1 ) ∝ −dT/dτ . Hence K ∼ τ −α with α = d/4 (model A) τ and α = (d + 2)/4 (model B). The time t = 0 dτ /K (τ ) grows as t ∼ τ 1+α , and therefore,   2d model A; d+4 −γ T ∼t (4.82) with γ=  2d model B. d+6 In either case, Haff’s cooling law is recovered only in the infinite-dimension limit, d → ∞. Otherwise, the appearance of spatial correlations slows down the temperature decay. As neighboring particles become correlated their relative velocity and consequently their collision rate is reduced. Qualitatively similar behavior was shown for freely cooling inelastic gases, namely, breakdown of the mean-field cooling law due to the formation of strong spatial correlations among particles velocities [24, 25].

4.7 Conclusions We have presented analytical results for random inelastic collision processes. In general, when the collision rate is uniform, the convolution structure of the collision integrals translates to products in Fourier space. While the governing equations are both non-linear and non-local, they are closed and amenable to analytical treatment. In the freely cooling case, a small wave number analysis of the Fourier transform displays both regular and singular terms. The regular terms yield the low-order moments, while the leading singular term gives the high-energy tail. In the forced case, the Fourier transform is an analytic function and complex residue analysis yields the high-energy tail. In the freely cooling case, the velocity distribution approaches a scaling form and displays an algebraic large-velocity tail. In one dimension, the scaling function is universal; otherwise, it depends on the degree of inelasticity. The exponent governing the high-energy tail is typically very large, and may be difficult to measure in practice. We have also shown that sufficiently large moments exhibit multi-scaling behavior and hence are not characterized by the temperature. The autocorrelation function decays algebraically, with an exponent that √ depends on the dissipation parameters. The spread of a tagged particle exhibits a universal ln t growth. We have demonstrated that an impurity immersed in a uniform fluid may or may not mimic the background. Fixing the dissipation parameters, for sufficiently low impurity mass, the impurity’s velocity distribution, velocity moments and extremal velocity statistics are all governed by the fluid. However, there is a series of phase transitions occurring at a series of increasing masses, where impurity moments of decreasing order decouple from the fluid.

112


These transitions indicate that sufficiently heavy impurities are very energetic and effectively, they experience a static fluid background. For binary mixtures, we have examined only primary velocity statistics and have shown that qualitatively, all components have the same temperature decay and extremal velocity statistics. It remains to be seen whether the different components may exhibit different asymptotic behaviors of more subtle velocity statistics such as the high-order moments. The impurity case demonstrates how, in certain limiting cases, mixtures may display anomalous behavior. In the forced case, injection of energy counters the energy dissipation and the system relaxes toward a steady state. We have considered white noise forcing, and have shown that the steady state distribution has an exponential high-energy tail. Steady state characteristics in the forced case are directly related to time dependent relaxation characteristics in the freely cooling case. Results obtained in the framework of the inelastic Maxwell model are exact for random collision processes where spatial structure is absent, and particles collide irrespective of their relative velocities. Such results are an uncontrolled approximation of the inelastic hard-sphere problem. Nevertheless, as a conceptual tool, the inelastic Maxwell model is powerful. For example, it demonstrates the development of correlations among the velocity components, as well as spatial correlations. It also raises doubts concerning the suitability of several widely used techniques such as perturbation expansions in the quasi-elastic limit, and expansions in terms of Sonine polynomials. We demonstrated that a lattice gas generalization of the Maxwell model remains tractable when the kinematic constraint (particles should collide only if they are moving toward each other) is ignored. Taking this constraint into account leads to shock-like structures in one dimension and to vortices in two dimensions [43]. Lattice models can therefore be used to study development of spatial correlations and spatial structures in inelastic gases. We presented in detail the most basic realization of the Maxwell model. Several other generalizations are feasible. Experiments can now measure the distributions of impact angles and of the effective restitution coefficients. Such phenomenological information can be incorporated into the rate equations. The integration measure can be redefined to give different weights to different angles. Moreover, a distribution of restitution coefficients can be introduced by integrating the collision integrals with respect to the dissipation parameters. For example, random collision processes were successfully used to model the role of the boundary in driven gases [83]. We restricted our attention to the case where all moments of the initial velocity distribution are finite. However, there are other infinite energy solutions of the Boltzmann equation. For example, any Lorentzian, F (k) = exp(−Ck), is a steady state solution of Eq. (4.6). A remaining challenge is to classify the evolution of an arbitrary velocity distribution. Also, it will be interesting to characterize the full spectrum of extremal behaviors, characterizing velocities far larger than the typical velocity. Finally, to model transport and other hydrodynamics problems one has to incorporate the spatial dependence explicitly. In the framework of the Maxwell model the simplest such problem – the shear flow – has been recently investigated by Cercignani [84]. For inelastic hard spheres, a number of unidirectional hydrodynamic flows were analyzed by Goldshtein and co-workers [85–87]. It would be interesting to investigate these problems in the framework of the Maxwell model.

References

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Acknowledgments We have benefited from discussions and correspondence with A. Baldassarri and M. H. Ernst. We also thank N. Brilliantov, A. Goldshtein, A. Puglisi, S. Redner, and H. A. Rose for useful discussions. This research was supported by DOE (W-7405-ENG-36) and NSF(DMR9978902).

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Article originally published: Eli Ben-Naim, Paul L. Krapivsky, The Inelastic Maxwell Model. In: Lecture Notes in Physics, Volume 624 (Granular Gas Dynamics), pp. 65–94 (Springer-Verlag, 2003).

5 Cluster Formation in Compartmentalized Granular Gases Ko van der Weele, René Mikkelsen, Devaraj van der Meer, and Detlef Lohse

Abstract A brief overview is given of the recent studies into cluster formation in compartmentalized gases, focusing upon the so-called Maxwell Demon effect. A common thread in these studies is that the clustering|( is related to the fact that the particle flux from a compartment, or the granular pressure, is a non-monotonic function of the number of particles in the compartment.

5.1 Introduction One of the characteristic features of granular gases is their tendency to spontaneously separate in dense and dilute regions [1–3]. This property, which makes them fundamentally different from any ordinary molecular gas, can be traced back to the fact that the collisions between the granular particles are inelastic. Every time two particles collide, their relative velocity is reduced proportional to the coefficient of normal restitution 0 ≤ e < 1. The case e = 1 corresponds to a standard elastic gas in which no clustering occurs. The clustering effect was first demonstrated in numerical studies of rapid granular shear flows [4] and freely cooling granular gases [5]. Figure 5.1, from the seminal paper by Goldhirsch and Zanetti [5], shows cluster formation in a simulated two-dimensional system consisting of 40,000 disks, colliding inelastically (like hockey pucks on a frictionless ice floor) with restitution coefficient e = 0.6. The particles start out from a spatially homogeneous state, with a Maxwellian velocity distribution, and are left to evolve without further energy input, which means that the mean kinetic energy (or equivalently, the granular temperature) decays with time due to the inelastic collisions. The clustering process can be understood as follows [5]: In a region where, due to some fluctuation, the density exceeds the average density of the gas, the collision rate is higher and the granular temperature will therefore drop faster than in the neighboring, less dense regions. In hydrodynamical terms this means that a pressure gradient is built up between high and low temperature (and thus pressure) regions, resulting in a migration of particles into denser regions from diluter ones. Hence the dense regions become denser, and the dilute regions diluter, and this self-enhancing process spontaneously leads to the formation of clusters (consisting of many slow particles) coexisting with almost empty regions (where the particles move much faster). In a freely cooling gas one may eventually even witness the phenomenon of inelastic collapse [6], in which strings of particles (within the clustered regions) come to a standstill after having experienced an infinite number of collisions in a finite time.

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5 Cluster Formation in Compartmentalized Granular Gases

Figure 5.1: Cluster formation in a simulated system of 40,000 disks colliding inelastically (with coefficient of restitution e = 0.6) on a frictionless floor, after a time corresponding to 500 collisions per particle. The area fraction covered by the disks is 0.05, and periodic boundary conditions are used in both directions (from [5]).

The above clustering mechanism also holds for granular gases that are kept at a constant granular temperature by an external energy input [7, 8]. In this review we focus on the cluster formation in such forced gases in compartmentalized systems. These are especially suited to get a clear-cut view of the clustering effect; moreover, they can be directly related to compartmentalized systems like sorting machines and conveyor belts, for which clustering is known to be a major source of problems [1, 9]. In Section 5.2 we will first describe the demonstration experiment [10] that was responsible for the current interest in these systems, and which is now known as the Maxwell Demon experiment (after the inspiring title of [11]). It consists of a vertically vibrated box divided in two connected compartments, filled with a few hundred small beads, and shows a beautiful counter-intuitive clustering effect. In Section 5.3 we sketch the theoretical model by Eggers [11], which quantitatively explains the effect. Section 5.4 describes the new phenomena that are observed, both in experiment and in theory, when the system is extended to more than two compartments [12–19]. Section 5.5 deals with an alternative analysis of the clustering effect, proposed by Lipowski, Droz, et al. [20–22], in terms of a modified Ehrenfest urn model. Section 5.6 describes a horizontally vibrated version of the Maxwell Demon system, analyzed by means of a hydrodynamic model by Brey et al. [23]. Section 5.7 again switches to the statistical mechanics level with a recent proposal by Cecconi et al. [24] to capture the essential features of the clustering effect in terms of a Kramers-type escape problem, featuring two particles in a double-well potential. Finally, in Section 5.8 we briefly sketch some further directions.

5.2

The Vertically Vibrated Experiment

119

0.08

0.1

2.5

0.06

0.08

2

F(n) 0.04

F(n)

0.02 0

0

0.2

0.4

a)

n

0.6

0.8

1

1.5

0.06 f(ξ) 0.04

1

0.02

0.5

0

0 0

0.2

0.4

n

0.6

0.8

1

b)

0

1

2

ξ

3

4

5

c)

Figure 5.2: The various functions governing the dynamic equilibrium between the compartments in three of the systems reviewed in this paper, each of them showing a non-monotonic behavior as function of the number of particles in the compartment: (a) the Eggers flux function for the vertically vibrated e = 5], with n representing the particle fraction (in the compartsystem [Eq. (5.3) with A = 1 s−1 and B ment) relative to the total number of particles in the system, (b) the flux function for the urn model of Lipowski and Droz [Eq. (5.11) with T0 = 0.1 and ∆ = 1], and (c) the particle-number dependency of the granular pressure for the horizontally vibrated system of Brey et al. [Eq. (5.14)], with ξ proportional to the number of particles in the compartment. The dashed line in (a) indicates that the flux from a relatively empty compartment (left intersection point) can be equal to the flux from a well-filled compartment (right intersection point), which is exactly the condition for a clustered state; the associated fractions add up to 1. The symmetric state n = 0.5 (corresponding to a flux just to the right of the e maximum of F (n)) is unstable at this value of B.

A unifying theme of the various models reviewed is that the spontaneous symmetry breaking of the particle distribution (i.e., the clustering effect) is related to a function that depends non-monotonically on the particle content of each compartment, see Figure 5.2. Figure 5.2a, from the Eggers theory, represents the particle flux F (nk ) from any of the two compartments (k = 1, 2) as a function of the fraction of the total number of particles in that compartment, nk = 1. Figure 5.2b, for the urn model, similarly represents the flux of particles leaving the kth urn. The third plot (Figure 5.2c), from the hydrodynamic theory for the horizontally vibrated system, depicts how the granular pressure in a compartment depends on the number of particles it contains. In each case, the one-humped form of the function makes it possible to have equal fluxes (or pressures) for the two compartments while the number of beads differ, which is precisely the requirement for a clustered state. This common feature of the three models is our main theme and will be elaborated in the next sections.

5.2 The Vertically Vibrated Experiment A striking illustration of the cluster formation in compartmentalized granular gases is provided by the experiment illustrated in Figure 5.3, which was first described by Schlichting and Nordmeier [10]. The setup consists of a box, mounted on a shaker, divided into two equally sized compartments by a wall extending from the bottom to height h. A few hundred beads are brought into a gaseous state by shaking the system vertically, and are able to jump from one compartment to the other.

120


a)

b)

c)

d)

Figure 5.3: Snapshots from the so-called Maxwell Demon experiment. During vigorous shaking a) the particles (glass beads with diameter 4 mm and restitution coefficient e = 0.95) are distributed uniformly over the two compartments. Reducing the shaking strength below a critical value results in the formation of a cluster in one of the compartments b)–d). Note that the particles in the dilute compartment jump higher than those in the dense compartment, i.e., the granular temperature is higher in the dilute compartment. The height of the wall is 60 mm.

If the shaking is vigorous enough, the inelasticity of the gas is overwhelmed by the energy input into the system, and the particles are distributed uniformly over the two compartments just as in any ordinary gas (Figure 5.3a). However, when the shaking strength is reduced below a critical level, the particles are seen to cluster into one of the compartments (Figure 5.3b–d). This goes on until a dynamical equilibrium is reached between the two compartments: the average outflow of rapid particles from the nearly empty compartment is balanced by the outflow of slow particles from the well-filled compartment. In terms of the granular temperature, one ends up with a “cold” compartment containing a lot of relatively slow particles, and a “hot” compartment containing only a few rapid particles. This spontaneous separation in cold and hot regions is reminiscent of Maxwell’s Demon [11, 25], a creature supposed to guard the door between two rooms filled with gas in thermodynamic equilibrium. Its task was to let slow molecules pass in one direction and fast molecules in the other, and thus create one cold and one hot room. The demon is notoriously powerless in any ordinary gas; and in fact, such a separation of cold and hot molecules (if it happened spontaneously) would be a violation of the second law of thermodynamics. However, Maxwell’s Demon rules in granular gases, thanks to the non-elasticity of the collisions, and thanks to the fact that such a gas is intrinsically far from equilibrium. The gas is not isolated from the rest of the world: it constantly receives energy from outside and, via the inelastic collisions constantly transfers this energy to the microscopic scales (thereby effectively losing it) in the form of heat, sound, and deformation energy. The granular Maxwell Demon effect is of course not violating any law of physics, but is instead a prime example of pattern formation in a non-equilibrium system [26, 27], like, e.g., Rayleigh–Bénard convection cells in a pan of oil heated from below [28], or sand ripples on the beach [29].

5.3

Eggers’ Flux Model

121

5.3 Eggers’ Flux Model A theoretical model for the Maxwell Demon effect was proposed by Eggers [11]. As a starting point, he took the condition for dynamic equilibrium between the two compartments, namely that the flux of particles from left to right must equal that from right to left: Fl→r

=

Fr→l ,

(5.1)

noting that an asymmetric equilibrium can only be explained if the particle flux from one compartment to the other is not a monotonously increasing function of the number of particles (as it would be for elastically colliding particles). Instead, it must show a maximum. For simplicity, Eggers considered a two-dimensional gas of colliding disks with radius r, in a slightly different setup from the one in Figure 5.3: The wall is taken to extend over the whole height of the system, with only a small opening (of width S) positioned at height h above the bottom. The bottom of the container is taken to move in a sawtooth manner, with amplitude a and frequency f , such that a colliding particle always finds it to move upward with the same velocity vb = af . Moreover, the amplitude a is very small compared to the mean free path of the particles, so the bottom is effectively stationary. Assuming the gas inside each compartment to be in a steady state, Eggers proceeds to derive an analytic expression for the particle outflow from each compartment, based on three equations from the kinetic theory of dilute granular gases [30]: (1) the equation of state relating the pressure, density, and temperature, (2) the force balance within the gas, which indicates how fast the pressure decreases with the height z above the bottom, and (3) the balance between the upward energy flux through the gas (fed by the vibrating bottom) and the dissipation due to the inelastic particle collisions. To minimize wall effects, which are not essential to the problem, all collisions with the walls and bottom are taken to be elastic. The temperature profile T (z) that is found on the basis of these equations, turns out to be close to constant, except for a narrow region of higher temperature near the bottom; this is the region where the energy is injected into the system and the particles have not yet had the opportunity to redistribute their kinetic energy via collisions. Taking for simplicity a constant temperature profile Tk (z) = Tk throughout the compartment, with k = 1, 2 labeling the two compartments, one finds that the number density is exponentially decaying with z (as in the standard barometric height distribution), in fair agreement with molecular dynamics simulations [11]: 2 ¯k gN af √ e−gz/Tk , with Tk = . (5.2) nk (z) = ¯k Tk 2 πr(1 − e2 )N ¯k denotes the number of particles divided by the width of the compartment (Nk /L), r Here N is the radius of a particle, and the expression for Tk (≡ 12 v2k ) is obtained by balancing the energy input and the energy dissipated in collisions [11]. The particle flux from compartment k through the hole is then given by nk (h) Tk /2πS, i.e., the product of the number density at the height of the hole, the velocity in the horizontal direction, and the extension of the hole S. This can be worked out to yield ¯k ) = F0 N¯k 2 e−bN¯k 2 , F (N

(5.3)

122


√ where F0 = 8π(Sgr/af )(1 − e2 ) and b = 4πghr 2 (1 − e2 )2 /(af )2 . This is indeed a non-monotonic function of the number Nk of particles in box k. In Figure 5.2a the above flux function is given in terms of nk (i.e., the fraction of the total ¯k = Nk /L = nk Ntot /L (with ¯k . Since N number of particles in compartment k) rather than N L the width of a compartment), the flux function then takes the form e

2

F (nk ) = An2k e−Bnk ,

(5.4)

with 2 √ SgrNtot (1 − e2 ) , A = 8π af L2

= 4π gh B (af )2

rNtot L

2

(1 − e2 )2 .

(5.5)

determines whether the system will end up in the uniform or in The dimensionless number B can be raised either the clustered state. For a given choice of granular beads (r and e fixed), B by increasing the height of the separating wall h or the total number of particles Ntot , or by decreasing the driving velocity af . → 0, the exponential term in Eq. (5.4) approaches unity, and F (nk ) thus In the limit B grows monotonically with nk (just as for an elastic gas with e = 1). This makes a balance between a well-filled and a nearly empty compartment impossible, and the system settles into the homogeneous state. is raised, however, the exponential term comes into play. This is depicted in FigAs B = 5. The function F (nk ) still starts out from zero at nk = 0 ure 5.2a for A = 1 s−1 and B the function goes down again, and initially increases with nk . However, beyond nk = 1/ B as a result of the dissipative effect of the increasingly frequent particle collisions. This enables a flux balance (see Figure 5.2a) between a well-filled and a dilute compartment, provided the > 4). Now the condition nk = 1 can be maximum of F (nk ) lies at a value nk < 12 (i.e., B satisfied not only for an equal pair n1 = n2 = 12 (corresponding to a flux just to the right of the maximum in Figure 5.2a) but also for an unequal pair n1 = n2 (corresponding to a smaller flux, indicated by the horizontal dashed line). The dynamics of the system is governed by the following balance equation, dn1 = −F (n1 ) + F (n2 ) + ξ1 = −F (n1 ) + F (1 − n1 ) + ξ1 , dt

(5.6)

(and analogously for dn2 /dt) which simply states that the time rate of change dnk /dt of the particle fraction in the kth compartment is equal to the inflow from its neighbor minus the outflow from the compartment itself. The term ξ1 models the noise which comes from statistical fluctuations in the particle flux [11]; without it, the above balance equation is to be interpreted as a mean-field description. In equilibrium, the two fluxes in Eq. (5.6) must cancel each other (F (n1 ) = F (1 − n1 )), ≥4 < 4 this yields one solution n1 = 1 (the symmetric state). For B see Eq. (5.1). For B 2 this solution becomes unstable, but simultaneously two asymmetric stable solutions come into existence; one representing a state with a cluster in the left compartment, and the second one its (equivalent) mirror image with a cluster in the right compartment. The clustering transition is

5.3

Eggers’ Flux Model

123 0.50 0.40 0.30

0.20 0.10 0.00 0.50

0.60

0.70

0.80

0.90

h Figure 5.4: The bifurcation of the time-averaged asymmetry parameter as function of h (the height of the hole above the bottom) in the Eggers model. The circles represent numerical simulations with Ntot = 360 disks (radius 0.01 m, restitution coefficient e = 0.95), compartment width L = 1.60 m, and velocity of the bottom af = 0.149 m s−1 . The dashed line is the result of the flux model defined by Eqs. (5.4)–(5.6), while the full line represents a more elaborate version of the model without the simplifying assumption that Tk (z) is independent of z (from [11]).

depicted in Figure 5.4, where the absolute value of the (time-averaged) asymmetry parameter

=

¯k − 1 N ¯ N 1 2 tot = nk − ¯tot 2 N

(5.7)

is plotted as function of h (the height of the hole above the bottom). The figure includes the results from molecular dynamics simulations of 360 inelastically colliding disks (open circles), the flux model defined by Eqs. (5.4)–(5.6) (dashed line), and the numerical result of the Eggers model without the simplifying assumption that T (z) is independent of z (solid line) [11]. Even though there is an offset between the constant-T theory (dashed line) and the simulation, captures the form of the bifurcation this theory using only one dimensionless parameter (B) fairly well. The transition to the clustered state is seen to be a second-order, continuous phase transition. Just above the critical point the asymmetric solutions according to the dashed line − 4)/16]β with a critical exponent β = 1/2. This is the are described by = ±[3(B common (mean-field) power-law behavior near a second-order phase transition [31]. In order to get an estimate for the amplitude of the fluctuations in the system, one may assume that the particles pass through the hole uncorrelated (which is equivalent to saying that ξ1 in Eq. (5.6) is a Gaussian white noise term, with zero mean and a δ correlation function). − 4)|−1/2 , tells The resulting variance of the asymmetry parameter, ( − )2 ∝ |Ntot (B us that the relative amplitude of the fluctuations decreases with growing particle number (as = 4 goes with the exponent 1/2 expected) and that the divergence near the critical point B typical of second order phase transitions [31]. Eggers notes that this model prediction is in reasonable agreement with numerical simulations [11].

124


5.4 Extension to More than two Compartments The transition to the clustered state was measured experimentally by van der Weele et al. [12], not only for the original system with two compartments but also for three compartments, and in a subsequent paper this was extended to an arbitrary number of K connected compartments [13]. The systems considered were 3-dimensional, as in Figure 5.3. This has no consequences for the general form of the flux function, which is still given by Eq. (5.4), but now read (cf. Eq. (5.5)): the two factors A and B 2 2 2 Sgr 2 Ntot r Ntot 2 = c2 gh (1 − e ) , B (1 − e2 )2 ≡ K 2 B . (5.8) A = c1 af Ω2 (af )2 Ω Here c1 and c2 are constants, which here are used as free parameters to set the timescale (c1 ) and the scale of the B-axis in the bifurcation diagram (c2 ). The opening S between the compartments is now a 2D surface (instead of a 1D length), and the compartment width L has been replaced by the ground area Ω. For a system consisting of K compartments arranged cyclically, such that the Kth and 1st compartments are neighbors, the balance equation Eq. (5.6) takes the form (disregarding the noise term): dnk = F (nk−1 ) − 2F (nk ) + F (nk+1 ) , dt

(5.9)

with k = 1, 2, . . . , K. For a non-cyclic arrangement the equation for the end compartments is modified, of course, but it turns out that this does not qualitatively change the results [13]. For the two-compartment system (K = 2), the experimental data confirm the secondorder transition predicted by the flux model. This can be seen in Figure 5.5 (top), where the measured nk (for both compartments k = 1, 2) are plotted as solid dots. In this figure, the has been rescaled to B, so the critical value at which the transition takes place now factor B lies at 1. The experiments were performed by changing the driving frequency f , while all the other quantities appearing in B were held fixed [12]. In contrast, for K = 3 compartments the clustering transition is found to be abrupt and hysteretic, i.e., a first-order phase transition.1 Figure 5.5 (bottom) shows the experimental results together with the flux model predictions for a cyclic three-compartment system [12]. The dots represent measurements from experiments that were started out from the uniform distribution { 13 , 13 , 13 }, and the crosses for those that were started from a single peaked distribution: They clearly show that there is an interval of B-values for which both the uniform and the clustered state are stable. The dashed curves for B > 1 that run above and below the horizontal line of the uniform state are associated with a transient state in which two of the compartments are competing for dominance, while the third compartment is already much more dilute. Staring out from the (unstable) uniform distribution, the system generally first goes through this transient state before it settles in the clustered equilibrium. No such transient states are encountered in the opposite transition for B < 0.73. 1

There is a close analogy with the K-state Potts model here, i.e., an assembly of Ntot connected spins with K possible orientations per spin. Also in the Potts model one finds a phase transition of second order if K = 2 (the Ising case) and a transition of first order if K ≥ 3 [21, 32].

5.4

Extension to More than two Compartments

125

a)

b) Figure 5.5: a) Bifurcation diagram for the Maxwell Demon experiment with K = 2 compartments (k = 1, 2). The dots are experimental data, and the lines are the stable (solid) and unstable (dashed) equilibria according to the flux model of Eqs. (5.4) and (5.8). The transition to the clustered state is a continuous one, i.e., a second-order phase transition. b) The same for the three-compartment experiment (k = 1, 2, 3). The dots and crosses are experimental data: dots for measurements that were started from the uniform distribution { 13 , 13 , 13 }, and crosses for those that were started from a single peaked distribution. The transitions to and from the clustered state are abrupt and hysteretic, typical of a firstorder phase transition (from [12]).

126


Figure 5.6: a) Bifurcation diagram for K = 5 compartments. The sketches on the right depict typical configurations associated with the solid (stable) and dashed (unstable) equilibria of the flux model. b) Four stages in a clustering experiment at B slightly above 1. The particles do not cluster directly into one compartment, but first go through a transient two-cluster state, which can be seen in the snapshots at t = 10 s and t = 25 s. c) Sudden collapse of a cluster at stronger shaking (B = 0.33): The cluster is clearly present until t = 42 s, then suddenly collapses, leaving no trace one second later (from [18]).

The same qualitative behavior, with a first-order transition, is found for all K ≥ 3 [13]. Quantitatively, the hysteretic behavior becomes more pronounced as the number of compartments is increased, and the transient states become more numerous and also more important. Figure 5.6a illustrates this for the case of K = 5 non-cyclic compartments. The region of hysteresis (where the uniform and the clustered state are both stable) now extends from B = 0.34 to B = 1 and the dashed lines of the transient states form a whole web, reaching even to the left of B = 1. They correspond to states with m = 2, 3, 4 clusters, respectively, of which one representative configuration is depicted. In Figure 5.6b four stages in the clustering process are shown, for a B-value slightly above 1, starting out from a nearly uniform distribution. A two-cluster transient state is clearly visible at t = 10 s and t = 25 s, and it takes about a minute before the single-cluster state is reached. For larger values of K the experiment can easily get stuck in such a transient state (especially for low driving frequencies, i.e., B 1) and it may take a very long time before

5.5

Urn Model

127

the single-cluster state is reached, even though mathematically speaking this is the only truly stable state [13, 14]. The clusters collapse one by one in an exceptionally slow coarsening process: The characteristic size of the surviving clusters is found to increase as [log(t)]1/2 only [15]. The opposite process of declustering, depicted in Figure 5.6c, is also of interest. This is not only because declustering is more desirable in practical applications (e.g., in sorting machines, where clustering is an unwanted and often costly effect) but also because the breakdown of a cluster turns out to be by no means the same as clustering in reverse time order. Van der Meer et al. [16] found a surprising phenomenon called “sudden collapse”: Starting out with all particles in one compartment, the cluster seems stable for a considerable time, spilling only a small number of particles to its neighbors. However, at a certain moment (between t = 42 s and 43 s in the experiment of Figure 5.6c) the cluster suddenly collapses and the particles spread out evenly over all compartments. The collapse, which can be delayed for extremely long times if B approaches the critical value where the single-cluster state becomes stable (with the cluster lifetime diverging as (Bcrit − B)−1/2 ), has been studied in detail in Refs. [16] and [17].

5.5 Urn Model A different analysis of the Maxwell Demon experiment was given by Lipowski and Droz [20], who pictured it as a modified version of the Ehrenfest urn model [33]. The particles are initially distributed over two urns, and can change urn according to a probabilistic rule that mimics the behavior of granular matter. This approach was introduced to have a better insight into the role played by the statistical fluctuations, which had not been covered by the deterministic Eggers model (excluding the noise term in Eq. (5.6)). In the model defined in [20], the particles in each urn are subject to thermal fluctuations and the granular temperature T of an urn is taken to depend on its particle content as follows: T (nk ) = T0 + (1 − nk )∆ ,

(5.10)

where nk = Nk /Ntot is the fraction of particles in the urn, and T0 and ∆ are positive constants. So T (nk ) decreases with nk , as it should for a granular gas, yet the linear decay (from T (0) = T0 + ∆ to T (1) = T0 ) is markedly different from the inverse square dependence derived by Eggers, cf. Eq. (5.2). The rule by which the particles change urn is defined as follows: (1) One of the Ntot particles is selected randomly, and (2) with probability exp(−1/T (nk )) it changes urn. This yields the following expression for the flux out of an urn: F (nk ) = nk e−1/T (nk ) ,

(5.11)

which is again a one-humped function of the particle fraction nk . This can be seen in Figure 5.2b, where F (nk ) is shown for T0 = 0.1 and ∆ = 1. Note that in the limit of infinite temperature (T0 = ∞ and ∆ ≥ 0, arbitrary) the original Ehrenfest model is recovered, for which every selected particle changes urn.2 2

More recently, Bena et al. [22] studied a closely related urn model with F (nk ) = nk exp(−Ank ). They describe the second-order clustering transition (at A = 2) in terms of the Yang–Lee theory of phase transitions.

128


Given the flux function, the dynamics of the system is governed by the same balance equation as in the Eggers model (cf. Eq. (5.6)): dn1 = −F (n1 ) + F (1 − n1 ). dt

(5.12)

The zeros of this equation are the equilibrium states of the system, and their stability can be found by plotting dn1 /dt as a function of n1 and examining its slope. If dn1 /dt goes through zero with a downward slope, the dynamics is directed (locally) towards this zero and the corresponding equilibrium is stable. Vice versa, an upward slope corresponds to an unstable equilibrium. In this way, the phase diagram of Figure 5.7a can be constructed. The uniform equilibrium n1 = 1/2 exists for all values of T0 and ∆, and it is stable everywhere except in region II. If one follows a path through this phase diagram (by changing the values of T0 and ∆ in very small steps, and very slowly, such that every time the system can adjust itself to its new equilibrium) one finds, upon crossing the line between regions I and II from the left, a secondorder phase transition. The uniform state n1 = 1/2 undergoes a pitchfork bifurcation, in which it becomes unstable and simultaneously gives birth to two stable asymmetric solutions (one corresponding to n1 > 1/2, and one to its mirror image with n1 < 1/2). This is analogous to what is found in the two-compartment Maxwell Demon experiment. However, if one follows the path further in the direction of region III, additional bifurcations are encountered that are not found in the original experiment. Upon entering region III, the symmetric state is re-stabilized via a second pitchfork bifurcation, simultaneously giving birth to two new, unstable asymmetric states. The already existing stable asymmetric states are not affected by this bifurcation. Through regions III and IV, the two unstable asymmetric states grow towards their stable counterparts and eventually, upon crossing the line between IV and I they coalesce and annihilate each other in a reverse saddle-node bifurcation. We then enter region I again, where only the stable symmetric state remains. The above three bifurcations can all be seen in Figs. 5.7b and c, where the absolute value of the time-averaged asymmetry parameter = (N1 − N2 )/2Ntot is plotted as a function of the control parameter; this is T0 in the first plot, and ∆ in the second. First, in Figure 5.7b we follow a vertical path through the phase diagram at a constant value of ∆ = 0.5, with a clear second-order phase transition at T0 = 0.25. The critical point corresponds precisely with the top of the boundary curve between regions I and II T0 = ∆/2 − ∆/2. Lipowski and Droz checked that in the vicinity of the critical point the asymmetry parameter || ∝ |T0 − 0.25|β , with the usual critical exponent β = 1/2, just as in the Eggers model. Second, in Figure 5.7c we follow a horizontal path through the phase diagram at a constant value of T0 = 0.2. If the horizontal axis would have √ started at ∆ = 0, one would also have seen the second-order phase transition at ∆ = 12 (1− 1 − 4T0 )2 = 0.153, associated with the border between regions I and II. As it is, we see the bifurcations at the lines between regions II and III (at ∆ ≈ 1.1) and IV and I (at ∆ ≈ 1.7). Together they form a hysteretic, first-order transition between the stable clustered state and the uniform distribution. There is no bifurcation between III and IV. Nevertheless, in the phase diagram of Figure 5.7a there is a dashed line between III and IV emanating from the tricritical point at

5.5

Urn Model

129

0.3 I

0.25 T0

0.2 IV

0.15 II

0.1

III

0.05 0

0

0.5

1

1.5 ∆

a) 0.6

|< ε>|

|< ε>|

0.3 0.2 0.1

b)

3

0.5

0.4

0.1

2.5

0.6

0.5

0

2

0.4 0.3 0.2 0.1

0.14

0.18

0.22

T0

0.26

0

0.3

c)

0.4

0.8

1.2

∆

1.6

2

2.4

Figure 5.7: a) Phase diagram of the urn model. The symmetric state ( = 0) is stable in regions I, III, and IV, and unstable in region II. In regions II, III, and IV a stable asymmetric state exists (|| = 0). In regions III and IV one can therefore end up in either the symmetric or the asymmetric state depending on the initial condition; the dashed line separating region III from IV is described in the text. b) Bifurcation diagram of the asymmetry parameter || as a function of T0 , for ∆ = 0.5 fixed, showing the secondorder phase transition between regions II and I. The solid curve corresponds to the numerical solution of Eq. (5.12). The symbols are the result of Monte Carlo simulations for Ntot =500 (+) and 5000 (∗), respectively. c) Hysteretic, first-order transition of || as a function of ∆, for T0 = 0.2 fixed, calculated from Monte Carlo simulations for Ntot = 2000. Symbols (+) correspond to increasing ∆, and () to decreasing ∆ (from [20]).

√ ∆ = 23 , T0 = ( 3 − 1)/3 = 0.244. This line has been determined by considering the probability distribution p(M, t), i.e, the probability that a given urn (say 1) contains M particles at time t, as outlined below. The evolution equations for p(M, t) follow directly from the dynamical rules of the urn model and can be solved numerically, starting from arbitrary initial conditions [20]. Figure 5.8 shows three typical probability distributions, expressed in terms of the asymmetry parameter , in the long-time limit t → ∞. The left plot, characteristic for region I, shows the single-

130


0.03 0.02 0.01 0

0.05

∆ =0.5, T0 =0.4

0.04 P(ε)

P(ε)

0.05

∆ =0.5, T0 =0.4

0.04

0.03 0.02 0.01

-0.4

-0.2

a)

0 ε

0.2

0.4

0

∆ =1.35, T0 =0.2

0.04 P(ε)

0.05

0.03 0.02 0.01

-0.4

b)

-0.2

0 ε

0.2

0.4

0

-0.4

-0.2

0 ε

0.2

0.4

c)

Figure 5.8: Probability distributions P () for the urn model in region I (a), region II (b), and region III (c) in the long-time limit for Ntot = 200 particles (from [20]).

peak distribution around = 0 corresponding to the symmetric state: p() ∝ exp(−2 ), see also [34, 35]. The middle plot, representing region II, shows two peaks away from the center, corresponding to the clustered states. These plots are for Ntot = 200 particles; the width of the peaks decreases with growing Ntot . The critical probability distribution on the border line between regions I and II (for ∆ < 1/3) has been determined in Refs. [34,35] to have the form p() ∝ exp(−4 ), i.e., a peak with a flattened top. At the tricritical point (for ∆ = 1/3) the top flattens even further to p() ∝ exp(−6 ). Going into regions III and IV, the probability distribution becomes more complicated and consists of three peaks, since both the symmetric and the asymmetric state are stable here. For comparatively small Ntot , as in the right plot of Figure 5.8, all three peaks are clearly visible, but in the thermodynamic limit Ntot → ∞ either the central peak vanishes (in region III) or the two outer peaks (in region IV) [20, 35]. This reflects the relative probability to end up in either the symmetric or the asymmetric state, starting from random initial conditions. On the line that separates regions III and IV the three peaks are equally strong; this is interpreted by Lipowski and Droz as a line of discontinuous transitions, based on a phenomenological analogy with the same kind of phase transition in equilibrium statistical mechanics. One of the main goals of the urn model was to study the fluctuations of the symmetry parameter close to the critical point. Making use of the variance of , and the calculated probability distribution in the long-time limit p(i, ∞), the susceptibility κ is defined as [20]:  N

2  Ntot tot 1  κ = Ntot ( − )2 = i2 p(i, ∞) − i p(i, ∞)  . (5.13) Ntot i=0 i=0 At a continuous phase transition the susceptibility is known to diverge, and indeed, that is exactly what is found at the transition between regions I and II. The measured data close to the critical point in Figure 5.7 (center) indicate that κ ∝ |T0 − 0.25|−γ , both in the symmetric and (albeit less clearly) in the asymmetric state, with the mean field exponent γ = 1 [20, 31]. Finally, the urn model has been extended to more than two urns by Coppex, Droz and Lipowski [21]. The K urns are arranged cyclically, and one allows only nearest neighbor interactions, according to the same selection rule as before. The resulting model is analogous to the K-compartment model described in Section 5.4, and despite the rather peculiar temper-

5.5

Urn Model

131

1

∆=0.3

0.9 0.169

ncl(t)

0.8 0.7 0.6

0.171

0.1705

0.1703

0.5 0.4 0.3 0

a)

500

1000

1500 t

2000

2500

b)

Figure 5.9: a) Sudden collapse of a cluster in the three urn system. Shown is the time evolution of the cluster fraction ncl (t) (corresponding to the urn initially containing all of the 50,000 particles) for various values of T0 close to the critical point at T0,c = 0.169829772, while ∆ = 0.3 is held fixed. The cluster lifetime τ grows as the critical point is approached; beyond this point τ becomes infinite as exemplified by the curve for T0 = 0.169, i.e, the cluster is stable there. (from [21]). b) The fractions nk (t), k = 1, . . . , 5 for a five-compartment system at B = 0.33, starting out with all the particles in compartment 3. The solid line is calculated from the flux model of Eqs. (5.4),(5.8)–(5.9), and the fluctuating lines are results from molecular dynamics simulations with input parameters corresponding to the actual experiment in Figure 5.6. The cluster collapse occurs at τ = 42 s (from [17]).

ature convention (which was at the root of the unprecedented first-order transition in the case of two urns) the results are very similar. In particular, for K = 3 it is found that the transition between the uniform and the clustered state is always of first order and accompanied by hysteresis. Also the unstable transient state (with two well-filled urns while the third one is nearly empty, n1 = n2 > n3 ) is recovered, and even the sudden collapse phenomenon has been re-examined in the context of this model. All particles are initially put in one urn, and the two parameters T0 and ∆ are chosen such that the system lies just outside the region of stability of the cluster. In Figure 5.9a the time evolution of the cluster fraction (ncl (t)) is shown for different values of T0 close to the critical value of T0,c = 0.1698, at a fixed value of ∆ = 0.3. The behavior is markedly similar to that of a real granular cluster, and for comparison we have included in Figure 5.9b an analogous plot for the five fractions in our five-compartment system of Figure 5.6c and Ref. [16]. After some initial spilling, in both cases a situation is reached where the cluster fraction decreases only slightly. At a certain point, however, it suddenly collapses and the particles are spread out uniformly over the K compartments. The first three curves in Figure 5.9a (for T0 = 0.1710, 0.1705 and 0.1703, respectively) show that the lifetime of the cluster grows strongly as T0 approaches the critical value, in agreement with the divergence found in [16, 17]; and beyond this point the lifetime becomes infinitely long, since the clustered state is stable there, as is indicated by the fourth curve (for T0 = 0.1690). There is only one dissimilarity between the urn model and the Eggers flux model for K compartments, and this concerns the diffusion towards the uniform state after the sudden collapse. In the urn model, the width of the density profile over the urns grows as t1/2 , with

132


the standard exponent 1/2 known from random-walker diffusion, whereas the Eggers model yields an anomalous diffusion exponent 1/3 [16]. These different exponents are related to the small-density behavior of the flux functions, which goes as F (nk ) ∝ nk in the urn model and as F (nk ) ∝ n2k in the Eggers model. Indeed, it can be shown that for the general case F (nk ) ∝ nα k the diffusion exponent is given by 1/(1 + α) [19].

5.6 Horizontally Vibrated System An elegant variation on the Maxwell Demon experiment was introduced by Brey et al. [23], who considered the horizontally vibrated system depicted in Figure 5.10a. It consists of a box, here seen in top view, divided into two equal compartments containing a number of inelastically colliding particles. Energy is injected into the system via the vibrating bottom wall. Just as in the Eggers model, the wall is taken to vibrate in a sawtooth manner (with an amplitude that is much smaller than the mean free path of the particles) and collisions with the walls are taken to be elastic.

a)

b)

Figure 5.10: a) Top view of the horizontally vibrated system. The bottom wall is vibrating in a sawtooth manner with very small amplitude. The two compartments are connected by a gap of height h, which is chosen to be considerably larger than the typical mean free path of the particles. b) Bifurcation diagram showing the asymmetry parameter as function of the dimensionless parameter ξm , which is proportional to the total number of particles Ntot . The open symbols are data from molecular dynamics simulations (for a box of dimensions L = 140d, S = 50d and h = 50d, with d the diameter of the particles), for various values of the restitution coefficient (here denoted by α); the filled symbols are obtained by a direct simulation Monte Carlo method [37]; and the full line is the theoretical prediction from the hydrodynamic model by Brey et al. discussed in the text (from [23]).

The bifurcation diagram for this system is shown in Figure 5.10b. Here the absolute value of the time-averaged asymmetry parameter is plotted as a function of a dimensionless parameter ξm , which is proportional to Ntot [23]. So the total number of particles is the control

5.6

Horizontally Vibrated System

133

parameter here, while all other parameters are kept fixed.3 For a sufficiently small number of particles, a steady state is reached with all particles distributed equally over the two compartments. But if Ntot is increased beyond a critical value, the symmetry is spontaneously broken and the vast majority of the particles clusters together in one compartment. To explain these observations, Brey et al. use a model based on a hydrodynamic description of a vibrated granular gas [23, 36]. They consider a dilute granular gas confined between a vibrating wall at x = 0 and a stationary one at x = L, and assume (for each compartment) that there are density gradients only in the x direction. Within this model, the temperature T (x) is a monotonously decreasing function of x, while the pressure p(x) = p is uniform throughout the whole compartment.4 Its value is found to depend non-monotonically on the number of particles in the compartment (i.e., on ξm,k ): p

∝

2ξm,k + sinh(2ξm,k ) cosh2 ξm,k

≡

f (ξm,k ) ,

k = 1, 2.

(5.14)

The function f (ξ) (i.e, the number-dependent part of the pressure) is depicted in Figure 5.2c. Just like the flux functions of Eggers and Lipowski and Droz in Figs. 5.2a,b, it is a nonmonotonous function, with a maximum at ξm,k = ξ0 ≈ 1.20 (the root of ξ0 tanhξ0 = 1). Again, the non-monotonic form is essential for the clustering phenomenon. For a steady state, the pressures in the two compartments are required to be equal: f (ξm,1 ) = f (ξm,2 ) ,

(5.15)

which may be compared with the analogous condition for the particle fluxes in Eq. (5.1). Now, by inspection of Figure 5.2c, it can be inferred that for a total particle number such that ξm,1 + ξm,2 < 2ξ0 , there is only one solution to the balance equation (5.15), namely the symmetric state ξm,1 = ξm,2 < ξ0 , corresponding to a point on the left flank of the function f (ξ). However, when ξm,1 + ξm,2 > 2ξ0 , apart from the symmetric state (which now corresponds to a point on the right-hand flank of f (ξ)), also an asymmetric solution ξm,1 = ξm,2 comes into existence, with one point lying to the left of ξ0 and the other one to the right. This yields the theoretical curve in the bifurcation diagram of Figure 5.10 (right). The asymmetry of the two populations, as measured by , is seen to grow very fast indeed as the total number of particles is increased beyond the critical value. Just above the critical point, Brey et al. checked that the asymmetric solutions are described by = ±0.31(ξm − ξ0 )1/2 , indicating again (just as in the models of the previous sections) a critical behavior with the standard exponent β = 1/2 for second order phase transitions. 3

This choice of control parameter could have been made in the original Maxwell Demon experiment as well (cf. e in Eq. (5.8)). Conversely, the control parameter used in that experiment (the velocity the dimensionless number B af of the vibrating wall) is not a valid control parameter here; Brey et al. find that the bifurcation diagram is not altered by modifying the velocity of the bottom, as long as it is large enough to keep the system fluidized [23].

4

There is an interesting difference compared with the system described by Eggers at this point. The model of Brey et al. has a constant pressure p, and a temperature T that decreases with the distance from the vibrating wall. By contrast, in Eggers’ model, p decreases with the height above the vibrating bottom, while T is constant.

134


As a final observation, in the limit for large particle numbers (ξ → ∞) the function f (ξ) takes on the value 2, and the equal-pressure point on the left flank of the function lies at ξ = 0.63923, corresponding to a certain small number of particles. Any further increase of Ntot is then absorbed by the dense compartment, while the particle number in the dilute compartment remains constant at its fixed value. Naturally, this picture ultimately breaks down when the particle number becomes so large that the dilute-gas approximation is no longer valid and excluded volume effects start to play an important role.

5.7 Double Well Model In this section we briefly touch upon an interesting idea put forward by Cecconi et al. [24], who portray the Maxwell Demon experiment as an escape problem, in the spirit of the Kramers model for reactions occurring via thermally activated barrier crossing [38, 39]. The two compartments are modelled as a double-well potential U (x) = −ax2 + bx4 /4, in which two particles are moving, driven by a stochastic heat bath and colliding inelastically. In the noninteracting case, without collisions, a particle is known to spend an average time τ ∝ exp(∆U/kB Tb ) in a well before it escapes to the other one, where ∆U is the height of the potential barrier it has to overcome, and Tb the temperature of the heat bath. The Boltzmann constant kB is in the present context set equal to unity. A similar expression for the escape time can be derived for the interacting case, with both ∆U and the granular temperature modified to take into account the particle interaction. Two different situations have to be distinguished: (1) the uniform state with one particle in each well, with escape time τ1 , and (2) the clustered state with both particles occupying the same well, with escape time τ2 . The two escape times can be written as [24]: Wn , n = 1, 2. (5.16) τn ≈ exp Tn The barrier height for single occupation of a well is unaltered, W1 = ∆U , but in the case of double occupation it is reduced to W2 = ∆U − δU < ∆U due to the excluded volume effect of the two particles. This effectively acts as a repulsive force. Similarly, the effective temperature T1 for a singly occupied well is still nearly equal to Tb , but the temperature T2 for double occupation is considerably smaller, due to the cooling effect of the inelastic particle collisions [24]. The reduction factor depends on the value of the restitution coefficient, and for a typical value e = 0.90 the ratio T2 /Tb is already found to be as low as 0.73. The inelastic collisions effectively act as an attractive force between the particles. Thus, the excluded volume effect and the inelasticity are two competing effects, the competition of which can be tuned by varying either the heat bath temperature Tb or the restitution coefficient e. If one increases Tb , keeping all other parameters fixed, both escape times τ1 and τ2 naturally decrease (following a linear behavior in an Arrhenius plot of logτn vs. 1/Tb [24]) but they do so at different rates. For high values of Tb (corresponding to vigorous shaking in the Maxwell Demon experiment) it is found that τ1 > τ2 , meaning that the system spends most of the time in the uniform state 1. However, when Tb is decreased below a certain critical

5.8

Further Directions

135

value, the situation is reversed and τ2 becomes larger than τ1 . This heralds the transition to the clustered regime, since now the particles spend most of the time in the clustered state 2. Also here the clustering can be related to a non-monotonicity in the behavior of the flux F (n) from a well, with n = 1, 2 the number of particles in the well. Recognizing that F (n) ∝ τn−1 , the flux for high Tb (when no clustering occurs) is an increasing function of n, whereas for low Tb (in the clustered regime) it decreases with n. A further bridge between the flux model and the stochastic heat bath description is given in Ref. [40]. The double-well system exhibits, apart from the clustering effect, also another intriguing phenomenon that we want to mention: At low temperature Tb and strong inelasticity, the two particles tend to synchronize their jumps from one well to the other. That is, the relative motion of the two particles becomes frozen out due to the repeated collisions, and together they behave as a two-particle “molecule” [24].

5.8 Further Directions There are many interesting generalizations and applications of the Maxwell Demon experiment. For example, with an eye to practical applications, where granular materials are rarely mono-disperse, one may replace the identical particles of the original experiment by a mixture of different particles. In a two-compartment system filled with a bi-disperse granular gas consisting of large and small particles of the same material, Mikkelsen et al. found experimentally that the clustering became competitive [41, 42]. Depending on the shaking strength, the clustering can be directed either towards the compartment initially containing the majority of the large particles, or to the one containing mainly small particles. The experimental observations are quantitatively explained by a bidisperse extension of the Eggers flux model [41]. In the same context, Barrat and Trizac [43] describe molecular dynamics simulations on a bidisperse extension of the horizontally vibrated system discussed in Section 5.6. The granular mixture in this case consists of heavy and light particles (of equal size but different mass density), with an equal number of each species. As in the monodisperse system of Brey et al. [23], the clustering transition is triggered by increasing the total particle number. It is found that the clustering of the heavier particles is considerably more complete than that of the lighter ones, since the latter are more mobile. The clustering effect can also be employed to generate spontaneous directed transport, as was recently shown by van der Meer et al. in two different compartmentalized systems [44]. The first system is the so-called “granular fountain”, which is created via a simple modification to the original Maxwell Demon setup depicted in Figure 5.3: A small hole is drilled, close to the bottom, in the wall between the compartments. As a result, in the clustered situation a net flow of particles goes through this hole back into the dilute compartment; and each particle that enters the dilute compartment soon picks up sufficient kinetic energy to jump over the wall again. This leads to a stable convection roll, with an upward particle flow in the dilute (hot) compartment and a downward flow in the dense (cold) one. In [44] this system is studied experimentally, numerically, and theoretically, and all observed phenomena are well reproduced by an adapted version of Eggers’ flux model. The second, related system is the “granular ratchet”. This is a cyclic array of K compartments connected alternately by walls of a certain height h (over which the particles can jump)

136


and very high walls, extending to the top lid of the setup, with a small hole near the bottom. The transport in this system takes the form of a net particle current, in either the clockwise or counterclockwise direction, arising spontaneously as a result of the granular clustering effect [44]. Finally, we want to mention the close analogy between granular clustering|) and the traffic jam problem [45–47]. If one divides the highway in cells of say 500 m (which in large parts of the Dutch highway network has actually been realized via induction loops in the asphalt, which permanently monitor the traffic) and considers the density ρk of cars in these cells, the traffic flow from one cell to the next can be described by a flux function F (ρk ). This function is in fact one of the basic notions in traffic analysis, known as the “fundamental diagram”, and has been measured innumerable times in real life [48, 49]: Just as the flux functions in Figure 5.2, F (ρk ) show a non-monotonic dependence on ρk . At low densities the cars flow freely from cell to cell and F (ρk ) increases with ρk , whereas at densities above 30 vehicles/km/lane the cars start to interact (braking, passing, and other manoeuvres all reduce the forward velocity) and F (ρk ) decreases. Even though the precise behavior of F (ρk ) in the congested regime needs further clarification, since the traffic data here do not follow a one-dimensional curve but instead are scattered over a two-dimensional region in the (F, ρ)plane [50, 51], the fundamental diagram approach with a one-humped function has already proved able to give an adequate description of traffic-jam formation on the highway. Recent reviews on this highly relevant application of the Maxwell Demon effect can be found in Refs. [48, 49].

Acknowledgments We thank J. Javier Brey, Jens Eggers, Isaac Goldhirsch, and Adam Lipowski for kindly permitting us to reproduce their figures. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), and RM and DvdM acknowledge financial support.

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[6] S. McNamara and W. R. Young, Inelastic collapse and clumping in a one-dimensional granular medium, Phys. Fluids A 4, 496 (1992); S. McNamara, Hydrodynamic modes of a uniform granular medium, Phys. Fluids A 5, 3056 (1993); S. McNamara and W. R. Young, Inelastic collapse in two dimensions, Phys. Rev. E 50, R28 (1994). [7] Y. Du, H. Li, and L.P. Kadanoff, Breakdown of hydrodynamics in a one-dimensional system of inelastic particles, Phys. Rev. Lett. 74, 1268 (1995). [8] A. Kudrolli, M. Wolpert, and J.P. Gollub, Cluster formation due to collisions in granular material, Phys. Rev. Lett. 78, 1383 (1997). [9] J. Duran, Sand, Powders, and Grains: An Introduction to the Physics of Granular Materials (Springer, New York, 2000). [10] H.J. Schlichting and V. Nordmeier, Strukturen im Sand, Math. Naturwiss. Unterr. 49, 323 (1996). [11] J. Eggers, Sand as Maxwell’s demon, Phys. Rev. Lett. 83, 5322 (1999). [12] K. van der Weele, D. van der Meer, M. Versluis, and D. Lohse, Hysteretic clustering in granular gas, Europhys. Lett. 53, 328 (2001). [13] D. van der Meer, K. van der Weele, and D. Lohse, Bifurcation diagram for compartmentalized granular gases, Phys. Rev. E 63, 061304 (2001). [14] U. Marini Bettolo Marconi and M. Conti, Dynamics of vibrofluidized granular gases in periodic structures, Phys. Rev. E 69, 011302 (2004). [15] D. van der Meer, K. van der Weele, and D. Lohse, Coarsening dynamics in a vibrofluidized compartmentalized granular gas, JSTAT 1, P04004 (2004). [16] D. van der Meer, K. van der Weele, and D. Lohse, Sudden collapse of a granular cluster, Phys. Rev. Lett. 88, 174302 (2002). [17] D. van der Meer and K. van der Weele, Breakdown of a near-stable granular cluster, Prog. Theor. Phys. Suppl. 150, 297–312 (2003). [18] K. van der Weele, D. van der Meer, and D. Lohse, Birth and sudden death of a granular cluster, in “Advances in Solid State Physics 42”, edited by B. Kramer, Proc. DPG Spring Meeting 2002 in Regensburg, Germany (Springer, Berlin, 2002) 371. [19] K. van der Weele, D. van der Meer, and D. Lohse, Maxwell’s demon in a granular gas, in “Order and Chaos 8”, edited by T. Bountis and Sp. Pnevmatikos, Proc. 14th Summerschool on Nonlinear Dynamics, Chaos, and Complexity, July–August 2001, Patras, Greece (K. Sfakianakis Publ., Thessaloniki, 2003) 239. [20] A. Lipowski and M. Droz, Urn model of separation of sand, Phys. Rev. E 65, 031307 (2002). [21] F. Coppex, M. Droz, and A. Lipowski, Dynamics of the breakdown of granular clusters, Phys. Rev. E 66, 011305 (2002). [22] I. Bena, F. Coppex, M. Droz, and A. Lipowski, Yang–Lee zeros for an urn model for the separation of sand, Phys. Rev. Lett. 91, 160602 (2003). [23] J. J. Brey, F. Moreno, R. García-Rojo, and M. J. Ruiz-Montero, Hydrodynamic Maxwell demon in granular systems, Phys. Rev. E 65, 011305 (2001). [24] F. Cecconi, A. Puglisi, U. Marini Bettolo Marconi, A. Vulpiani, Noise activated granular dynamics, Phys. Rev. Lett. 90, 064301 (2003).

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[25] A.S. Leff and A.F. Rex, Maxwell’s Demon: Entropy, Information, Computing (Adam Hilger, Bristol, 1990). [26] J.P. Gollub and J.S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys. 71, S396 (1999). [27] T. Shinbrot and F.J. Muzzio, Noise to Order, Nature 410, 251 (2001) [28] E. Bodenschatz, W. Pesch, and G. Ahlers, Recent developments in Rayleigh–Bénard convection, Annu. Rev. Fluid Mech. 32, 709 (2000). [29] K.H. Andersen, M.-L. Chabanol, and M. van Hecke, Dynamical models for sand ripples beneath surface waves, Phys. Rev. E 63, 066308 (2001); K.H. Andersen, M. Abel, J. Krug, C. Ellegaard, L.R. Søndergaard, and J. Udesen, Pattern dynamics of vortex ripples in sand: Nonlinear modeling and experimental validation, Phys. Rev. Lett. 88, 234302 (2002). [30] J. T. Jenkins and S. B. Savage, A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles, J. Fluid Mech. 130, 187 (1983); J. T. Jenkins and M. W. Richman, Boundary conditions for plane flows of smooth, nearly elastic, circular disks, J. Fluid Mech. 171, 53 (1986); S. McNamara and S. Luding, Energy flows in vibrated granular media, Phys. Rev. E 58, 813 (1998). [31] P.M. Chaikin and T.C. Lubensky, Principles of condensed matter physics (Cambridge University Press, Cambridge, 1995). [32] F.Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982). [33] P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach to Mechanics (Dover, New York, 1990). [34] A. Lipowski and M. Droz, Moment ratios for an urn model of sand compartmentalization, Phys. Rev. E 66, 016118 (2002). [35] G.M. Shim, B.Y. Park, and H. Lee, Analytic study of the urn model for separation of sand, Phys. Rev. E 67, 011301 (2003). [36] J.J. Brey, M.J. Ruiz-Montero, and F. Moreno, Boundary conditions and normal state for a vibrated granular fluid, Phys. Rev. E 62, 5339 (2000). [37] G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Clarendon Press, Oxford, 1994). [38] H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7, 284 (1940). [39] P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990). [40] U. Marini Bettolo Marconi and A. Puglisi, Statistical mechanics of granular gases in compartmentalized systems, Phys. Rev. E 68, 031306 (2003). [41] R. Mikkelsen, D. van der Meer, K. van der Weele, and D. Lohse, Competitive clustering in a bidisperse granular gas, Phys. Rev. Lett. 89, 214301 (2002). [42] R. Mikkelsen, K. van der Weele, D. van der Meer, M. Versluis, and D. Lohse, Competitive clustering in a granular gas, Phys. Fluids 15(9), S8 (2003). [43] A. Barrat and E. Trizac, A molecular dynamics ’Maxwell Demon’ experiment for granular mixtures, Molecular Physics 101, 1713 (2003).

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[44] D. van der Meer, P. Reimann, K. van der Weele, and D. Lohse, Spontaneous ratchet effect in a granular gas, Phys. Rev. Lett. 92, 184301 (2004). [45] O. Biham, A.A. Middleton, and D. Levine, Self-organization and a dynamical transition in traffic-flow models, Phys. Rev. A 46, R6124 (1992). [46] B.S. Kerner and P. Konhäuser, Cluster effect in initially homogeneous traffic flow, Phys. Rev. E 48, R2335 (1993); Structure and parameters of clusters in traffic flow, Phys. Rev. E 50, 54 (1994). [47] E. Ben-Naim, P.L. Krapivsky, and S. Redner, Kinetics of clustering in traffic flow, Phys. Rev. E 50, 822 (1994). [48] D. Chowdhury, L. Santen, and A. Schadschneider, Statistical physics of vehicular traffic and some related systems, Phys. Rep. 329, 199 (2000). [49] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73, 1067 (2001). [50] B.S. Kerner, S.L. Klenov, and D.E. Wolf, Cellular automata approach to three-phase traffic theory, J. Phys. A: Math. Gen. 35, 9971 (2002). [51] K. van der Weele, W. Spit, T. Mekkes, and D. van der Meer, From granular flux model to traffic flow description, in “Traffic and Granular Flow ’03”, S. Hoogendoorn, S. Luding, and D.E. Wolf (Eds.) (Springer, Berlin, 2004).

Part III

Dense Granular Flow

6 Continuum Modeling of Granular Flow and Structure Formation Igor S. Aranson and Lev S. Tsimring

6.1 Introduction A quantitative description of large-scale behavior of granular media requires reliable continuum models. For the dilute granular flows such models are derived from the kinetic theory of inelastically colliding particles with hard-core interaction [1, 2]. The kinetic theory is capable of predicting correct behavior even for moderately dense granular flows (up to 60% of random close-packed density), however, it fails to describe slow dense flows, especially in the vicinity of fluidization transition. In the last few years there have been many experimental [3–9] and theoretical [10–16] studies that explored a broad range of granular flow conditions from rapid dilute flows to slow dense flows, as well as the details of the shear-driven fluidization transition. There are several phenomenological approaches to the description of slow dense flows and the fluidization transition. In Ref. [17], Savage proposed a continuum theory based on the so-called associated flow rule that relates the strain rate and the shear stress in plastic frictional systems. Averaging strain-rate fluctuations yields a Bingham-like constitutive relation in which the shear stress has a viscous as well as strain-rate independent parts. According to this theory, the stress and strain-rate tensors are always co-axial and, furthermore, it also postulates that the viscosity diverges as the density approaches the close-packing limit. Losert et al. [5] (see also [18]) proposed a similar hydrodynamic model based on a Newtonian stress–strain constitutive relation with density-dependent viscosity, without a strain-rate independent component. As observed in Ref. [5], the ratio of the full shear stress to the strain rate diverges at the fluidization threshold. This was also interpreted in Ref. [5] as a divergence of the viscosity coefficient when the volume fraction approaches the randomly packed limit. This description works only in a fluidized state and cannot properly account for hysteretic phenomena in which static and fluidized states co-exist under the same external load, such as stick-slip oscillations [4], avalanching [8], or shear band formation. In many granular flows of interest, static and dynamics regions co-exist under the same external load conditions. Examples of such hysteretic phenomena include stick-slip oscillations [4], avalanching [8], or shear band formation. This calls for a unified theory which would be applicable both in the flowing regime and in the static regime. Another popular approach to this class of problem is based on representation of the granular flow as an effective two-phase system. This kind of theory for the dense near-surface flows was pioneered by Bouchaud, Cates, Ravi Prakash and Edwards (BCRE) [19] and subsequently

144

6 Continuum Modeling of Granular Flow and Structure Formation

developed by de Gennes, Boutreux and and Raphaël [20–22]. This model only applies to a class of granular systems which are spatially separated into two phases, static bulk and thin rolling near-surface layer. In our recent papers [23–27] we proposed a different approach based on the order parameter description of granular matter. The value of the order parameter specifies the ratio between static and fluid parts of the stress tensor. The order parameter was assumed to obey dissipative dynamics governed by a free energy functional with two local minima. This description based on the separation of static and fluid components of the shear stress, calls for an alternative definition of viscosity as a ratio of the fluid part of the shear stress to the strain rate. Since the fluid shear stress vanishes, together with the strain rate, the viscosity coefficient in our theory is expected to remain finite at the fluidization threshold. We assumed the simplest Newtonian friction law, so the viscosity coefficient is a constant. This model yields a good qualitative description of many phenomena occurring in granular flows, such as hysteretic transition to chute flow, stick-slip regime of a driven near-surface flow, structure of avalanches in shallow chute flows, etc. In this chapter we review this continuum theory and illustrate its applicability to the avalanche dynamics and shear granular flows. The chapter is organized as follows. Section 6.2 introduces the continuum description of partially fluidized flows based on the relaxational dynamics of the order parameter. In Section 6.3 we apply our theory to the description of avalanches in granular layers. In Section 6.4 we validate our continuum model by 2D soft-particle molecular dynamics simulations. In Section 6.5 we calculate the stress and velocity distributions in a thick granular layer under non-zero gravity driven by a moving heavy upper plate. In Section 6.6 we apply our model to the problem of stick-slip behavior in granular friction.

6.2 Order Parameter Description of Partially Fluidized Granular Flows In this section we review our continuum theory [23–26] which provides an alternative approach to the formulation of the constitutive relations in partially fluidized granular flows. The mass, momentum and energy conservation equations have the usual form Dν Dt Du ν Dt DT ν Dt

= −ν∇ · u,

(6.1)

= −∇ · σ + νg,

(6.2)

= −σ : γ˙ − ∇ · q − ε,

(6.3)

where ν is the density, u is the velocity field, T = (uu−u2 )/2 is the granular temperature, D/Dt = ∂t + (u · ∇) is the material derivative, g is the gravitational acceleration, σαβ is the stress tensor, q is the energy flux vector, γ˙ αβ = ∂α uβ + ∂β uα is the strain-rate tensor, and ε is the energy dissipation rate. These three equations have to be supplemented by the constitutive relations for the stress tensor σ, energy flux q, and the energy dissipation rate ε. For dilute systems, there is a linear

6.2

Order Parameter Description of Partially Fluidized Granular Flows

145

˙ relationship between stress σ and strain rate γ, ˙ αβ − µγ˙ αβ σ αβ = [p + (µ − λ)Trγ]δ

(6.4)

where p is pressure, and µ and λ are shear and bulk viscosities [1]. In the dense flow regime, the density of the granular matter is close to the random closed packed limit, ν = νc , and the granular matter can be considered incompressible ∇ · u = 0. This also allows us to drop the energy equation (6.3). Next, we separate the stress tensor σ into two parts, a static (contact) part σ s , and a fluid part σ f . The latter is taken in a purely Newtonian form f = pf δαβ − µf γ˙ αβ σαβ

(6.5)

where pf is the “partial” fluid pressure, µf is the viscosity coefficient associated with the fluid stress tensor which is different from µ0 introduced for the full stress tensor. As we shall see in the following, unlike µ0 , µf does not diverge as ν → νc . In our original model [23, 24] we simply take µf = const. We postulated that the fluid part of the off-diagonal components of the stress tensor is proportional to the off-diagonal components of the full stress tensor with the proportionality coefficient being a function of the order parameter ρ, f σyx = q(ρ)σyx .

(6.6)

This formula implies that the off-diagonal parts of stress tensors σ, σ f , σ s are collinear, which as we see in Section 6.4, is a reasonable assumption for our 2D simulations. However, it does not have to hold in all cases. In those circumstances, the order parameter ρ would become a tensor itself. Equation (6.6) stipulates the equation for the static part of the off-diagonal stress components, s = (1 − q(ρ))σyx . σyx

(6.7)

f,s f,s Both fluid and solid parts of the stress tensor are assumed symmetric, σyx = σxy . This assumption is confirmed by our numerical simulations (see below). We choose a fixed range for the order parameter such that it is zero in a completely fluidized state and one in a completely static regime. Thus the function q(ρ) has the property q(0) = 1, q(1) = 0. In Refs. [23, 24] for simplicity we took q(ρ) = 1 − ρ. Similar relationships can be postulated for the diagonal terms of the stress tensor, f σxx s σxx

f = qx (ρ)σxx , σyy = qy (ρ)σyy , s = (1 − qx (ρ))σxx , σyy = (1 − qy (ρ))σyy ,

(6.8) (6.9)

where the scaling functions qx,y (ρ) can differ from q(ρ). Combining Eqs. (6.5)–(6.9), we obtain the constitutive relation in the closed form, σαβ = pf δαβ /qα (ρ) − µf γ˙ αβ /q(ρ), α, β ∈ {x, y}.

(6.10)

146


Figure 6.1: Free energy density F vs the order parameter ρ for three different values of δ.

We assume that, because of strong dissipation in dense granular flows, the order parameter ρ has purely relaxational dynamics controlled by the Ginzburg–Landau equation, Dρ ∂F (ρ) = D∇2 ρ − Dt ∂ρ

(6.11)

Here D is the diffusion coefficient and F (ρ) is the free energy density. We postulate that F (ρ) has two local minima at ρ = 1 (solid phase) and ρ = ρf < 1 (fluid phase) to account for the bistability near the solid–fluid transition. The relative stability of the two phases is controlled by the parameter δ. For small δ the solid phase is more favorable, and vice versa. Typical profiles of the free energy density as a function of ρ for different δ are shown in Figure 6.1. The control parameter δ is determined by the stress tensor which in Ref. [23, 24] was taken to be a linear function of φ∗ = max |σmn /σnn |, where the maximum is sought over all possible orthogonal directions m and n. In Refs. [23, 24] we assumed the simple quartic form for the free energy density, ρ ρ(ρ − 1)(ρ − δ)dρ. (6.12) F (ρ) = It is easy to see that in the interval 0 < δ < 1, Eq. (6.11) has two stable uniform solutions ρ = 0, 1 corresponding to fluid and solid states, and one unstable solution ρ = δ. In subsequent work [25, 26] we extracted the specific form of the free energy from the 2D soft particle molecular dynamics simulations (see below Section 6.4). Further, according to observations, there are two angles which characterize the fluidization transition in the bulk granular material: an internal friction angle tan−1 φ1 such that, if φ ≤ φ1 , the static equilibrium is unstable, and the “dynamic repose angle” tan−1 φ0 such that at φ < φ0 , the “dynamic” phase ρ = 0, is unstable. Values of φ0 and φ1 depend on microscopic properties of the granular material, and in general they do not coincide. Typically there is a

6.3

Avalanches on an Inclined Plane

147

range in which both static and dynamic phases co-exist (this is related to the so-called Bagnold hysteresis [28]). The simplest form of the control parameter δ is given by δ = (φ2 − φ20 )/(φ21 − φ20 ).

(6.13)

The momentum conservation equation (6.2) together with Eqs. (6.10)–(6.12) represent a closed set of continuous equations which after being augmented by appropriate boundary conditions, can describe a variety of interesting granular flows such as avalanches in thin chute flows, drum flows, stick-slip oscillation in surface-driven flows, etc. [23, 24]. Since a rigorous derivation of the continuum equations and constitutive relations for dense partially fluidized flows from first principles is not possible at the moment, we tested the assumptions and fitted the parameters of the model (6.2), (6.10)–(6.12) using available experimental and molecular dynamics simulations.

6.3 Avalanches on an Inclined Plane In this section we will apply our theory to the description of avalanches in thin granular layers on a rough inclined plane. This problem has been studied experimentally in Ref. [8]. We consider an initially flat layer of dry cohesionless grains of thickness h on a sticky surface tilted by angle ϕ to the horizon. We introduce a Cartesian coordinate frame aligned with the unperturbed (flat) surface of the tilted layer with the z axis normal to the surface, and the x axis oriented downhill (see inset in Figure 6.2). Coordinate z = 0 corresponds to the position of the (unperturbed) free surface where the stress is absent, and z = −h corresponds to the bottom of the layer. In the case of stationary shear flow in a flat layer, the force balance conditions yield: σzz,z + σxz,x = −g cos ϕ , σxz,z + σxx,x = g sin ϕ,

(6.14)

where the subscripts after the commas mean partial derivatives. The solution to Eqs. (6.14) in the absence of lateral stresses σyy = σyx = σyz = 0, is given by σzz = −g cos ϕ z , σxz = g sin ϕ z , σxx,x = 0.

(6.15)

Thus, in a stationary flow there is a simple relationship between shear and normal stresses, σxz = − tan ϕσzz independent of the flow profile. In static equilibrium, the force balance 0 0 0 0 = − tan ϕσzz . Since by assumption σzz = σzz , we obtain σxz = σxz . also gives σxz In the case of a thin layer on an inclined plane, the most “unstable” yield direction is 0 0 /σzz | = tan ϕ. In the following, we parallel to the plane, so we can simply write φ∗ = |σxz will consider spatially and temporally inhomogeneous granular flows. In such flows the layer thickness varies, and the parameter φ becomes a variable determined by the local slope of the free surface. We will limit ourselves to the case of small deviations of the local slope from the unperturbed value φ∗ , i.e., |∂x h| tan ϕ. Let us now discuss the boundary conditions for the order parameter and velocity. At the bottom z = −h we set ρ = 1, since the granular medium should be in a solid phase near the no-slip surface. The boundary condition at the free surface is less obvious. For simplicity, we choose the no-flux boundary condition for the order parameter ρz = 0.

148


All components of velocity should be zero at the bottom z = −h. The kinematic boundary condition on the free surface for an incompressible medium can be expressed in the form of the mass conservation law (6.16) ∂t h = − (∂x Jx + ∂y Jy ) , 0 where Jx,y = −h vx,y dz are in-plane components of the flux of the granular material. In a typical situation of the chute flows, the downhill velocity vx is much larger than the orthogonal y-component vy , so the mass conservation constraint can simply be expressed as ∂t h = −∂x J.

(6.17)

The velocity vx is determined from the constitutive relation Eq. (6.10) with the no-slip boundary condition vx = 0 at z = −h. The mass conservation law Eq. (6.17) can be rewritten in terms of the variable δ which is related to the gradient of the local thickness ∂x h = φ − φ∗ . If we assume that the difference between the critical values φ0,1 is small, (φ1 − φ0 )/φ1 1, which is the case for most granular flows, and the plane tilt is close to critical, φ ≈ φ0,1 , then from Eq. (6.13) we obtain (see also Figure 6.2, inset) 1 φ = ∂x h ≈ − (δ − δ0 ). β

(6.18)

Here β = 1/(φ1 − φ0 ) > 0, δ0 = const corresponds to the flow with constant thickness h. Because of the no-slip boundary condition at the bottom of the chute, for shallow layers the flow velocity is small, so the convective flux of the order parameter can be neglected, and the material derivative Dρ/Dt in Eq. (6.11) can be replaced by ∂t ρ, ∂t ρ = ∇2 ρ + ρ(1 − ρ)(ρ − δ).

(6.19)

Here we used for simplicity the quartic form of the free energy Eq. (6.12).

6.3.1 Stability of Simple Solution The stability of the granular flow in a uniform layer of constant thickness h (see Figure 6.2, inset) can be studied assuming that δ = δ0 is a constant, compare Eq. (6.18). Eq. (6.19) always has a stationary solution ρ = 1 satisfying boundary conditions ρ(−h) = 1, ∂z ρ(0) = 0. This solution corresponds to a flat layer of constant thickness h at static equilibrium. For δ > 1 this solution is stable at small h, but loses stability at a certain critical thickness hc > 1. The most unstable mode satisfying the above boundary conditions, has the form ρ = 1 − Aeλt cos(πz/2h), A 1. The corresponding eigenvalue is λ(h) = δ − 1 − π 2 /4h2 .

(6.20)

From this formula we find the neutral curve λ = 0 for the linear stability of the “solid phase” solution ρ = 1 π . hc = √ 2 δ−1

(6.21)

6.3


149

For large enough h, in addition to the trivial state ρ = 1, there exist non-trivial stationary solutions satisfying the above boundary conditions which describe granular flows supported by a constant supply of granular material up-stream. The velocity profile corresponding to a solution ρ(z) can be found from Eq. (6.10): ∂vx 0 = (1 − ρ)σxz = −µ(1 − ρ)z, ∂z

(6.22)

where µ = g sin ϕ/η. The flux of grains in the stationary flow J is given by 0 0 z 0 J= vx (z)dz = −µ (1 − ρ(z ))z dz dz = µ z 2 (1 − ρ)dz. −h

−h

−h

(6.23)

−h

The non-trivial flow solution exists only if the thickness of the layers exceeds the threshold value hs . The value of hs can be found as a minimum of the following integral as a function of ρ0 , the value of ρ at the surface z = 0, hs = min

1

ρ0

dρ ρ4 2

−

2(δ+1)ρ3 3

,

(6.24)

+ δρ2 − c(ρ0 )

where c(ρ0 ) = ρ40 /2 − 2(δ + 1)ρ30 /3 + δρ20 . This integral can be calculated analytically for δ → ∞ and δ → 1/2. It is easy to show that for large δ, the critical solution of Eq. (6.19) has a form ρ = 1 + A cos(kz) with A 1 and k = (δ − 1)1/2 , and therefore, hs (δ) → hc (δ). For δ → 1/2, the critical phase trajectory comes close √ to two saddle points ρ = 0 and ρ = 1, and an asymptotic evaluation of (6.24) gives hs = − 2 log(δ − 1/2)+ const. This expression agrees qualitatively with the empirical formula φ−φ0 ∼ exp[−hs /h0 ] proposed in Ref. [7,8]. The neutral stability curve hc (δ) and the critical line hs (δ) are shown in Figure 6.2. They separate the parameter plane (δ, h) into three regions. At h < hs (δ), the trivial static equilibrium ρ = 1 is the only stationary solution of Eq. (6.19) for chosen BC. Within the region hs (δ) < h < hc (δ) the static equilibrium state co-exists with the stationary flow. For h > hc (δ), the static regime is linearly unstable, and the only stable regime is that of the granular flow. These findings agree qualitatively and even quantitatively with experimental results [7, 8] (see Section 6.3.3 below).

6.3.2 Avalanches in a Single-mode Approximation In the vicinity of the neutral curve (6.21) we may look for a solution of Eqs. (6.10), (6.12) in the form π z + w, (6.25) ρ = 1 − A(x, y, t) cos 2h where A 1 is a slowly varying function of t, x, and y, and w A is a small correction to the single-mode solution. At the neutral curve λ(δ, h) = δ − 1 − π 2 /4h2 = 0 the expression (6.25) with A = const, h = const is the exact solution of the linearized Eq. (6.12). In the vicinity of the neutral curve defined by the condition |λ| 1, the ansatz (6.25) with the slowly-varying functions A, h gives an approximate solution to the full Eq. (6.12).

150


After substituting ansatz (6.25) into Eq. (6.12) we obtain to first order in w the expression ˆ does not contain terms ˆ = H, ˆ where L ˆ = ∂z2 + δ − 1 is the linearized operator, and H Lw linear in w and depends only on A and its derivatives. Applying the solvability conditions 0 ˆ cos(πz/2h)dz = 0, we arrive at H −h At = λA + ∇2⊥ A +

8(2 − δ) 2 3 3 A − A 3π 4

(6.26)

¯ = µ(3π 2 − 16)/3π 3 = 0.146µ. Equation (6.26) should be where ∇2⊥ = ∂x2 + ∂y2 , and α augmented by the mass conservation equation (6.17) ∂J ∂h3 A ∂h =− = −α , ∂t ∂x ∂x

(6.27)

where J was calculated from Eq. (6.23) using ansatz (6.25) and α = 2µ(π 2 − 8)/π 3 = 0.12µ. Since variations of h also change the local surface slope, we replace δ in (6.26) by δ0 − βhx , see Eq. (6.18). In deriving these equations we implicitly assumed that (2 − δ)A2 and A3 are of the same order, i.e., δ ≈ 2. However, qualitatively, a similar equation with a different nonlinearity can be obtained for any δ and h. In order to study the avalanches in two dimensions (x, y) we performed simulations in a system of Lx = 400 dimensionless units in the x direction and Ly = 200 units in the y direction, with the number of grid points 1200 × 600. We triggered avalanches by a localized perturbation introduced near the point (x, y) = (Lx /4, Ly /2) against a uniform static layer h = h0 , A = 0. Close to the solid line in Figure 6.2 we indeed observed avalanches propagating only downhill, with a shape similar to the experimental one [8], see Figure 6.3 a)–c). The avalanche has a triangular shape with the opening angle ψ in which the layer thickness h is decreased with respect to the original value h0 . Because of mass conservation, near the front of the avalanche the layer thickness is greater than h0 , as in experiments. For thicker layers or larger values of δ we obtained avalanches which expand both upand downhill, Figure 6.3 d)–f). Unlike the previous case, the whole avalanche zone is in motion, as new rolling particles constantly arrive from the upper boundary of the avalanche zone. For certain parameters, we observed small secondary avalanches in the wake of the large primary avalanche, see Figure 6.3f). The line separating the triangular/uphill avalanches in the (δ, h) plane can be found by tracking the existence of the uphill front in the 1D system, see Figure 6.2.

6.3.3 Comparison with Experiment To compare the theoretical results with experiments quantitatively, we have to specify the parameters φ0 and φ1 , characteristic length l and time τ , and the viscosity η. The repose angle φ1 can be easily determined from the value of the chute angle corresponding to the vertical asymptote of the stability curve on the experimental bifurcation diagram of Ref. [8]. While the value of φ0 cannot be directly read from the bifurcation diagram, the vertical asymptote to the line bounding the region of existence of avalanches in Ref. [8] gives the value of the angle φ˜0 at which the front between granular solid and fluid does not move, i.e., δ = 1/2.

6.3


151

Figure 6.2: Comparison of theoretical and experimental phase diagrams. Lines obtained from theory, symbols depict experimental data from Ref. [8]. Solid line and circles limit the range of existence of avalanches, line and triangles correspond to the linear stability boundary of the static chute, and the line and crosses denote the boundary between triangular and uphill avalanches for β = 3.15, α = 0.025 (or, correspondingly, µ = 0.2).Inset: Schematic representation of a chute geometry.

a

b

c

d

e

f

Figure 6.3: Grey-coded images demonstrating the evolution of a triangular avalanche for t = 50, a) t = 200, b) and 250, c) and up-hill avalanche for t = 40, a); t = 100, b) and 180, c). White color corresponds to maximum height of the layer, and black to minimum height. Parameters of Eqs. (6.26), (6.27) are: α = 0.15, β = 0.25, δ = 1.2 and h0 = 3 for a)–c) and α = 0.05, β = 0.25, δ = 1.07 and h0 = 5.5 for d)–f). A small secondary avalanche is seen on the image f).

152


Thus we can express our parameter δ through φ˜0 , φ1 . For the experimental parameters of Ref. [8], tan−1 φ˜0 ≈ 25◦ and tan−1 φ1 ≈ 32◦ . It gives β ≡ 1/2(φ1 − φ˜0 ) ≈ 3.15. Based on the comparison with experimental results for velocity decay in stationary flow Ref. [29], as a characteristic length l we can take the mean diameter of the grain d which for experiment Ref. [8] was 0.24 mm. Solid lines in Figure 6.2 indicate theoretical stability boundaries, which correspond well to the experimental findings. The line separating the triangular and uphill avalanches is not universal, and depends on the parameter α in Eq. (6.27). From the numerical solution of Eqs. (6.26) and (6.27) we find that the best fit to experimental data occurs for α ≈ 0.025 (correspondingly µ ≈ 0.2).

6.4 Fitting the Theory with Molecular Dynamics Simulations The constitutive relations introduced above in an ad hoc manner, can be specified on the basis of molecular dynamics simulations. To model the interaction of individual grains we use the so-called soft-contact approach. The grains are assumed to be non-cohesive, dry, inelastic disk-like particles. Two grains interact via normal and shear forces whenever they overlap. For the normal impact we employ the spring-dashpot model [11]. This model accounts for repulsion and dissipation; the repulsive component is proportional to the degree of overlap, and the velocity dependent damping component simulates the dissipation. The model for shear force is based upon the technique developed by Cundall and Strack [30]. It incorporates tangential elasticity and Coulomb laws of friction. The elastic restoring force is proportional to the integrated tangential displacement during the contact and limited by the product of the friction coefficient and the instantaneous normal force. The grains possess two translational and one rotational degrees of freedom. The motion of a grain is obtained by integrating Newton’s equations with the forces and torques produced by its interactions with all the neighboring grains and walls of the container. The advantages and limitations of the employed contact force model were thoroughly studied by a number of authors [10–12]. In fact this is the simplest model which allows us to account for both static and dynamic friction.

6.4.1 Order Parameter for Granular Fluidization: Static Contacts vs. Fluid Contacts At any moment in time all contacts are classified as either “fluid” or “solid”. A contact is considered “solid” if it is in a stuck state (Ft < µt Fn ) and its duration is longer than a typical time of collision t∗ . The first requirement eliminates long-lasting sliding contacts, and the second requirement excludes short-term collisions pertinent for completely fluidized regimes. We choose a typical collision to last t∗ = 1.1tc . When either of the requirements is not fulfilled, the contact is assumed “fluid”. We define the order parameter as the ratio between space–time averaged numbers of “solid” contacts Zs and all contacts Z within a sampling area [31], ρ(y) = Zsi /Z i .

(6.28)

6.4

Fitting the Theory with Molecular Dynamics Simulations

153

This definition includes at least two limiting cases: when a granulate is in a static state and when it is strongly agitated, i.e., completely fluidized. In fact, in the former quiescent state all contacts are stuck and ρ = 1. In the fluidized case Zs is zero and Z is small but finite, therefore ρ = 0. Let us note that the order parameter ρ just introduced is expected to be as sensitive to the degree of fluidization as the stress tensor. A small rearrangement of the force network may result in strong fluctuations of either field, while such quantities as the solid fraction or granular temperature will remain virtually constant.

6.4.2 Stress Tensor The full stress tensor σ consists of a contact (virial) part and the Reynolds part. In turn, the contact stress tensor can be split into the “solid contact” component, σ s , and the “fluid contact” component, (σ f ), in the same way as was done with the contacts themselves. Combining the “fluid contact” component with the Reynolds stress, we obtain the full stress tensor as a sum of two parts σ = σf + σs .

(6.29)

The “fluid” part of the stress tensor σ f is due to short-term collisional stresses and the Reynolds stresses, whereas the solid part σ s accounts for persistent force chains. The Reynolds contribution to the stress is negligibly small in the vicinity of the phase transition, but comes into play when the granulate is highly agitated. In the system which is neither completely rigid nor completely fluidized we expect the coexistence (in time and space) of both phases. A particular grain may have both types of contact, thus contributing to both σ f and σ s .

6.4.3 Couette Flow in a Thin Granular Layer We determined the form of the free energy potential and the constitutive relations by simulating the fluidization transition in a thin (50 × 10) granular layer between two “rough plates” under fixed pressure P and zero gravity conditions (Figure 6.4, inset). The layer was chosen narrow enough, so that the properties of the granular layer were constant across the layer. We slowly increased two opposite forces F1 = −F2 applied to the plates along the horizontal x axis beyond the fluidization threshold and then reduced them back to zero. For the applied pressure P = 40 the strain rate remains zero, and the order parameter is close to one until the shear stress reaches a certain critical value σ1 ≈ 12.6. Above the yield stress, the strain rate jumps to ≈ 0.35, and the order parameter drops to ≈ 0.15. At larger |σyx |, the strain rate increases faster than linearly, and the order parameter rapidly approaches zero. The return curve corresponding to the diminishing of the shear stress follows roughly the same path, and then continues to another (smaller) value of the shear stress (σ2 ≈ 9.4). At this value the layer jams, the strain rate returns to zero, and the order parameter jumps back to one. These simulations were performed at several different values of the compressing pressure P . Data for different pressure values in the flow regime fall onto the same universal curve if one normalizes the shear stress by the pressure (see Figure 6.4). Based on this data, we can make a

154


1

F

0.8

P Pext=40

ρ

Ly

30 20

0.6

Lx

-F

P

0.4

0.2

0 0.1

0.2

0.3

δ=−σxy/Pext

0.4

0.5

Figure 6.4: The order parameter as a function of the normalized shear stress in a thin Couette geometry. Open symbols correspond to the decrease in the applied shear stress, closed symbols correspond to the increase in the stress. Inset: Sketch of the granular shear flow model.

simple analytic fit of this curve as f (ρ, δ) = (1 − ρ) ρ2 − 2ρ∗ ρ + ρ2∗ exp[−A(δ 2 − δ∗2 )] = 0

(6.30)

with ρ∗ = 0.6, A = 25, δ∗ = 0.26 (see Figure 6.4, line) and use it in the polynomial expansion of the free energy density which enters the order parameter equation (6.11): ρ f (ρ, δ)dρ. (6.31) F (ρ) =

6.4.4 Fitting the Constitutive Relation The next step is to fit the constitutive relation from MD simulations. To this end, we use f and the “static the same Couette flow simulations, but now we analyze the “fluid stress” σαβ s stress” σαβ separately during our ramp-down simulations at three different values of P . Plotf ting σyx /σyx as a function of the order parameter ρ for different P (Figure 6.5a), we observe that all data collapse onto a single curve which is well fitted by q(ρ) = (1 − ρ)2.5 . The corresponding expression for the function q(ρ) adapted in Section 6.3 was simply q = 1 − ρ which is only slightly different from that extracted from MD simulations, see Figure 6.5a.

6.5

Surface-driven Shear Granular Flow Under Gravity

155

f,s f,s The fluid as well as solid parts of the stress tensor are nearly symmetric, σyx = σxy , f therefore the ratio σxy /σxy is described by the same scaling function q(ρ). On the other hand, the same procedure for the diagonal elements of the stress tensor yields a noticeably different functional form (see Figure 6.5b). Furthermore, a small but noticeable difference is evident f f /σxx and σyy /σyy . More detailed analysis shows that in fact fluid parts of the between σxx f f diagonal components of the stress tensor σxx and σyy are nearly identical, and the difference is due to the solid part of the normal stresses. This observation is consistent with the fact that the diagonal terms of the static stress tensor are determined by the details of the external loading. On the other hand, in a completely fluidized state the diagonal terms are all equal to the hydrodynamic pressure pf . In a partially fluidized regime, the diagonal terms of the shear stress can be expressed as

σxx = pf /qx (ρ), σyy = pf /qy (ρ).

(6.32)

Both functions qx,y (ρ) should approach 1 as ρ → 0, but they may have a different functional form to reflect the anisotropy of the static stress tensor. If the fluid pressure pf approaches α . In zero in the solid state as pf ∝ (1 − ρ)α , then qx,y ∝ (1 − ρβx,y )α , so σxx,yy ∝ βx,y our simple Couette flow, the diagonal stress tensor components can be well fitted by qx (ρ) ≈ (1 − ρ)1.9 and qy (ρ) ≈ (1 − ρ1.2 )1.9 (see Figure 6.5b). We observe that even in a partially fluidized regime, the “fluid phase” component indeed behaves as a real fluid with a wellbehaved “partial” pressure pf which is zero in a solid state at ρ = 1 and is becoming the full pressure in a completely fluidized state ρ = 0. Plotting the fluid shear stress versus the strain rate, we can test the validity of the Newf vs γ˙ for three diftonian model for the stress–strain relation (6.5). Figure 6.6 shows −σyx ferent pressures P = 20, 30, 40. At small γ˙ all three lines are close to the same straight line f = 12γ, ˙ which indicates that the Newtonian scaling for fluid shear stress holds reasonably σyx well. The deviations at large γ˙ are evidently caused by variations of temperature and density in the dilute regime. Note that in contrast, the full shear stress does not go to zero as γ˙ → 0 (see Figure 6.6, inset), so a viscosity coefficient conventionally defined as the ratio of the full shear stress to the strain rate diverges at the fluidization threshold as observed in Ref. [5]. Combining the Newtonian law for the fluid stress–strain dependence with the order parameter scaling of the fluid stress tensor, we arrive at the relationship between the full stress tensor and the strain-rate tensor (6.10) with µf = 12, qx (ρ) = (1 − ρ)1.9 , qy (ρ) = (1 − ρ1.2 )1.9 , q(ρ) = (1 − ρ)2.5 .

6.5 Surface-driven Shear Granular Flow Under Gravity Now we can apply the theoretical model (which was validated above on the basis of molecular dynamics simulations of a thin Couette flow with no gravity) to a description of the shear granular flow in a thick granular layer under gravity, driven by the upper plate which is pulled in a horizontal direction. A similar system has been studied experimentally in Refs. [4, 9]. We simulated up to 20,000 particles in a rectangular 2D box under a heavy plate which was moved either with a constant speed Vx or a constant force Fx . Periodic boundary conditions were imposed in a horizontal direction. After a transient, a quasi-stationary fluidization and

156


1 0.8

a

q=1-ρ

f

σ /σ

0.6 0.4 0.2

q(ρ)

0 1 0.8

P=20 P=30 P=40

qy(ρ)

f

σ /σ

0.6

b

0.4 0.2 0 0

qx(ρ) 0.2

0.4

ρ

0.6

0.8

1

f Figure 6.5: Ratios of the fluid stress components to the corresponding full stress components σαβ /σαβ vs. ρ for three different pressures P : a) Shear stress components, closed symbols σyx , open symbols σxy , line is a fit q(ρ) = (1 − ρ)2.5 . b) Normal stress components, closed symbols σxx , open symbols σyy , lines are the fits qx (ρ) = (1 − ρ)1.9 , qy (ρ) = (1 − ρ1.2 )1.9 . Dashed line indicates q(ρ) = 1 − ρ used in Section 6.3.

shear flow established in the near-surface layer, while near the bottom, grains remained in a nearly static jammed regime. Typical vertical profiles of the density, flow velocity, and the order parameter are given in Figure 6.7. The density of grains remains nearly constant very close to the maximum random packing density value, except for a narrow near-surface boundary layer. The horizontal velocity decays exponentially from the plate in agreement with experimental data [4,9]. The vertical profiles of the order parameter demonstrate a continuous transition from a fluid state near the upper plate to a solid state below. We can compare the stationary vertical profiles of the order parameter and the horizontal velocity with theoretical predictions. In most of our numerical simulations we specified the velocity of the upper plate rather than the applied force. That allowed us to study the regimes of slow dense flows which would be unstable had we applied a constant shear force. The shear stress tensor component σyx in the stationary regime was indeed constant across the layer. However, due to slippage near the moving plate, the relation between the plate speed and the shear stress is complicated. We do not address the issue of boundary conditions here as it is the subject of a separate study (see for example [32]). Here we simply use the values which are obtained in numerical simulations.

6.5

Surface-driven Shear Granular Flow Under Gravity

15

157

15

-σxy

10

5

f

-σxy

10

0

0

0.2

0.4.

γxy

0.6

0.8

5 P=20 P=30 P=40 0

0

0.2

0.4

.

γxy

0.6

0.8

Figure 6.6: Fluid part of shear stress vs strain rate for three different external pressures, the straight line ˙ Inset: full shear stress vs strain rate. is a constant viscosity fit σ f = 12γ.

In the stationary regime, the relevant stress tensor components are specified as follows: σyy = P + (H − y),

(6.33)

σyx = σxy = const

(6.34)

where P is the external pressure applied to the upper wall. Both at the top and the bottom plate we impose no-flux boundary conditions for the order parameter, ∂y ρ(0) = ∂y ρ(H) = 0. The stationary shear flow solution of the continuum equations can be found numerically as follows. Since the components of the full stress tensor are assumed known, we solve the time-dependent Eq. (6.11) until the solution reaches a stationary state. The resulting solution for the order parameter is then used to obtain the velocity profile by integrating the constitutive relation (6.10) with the specified function q(ρ) and the viscosity µf from the bottom (y = 0) up. Since the grains are strongly compressed near the rough bottom plate due to gravity, we assume the no-slip boundary condition for the horizontal velocity at y = 0. The momentum conservation Eq. (6.2) is satisfied automatically. The obtained profiles of velocity and the order parameter were compared with our 2D molecular dynamics simulations. The results of the comparison between the velocity and order parameter profiles obtained in simulation and by using the continuum theory, are shown in Figure 6.8. The only fitting

158


1

ρ

0.8 0.6 0.4 0.2

Vx

0 40 P=10 V=5 P=50 V=5 P=10 V=50 P=50 V=50

30 20 10 0

ν

0.5

0

0

20

40

y

60

80

Figure 6.7: Density ν, horizontal velocity Vx , and the order parameter ρ profiles in a deep granular layer driven by upper moving plate for a system of 5, 000 particles in the cell of Lx ×Ly = 50×100 units, for P = 10, Vx = 5 (line with squares), P = 10, Vx = 50 (line with upward triangles), P = 50, Vx = 5 (line with circles), and P = 50, Vx = 50 (line with downward triangles). See Ref. [26] for other parameter values.

parameter is the diffusion constant D in the order parameter equation that has not been determined in our test-bed analysis. From our simulations we have concluded that the diffusion coefficient depends on the local stress tensor. However, more elaborate numerical experiments are required to quantify this dependence. The vertical profiles of the order parameter and the horizontal velocities agree reasonably well with theory. However, for low pressures, the horizontal velocity profiles deviate from the numerical data presumably because the viscosity coefficient is no longer a constant in a dilute region near the top plate.

6.6 Stick-Slips and Granular Friction In this section we focus on the essentially non-stationary phenomenon, a stick-slip motion of the heavy plate pulled with a constant speed V0 via a soft linear spring with elasticity constant K on the top of a granular layer. This problem was studied experimentally in Ref. [4]. The force acting on a plate from the spring is F = K(x − V0 t − x0 ), where x is the position of

6.6

Stick-Slips and Granular Friction

159

ρ,Vx

3 2

b

1

ρ,Vx

0 3

a

2 1 0 0

20

40

y

60

80

Figure 6.8: Profiles of the order parameter and velocity in a thick granular layer driven at the surface by a heavy moving plate for P = 10, Vx = 5, a) and P = 10, Vx = 50, b) Circles show profiles of ρ and squares show velocity Vx profiles correspondingly. Lines show the theoretical results obtained from the continuum model(6.11), (6.30), empty symbols indicate numerical data.

the plate front and x0 is its initial position (at t = 0 the spring is unloaded). All variables and parameters are scaled by the gravity acceleration g, particle size dp and mass mp . The equations of motion for the plate read x˙ = V, mV˙ = σ − κ(x − V0 t − x0 )

(6.35)

where m = M/Lx , κ = K/Lx , and σ is the (negative) yx shear stress component (we assume that it does not depend on y). For a given σ, the order parameter can be found from the GL equation (compare Eq. (6.30)): 2 2 (6.36) ∂t ρ = D∂y2 ρ − (ρ − 1) ρ2 − 2ρ∗ ρ + ρ2∗ e−A(δ −δ∗ ) with ρ∗ = 0.6, δ∗ = 0.25, A = 25, D = 2, = 0.02, where δ(y) = −σ/p(y) = −σ/(m − y). We assume no-flux boundary conditions for the order parameter ρ (0) = ρ (−Ly ) = 0 and no-slip condition V (0) = V (−Ly ) = 0 for the velocity. The characteristic feature of this equation is the bistability in a certain range of δ, 0.25 < δ < 0.3. This feature in fact gives rise to the stick-slip oscillations of the plate. To close the system, we can find the shear stress σ by integrating the constitutive relation for shear component of the stress σij (6.5), see [26]

160


and Section 6.4. Using the boundary condition V (−Ly ) = 0 from Eq. (6.5) one derives σ = −µf V

0 −Ly

−1 (1 − ρ)

2.5

dy

(6.37)

These equations can be used only within the slip event, because during the stick phase the OP ρ → 1, and velocity V (0) → 0, so Eq. (6.37) becomes indeterminate. In this case, the inertia of the plate can be ignored, the shear stress coincides with the spring force, σ = −κ(x − V0 t − x0 ),

(6.38)

and V can be found from the constitutive relation (6.5), V = V (0) = −µ−1 f σ

0

−Ly

(1 − ρ)2.5 dy

(6.39)

These equations were integrated numerically using the finite difference method. When the plate velocity reaches a certain small threshold value Vtr = 10−3 , we switch from Eqs. (6.36), (6.38), (6.39) to Eqs. (6.35)-(6.36), (6.37). We should note that in a long stick phase the plate velocity turns into machine zero and the order parameter into the fixed point ρ = 1, from which it cannot escape without perturbations. To avoid this behavior, we added a small random perturbation (of amplitude ξ = O(10−6 )) to the order parameter at every time-step. We chose the parameters of our model (mass per unit length, m, and the spring constant per unit length κ) based on experimental values [4]. A direct comparison between our 2D system and the 3D experiment is difficult. However, we can assume that our 2D system represents a one-particle diameter “slice” of the full 3D system. In this case, the mass and the spring constant in the experimental system should be scaled by the area of the top plate. Using nondimensional units based on the gravitational acceleration, particle mass and diameter, we find that the mass of the plate 10 g corresponds to m ≈ 20, and the spring constant k = 135 N/m corresponds to κ ≈ 2.7. These values of m and κ have been used in most of our calculations. We also ran simulations for stiffer springs with κ = 10 and 20. The velocities are scaled by (gdp )1/2 . Thus for smooth particles explored in Ref. [4] the velocity scale is approximately 30 mm s−1 . The main control parameter is the pulling velocity V0 . At large V0 we obtain a stationary near-surface shear flow. At small velocities V0 → 0 the model exhibits relaxation oscillations, reminiscent of the normal dry friction between two solids. The spring deflection ∆ = σ/κ grows almost linearly with no flow until it reaches a certain threshold value after which the near-surface layer fluidizes, and the ensuing shear flow relieves the accumulated stress. After that the layer “freezes” again, and the process repeats (Figure 6.9). In agreement with experiment, in the inertia-dominated regime at larger pulling speeds, the deflection of the spring becomes almost sinusoidal. Our study indicates that the transition from sliding to stick-slips is discontinuous, and there is a range of velocities V0 at which sliding and stick-slips co-exist. We find that individual slip events bear strong similarity to experiments (compare Figure 6.10 with Figure 13 of [4]).

6.6

Stick-Slips and Granular Friction

161

8

∆

7 6 5 4 0.4

0

50

0

50

100

150

100

150

V0=0.01 V0=0.1200

V

0.3 0.2 0.1 0

time

200

Figure 6.9: Oscillations of the deflection ∆, (top plot) and the plate velocity V , (bottom plot) in GL theory for m = 20, κ = 2.7, ξ = 10−6 : relaxation oscillations for V0 = 0.01, quasi-sinusoidal oscillations at V0 = 0.1. 0.5

0.45

|σ|/m

0.4

0.35

0.3

0.25

0.2

0

0.1

V

0.2

0.3

Figure 6.10: Structure of an individual slip event m = 20, K = 2.7, V0 = 0.001, = 0.02: normalized shear stress |σ|/M vs. plate velocity V .

162


6.7 Conclusions We have presented the phenomenological continuum theory of dense granular flows in which the constitutive relations depend on the order parameter describing the fluidization transition. This theory allowed us to describe such phenomena as avalanches in thin granular layers and near-surface shear flows in deep granular layers. We performed a series of numerical simulations of 2D wall-driven granular flows in order to quantify the order parameter dynamics and the resulting constitutive relations. We defined the order parameter as a ratio of the number of static contacts to the total coordination number averaged over a small mesoscopic volume. Using simulations of a thin Couette flow between two rough plates, we calculated the free energy density for the order parameter. Simulations confirmed that the ratio of the shear to the normal stress in the bulk of the granular flow can parameterize the stationary states of the order parameter equation. The same simulations allowed us to determine the detailed structure of the constitutive relation. We split the total stress tensor into fluid and solid components, in which the former are comprised of the Reynolds stresses and the stresses transmitted through shortterm collisions, while the latter are formed by the force chains through persistent contacts. The ratio of fluid and solid stress components is indeed determined by the order parameter through scaling functions q(ρ), qx,y (ρ). Remarkably, the fluid component of the stress tensor is a linear function of the strain rate γ˙ in the slow dense-flow regime. This justifies the Newtonian scaling of the stress–strain relationship adopted in the theory. Our results suggest that the granular material under shear stress is similar to a multi-phase system with the fluid phase “immersed” into the solid phase. The fluid phase behaves as a simple Newtonian fluid for small shear rates when the density is almost constant, but exhibits shear thinning at larger shear rates when the density begins to drop. We observed that the Reynolds part of the fluid R ∼ γ˙ 2 . We anticipate that for very large shear rates, shear stress obeys the Bagnold scaling σyx when the Reynolds stress becomes dominant, the overall stress tensor should exhibit Bagnold scaling locally. Many issues still remain open. The spatially non-uniform dynamics of the order parameter requires a more detailed study. We found that the diffusion constant postulated in Eq. (6.11) appears to be a function of the normal shear stress as well as the local strain rate. However, we do not have sufficient numerical data to provide a quantitative description of this dependence. It would be of interest to analyze the propagation of a fluidization front in a granular layer prepared in a meta-stable static regime. Such simulations could provide an insight into the mechanisms of the local coupling of the order parameter. The theory presented in this chapter is essentially a two-dimensional one. The generalization of the order parameter concept to the 3D case can be a very challenging task. In particular, one may expect that in certain situations the scalar order parameter field will be insufficient, and some more general description has to be applied. The molecular dynamics algorithm employed is based on a number of approximations. These approximations, however well tested and widely accepted [10–12], directly affect the results of our fitting the continuum model. For example, if one replaces the Hookean model of particle interaction with a Hertzian one, an appreciable difference in the structure of the order parameter may be observed. More numerical work is needed to quantify the relationships between the microscopic parameters of the system (nature of collisions, restitution coefficient, friction, etc) and the parameters of the continuum model.

References

163

Finally, our simulations were limited to 2D systems so the resulting continuum theory can only be directly applicable to 2D systems. While we anticipate that the structure of the model remains similar in the 3D case, the specific form of the fitting functions should change. Our future work should allow us to perform a comparison of the 3D model not only with numerical simulations, but also with experimental data. We applied our theory to the essentially non-stationary phenomenon of stick-slip oscillations in a thin granular layer driven by a heavy plate attached to the elastic spring [27]. The model parameters were extracted from the two-dimensional “testbed” simulations described in previous sections. The time evolution of spring deflection obtained from the solution of non-stationary equation (6.11) was found to be quite similar to experimental results: it has a form of well-separated slip events at small pulling speeds and quasi-sinusoidal oscillations near the onset of sliding motion at large speeds.

Acknowledgments The authors are indebted to Dmitri Volfson for carrying out numerical simulations and to B. Behringer, J. Gollub, T. Halsey, and P.G. de Gennes for useful discussions. This work was supported by the Office of the Basic Energy Sciences at the US Department of Energy, grants W-31-109-ENG-38, and DE-FG03-95ER14516. Simulations were performed at the National Energy Research Scientific Computing Center.

References [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10] [11] [12]

[13] [14]

J.T.Jenkins and M.W.Richman, Phys. Fluids, 28, 3485 (1985). I. Goldhirsch, Ann. Rev. Fluid Mech. 35, 267 (2003). C.T. Veje, D.W. Howell, and R.P. Behringer Phys. Rev. E 59, 739 (1999). S. Nasuno, A. Kudrolli, A. Bak, and J. P. Gollub, Phys. Rev. E. 58, 2161 (1998). W. Losert, L. Bocquet, T.C. Lubensky, J.P. Gollub, Phys. Rev. Lett. 85, 1428 (2000); L. Bocquet, W. Losert, D. Schalk, T.C. Lubensky, J.P. Gollub, Phys. Rev. E 65, 011307 (2002). D.M. Mueth et al., Nature (London) 406, 385 (2000); D.M. Mueth, Phys. Rev. E 67, 011304 (2003). O. Pouliquen, Phys. Fluids 11, 542 (1999). A. Daerr and S. Douady, Nature (London) 399, 241 (1999); A. Daerr, Phys. Fluids 13, 2115 (2001) J.C. Tsai, G.A. Voth, and J.P. Gollub, Phys. Rev. Lett. 91, 064301 (2003) P. A. Thompson and G. S. Grest, Phys. Rev. Lett. 67, 1751 (1991). J. Schäfer, S. Dippel, and D. E. Wolf, J. Phys. I France 6, 5–20 (1996). D. Ertas, G.S. Grest, T.C. Halsey, D. Levine, and L.E. Silbert, Europhys. Lett. 56, 214 (2000); L. E. Silbert, D. Ertas, G. S. Grest, T. C. Halsey, D. Levine, and S. J. Plimpton, Phys. Rev. E 64, 051302 (2002). E. Aharonov, D. Sparks, Phys. Rev. E 60, 6890–6896 (1999). E. Aharonov, D. Sparks, Phys. Rev. E 65, 051302 (2002).

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[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

[32]


A. Lemaitre, Phys. Rev. Lett. 89, 064303 (2002); Phys. Rev. Lett. 89, 195503 (2002). D. Ertas and T.C. Halsey, Europhys. Lett. 60, 931 (2002) S.B.Savage, J. Fluid Mech., 377, 1 (1998). L. Bocquet, J. Errami, and T. C. Lubensky Phys. Rev. Lett., 89 184301 (2002). J.-Ph. Bouchaud, M.E. Cates, J. Ravi Prakash, and S.F. Edwards, J. Phys. I France 4, 1383 (1994); Phys. Rev. Lett. 74, 1982 (1995). P.G. de Gennes, in Powders and Grains, R. Behringer and Jenkins (eds), p. 3, Balkema, Rotterdam (1997). T. Boutreux, E. Raphaël, and P.G. de Gennes, Phys. Rev. E 58, 4692 (1998). T. Boutreux and E. Raphaël, Phys. Rev. E 58, 7645 (1998). I.S. Aranson and L.S. Tsimring, Phys. Rev. E, 64, 020301 (2001). I.S. Aranson and L.S. Tsimring, Phys. Rev. E, 65, 061303 (2002). D. Volfson, L.S. Tsimring, and I.S. Aranson, Phys. Rev. Lett. 90, 254301 (2003). D. Volfson, L.S. Tsimring, and I.S. Aranson, Phys. Rev. E, 68, 021301 (2003). D. Volfson, L.S. Tsimring, and I.S. Aranson, Phys. Rev. E, 69, 031302 (2004). R.A. Bagnold, Proc. Roy. Soc. London A 225, 49 (1954); ibid., 295, 219 (1966). T.S. Komatsu, S. Inagaki, N. Makagawa, S. Nasuno, Phys. Rev. Lett. 86, 1757 (2001). P. A. Cundall and O. D. L. Strack, G´ eotechnique 29, 47 (1979). In L.E. Silbert, D. Ertas, G.S. Grest, T.C. Halsey, and D. Levine, Phys. Rev. E 65, 051307 (2002), a similar quantity, a fraction of sliding contacts averaged over the whole flow region, was introduced. For a chute flow, it exhibited a first-order transition as a function of the chute angle. K. Hui, P.K. Haff, J.E. Ungar, and R. Jackson, J. Fluid Mech. 145, 223 (1984); J.T. Jenkins and M.W. Richman, J. Fluid Mech. 171, 53 (1986).

7 Contact Dynamics Study of 2D Granular Media: Critical States and Relevant Internal Variables Farhang Radjaï and Stéphane Roux

7.1 A Geometry–Mechanics Dialogue Cohesionless granular materials share marked plastic properties which reflect their common granular structure. Flow and irrecoverable deformations of these materials are to be understood as consequences of the relative displacements of the grains (seen as rigid particles interacting via contact and friction) caused by an external mechanical action. Moreover, the contact and friction law at the grain level can faithfully be described as a simple Coulomb friction law. Yet, this incredibly simple (and correct) picture of granular plasticity cannot be described by an equally simple model entirely based on the properties of the grains and their organization in space. Recent interdisciplinary research on the matter has even transformed the status of sand as a rather old poetic symbol of simplicity into a paradigm of complexity! [1, 2] Obviously, a complex behavior emerges from the geometry of discrete grains, geometry which controls both the “state” of the medium (motivating the geometrical nature of internal variables), and its evolution, since the velocity field has to be compatible with the local steric constraints of non-penetrable particles. Thus, although we know the local behavior of the grains and their interactions very well — it could hardly be made simpler — still, proposing a macroscopic continuum description appears to be a challenge.

7.2 A Granular Model The experimental observation, about four decades ago, of force inhomogeneity and structural anisotropy in model particle assemblies suggested that a detailed description of granular microstructure should provide a key for understanding the quasi-static rheology of granular materials [3, 4]. Numerical simulations thus provide a natural tool to address this question [5, 6]. The elements are idealized grains (discs or polygons in two dimensions, spheres or polyhedra in three dimensions). The equations of motion of each grain are integrated by taking into account contact interactions and body or boundary forces and displacements. It happens that, even with the most basic ingredients, a complex behavior is observed, as in experiments. The simulations provide detailed information about grain motions and contact forces. Fascinating phenomena, such as the bimodal transmission of stresses [7] and collective particle motions [8, 9] at intermediate scales between the grain size and the system size, can be ob-

166

7 Contact Dynamics Study of 2D Granular Media

served. Yet, such, often intriguing, phenomena have in many ways helped more to reinforce the mystery of sand than to unravel it [2]. In the following, we will consider only an assembly of rigid discs in two dimensions, with a narrow size distribution so that the packing does not naturally crystallize, and yet can be compared to an ideal single-sized particle system as considered theoretically. The grains are treated as rigid and frictional. Although this idealization corresponds to typical experimental systems such as sand with a stress level which does not induce grain crushing and plastic deformation at contacts, it introduces some difficulties in the numerical simulations because of the so-called non-smooth character of the contact law. Nevertheless suitable numerical algorithms have been proposed recently to faithfully account for such contact laws, as discussed in the following section. Such algorithms are known under the generic name of “contact dynamics” [10–12]. Moreover, even for quasi-static loading, unstable configurations occur quite frequently (see Section 7.4.6). A global energetic analysis of the simulation reveals a total energy dissipation which is not consistent with an analysis based on the succession of only stable configurations. A considerable amount of energy is dissipated through micro-instabilities. This shows the importance of a faithful account of the dynamics even for quasi-static studies. In other words, we need to tackle inelastic collisions with multi-contact aggregates. Although the mathematical problem of such collisions is ill-posed, again contact dynamics is a technique which provides mechanically acceptable solutions to the problem, allowing thereby to carry out the calculations past these dynamic micro-events. A restitution coefficient has yet to be introduced as an additional internal parameter. In all the illustrations presented in this work, perfectly inelastic collisions, together with a coefficient of friction equal to 0.5, have been assumed although other choices could have been made. The influence of these parameters on the stress–strain and volume-change behavior will not be discussed in this paper.

7.2.1 Contact Dynamics In order to address the question of the transition from a micro to a macro scale description, it is important to be able to resort to a reference system where all detailed information is available. For two decades, such a description has been offered by distinct element numerical simulations. However, different numerical algorithms may be used to describe the grain contacts. One of the most often used schemes introduces an elastic-frictional contact law accounting for the grain elasticity [5]. Such a description requires a very precise contact law to approach the rigid-particle limit. The price to pay is a very fine time discretization and hence a rather time-consuming computation. This is even more true when dynamic collisions are to be described. Recently, however, an alternative approach, called “contact dynamics”, has been proposed that deals directly with the infinitely stiff contact law, and more generally, non-smooth laws [10–12]. Moreover, this approach naturally takes into account collisions with a genuine dynamics (and with neither viscous regularization nor fictitious inertia adjusted to speed up convergence). In all illustrations presented in the following a contact dynamics algorithm has been used. In numerical simulations performed up to large cumulative deformation, the boundary conditions tend to introduce spurious effects in the vicinity of the walls, and may even lead

7.2

A Granular Model

167

Figure 7.1: An example of a granular packing considered in this work. Bi-periodic boundary conditions are introduced in order to limit wall effects. The simulation cell is shown by solid lines.

to a strongly inhomogeneous strain field. The latter should, however, be as homogeneous as possible to extract a meaningful intrinsic mechanical response comparable to a macroscopic behavior law. In order to reduce these edge effects, we have introduced bi-periodic boundary conditions (as shown in Figure 7.1) which consists of imposing such a periodicity along both directions of the simulation cell for the stress and the strain fields. The displacement field, however, is not periodic if the mean strain is non-zero. It contains an affine component in addition to a bi-periodic field. This requires a special treatment of these two components with additional characteristics associated with the non-periodic part of the displacement, such as a fictitious inertia. However, the latter can be shown to play a negligible role in the results obtained. The origin of space being now an arbitrary parameter, the stress–strain and more generally any geometrical observable, has to be translationally invariant, and hence the strain remains spatially homogeneous on a large scale.

7.2.2 Driving Modes We introduce here different types of kinematics which all include a shear component which can be extended to large-amplitude strains. In the following, the macroscopic (large-scale) velocity field will have the following expression: Dx + (γ − R)y u˙ = (7.1) Dy + (γ + R)x In this expression, γ is the shear rate and 2D is the volumetric strain rate, tr(∇u). R is the ˙ rotation rate Ω. 0 γ 0 −R D γ ˙ (7.2) Ω= ˙ = ˙dev = γ 0 R 0 γ D

168


In our simulations, the confining pressure is set to a fixed value p, so that the volumetric strain rate 2D results from computation and it evolves together with the solid fraction ρ. The volumetric strain rate scaled by the deviatoric strain rate defines the dilation angle: tan ψ =

2D γ

(7.3)

The cumulative deviatoric strain ε = γ dt will be used as a control parameter. The rotation parameter R may appear to be a gratuitous parameter whose role may be anticipated to vanish because of Galilean invariance. In fact, this will be shown not to be true because of the naturally developing anisotropy of the medium resulting from privileged contact orientations. Therefore, the rotation can be considered as a tool which can be used in order to study how this anisotropy follows the macroscopic strain. Thus we will study the role of this global rotation rate as compared to the deviatoric strain, measured by R. Among other possible choices, two of them correspond to standard cases: (i) R = 0 corresponds to pure shear or biaxial test; (ii) R = γ corresponds to simple shear. However, we will see that even for large values of the ratio R/γ, the effect of rotations is weak. Hence, we will resort to instantaneous changes in strain orientation through load reversal and cyclic shear in order to reveal off-critical states. The deviatoric stress state adjusts itself so as to accommodate the imposed strain. The chosen description of rigid frictional particles does not set any stress scale. This is the reason why part of the stress has to be set through the boundary conditions (spherical component p of the stress tensor in this work). The internal angle of friction φ for a given stress state is defined from the deviatoric component q of the stress tensor by sin φ = q/p .

(7.4)

7.3 Macroscopic Continuum Description In the following, we will describe the results of numerical simulations on our model system. Prior to this presentation, it is useful to discuss the framework which is expected to host the constitutive law of the continuum description.

7.3.1 Constitutive Framework The particles are considered as rigid, and no elastic behavior is introduced at the contact level (assumed to be purely frictional). Therefore, no elasticity is anticipated in the continuum description. The Coulomb friction law itself can be seen as a perfectly rigid plastic law, with forces and displacements playing the role of stress and strain. The macroscopic strain will thus be entirely plastic, and hence the behavior will be rigid-plastic. Moreover for all contacts, at yield, the orientation of the velocity is not normal to the Coulomb force cone. This deviation from the normality rule, is not expected to be be removed when going to the continuum limit. Therefore, the continuum plastic law will be non-associated. This implies that the direction of the incremental plastic strain will have to be specified by an additional law, the plastic flow rule, independently from the description of the rigid domain in stress

7.3

Macroscopic Continuum Description

169

space where no plastic strain is expected. This additional information only gives access to the direction of the strain (its magnitude resulting from the fact that the state remains on the yield surface). A convenient way of characterizing this direction is the relative weight of volumetric to deviatoric strain rates, which is given by the dilation angle as defined in Eq. (7.3). We also note that the rigid character of the particles as well as the Coulomb friction law do not introduce any stress scale. As a result, the continuum description should obey the same scale-free property. Hence, the yield surface in stress space should be invariant through an arbitrary dilation, i.e., it has to be a cone. In the test cases used in the computation, only the ratio q/p, or the friction angle φ defined through Eq. (7.4) will be meaningful. Plasticity implies that the incremental behavior may be history dependent, and obviously that will be the case for granular media. Thus, the next question to address is how to characterize the internal state of the medium. At the level of individual particles, the problem appears as deterministic if we know the geometry of the system. Therefore, all internal state variables defined at a continuum level must have a geometrical interpretation. For example, the cumulative plastic strain is disqualified by the previous argument as a legitimate internal variable, in contrast to several phenomenological plastic laws used in this field [13]. In the simplest case with no hardening, the friction and dilation angles are sufficient to entirely characterize the behavior. Although φ and ψ appear to play a dual role (for instance, “normality” or “associatedness” can simply be written φ = ψ), it is worth noting that φ is defined as a secant property (ratio of two total stress components) whereas ψ is a tangent property (ratio of incremental (or rate of) strain components). In our case, we do expect to observe some hardening and anisotropy developing in the medium. Nevertheless, we will keep the same vocabulary. However, it is important to note that some simple constructions based on these two quantities will no longer be operational (e.g. the use of Mohr’s plane for anisotropic media is pointless).

7.3.2 Relation Between Micro- and Macro-descriptors As the medium is discrete, the passage from the velocity field defined at particle centers and continuum fields from which strain rates can be computed, is not direct. However, a number of authors have contributed to the solution of this problem and defined a few mathematical prescriptions which provide a satisfactory answer. Slight variations may appear between different formulations. However, they concern only small-scale estimates, and at a large scale they all point to the same answer. An analogous passage from discrete forces, transmitted at contact points, to a global stress field is also to be performed. In this passage, once the boundary ∂Σ of a domain Σ of area A is chosen, the average strain over the domain can be computed according to [14] 1 (n ⊗ u + u ⊗ n)dS (7.5) = 2A ∂Σ where n is the normal to the boundary ∂Σ, and u is the displacement vector. The latter, however, is not yet defined apart from a discrete set of points. One suitable choice (although not the only one) is to construct a linear interpolation of the velocity all along each segment connecting the centers of particles in contact. With those segments, one can form elementary

170


polygons, which are the irreducible loops of particles in contact. The velocity interpolation within the cell can easily be shown not to affect the expression of . Thus we arrive at an operational definition of the macroscopic strain over these polygonal domains and, by construction, over a collection of an arbitrary number of them. Algebraic manipulations of the above formula using the linear interpolation of the velocities along each edge, allows one to arrive at different expressions for the strain. For the stress, a similar procedure can be introduced at the particle level, based on the following expression [15–17] r (i) (i) (n ⊗ f + n(i) ⊗ f(i) ) (7.6) σ= A i where f(i) is the force transmitted at contact (i) on the particle, and n(i) is the corresponding normal, r is the grain radius and A is the area of the Voronoï cell (based on the contact network) centered on the particle. The sum extends over all contacts surrounding a particle. It is interesting to note that the elementary irreducible polygons over which stress and strains can be defined are dual to each other: it is a loop of contacting particles for the strain, and a Voronoï cell for the stress. Each of these two elementary structures can be characterized by statistical distributions which will give some information about the mean number of contacts per particle (the coordination number z), the mean number of particles involved in each irreducible loop, the orientation of these contacts (and in particular, a moment of the distribution known as the fabric tensor), and so on. If these two structures are obviously different, they are also intimately related to each other, but in a non-trivial fashion.

7.3.3 Internal Variables Knowing the position {xi } of each particle (i), together with the boundary conditions, makes the problem well-posed, and suited to a direct numerical simulation. Thus, the internal variables are contained in the set of variables {xi }i=1,N . The difficulty is to coarse-grain our description and thus retain only a few geometrical parameters to characterize the internal state, so that the stress–strain relation can be captured. The phase volume fraction provides simple first-order information. In the case of a dry granular material, we distinguish the solid phase from the pore phase. The solid fraction ρ (volume fraction occupied by the grains) is known to influence strongly the shear strength and stress–strain behavior [18]. As an alternative to the solid fraction, considering that particles can only interact at contact, we may envisage another parameter, namely the mean coordination number, z, defined as the average number of contact neighbors of a grain. It can also be used as a descriptor for the average compactness of the structure. In fact, each of these two indicators has strengths and weaknesses. The coordination number is fragile in the sense that contacts may be created or opened by minute displacements. For instance, a uniform dilation of an arbitrary small amplitude will open all contacts. This is in contrast with the solid fraction which appears to be a more robust parameter. On the other hand, one also expects that a change in coordination number, even with a moderate impact on the solid fraction, is sufficient to alter significantly the tangent mechanical properties. Moreover, considering those two quantities in sufficiently large-scale samples, or over large enough

7.4

Numerical Results

171

strains, with slowly varying strain fields, will presumably link ρ and z, so that this choice may be a matter of convenience. Those two parameters are, however, simple scalars. In the following, we will see that scalar parameters are unable to describe the state of the medium which appears to be sensitive to the orientation of the shear, and not only its magnitude. This implies that at least a secondorder tensor has to be introduced as an internal variable. Along the same lines as those which lead to the introduction of the coordination number, the higher-order microstructural information based on contacts will be the fabric tensor defined as the volume average of the diadic tensor product of contact normals F = n ⊗ n. The tradition is to construct such a tensor normalized through the contacts [17, 19]. This provides naturally a tensor of unit trace (tr(F) = 1), so that only the deviatoric component contains information. It is to be noted that we may normalize the fabric tensor by the number of particles rather than that of contact. This is mathematically equivalent, but now the trace of the fabric tensor is the coordination number, tr(F) = z, so that the 0th and the 1st order information are contained in the same object. The introduction of the fabric tensor can be described in an equivalent way by introducing the statistical distribution of contact orientations P (θ) [14, 16, 17]. By symmetry, P is a πperiodic function. Expansion of P (θ) on a Fourier basis P (θ) =

1 {1 + a cos(2θ − 2θF )} + h.o.t. π

(7.7)

with a truncation at second order gives an equivalent information to the fabric tensor: the major principal direction of F is θF , and the deviatoric part of F is (a/2)tr(F). The tensor F might still appear insufficient to characterize the internal state, although it appears as the strict minimum. In fact, the exact number of internal variables is a matter of judgment of the accuracy and generality of the behavior law to be extracted. For instance, for cyclic loading, it is conceivable that a finer-scale description of the environment of particles may be required because of memory effects (“echo” of contact opening or closing). In contrast, if only simple shear were to be considered in a single direction (monotonic shear), a simple scalar internal variable may be sufficient to parameterize the system evolution. It is also important to emphasize that, as the description is coarse-grained, some details are naturally lost. Hence in any theoretical attempt to relate micro and macro scales, it will be important to be able to generate representative volume elements showing correct statistics from only partially available information. Below, we will document the evolution of the fabric tensor and coordination number (through z, a and θF ) along various stress paths.

7.4 Numerical Results 7.4.1 Critical States The mechanical behavior of a granular packing depends strongly on its initial preparation (solid fraction, fabric, . . . ). However, when sheared over large enough strains (and without persistent localization of strain), it may lose the memory of its initial state and reach a state which only depends on the imposed strain, the so-called “critical” state [20,21]. In Figure 7.2a

172


0.5 (a)

q/p

0.4 0.3 0.2

dense loose

0.1 0.0 0.00

0.05

0.85

0.10 ε

0.15

0.20

(b)

0.84 ρ

0.83 0.82 dense loose

0.81 0.80 0.00

0.05

0.10 ε

0.15

0.20

Figure 7.2: Evolution of stress ratio q/p, a) and solid fraction ρ, b) as a function of the cumulative shear strain ε for initially dense and loose systems. Notice the jump of the shear stress from zero to 0.4 in the dense case at the onset of loading.

we show the stress ratio q/p as a function of the cumulative shear strain ε in a simple shear test for two initially different states (one dense and one loose, and both isotropic). Up to intrinsic statistical fluctuations, the stress ratio reaches a plateau and its value is the same for both simulations beyond ε 15%. Other characteristics of these two systems converge to identical values. Figure 7.2b shows the evolution of the solid fraction ρ as a function of ε in the same tests. Again, the same value of solid fraction is reached beyond ε 15%. The critical state stress ratio and solid fraction for our granular system are ρc 0.83 and sin φc 0.3, respectively. In order to properly identify the internal state variables, it is essential to be able to capture the critical state, and hence it is important first to study whether there is only one such critical state or more, and to find the dimensionality of the critical region in the state parameter space.

7.4

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173

The critical-state properties are well documented in soil mechanics literature [18,21]. Both in experiments and simulations with elastic contacts, the critical state density is found to increase with the confining pressure p. However, as discussed in Section 7.3.1, the rigid character of the particles in our simulations, as well as the Coulomb friction law, do not introduce any stress scale. For this reason, in our system the critical-state density is independent of p. However, both with elastic and rigid contacts and in contrast to what is often assumed, the critical state cannot be characterized by a single scalar parameter, namely the solid fraction. To demonstrate this property, it suffices to consider a system sheared over large strains, so that a critical state is reached, and then to reverse the loading. If the state could be characterized by a scalar (e.g., solid fraction), then the direction of the shear would be inessential, and the system would remain at criticality after shear reversal. But, as can be seen in Figure 7.3a, a transient regime occurs. Since the major principal stress direction θσ rotates by π/2 (from 3π/4 to π/4 in our reference system of axes) upon unloading, the quantity shown in Figure 7.3a is q = q cos 2(3π/4 − θσ ). For θσ = 3π/4, we have q = q whereas for θσ = π/4 one has q = −q. In fact, p + q is simply the normal stress along θ = 3π/4. We see that the stress ratio reaches its critical state value qc /p = sin φc 0.3 upon shear reversal (after nearly 10% deformation) in a new direction (θσ π/4 so that q = −q). The corresponding behavior for the solid fraction is shown in Figure 7.3b. We observe that, upon shear reversal, the solid fraction first increases during ∆ε = −10% to reach a peak. Then, dilation takes place and the density reverts to the critical value ρc 0.83 after more than 40% reverse deformation! Hence, while the transient following shear reversal is rather short for the stress ratio, it is surprisingly long-lasting for the solid fraction. These observations motivate the introduction of the fabric tensor as one of the possible candidates for the internal variable. Indeed, the fabric tensor does capture the strong structural anisotropy which progressively sets in under shear, as will be shown later below (Section 7.4.4). Finally, because of the fact that the internal state depends on the privileged contact orientation, it appears useful to try to quantify the effect of an overall rotation imposed in the displacement field, parameterized here by R. Indeed, because of the fabric anisotropy which reflects the distribution of contact normal orientations, the presence of rotation in the overall displacement field, and its amplitude, as compared to the deviatoric strain, will matter: the fabric will give rise to anisotropic tangent mechanical properties. Thus the angle between the incremental strain and the fabric major principal axes, plays a role in as much as the rotation over the characteristic strain required for fabric adjustment is appreciable. Figure 7.4a shows the evolution of the stress ratio for three tests where the initial conditions are the same but the rotation rates are alternatively R = γ, R = 4γ and R = 10γ. Initially, the system is in a critical state reached by means of a simple shear path. Then, R is set to the desired value with the same shear rate γ in all cases, and the simulation is pursued. In the case R = γ, the system remains in the same critical state. In the two other cases (R = 4γ and R = 10γ), as we can observe in Figure 7.4a, that the stress ratio is mostly below that of the path R = γ, although the fluctuations are large compared to the systematic difference. Clearly, we need even larger rotation rates in order to get a different critical state. This point was not checked, but the signature of the rotation rate appears clearly in other parameters such as the stress directions compared to the fabric directions. In Figure 7.4b the difference θσ − θF , where θF is the major principal direction of the fabric tensor, is

174


0.6 0.4

+

(a)

q’ / p

0.2 0.0 -0.2

-

-0.4 -0.6 -0.4

-0.2

0 ε

0.85 (b)

0.2

0.4

0.2

0.4

+

ρ

0.84 -

0.83 0.82 -0.4

-0.2

0 ε

Figure 7.3: Evolution of the “signed” stress ratio q /p, a) and solid fraction ρ, b) in an initially dense packing as a function of the cumulative shear strain ε in a simple shear deformation (+) and in the reverse deformation (-). Notice that strain reversal (marked by an updown arrow) involves instantaneous jump of the stress ratio.

shown for the above tests. We see that, the fabric direction is by a few degrees below the stress direction for large rotation rates. This phase difference has a weak impact on the stress ratio. Nevertheless, this observation shows that the critical state does depend on the rotation parameter R/γ.

7.4.2 Stress–Strain Relation Starting with an initially isotropic system, the stress ratio q/p increases almost monotonically (ignoring short-scale fluctuations) with shear strain, as shown in Figure 7.2a. In the case of perfectly rigid particles, which is also the case in our simulations, this increase in shear resistance is a purely hardening effect. In other words, the initial elastic regime generally

7.4

Numerical Results

175

0.40 (a)

q/p

0.35 0.30 R=γ R = 4γ R = 10γ

0.25 0.20 0.00

0.05

8

0.10 ε

0.15

0.20

θσ − θF (deg)

(b)

0 -8 -16 0.00

R=γ R = 4γ R = 10γ

0.05

0.10 ε

0.15

0.20

Figure 7.4: a) Stress ratio q/p as a function of the cumulative shear strain ε for three different values of rotation rate R starting with a system in critical state with R = γ, where γ is the shear rate, b) Corresponding evolution of the phase difference between major principal directions θσ and θF of the stress tensor and fabric tensor, respectively.

observed in simulations with elastic contacts (by means of other distinct element methods of “molecular dynamics” type) is totally absent from our results. When the packing is initially dense (ρ > ρc ), the stress ratio reaches a peak before declining to its critical-state value (shear softening). In the initially loose case (ρ < ρc ), no peak occurs and the increase in q/p is slow. Indeed, for an initially dense system the stress ratio undergoes a huge jump over the first time-step. This is reminiscent of a rigid-plastic behavior without internal variables. However, particle rearrangements take over afterwards and the behavior is then governed by the evolution of the microstructure. A similar jump occurs also at the moment of shear reversal (see Figure 7.3a) but the particle rearrangements are again responsible for the long transient towards the critical state. These jumps also set the principal stress directions after shear reversal.

176


0.5 biaxial compression simple shear

q/p

0.4 0.3 0.2 0.0

0.1

ε

0.2

0.3

Figure 7.5: Stress ratio q/p as a function of the cumulative shear strain ε for a simple shear test and a biaxial compression test.

The above features are common to all initial states and driving modes. For instance, biaxial compression leads to the same behavior as with simple shear as shown in Figure 7.5. Indeed, these two driving modes differ only in the rigid rotation rate (R = 0 in biaxial compression, R = γ in simple shear) as discussed in Section 7.2.2. The effect of a non-zero rotation rate seems thus to be negligible with respect to these tests. Let us mention here that in simple shear tests with rigid walls, it is basically impossible to achieve homogeneous shearing for large strains. For this reason, the critical state values of the shear stress might turn out to be different from that in a biaxial compression test when wall-type boundaries are used [22]. The stress–strain relation, under cyclic driving, depends on the strain amplitude. When this amplitude is large enough to erase the memory of the microstructure, the behavior is a simple switch between critical states with long transients as shown, e.g., in Figure 7.3a. Otherwise, the system does not reach the critical state. This behavior is shown in Figure 7.6a. The strain amplitude is ∆ε = 0.04. We see that a stable cycle is established between q /p = 0.4 and q /p = −0.4 after two oscillations. However, as shown in Figure 7.6b, the solid fraction continues to increase periodically well beyond the critical state value ρc . The indefinite evolution of the solid fraction in cyclic deformations means that, if cyclic deformations below the critical strain amplitude are to be modelled, internal variables pertaining to the memory of the packing should be introduced.

7.4.3 Dilatancy The evolution of the solid fraction was illustrated in Figs. 7.2b, 7.3b and 7.6b. Correspondingly, the dilation angle ψ evolves with the strain and tends to zero (on average) in the critical state. Figure 7.7 shows the dilation angle in a monotonic simple shear test for two samples: initially dense (ρ > ρc and initially loose (ρ < ρc ). Note the strong contraction of the loose sample (negative dilation angle), as well as the dilation of the dense sample (positive dilation angle). It is also remarkable that the dilation angle for the dense sample reaches a stable

7.4

Numerical Results

177

0.4

q’ / p

0.2 0.0 -0.2 (a)

-0.4 0.00

0.01

0.85

0.02 ε

0.03

0.04

0.84 ρ

0.83 0.82 0.81 0.80 0.00

(b)

0.01

0.02 ε

0.03

0.04

Figure 7.6: Evolution of the “signed” stress ratio q /p, a) and solid fraction ρ, b) for an initially loose packing as a function of the cumulative shear strain ε in a cyclic simple shear deformation with strain amplitude 0.04.

plateau with ψ 10 degrees up to ε = 7% before decaying to zero (on average) at larger strains. Figure 7.8 shows the dilation angle ψ as a function of the internal friction angle φ for dense and loose systems. The observed “stress–dilatancy” diagram is close to linear over a wide range of angles (basically in the range of negative dilation angles). The behavior is clearly non-associated (associated flow rule implying ψ = φ). Very large dilation angles occur at the beginning of shear both in loose and dense cases. The stress–dilatancy relation for cyclic shearing is shown in Figure 7.9. Interestingly, while the range of variations of tan ψ declines progressively, all cycles pass through a fixed point corresponding to φ = 0 and ψ −25 degrees. In soil mechanics, several existing models define the mechanism of plastic deformation from a stress–dilatancy diagram (see, e.g., [21]). Well known examples are the classical relations proposed by Rowe and Taylor. Our results show that this relation

178


40

ψ (deg)

20 0 -20 ρ > ρc ρ < ρc

-40 -60 0.00

0.05

0.10 ε

0.20

0.15

Figure 7.7: Dilation angle ψ as a function of the cumulative shear strain ε for initially dense (ρ > ρc ) and loose (ρ < ρc ) systems.

40

ψ (deg)

20 0 -20 dense loose

-40 -60 0

5

10

15 20 φ (deg)

25

30

Figure 7.8: Relation between the internal angle of friction φ and dilation angle ψ in initially dense and loose systems.

may be cast into simple forms, e.g., linear relation between φ and ψ, both for monotonic and cyclic deformations. In the latter case the relation involves a hysteresis. It is important to note that the dilation angle can be quite large (in absolute value) for small values of the internal angle of friction (basically for negative dilatancies), but it is generally low for large values of the internal angle of friction (for positive dilatancies). These features are basically difficult to measure in experiments where the contribution of the elastic strains cannot be easily isolated from that of the plastic strains.

7.4

Numerical Results

179

20

ψ (deg)

10 0 -10 -20 -30 -40 -30 -20 -10

0 10 φ (deg)

20

30

Figure 7.9: Relation between the internal angle of friction φ and dilation angle ψ in an initially loose system during a cyclic simple shear deformation with strain amplitude 0.04.

7.4.4 Internal Variables In this section, we would like to show that the stress ratio and the dilation angle are closely controlled by lowest-order geometrical descriptors of the system represented by the fabric variables z (coordination number), a (fabric anisotropy), and θF (principal fabric direction); see Section 7.3.3. Since both z and ρ are descriptors of the compactness in a granular medium, it is interesting to see how they are related together along different stress paths. Figure 7.10a shows solid fraction versus coordination number for a simple shear test with an initially dense system and an initially loose system. We see that there is a one-to-one relation between z and ρ only in the case of an initially loose system and in the loading period. Otherwise, in all other cases, they show very different behaviors. In particular, at some stages of deformation they evolve in opposite directions and they vary in very different proportions. For example, starting with a dense sample, z decreases very fast due to contact loss in the direction of extension, whereas the solid fraction decreases to a much lesser extent. In fact, density is linked not only with z but also with the anisotropy a in an intricate way that depends on the particle size distribution and the geometrical texture. In Figure 7.10b, we have displayed the evolution of the solid fraction with a = a cos 2(3π/4 − θF ) in four simple shear tests. The use of a instead of a is motivated (as for stress deviator q) by the need for a “signed” anisotropy which reflects the changes of principal fabric directions. It is interesting to observe that the reverse paths (starting with the first critical state reached from different initial conditions) overlap nicely and the loading path for the dense case comes in the continuity of the loading path for the loose case with nearly the same slope. Figures 7.11a and 7.11b show the variation of stress ratio q /p with z and a , respectively, both for initially dense and loose systems. The jumps of q /p upon strain reversal and at the beginning of the test for the initially dense sample are not shown. Apart from this initial behavior of very dense samples, the stress ratio varies monotonically with a . The relation

180


0.85

(a)

0.84 ρ

0.83 0.82 dense + loose + dense loose -

0.81 0.80 3.2

0.85

3.4

3.8

3.6

4.0

4.2

z (b)

0.84 ρ

0.83 0.82 0.81

dense + loose + dense loose -

0.80 -0.3 -0.2 -0.1 0.0 a’

0.1 0.2

0.3

Figure 7.10: Solid fraction ρ as a function of the coordination number z, a) and the “signed” anisotropy a , b) for an initially dense system (dense+), an initially loose system (loose+) and along corresponding reverse deformations (dense- and loose-).

may even be approximated as piecewise linear. This property appears more clearly for the cyclic deformation with amplitude ∆ε = 0.04. It is also interesting that the stress principal directions follow closely the fabric principal directions, up to a phase difference occurring after the reversal of shear strain. These observations show that the strength properties of a granular system are more directly dependent on structural anisotropy and fabric direction than on the coordination number. On the other hand, the coordination number, as we will see below, controls the range of variations of the anisotropy. The relation of the dilation angle ψ with z and a is shown in Figs. 7.12a and 7.12b. This relation is simpler for the initially loose system where a nearly linear dependence with a and z can be approximated. In cyclic deformation, the relation between tan ψ and a takes a “butterfly” form.

7.4

Numerical Results

181

0.6

q’ / p

0.4 0.2 0.0 -0.2 -0.4 3.2

dense + loose + reverse

3.4

(a)

q’ / p

4.0

4.2

z

0.6 0.4

3.8

3.6

dense + loose + reverse cyclic

0.2 0.0 -0.2 -0.4 -0.3 -0.2 -0.1 0.0 a’

(b)

0.1 0.2

0.3

Figure 7.11: Signed stress ratio q /p as a function of the coordination number z, a) and the “signed” anisotropy a , b) for an initially dense system (dense+), an initially loose system (loose+), along a reverse deformation (reverse) from the critical state, and for the cyclic simple shear deformation with strain amplitude 0.04 in the case of stress ratio.

7.4.5 Evolution of Internal Variables While the internal variables determine the friction angle (Section 7.4.4), a theory of granular plasticity requires that the evolution of internal variables with the (plastic) strain be specified. In Figure 7.13a and 7.13b the evolution of z and a are shown as a function of the cumulative shear strain ε. Roughly speaking, z evolves like the solid fraction ρ (see Figure 7.2a and 7.3a), whereas a shows the same evolution as the stress ratio q /p (see Figure 7.2b and 7.3b). But, the evolution of internal variables, which are all of geometric nature, might be linked with particle motions induced by the imposed strain. The initial rapid increase of z in the initially dense case is a consequence of the loss of contacts in the direction of extension. This leads at the same time to an increasingly larger structural anisotropy with an excess of contact normals in the major principal direction of

182


40

ψ (deg)

20


0 -20 -40 -60 3.2

(a)

3.4

ψ (deg)

4.0

4.2

z

40 20

3.8

3.6


0 -20 -40 -60 -0.3 -0.2 -0.1 0.0 a’

(b)

0.1 0.2

0.3

Figure 7.12: Dilation angle ψ as a function of the coordination number z, a) and the “signed” anisotropy a , b) for an initially dense system (dense+), an initially loose system (loose+) and along a reverse deformation (reverse) from the critical state.

the strain-rate tensor. A similar mechanism is at work in the initially loose system with gain of contacts along the major principal direction of the strain-rate tensor, leading to a larger anisotropy with an increase in the coordination number. The slower variation observed in the latter case can be explained by the unilateral character of contact interactions: a contact may be lost due to a minute relative displacement, whereas contact gain requires displacements of the order of the distance between particles. The same effect is at play, as the shear strain is reversed, when z first decreases before increasing later to its critical-state value ( 3.6). A similar effect occurs for ρ with the important difference that ρ increases upon shear reversal. What happens is that, as in an initially dense system, contacts are lost along the new direction of extension (as soon as shear reversal is imposed) without significant deformation. This is reflected in the decrease of anisotropy. The contact gain in the new direction of compression occurs but is late due to loss in the direction

7.4

Numerical Results

183

4.0 dense + loose + reverse

z

3.8 3.6 3.4

(a)

3.2 -0.2

0.0

0.3 0.2

ε

0.2

0.4


a’

0.1 0.0 -0.1 -0.2 -0.3 -0.4

(b)

-0.2

0.0 ε

0.2

0.4

Figure 7.13: Evolution of the coordination number z, a) and the “signed” anisotropy a , b) for an initially dense system (dense+), an initially loose system (loose+) and along a reverse deformation (reverse) from the critical state.

of extension since it requires a significant deformation. This delay leads to a lower value for z but a larger value of ρ. For this reason, the evolution of z with the cumulative shear strain in cyclic deformation takes a “butterfly” shape. In the same way, the evolution of a in the cyclic case falls on a limit cycle similar to that of q /p (Figure 7.6a). The last, but not the least, point that merits particular attention in the phenomenology of internal variables, is that all geometrical states in the space of fabric parameters (z,a) are not accessible. The coordination number z cannot exceed an upper limit as a result of steric constraints and it cannot be too weak as a consequence of the requirement of static equilibrium of each particle. For the same reason, the anisotropy a has an upper bound that depends on z. This means that the evolution of internal variables is not simply controlled by the mean strainrate tensor (as is sometimes assumed in homogenization theories for quasi-static loading). The fabric evolves as a result of local particle displacements leading to gain and loss of contacts,

184


and these local displacements are dependent not only on the mean-strain rate but also on the local geometrical constraints and the requirement of equilibrium. In Figure 7.14 we have displayed the variations of a with z for several simulations. The peak corresponds to the critical state. Apart from the overshoot for the dense case that goes beyond the peak (the loop), all points corresponding to the cyclic and monotonous deformation lie below the two limit curves defined by the very dense and very loose systems.

0.3 loose + dense + cyclic reverse

a

0.2 0.1 0.0 3.2

3.4

3.8

3.6

4.0

4.2

z Figure 7.14: Relation between the coordination number z and the anisotropy a along different deformations paths.

7.4.6 Frictional/Collisional Dissipation A somewhat intriguing behavior observed in the simulations is that the evolution of macroscopic observables is not smooth. Strong fluctuations occur at small timescales. These fluctuations disappear partially as the system size is increased or they can be integrated by averaging over a small strain interval. But these small-scale fluctuations reflect real dynamic instabilities occurring in the form of collective restructuring events that dissipate a non-negligible amount of energy. Figure 7.15 displays the dissipation rate due to friction at sliding contacts normalized by the rate of strain energy supplied to the system in a simple shear deformation. We observe that 75% of the strain work is on average dissipated by friction, the rest being dissipated through inelastic collisions occurring during instabilities. These instabilities may have paradoxical effects. In particular, they can produce an effective mean dissipation that is “Coulomb-like”. Indeed, the dynamics of restructuring being much faster than any external loading (quasi-static condition), the time-averaged dissipation appears to be rate-independent. Moreover, the dissipation at each event is equal to the strain work accumulated prior to the event. Therefore, the dissipation is simply proportional to the mean stress. Both these features characterize a solid “Coulomb friction”. Hence, a macroscopic description of the fluctuations, as well as the mean, appears to be a necessary step towards a quantitative account of the behavior [13].

7.5

Conclusion

185

Wfric / Wext

1.0 0.8 0.6 + -

0.4 -0.4

-0.2

0.0 ε

0.2

0.4

Figure 7.15: Rate of energy dissipated by friction Wf ric at sliding contacts normalized by the rate of strain energy Wext fed into the system during a simple shear deformation (+) and a reverse deformation from the critical state (-).

7.5 Conclusion The contact dynamics approach with bi-periodic boundary conditions was used to investigate the plastic properties (yield, flow, hardening) of granular media composed of perfectly rigid particles interacting via contact and friction in two dimensions. The implemented numerical procedure allows us to control the deformations of the system along arbitrary paths, e.g. applying separately shear and rotation rates, and with no wall effects and strain localization so that the system remains macroscopically homogeneous during deformations. On the other hand, since no contact elasticity is introduced, we have direct access to plastic deformations which would otherwise require specific methods for separation from elastic strains. We studied the stress–strain and volume-change behavior for different types of loading and initial conditions and in relation to the granular fabric. It was shown that the critical state depends on the level of rotation rate (compared to the shear rate) which influences the fabric principal directions, with respect to those of the stress. In this respect, the critical state is not, strictly speaking, a unique state, although the effect of rotations can be observed only at high rotation rates that are of little practical interest. On the other hand, the stress–dilatancy diagrams obtained from the simulations reveal simple relationships between the dilation angle and the internal angle of friction as suggested by some models proposed in the past in the field of soil mechanics. Nevertheless, it was shown that this relation involves hysteretic effects. We also observe that, although the dilation angle is small at high levels of the stress ratio, it is quite large at low stress ratios. The results suggest also very simple links between the evolution of the fabric and that of the shear stress and dilatancy. In monotonous deformations, the mechanical state of the system is basically encoded in the granular microstructure so that it does not seem to be necessary to

186


use, for example, the cumulative shear strain as an internal variable. The behavior is simply dictated by the absolute distance of the present fabric state (described at the lowest level by the coordination number and the structural anisotropy) from the critical state. In cyclic deformations with low strain amplitude, however, the system does not tend to the critical state and the description of the state requires a different approach. We argued that the macroscopic framework is basically a hardening rigid-plastic behavior. The rather simple relationships observed between the fabric, on one hand, and stress-ratio and dilation angle, on the other hand, suggest that a continuum description can be set up in which the fabric parameters at the lowest level may be used as internal variables. This description can then be enriched by including higher-order microstructural information. It is transparent to the microstructure and it differs from a purely phenomenological approach in which the physical sense of the internal variables may not be easy to establish. In fact, the numerical results briefly commented in this paper can be seen as the basic information that should guide possible strategies for building such a continuum model.

References [1] P. G. de Gennes, Rev. Mod. Phys. (1999), 71, S374. [2] H. M. Jaeger and S. R. Nagel, Rev. Mod. Phys. (1996), 68, 1259. [3] P. Dantu, Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering, Butterworths Scientific Publications, London, (1957), 1, 144. [4] M. Oda, Soils and Foundations, (1972), 12, 17; ibid., (1972), 12, 1. [5] P. A. Cundall and O. D. L. Stack, Geotechnique, (1979), 29, 47. [6] P. A. Cundall, A. Drescher, and O. D. L. Strack, IUTAM Conference on Deformation and Failure of Granular Materials, Delft, (1982), 355. [7] F. Radjai, D. E. Wolf, M. Jean, and J. J. Moreau, Phys. Rev. Lett. (1998), 80, 61. [8] M. R. Kuhn, Mechanics of Materials, (1999), 31, 407. [9] F. Radjai and S. Roux, Phys. Rev. Lett., (2002), 89, 064302. [10] J. J. Moreau, in “Nonsmooth Mechanics and Applications”, CISM Courses and Lectures, (1988), 302, 1. [11] J. J. Moreau, Eur. J. Mech. A, (1994), 13, 93. [12] M. Jean and J. J. Moreau, Proceedings of Contact Mechanics International Symposium, Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, (1992), 31. [13] S. Roux and F. Radjai, in “Mechanics for a New Millennium”, ed. H. Aref H. and J. Philips, Kluwer, Netherlands, (2001), 181. [14] F. Calvetti, G. Combe, and J. Lanier, Mech. Coh. Frict. Materials, (1997), 2, 121. [15] J. Christoffersen, M. M. Mehrabadi, and S. Nemat-Nasser, J. Appl. Mech., (1981), 48, 339. [16] L. Rothenburg and R. J. Bathurst R. J., Geotechnique, (1989), 39, 601; R. J. Bathurst and L. Rothenburg, Mechanics of Materials, (1990), 9, 65. [17] B. Cambou, in “Powders and Grains 93”, ed. C. Thornton, Balkema, Amsterdam, (1993), 73.

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[18] J. K. Mitchell, “Fundamentals of Soil Behavior”, Wiley, New York, (1993). [19] M. Satake, in “Proceedings of the IUTAM Symposium on Deformation and Failure of Granular Materials”, Delft, ed. P. A. Vermeer and H. J. Luger, Balkema, Amsterdam, (1982), p. 63. [20] A. Schofield and P. Wroth, “Critical State Soil Mechanics”, McGraw-Hill, London, (1967). [21] D. M. Wood, “Soil Behavior and Critical State Soil Mechanics”, Cambridge University Press, Cambridge, (1990). [22] C. Thornton and L. Zhang, “Powders and Grains 2001”, edited by Y. Kishino, Balkema, Tokyo, (2001).

8 Collision of Adhesive Viscoelastic Particles Nikolai V. Brilliantov and Thorsten Pöschel

Abstract The collision of convex bodies is considered for small impact velocity, when plastic deformation and fragmentation may be disregarded. In this regime the contact is governed by forces according to viscoelastic deformation and by adhesion. The viscoelastic interaction is described by a modified Hertz law, while for the adhesive interactions, the model by Johnson, Kendall and Roberts (JKR) is adopted. We solve the general contact problem of convex viscoelastic bodies in quasi-static approximation, which implies that the impact velocity is much smaller than the speed of sound in the material and that the viscosity relaxation time is much smaller than the duration of a collision. We estimate the threshold impact velocity which discriminates restitutive and sticking collisions. If the impact velocity is not large as compared with the threshold velocity, adhesive interaction becomes important, thus limiting the validity of the pure viscoelastic collision model.

8.1 Introduction The large set of phenomena observed in granular systems, ranging from sand and powders on Earth to granular gases in planetary rings and protoplanetary discs, is caused by the specific particle interaction. Besides elastic forces, common for molecular or atomic materials (solids, liquids and gases), colliding granular particles also exert dissipative forces. These forces correspond to the dissipation of mechanical energy in the bulk of the grain material as well as on their surfaces. The dissipated energy transforms into energy of the internal degrees of freedom of the grains, that is, the particles are heated. In many applications, however, the increase in temperature of the particle material may be neglected (see, e.g. [6]). The dynamical properties of granular materials depend sensitively on the details of the dissipative forces acting between contacting grains. Therefore, choosing the appropriate model of the dissipative interaction is crucial for an adequate description of these systems. In real granular systems the particles may have a complicated non-spherical shape, they may be nonuniform and even composed of smaller grains, kept together by adhesion. The particles may differ in size, mass and in their material properties. In what follows we consider the contact of granular particles under simplifying conditions. We assume that the particles are smooth, convex and of uniform material. The latter assumption allows us to describe the particle deformation by continuum mechanics, disregarding their molecular structure. It is assumed that particles exert forces on each other exclusively via pairwise mechanical contact, i.e., electromagnetic interaction and gravitational attraction are not considered.

190

8 Collision of Adhesive Viscoelastic Particles

8.2 Forces Between Granular Particles 8.2.1 Elastic Forces When particles deform each other due to a static (or quasi-static) contact they experience an elastic interaction force. Elastic deformation implies that, after separation of the contacting particles, they recover their initial shape, i.e., there is no plastic deformation. The stress tensor ij (r ) describes the i-component of the force, acting on a unit surface which is normal to σel the direction j (i, j = {x, y, z}). In the elastic regime the stress is related to the material deformation 1 ∂ui ∂uj , (8.1) + uij (r ) = 2 ∂xj ∂xi where u(r ) is the displacement field at the point r in the deformed body, via the linear relation 1 ij σel (r ) = E1 uij (r ) − δij ull (r ) + E2 δij ull (r ) . (8.2) 3 Repeated indices are implicitly summed over (Einstein convention). The coefficients E1 and E2 read E1 =

Y , (1 + ν)

E2 =

Y , 3(1 − 2ν)

(8.3)

where Y is the Young modulus and ν is the Poisson ratio. Let the pressure P (x, y) act on the surface of an elastic semispace, z > 0, leading to a displacement field in the bulk of the semispace [18]: Gik (x − x , y − y , z) Pk (x , y ) dx dy , (8.4) ui = where Gik (x, y, z) is the corresponding Green function. For the contact problem addressed here we need only the z-component of the displacement on the surface z = 0, that is, we need only the component Gzz (x, y, z = 0) =

(1 − ν 2 ) (1 − ν 2 ) 1 1 = πY πY r x2 + y 2

(8.5)

of the Green function [18]. Consider a contact of two convex smooth bodies labeled as 1 and 2. We assume that only normal forces, with respect to the contact area, act between the particles. In the contact region their surfaces are flat. For the coordinate system centered in the middle of the contact region, where x = y = z = 0, the following relation holds true: B1 x2 + B2 y 2 + uz1 (x, y) + uz2 (x, y) = ξ ,

(8.6)

where uz1 and uz2 are respectively the z-components of the displacement in the material of the first and of the second bodies on the plane z = 0. The sum of the compressions of both

8.2

Forces Between Granular Particles

191

bodies in the center of the contact area defines ξ. The constants B1 and B2 are related to the radii of curvature of the surfaces in contact [18]: 1 1 1 1 + + + R1 R2 R1 R2 2 2 (8.7) 1 1 1 1 1 1 1 1 2 . 4 (B1 −B2 ) = − + − + 2 cos 2ϕ − − R1 R2 R1 R2 R1 R2 R1 R2 2 (B1 + B2 ) =

Here R1 , R2 and R1 , R2 are respectively the principal radii of curvature of the first and the second body at the point of contact and ϕ is the angle between the planes corresponding to the curvature radii R1 and R1 . Equations (8.6), (8.7) describe the general case of the contact between two smooth bodies (see [18] for details). The physical meaning of (8.6) is easy to see for the case of a contact of a soft sphere of a radius R (R1 = R2 = R) with a hard, undeformed plane (R1 = R2 = ∞). In this case B1 = B2 = 1/R, the compressions of the sphere and of the plane are respectively uz1 (0, 0) = ξ and uz2 = 0, and the surface of the sphere before the deformation is given by z(x, y) = (x2 + y 2 )/R. Then (8.6) reads in the flattened area uz1 (x, y) = ξ − z(x, y), that is, it gives the condition for a point z(x, y) on the body’s surface to touch the plane z = 0. The displacements uz1 and uz2 may be expressed in terms of the normal pressure Pz (x, y) which acts between the compressed bodies in the plane z = 0. Using (8.4) and (8.5) we rewrite (8.6) as 1 π

1 − ν22 1 − ν12 + Y1 Y2

Pz (x , y ) dx dy = ξ − B1 x2 − B2 y 2 , r

(8.8)

where r = (x − x )2 + (y − y )2 and integration is performed over the contact area. Equation (8.8) is an integral equation for the unknown function Pz (x, y). We compare this equation with the mathematical identity [18]

dx dy r

x 2 y 2 πab 1− 2 − 2 = a b 2

∞ y2 dt x2 − 2 , 1− 2 (8.9) 2 a +t b +t (a + t)(b2 + t)t 0

where integration is performed over the elliptical area x 2 /a2 + y 2 /b2 = 1. The left-hand sides of both equations contain integrals of the same type, while the right-hand sides contain quadratic forms of the same type. Therefore, the contact area is an ellipse with the semi-axes a and b and the pressure is of the form Pz (x, y) = const 1 − x2 /a2 − y 2 /b2 . The constant may be found from the total elastic force Fel acting between the bodies. Integrating Pz (x, y) over the contact area we obtain 3Fel x2 y2 Pz (x, y) = 1− 2 − 2 . (8.10) 2πab a b We substitute (8.10) into (8.8) and replace the double integration over the contact area by integration over the variable t, according to the identity (8.9). Thus, we obtain an equation

192


containing terms proportional to x2 , y 2 and a constant. Equating the corresponding cients we obtain Fel D ∞ dt Fel D N (x) ξ= = 2 2 π π b (a + t)(b + t)t 0 ∞ Fel D dt Fel D M (x) B1 = = 2 2 2 π π a2 b (a + t) (a + t)(b + t)t 0 ∞ Fel D dt Fel D M (1/x) B2 = , = 2 2 2 π π ab2 (b + t) (a + t)(b + t)t 0 where 3 D≡ 4

1 − ν22 1 − ν12 + Y1 Y2

coeffi(8.11) (8.12) (8.13)

(8.14)

and x ≡ a2 /b2 is the ratio of the contact ellipse semi-axes. In (8.11)-(8.13) we introduce the short-hand notations1 ∞ dt N (x) = (8.15) (1 + xt)(1 + t)t 0 ∞ dt M (x) = . (8.16) (1 + t) (1 + t)(1 + xt)t 0 From these relations will follow the size of the contact area, a, b, and the compression ξ as functions of the elastic force Fel and the geometrical coefficients B1 and B2 . The dependence of the force Fel on the compression ξ may be obtained from scaling arguments. If we rescale a2 → αa2 , b2 → αb2 , ξ → αξ and Fel → α3/2 Fel , with α constant, Eqs. (8.11)–(8.13) remain unchanged. That is, when ξ changes by the factor α, the semi-axis a and b change by the factor α1/2 and the force by the factor α3/2 , i.e., a ∼ ξ 1/2 , b ∼ ξ 1/2 and Fel = const ξ 3/2 .

(8.17)

The dependence (8.17) holds true for all smooth convex bodies in contact. To find the constant in (8.17) we divide (8.13) by (8.12) and obtain the transcendental equation √ xM (1/x) B2 (8.18) = B1 M (x) for the ratio of semi-axes x. Let x0 be the root of Eq. (8.18), then a2 = x0 b2 and we obtain Fel D N (x0 ) π b Fel D M (x0 ) B1 = , π x0 b3 ξ=

1

(8.19) (8.20)

The function N (x) and M (x) may be expressed as a combination of the Jacobian elliptic functions E(x) and K(x) [1].

8.2


193

where N (x0 ) and M (x0 ) are pure numbers. Equations (8.19), (8.20) allow us to find the semi-axes b and the elastic force Fel as functions of the compression ξ. Hence we obtain the force, i.e., we get the constant in (8.17): 1/2 π M (x0 ) ξ 3/2 = C0 ξ 3/2 . (8.21) Fel = D B1 x0 N (x0 ) For the special case of contacting spheres (a = b), the constants B1 and B2 read 1 1 1 1 1 = B1 = B2 = + . 2 R1 R2 2 Reff

(8.22)

In this case x0 = 1, N (1) = π, and M (1) = π/2, leading to the solution of (8.19), (8.20): a2 = Reff ξ

(8.23) √ 2Y Fel = ρξ 3/2 ; ρ≡ Reff , (8.24) 3(1 − ν 2 ) where we use the definition (8.14) of the constant D. This contact problem was solved by Heinrich Hertz in 1882 [14]. It describes the force between elastic particles. For inelastically deforming particles it describes the repulsive force in the static case.

8.2.2 Viscous Forces When the contacting particles move with respect to each other, i.e., the deformation changes with time, an additional dissipative force arises, which acts in the opposite direction to the relative particle motion. The dissipative processes occurring in the bulk of the body cause a viscous contribution to the stress tensor. For small deformation the respective component of the stress tensor is proportional to the deformation rate u˙ ij (r ), according to the general relation [8]: ij (r, t) σdis

t t 1 = E1 dτ ψ1 (t − τ ) u˙ ij (r, τ ) − δij u˙ ll (r, τ ) + E2 dτ ψ2 (t − τ )δij u˙ ll (r, τ ) , 3 0

0

(8.25) where the (dimensionless) functions ψ1 (t) and ψ2 (t) are the relaxation functions for the distortion deformation and ψ2 (t) for the dilatation deformation. In many important applications the viscous stress tensor may be simplified significantly. If the relative velocity of the colliding bodies is much smaller than the speed of sound in the particle material and if the characteristic relaxation times of the dissipative processes τvis, 1/2 are much smaller than the duration of the collision tc , ∞ ψ1/2 (τ )dτ tc , (8.26) τvis, 1/2 ≡ 0

the viscous constants η1 and η2 may be used instead of the functions ψ1 (t) and ψ2 (t). Thus ∞ η1/2 = E1/2 τvis, 1/2 = E1/2 ψ1/2 (τ )dτ (8.27) 0

194


and the dissipative stress tensor reads (see [8] for details) 1 ij (r, t) = η1 u˙ ij (r, τ ) − δij u˙ ll (r, τ ) + η2 δij u˙ ll (r, τ ) . σdis 3

(8.28)

It may be also shown that the above conditions are equivalent to the assumption of quasi-static deformation [7,8]. When the material is deformed quasi-statically, the displacement field u(r ) in the particles coincides with that for the static case uel (r ), which is the solution of the elastic contact problem. The field uel (r ), in its turn, is completely determined by the compression ξ, which varies with time during the collision, i.e., uel = uel (r, ξ). Therefore, the corresponding displacement rate may be approximated as ∂ u˙ (r, t) ξ˙ uel (r, ξ) ∂ξ

(8.29)

and the dissipative stress tensor reads, respectively ∂ 1 ij el el η1 uel δ + η = ξ˙ − u δ u σdis ij 2 ij ij ll ll . ∂ξ 3

(8.30)

From (8.30) and (8.2) follows the relation between the elastic and dissipative stress tensors within the quasi-static approximation, ∂ ij ij = ξ˙ σel (E1 ↔ η1 , E2 ↔ η2 ) , σdis ∂ξ

(8.31)

where we emphasize that the expression for the dissipative tensor may be obtained from the corresponding expression for the elastic tensor after substituting the elastic constants by the ˙ relative viscous constants, and application of the operator ξ∂/∂ξ. zz of the elastic stress is equal to the normal pressure Pz at the plane The component σel z = 0 of the elastic problem, Eq. (8.10) ∂uy ∂uz E1 ∂uz ∂ux zz + E2 − + + σel (x, y, 0) =E1 ∂z 3 ∂x ∂y ∂z (8.32) 2 2 y x 3Fel 1− 2 − 2 . = 2 πab a b Now we compute the total dissipative force acting between the bodies. Instead of a direct computation of the dissipative stress tensor, we employ the method proposed in [7, 8]: We transform the coordinate axes as x = αx ,

y = αy ,

z = z

(8.33)

(η2 − 13 η1 ) α(E2 − 13 E1 )

(8.34)

with α=

η2 − 13 η1 η2 + 23 η1

a = αa

E2 + 23 E1 E2 − 13 E1

β=

b = αb .

(8.35)

8.2


195

and perform the transformations ∂uy ∂uz η1 ∂ux ∂uz + η2 − + + η1 ∂z 3 ∂x ∂y ∂z E1 ∂uy ∂uz ∂uz ∂ux + + = β E 1 + E2 − ∂z 3 ∂x ∂y ∂z x 2 y 2 x2 y2 3Fel 2 3Fel 1 − − = βα 1 − − . =β 2 πa b a 2 b 2 2 πab a2 b2

(8.36)

˙ Applying the operator ξ∂/∂ξ to the last expression on the right-hand side we obtain the dissipative stress tensor. Subsequent integration over the contact area yields, finally, the total dissipative force acting between the bodies: ∂ Fdis = Aξ˙ Fel (ξ) , ∂ξ where 1 (3η2 − η1 )2 A≡α β= 3 (3η2 + 2η1 ) 2

(8.37)

1 − ν 2 (1 − 2ν) . Y ν2

(8.38)

Using the scaling relations for the elastic force, Eq. (8.17), and for the semi-axes of the contact ellipse, we obtain 3 Fel ∂Fel = , ∂ξ 2 ξ

∂a 1a = , ∂ξ 2ξ

∂b 1b = . ∂ξ 2ξ

(8.39)

Then from (8.36) and (8.21), the distribution of the dissipative pressure in the contact area may be found: −1/2 3A AC0 ˙ x2 y2 Pzdis (x, y) = , (8.40) ξ ξ 1− 2 − 2 4π ab a b where the constant C0 is defined in (8.21). We wish to stress that, to derive the above expressions, we assumed only that the surfaces of the two bodies in the vicinity of the contact point before the deformation, are described (1) (1/2) by the quadratic forms z1 = κij xi xj and z2 = κ(2) xi xj (i, j = x, y, z), where κij are symmetric tensors [18]. Therefore, the relations obtained are valid for a contact of arbitrarily shaped convex bodies. For spherical particles of identical material, (8.37) and (8.24) yield [7, 8] 3 (8.41) Fdis = Aρξ˙ ξ , 2 with ρ as defined in (8.24). Hence, the total force acting between viscoelastic spheres takes the simple form [7, 8] 3 ˙ 3/2 F =ρ ξ + A ξξ . (8.42) 2

196


The range of validity of (8.42) for the viscoelastic force is determined by the quasi-static approximation. The impact velocity must be significantly smaller than the speed of sound. On the other hand, the impact velocity must not be too small in order to neglect adhesion. We also neglect plastic deformation in the material.

8.2.3 Adhesion of Contacting Particles 8.2.3.1 Models of Adhesive Interaction The Hertz theory has been derived for the contact of non-adhesive particles. Adhesion becomes important when the distance of the particle surfaces approaches the range of molecular forces. Johnson, Kendall and Roberts (JKR) [17] extended the Hertz theory by taking into account adhesion in the flat contact region. They show that the contact area is enlarged by the action of the adhesive force. Therefore, they introduced an apparent Hertz load FH which would cause this enlarged contact area. To simplify the notation, we consider the contact of identical spheres. The contact area is then a circle of radius a, which corresponds to the compression ξH for the Hertz load FH . In reality, however, this contact radius occurs at the compression ξ which is smaller than ξH . In the JKR theory it is assumed that the difference between the Hertz compression ξH and the actual one, ξ, may be attributed to the additional stress −1/2 r2 FB 1 − , (8.43) PB (x, y) = 2πa2 a2 which is the solution of the classical Boussinesq problem [32]. This distribution of the normal surface traction gives rise to a constant displacement over a circular region of an elastic body. The displacement ξB corresponding to the contact radius a and the total load FB are related by ξB =

2 FB D , 3 a

(8.44)

where the constant D is defined in (8.14). The value of FB < 0 mimics the additional surface forces, such that the pressure is positive (compressive) in the center of the contact area, while it is negative (tensile) near the boundary [17]. Hence, the shape of the body is determined by the action of two effective forces FH and FB . The total force between the particles is their difference, F = FH − FB . Johnson et al. assumed that the elastic energy stored in the deformed spheres may be found as a difference of the elastic energy corresponding to the Hertz force FH and that due to the force FB [17]. Using Us = −πγa2

(8.45)

for the surface energy, where γ > 0 is twice the surface free energy per unit area of the solid in vacuum or gas, and minimizing the total energy, we find [17] 3γ 2 , (8.46) FB = −2πa 2πDa

8.2


and, thus, the contact radius corresponding to the total force F :   2 1 3 3 πγR  a3 = D R F + πγR + 3πγRF + 2 2 2 and also the compression 8πγDa 2a2 − . ξ= R 3

197

(8.47)

(8.48)

The first term in (8.48) is the Hertz compression ξH , which coincides with (8.23) for Reff → R/2. Equation (8.47) may be solved to express the total force as a function of the contact radius: 6πγ 3/2 2a3 − a . (8.49) F (a) = DR D For vanishing applied load the contact radius a0 is finite: a30 =

3 DπγR2 . 2

(8.50)

For negative applied load the contact radius decreases and the condition for a real solution of (8.47) yields the maximal negative force which the adhesion forces can resist, 3 Fsep = − πγR , 4

(8.51)

corresponding to the contact radius a3sep =

3 1 DπγR2 = a30 . 8 4

(8.52)

For a larger (in the absolute value) negative force, the spheres separate. For spheres of dissimilar radii, in (8.47)–(8.52) R should be substituted by 2Reff . Another approach to the problem of the adhesive contact was developed by Derjaguin, Muller and Toporov (DMT). They assumed that the Hertz profile of the pressure distribution on the surface stays unaffected by adhesion and obtained the pull-off force Fsep = −2πγReff [9]. The assumption of the Hertz profile allows one to avoid the singularities of the pressure distribution (8.43) on the boundary of the contact zone. Since the experimental measurement of γ is problematic, it is not possible to check the validity of the JKR and DMT theories, i.e., to resolve their disagreement. In later studies [20, 21] a more accurate theoretical analysis has been performed. The elastic equations have been solved numerically for a simplified microscopic model of adhesive surfaces with Lennard–Jones interaction. Within this microscopic approach, the relative accuracy of different theories has been estimated for a wide range of model parameters. It was found that the DMT theory is valid for small adhesion and for small, hard particles. JKR

198


theory is more reliable for large, soft particles with large adhesion forces, which, however, should be short-ranged. In [2] the Lennard–Jones continuum model of solids was studied. The adhesive forces between the surfaces then read 6 H z0 − 1 . (8.53) Ps (h) = 6πh3 h6 Here Ps (h) describes the forces acting per unit area between the surfaces, h = h(r) is the actual microscopic distance between them. H is the Hamaker constant, characterizing the van der Waals attraction of the particles in a gas or vacuum and z0 is the equilibrium separations of the surfaces. The surface energy in this model is defined by γ=

H . 16πz02

(8.54)

It was observed in [2] that the accuracies of different theories vary depending of the value of the Tabor parameter µ, [29] 2 (8.55) µ3/2 ≡ γD Reff /z03 . 3 In agreement with [20, 21] it has been shown [2] that small values of µ (small hard particles with low surface energies) favor the DMT theory (µ < 10−2 ) while for µ ∼ 1–10 the JKR theory proves to be rather more accurate. Both JKR and DMT fail for large µ when the strong adhesion is combined with the soft material of the contacting bodies. In this limit, the surfaces jump into contact, which corresponds to a spontaneous non-equilibrium transition (see e.g. [27]). Similar analysis has been performed later [11], where the author concluded that the DTM theory generally fails, both in original and corrected forms. One of the main conclusions of [2,11] is that the JKR theory, albeit simple, gives relatively accurate predictions for basic quantities in the range of its validity (µ ∼ 1–10). Among the theories developed to cover the DMT–JKR transition regimes [11, 16, 19–21, 29] the theory by Maugis [19] is the most frequently used. It is based on a simplified model of adhesive forces. The adhesive force of constant intensity PD is extended over a fixed distance hD above the surface, yielding the surface tension γ = PD hD . The description of a contact in this model is based on two coupled analytical equations which are to be solved numerically. The recently developed double-Hertz model [12, 13] constructs the solution for the adhesive contact as a sum of two Hertzian solutions, which make the theory analytically more tractable than the Maugis model. Combining, in the adopted manner, the successful assumptions of the JKR and the modified DMT model, a generalized analytical theory for the adhesive contact has been proposed [26]. In what follows we assume that the parameters of our system belong to the range of validity of the JKR model, µ ∼ 1 − 10, and will use this simple analytical theory to describe the adhesive contacts between spheres. Moreover, we assume that the adhesive force is small. To estimate the influence of the adhesive force, we approximate ξ ≈ 2a2 /R in (8.48) and substitute it into (8.49) to obtain (see also [28]), 3/4 3/4 ξ . (8.56) F ≈ ρξ 3/2 − 6πγ/D Reff

8.3

Collision of Granular Particles

199

8.2.3.2 Viscoelasticity in Adhesive Interactions The adhesive forces between particles cause the additional deformation in the contacting bodies as compared to a pure Hertzian deformation, hence in the corresponding dynamical problem an additional deformation rate arises. Therefore, the dissipative forces must have an additional component attributed to the adhesive interactions. The adhesive contact of viscoelastic spheres has been studied numerically in [13, 28]. In [5] the quasi-static condition for the colliding viscoelastic adhesive spheres was used and an analytical expression for the interaction force has been derived for the JKR model. Similar to the case for non-adhesive particles, it was assumed that in the quasi-static approximation, the deformation field may be parameterized by the value of the compression ξ. (Note that this assumption neglects the possible hysteresis which can happen for the negative total force [2]). Performing the same transformation which lead to the expression (8.37) for the case of non-adhesive contact, and using the approximation (8.56) we obtain the estimate for the dissipative forces [5] 3/4 −1/4 3 3 ˙ Fdis = Aρξ˙ ξ + B 6πγ/D Reff (8.57) ξξ 2 4 (3η2 − η1 )Y ν . (8.58) B ≡ αβ = 3(1 + ν)(1 − 2ν) Note the singularity in the second term of (8.57) at ξ = 0 2 . It is attributed to the quasi-static approximation for JKR theory and physically reflects the fact that the adhesive particles can jump into contact [27] with the discontinuous change of the compression ξ. Consider now how the above forces determine the particle dynamics.

8.3 Collision of Granular Particles 8.3.1 Coefficient of Restitution Based on the particle interaction forces discussed so far, we turn now to the description of the particle collisions. It is assumed that the colliding particles do not exchange tangential forces3 , hence, only normal motion is considered. Let the particles be spheres of the same material, which start to collide at time t = 0 at relative normal velocity g (impact rate). The time-dependent compression then reads ξ(t) = Ri + Rj − |ri (t) − rj (t)| ,

(8.59)

where ri (t) and rj (t) are positions of the particle centers at time t (see Figure 8.1). The relative normal motion of particles at a collision is equivalent to the motion of a point particle with the effective mass mi m j . (8.60) meff = mi + mj 2

3

R This is a weak or integrable singularity, that is 0 ξ −1/4 dξ ∼ 3/4 → 0 for → 0. Hence for practical application of (8.57) one can use ξ > , where may be very small but a finite number. See [6] for a discussion of the consistency of this assumption.

200


ξ R v1

v2

Figure 8.1: Head-on collision of identical spheres. The time-dependent state is characterized by the ˙ = v1 (t) − v2 (t). compression ξ(t) ≡ 2R − |r1 (t) − r2 (t)| and the compression rate ξ(t)

For the moment let us neglect adhesion and consider the collision of viscoelastic particles interacting via the force (8.42). The equation of motion and the initial conditions read ρ 3 ˙ 3/2 ¨ ˙ ξ + eff ξ + A ξξ = 0 , ξ(0) = g, ξ(0) = 0 . (8.61) m 2 When granular particles collide, part of the energy of the relative motion is dissipated. The coefficient of (normal) restitution quantifies this phenomenon: ˙ ˙ c )/g , = −ξ(t ε = −ξ˙ (tc ) /ξ(0)

(8.62)

˙ where ξ(0) = g is the pre-collision relative velocity and tc is the duration of the collision. In general, ε is a function of the impact velocity. It can be obtained by integrating (8.61) numerically [7, 8, 15] or analytically [24].

8.3.2 Dimensional Analysis The analytical solution [24] requires considerable efforts, here we give a simplified derivation which is based on a dimensional analysis of the equation of motion (8.61) [23]. This method was employed before [31] to prove that the frequent assumption ε =const. is inconsistent with mechanics of materials. For the dimensional analysis, the elastic and dissipative forces are represented in the more general form Fel = meff D1 ξ α ,

Fdiss = meff D2 ξ γ ξ˙β ,

(8.63)

with D1/2 being material parameters. With these notations the equation of motion for colliding particles reads ξ¨ + D1 ξ α + D2 ξ γ ξ˙β = 0 ,

˙ ξ(0) = g,

ξ(0) = 0 .

(8.64)

For the case of pure elastic deformation (D2 = 0) the maximal compression ξ0 is obtained by equating the initial kinetic energy, meff g 2 /2 and the elastic energy meff D1 ξ0α+1 /(α + 1): ξ0 ≡

α+1 2 D1

1/(1+α)

g 2/(1+α) .

(8.65)

8.3


201

We chose ξ0 as the characteristic length of the problem. The time needed to cover the distance ξ0 when traveling at velocity g defines the characteristic time: τ0 ≡ ξ0 /g. Thus, the dimensionless variables read ˙ ¨ ˙ , (8.66) ξˆ ≡ ξ/g ξˆ = ξ0 /g 2 ξ¨ . ξˆ ≡ ξ/ξ0 , In dimensionless form, (8.61) reads 1 + α ˆα ˙ ¨ ξ = 0, ξˆ + κ ξˆγ ξˆβ + 2

ˆ˙ ξ(0) = 1,

ˆ =0 ξ(0)

(8.67)

with κ = κ(g) = D2

1+α 2 D1

(1+γ)/(1+α)

g 2(γ−α)/(1+α)+β .

(8.68)

None of the terms in (8.67) depends either on material properties or on impact velocity, except for κ. Therefore, if the motion of the particles depends on material properties and on impact velocity, it may depend only via κ, i.e., in the combination of the parameters as given by (8.68). Hence, any function of the impact velocity, including the coefficient of restitution must be of the form ε(g) = ε[κ(g)]. A similar result for ε → 0, β = 1 and α = 3/2 has been obtained in [10]. Hence, if the coefficient of restitution does not depend on the impact velocity g, it is implied that 2(γ − α) + β (1 + α) = 0 .

(8.69)

For small ξ˙ a linear dependence of the dissipative force on the velocity seems to be realistic, i.e., β = 1. Then ε =const. holds true for the following cases: • For the linear elastic force Fel ∝ ξ, (i.e. α = 1) condition (8.69) implies the linear ˙ (γ = 0). dashpot force Fdis ∝ ξ, • For the Hertz law for 3D-spheres (8.24), (i.e. α = 3/2), condition (8.69) requires Fdis ∝ ξ˙ ξ 1/4 , (γ = 14 ). As far as we can see there is no physical argument to justify this functional form of the dissipative force. Therefore, we conclude that the assumption ε =const. is in agreement with mechanics of materials only in the case of (quasi-)one-dimensional systems. For three-dimensional spheres it disagrees with basic mechanical laws. For viscoelastic spheres, according to (8.42), the coefficients are α = 3/2, β = 1, and γ = 1/2. From (8.68) it follows that 3/5 2/5 3 5 ρ eff A g 1/5 (8.70) κ= 2 4 m and, therefore,

2/5 ρ eff 1/5 . ε=ε A g m

(8.71)

202


If we assume that the function ε(g) is sufficiently smooth and can be expanded into a Taylor series, and with ε(0) = 1, for small impact velocity the coefficient of restitution reads

where

ε = 1 − C1 Aκ2/5 g 1/5 + C2 A2 κ4/5 g 2/5 ∓ . . .

(8.72)

√ 3/2 5/2 Y Reff ρ 3 3 . = κ= 2 meff 2 meff (1 − ν 2 )

(8.73)

The coefficients C1 , C2 , . . . are pure numbers which are given analytically in [24]. Here we give a simple derivation of these coefficients (which is correct for C1 and C2 and approximately correct for C3 and C4 , using the method proposed in [23]).

8.3.3 Coefficient of Restitution for Spheres 8.3.3.1 Small Inelasticity Expansion ˆ˙ Using d/dξ = ξd/d ξˆ the equation of motion for a collision adopts the form ˆ dE(ξ) d 1 ˆ˙2 1 ˆ5/2 ˙ ˆ = 0, ˆ˙ ξ + ξ = −κ ξˆ ξˆ = , ξ(0) ξ(0) = 1, 2 dξˆ 2 dξˆ

(8.74)

where we introduce the mechanical energy E=

1 ˆ˙2 1 ˆ5/2 ξ + ξ . 2 2

(8.75)

The first stage of the collision starts at ξˆ = 0 and ends in the turning point of maximal compression ξˆ0 . During the second stage, the particles return to ξˆ = 0. The energy dissipation during the first stage is given by 0

ξˆ0

dE ˆ dξ = −κ dξˆ

ξˆ0

˙ ˆ ˆ ξ. ξˆ ξd

(8.76)

0

˙ ˆ˙ ξ) ˆ is needed. In For the evaluation of the right-hand side of (8.76), the dependence ξˆ = ξ( ˆ the case of an elastic collision where the maximal compression is ξ0 = 1 (according to the definition of the dimensionless variables) from energy conservation, it follows that ˆ˙ ξ) ˆ = 1 − ξˆ5/2 , (8.77) ξ( ˙ i.e., ξˆ vanishes at the turning point ξˆ = 1. For inelastic collisions ξˆ0 1, therefore, ˙ ˆ ˆ ˆ ξˆ0 )5/2 . ξ(ξ) ≈ 1 − (ξ/

(8.78)

Using (8.78) the integration in (8.76) may be performed yielding 1 ˆ5/2 1 3/2 − = −κ b ξˆ0 ξ 2 0 2

(8.79)

8.3


203

where we take into account E(ξˆ0 ) =

1 ˆ5/2 ξ , 2 0

E(0) =

1 ˆ˙2 1 ξ (0) = 2 2

and introduce the constant √ 1 √ π Γ (3/5) b≡ . x 1 − x5/2 dx = 5 Γ (21/10) 0

(8.80)

(8.81)

Let us define the inverse collision, the collision that starts with velocity ε g and ends with velocity g. During the inverse collision, the system gains energy. The maximal compression ξˆ0 is naturally the same for both collisions, since the inverse collision equals the direct collision, except for the fact that time runs in the reverse direction, hence, ˆ dE(ξ) ˙ = +κ ξˆ ξˆ , dξˆ

ˆ˙ ξ(0) = ε,

ˆ = 0. ξ(0)

(8.82)

This suggests an approximative relation for the inverse collision, ˆ˙ ξ) ˆ ≈ε ξ(

ˆ ξˆ0 )5/2 , 1 − (ξ/

(8.83)

with the additional pre-factor ε, which is the initial velocity for the inverse collision. Integration of the energy gain for the first phase of the inverse collision (which equals, up to its sign, the energy loss in the second phase of the direct collision [24]) may be performed just in the same way as for the direct collision, yielding 1 ˆ5/2 ε2 3/2 ξ = +ε κ b ξˆ0 , − 2 0 2

(8.84)

5/2 where again E(ξˆ0 ) = ξˆ0 /2 and E(0) = ε2 /2 is used. Multiplying (8.79) by ε and summing 5/2 it with (8.84), the maximal compression is ε = ξˆ0 . Substituting this into (8.79) we arrive at an equation for the coefficient of restitution

ε + 2κ b ε3/5 = 1 .

(8.85)

The formal solution to this equation may be written as a continuous fraction (which does not diverge in the limit g → ∞): ε−1 = 1 + 2κ b(1 + 2κ b(1 + · · · )2/5 · · · )2/5

(8.86)

For practical applications the series expansion of ε in terms of κ is more appropriate. We return to dimensional units and define the characteristic velocity g ∗ such that 1 κ≡ 2b

g g∗

1/5 ,

(8.87)

204


with b being defined in (8.81). Using, moreover, the definition (8.68) together with (8.42), which provides the values of D1 and D2 , the characteristic velocity reads √ π Γ (3/5) 3 ρ 2/5 −1/5 A (g ∗ ) = 1/5 2/5 . (8.88) meff 2 5 Γ (21/10) 2 With this new notation the coefficient of restitution adopts a simple form: ε = 1 − a1

g g∗

1/5 2/5 3/5 4/5 g g g + a2 − a + a ∓ ··· , 3 4 g∗ g∗ g∗

(8.89)

with a1 = 1, a2 = 3/5, a3 = 6/25 = 0.24, a4 = 7/125 = 0.056. Rigorous but elaborated calculations [24] show that, while the coefficients a1 and a2 are exact, the correct coefficients a3 and a4 are: a3 ≈ 0.315 and a4 ≈ 0.161. The coefficients Ci of the expansion (8.72) can be obtained via Ci = ai C1i = ai (g ∗ )−i/5 .

(8.90)

In particular, √

C1 =

π Γ (3/5) , 21/5 52/5 Γ (21/10)

C2 =

3 2 C 5 1

(8.91)

and respectively, C3 ≈ −0.483, C4 ≈ 0.285. The convergence of the series is rather slow, and accurate results can be expected only for small enough g/g ∗ . Let us briefly mention a complication of the quasi-static approximation (QSA). During the expansion phase it may happen that the repulsive force according to (8.42) becomes negative, i.e., seemingly the particles attract each other. For the interaction of non-cohesive particles we had, however, excluded attractive forces. This is an artefact, since in reality the particles lose contact already, before completely recovering their spherical shape, i.e., before ξˆ = 0 (see [22] for a detailed explanation of this problem). This effect, however, is not in agreement with the QSA. Obviously, (8.42) which is a result of the QSA, derived in [8], is not appropriate to describe the very end of the particle contact. Taking this effect into account we obtain a larger coefficient of restitution as compared with the presented computation [25]. For small dissipation, the correction is rather small. This small correction is neglected here. 8.3.3.2 Padé Approximation For practical applications, such as molecular dynamics simulations, the expansion (8.89) is of limited value, since it diverges for large impact velocities, g → ∞. It is possible, however, to construct a Padé approximant for ε, based on the above coefficients, which reveals the correct limits, ε(0) = 1 and ε(∞) = 0. The dependence of ε(g) is expected to be a smooth monotonically decreasing function, which suggests that the order of the numerator must be smaller than the order of the denominator. The 1-4 Padé-approximant ε=

1 + d1 (g/g ∗ )1/5 1 + d2 (g/g ∗ )

1/5

+ d3 (g/g ∗ )

2/5

+ d4 (g/g ∗ )

3/5

+ d5 (g/g ∗ )

4/5

(8.92)

8.3


205

0.5

Padé approximation Experiment by Bridges et al. (1984)

ε

0.4

0.3

0.2

0

1

2

3

g (cm/s)

4

5

Figure 8.2: Dependence of the coefficient of normal restitution on the impact velocity for ice particles. The dashed line is experimental [4], the solid line is the Padé-approximation (8.92) with the constants given by (8.93) and with the characteristic velocity for ice g ∗ = 0.32 cm s−1 .

satisfies these conditions. Standard analysis (e.g. [3]) yields the coefficients dk in terms of the coefficients ak d0 = a4 − 2a3 − a22 + 3a2 − 1 d1 = [1 − a2 + a3 − 2a4 + (a2 − 1)(3a2 − 2a3 )] /d0 d2 = [(a3 − a2 )(1 − 2a2 ) − a4 ] /d0 d3 = a3 + a22 (a2 − 1) − a4 (a2 + 1) /d0 d4 = a4 (a3 − 1) + (a3 − a2 )(a22 − 2a3 ) /d0 d5 = 2(a3 − a2 )(a4 − a2 a3 ) − (a4 − a22 )2 − a3 (a3 − a22 ) /d0

(8.93) ≈ 2.583 ≈ 3.583 ≈ 2.983 ≈ 1.148 ≈ 0.326

Using the characteristic velocity g ∗ = 0.32 cm s−1 for ice at very low temperature as a fitting parameter, we compare the theoretical prediction of ε(g), given by (8.92), with the experimental results [4], see Figure 8.2. The discrepancy with the experimental data at small g follows from the fact that the extrapolation expression, ε = 0.32/g 0.234 used by [4] to fit the experimental data has an unphysical divergence at g → 0 and does not imply the failure of the theory for this region. The scattering of the experimental data presented by [4] is large for small impact velocity, according to experimental complications, therefore the fit formula of [4] cannot be expected to be accurate enough for velocities that are too small. For very high velocities the effects, such as brittle failure, fracture and others, may contribute to the dissipation, so that the mechanism of the viscoelastic losses could not be the primary one. In the region of very small velocity, other interactions than viscoelastic ones, e.g., adhesive interactions, may be important.

8.3.4 Coefficient of Restitution for Adhesive Collisions For very small velocities, when the kinetic energy of the relative motion of colliding particles is comparable with the surface interaction energy at the contact, the adhesive forces play an important role in collision dynamics – they may change the coefficient of restitution qualitatively. Indeed, as described above, adhesive particles in contact are compressed even for

206


vanishing external load, i.e., a tensile force must be applied to separate the particles. Therefore, at the second stage of the collision, the separating particles must overcome a barrier due to the attractive interaction, which keeps them together. The work against this tensile force reduces the kinetic energy of the relative motion after the collision, that is, it reduces the effective coefficient of restitution. For small impact velocity the kinetic energy of the relative motion may be too small to overcome the attractive barrier, i.e., the particles stick together after the collision, corresponding to ε = 0. From these arguments it follows that the description of particle collisions by pure viscoelastic interaction has a limited range of validity, not only for large impact rate when plastic deformation becomes important, but also for small impact rata due to adhesion. A simplified analysis of adhesive collisions is presented in [5] to estimate the influence of adhesive forces on the coefficient of restitution. It allows to estimate the range of validity of the viscoelastic collision model. We assume that the JKR theory is adequate for the given system parameters. We also assume that the adhesion is small and that the adhesive interactions may be neglected when the force between the particles is purely repulsive. Hence, we take into account the influence of adhesive interaction only when the total force is attractive, that is when the force is mainly determined by adhesion. This happens at the very end of the collision. We also neglect the additional dissipative forces, which arise due to the adhesive interaction and assume that all dissipation during the collision may be attributed to the viscoelastic interactions. At the second stage of a collision, when the particles move away from each other they pass the point where the contact area is a0 and the total force vanishes. As the particles move away further, the force becomes negative, until it reaches, at a = asep , the maximum negative value F = Fsep , Eq. (8.51). At this point the contact of the particles is terminated and they separate. According to our assumption, the work of the tensile force which acts against the particles, separation reads W0 =

ξ(asep ) ξ(a0 )

F (ξ)dξ =

asep

a0

F (a)

dξ da . da

(8.94)

Using (8.49) for the total force F (a), (8.48) for the compression, which allows one to obtain dξ/da, and (8.50), (8.52) for a0 and asep , we obtain the work of the tensile forces, 1/3 , W0 = q0 π 5 γ 5 D2 R4 with the constant 1 1/3 2 3 − 1 32/3 . q0 = 10

(8.95)

(8.96)

Close to the end of the collision, just before the tensile forces start to act, the relative velocity is g = εg. The final velocity g , when the particles completely separate from each other, may be found from the conservation of energy: 1 eff 2 1 eff 2 m (g ) − m (g ) = W0 . 2 2

(8.97)

8.4

Conclusion

207

From the latter equation we obtain the coefficient of restitution for the adhesive collision, εad , ε2 (g)g 2 − 2W0 /meff g = , (8.98) εad (g) = g g where ε(g) is the coefficient of restitution without the adhesive interaction. Hence we obtain the condition for the validity of the viscoelastic collision model, 2W0 ε(g)g . (8.99) meff The threshold impact velocity gst which separates the restitutive (g > gst ) from the sticking (g < gst ) collisions, may be obtained from the solution of the equation 1 eff 2 m ε (g)g 2 = W0 . 2

(8.100)

Using (8.72) we obtain for viscoelastic spheres, in the leading-order approximation, with respect to the small dissipative parameter A:

1/10 2W0 2W0 2/5 1 + C1 Aκ . (8.101) gst = meff meff For head-on collisions (vanishing tangential component of the impact velocity) the colliding particles stick together if g < gst . In this case, after the collision, the particles form a joint particle of mass m1 + m2 .

8.4 Conclusion We have considered the collision of particles in granular matter with respect to viscoelastic and adhesive interaction. Thus, the elastic contribution due to the classical Hertz theory is complemented by the simplest model for dissipative material deformation, where the viscous stress is linearly related to the strain rate. Moreover, quasi-static approximation was assumed, i.e., the impact velocity is much smaller than the speed of sound in the material and the viscosity relaxation time is much smaller than the duration of the collision. Using these approximations, we obtained the general solution for the contact problem for convex viscoelastic bodies. The validity of this model is violated for large impact velocity due to plastic deformations and also for very small impact velocity due to surface forces. We have discussed the available models of adhesive interaction. For the model by Johnson, Kendall and Roberts [17] which has been shown to be accurate in a range of parameters of practical interest, the additional dissipation arising due to adhesive forces have been estimated. From the comparison of the force contribution due to pure viscoelastic interaction and the contribution due to adhesion, we have estimated the range of validity of the viscoelastic model. For head-on collisions we have also estimated the marginal value of the impact velocity, which discriminates restitutive and sticking collisions.

208


References [1] M. Abramowitz and A. Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1965. [2] P. Attard and J. L. Parker. Deformation and adhesion of elastic bodies in contact. Phys. Rev. A, 46:7959, 1992. [3] G. A. Baker. The Padé Approximant in Theoretical Physics. Academic Press, New York, 1970. [4] F. G. Bridges, A. Hatzes, and D. N. C. Lin. Structure, stability and evolution of Saturn’s rings. Nature, 309:333, 1984. [5] N. V. Brilliantov. Viscoelastic interactions of adhesive particles. Preprint, 2004. [6] N. V. Brilliantov and T. Pöschel. Kinetic Theory of Granular Gases. Oxford University Press, Oxford, 2004. [7] N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, and T. Pöschel. The collisions of particles in granular systems. Physica A, 231:417, 1996. [8] N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, and T. Pöschel. Model for collisions in granular gases. Phys. Rev. E, 53:5382, 1996. [9] B. V. Derjaguin, V. M. Muller, and Yu. Toporov. Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci., 53:314, 1975. [10] E. Falcon, C. Laroche, S. Fauve, and C. Coste. Behavior of one inelastic ball bouncing repeatedly off the ground. Eur. Phys. J. B, 3:45, 1998. [11] J. A. Greenwood. Adhesion of elastic spheres. Proc. R. Soc. Lond. A, 453:1277, 1977. [12] J. A. Greenwood and K. L. Johnson. An alternative to the maugid model of adhesion between elastic spheres. J. Phys. D: Appl. Phys., 31:3279, 1998. [13] G. Haiat, M. C. Phan-Huy, and E. Barthel. The adhesive contact of viscoelastic spheres. J. Mechanics and Physics of Solids, 51:69, 2003. [14] H. Hertz. Über die Berührung fester elastischer Körper. J. f. reine u. angewandte Math., 92:156, 1882. [15] J.-M. Hertzsch, F. Spahn, and N. V. Brilliantov. On low-velocity collision of viscoelastic particles. J.Phys. II France, 5:1725, 1995. [16] B. D. Hughes and L. R. White. The implications of elastic deformation on the direct measurement of surface forces. J. Chem. Soc., Faraday Transactins I, 76:963, 1980. [17] K. L. Johnson, K. Kendall, and A. D. Roberts. Surface energy and contact of elastic solids. Proc. R. Soc. London Ser. A, 324:301, 1971. [18] L. D. Landau and E. M. Lifshitz. Theory of Elasticity. Oxford University Press, Oxford, 1965. [19] D. Maugis. Adhesion of spheres: the JKR-DMT transition using a Dugdale model. J. Colloid Interface Sci., 150:243, 1992. [20] V. M. Muller, V. S. Yuschenko, and B. V. Derjaguin. On the influence of molecualar forces on the deformation of an elastic sphere and its sticking to a rigid plane. J. Colloid Interface Sci., 77:91, 1980.

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[21] V. M. Muller, V. S. Yuschenko, and B. V. Derjaguin. General theoretical consideration of the influence of surface forces on contact deformation. J. Colloid Interface Sci., 92:92, 1983. [22] T. Pöschel and T. Schwager. Computational Granular Dynamics: Models and Algorithms. Springer, Berlin, 2004. [23] R. Ramírez, T. Pöschel, N. V. Brilliantov, and T. Schwager. Coefficient of restitution for colliding viscoelastic spheres. Phys. Rev. E, 60:4465, 1999. [24] T. Schwager and T. Pöschel. Coefficient of restitution of viscous particles and cooling rate of granular gases. Phys. Rev. E, 57:650, 1998. [25] T. Schwager and T. Pöschel. Duration of contact and coefficient of restitution. preprint, 2004. [26] U. D. Schwarz. A generalized analytical model for the elastic deformation of an adhesive contact between a sphere and a flat surface. J. Colloid Interface Sci., 261:99, 2003. [27] J. R. Smith, G. Bozzolo, A. Banerjea, and J. Ferrante. Avalanche in adhesion. Phys. Rev. Lett., 63:1269, 1989. [28] F. Spahn, N. Albers, M. Sremcevic, and C. Thornton. Kinetic description of coagulation and fragmentation in dilute granular particle ensembles. Europhys. Lett., 67:545, 2004. [29] D. Tabor. Surface forces and surface interactions. J. Colloid Interface Sci., 58:2, 1977. [30] Y. Taguchi. Powder turbulence: Direct onset of turbulent flow. J. Physique, 2:2103, 1992. [31] T. Tanaka, T. Ishida, and Y. Tsuji. Direct numerical simulation of granular plug flow in a horizontal pipe: The case of cohesionless particles (in japanese). For an english presentation of this work see [30]. Trans. Jap. Soc. Mech. Eng., 57:456, 1991. [32] S. P. Timoshenko. Theory of Elasticity. McGraw-Hill, New York, 1970.

Part IV

Hydrodynamic Interactions

9 Fluidized Beds: From Waves to Bubbles Élisabeth Guazzelli

9.1 Introduction A fluidized bed consists of a suspension of particles in a vertical channel, submitted to an upward flowing fluid (gas or liquid), forced through a porous base-plate, as can be seen in Figure 9.1. At low flow rates, the bed is packed. As the flow is increased, the drag force on the particles increases until it is sufficient to balance their net weight. After this point, referred to as minimum fluidization, the particles become free to move and the bed is said to be fluidized. When the flow is further increased above minimum fluidization, the bed expands uniformly and takes up a smaller concentration to accommodate the upward flow. This is rather similar to a sedimenting suspension of particles where the mean velocity adjusts itself to suit the mean concentration, being smaller for larger concentration. This is why a fluidized bed has been considered, sometimes as an ensemble of sedimenting particles referred to different reference frames, and hence as a stationary sedimenting suspension. However, uniform expansion is just one of the possibilities for fluidization and usually fluidized beds are unstable and contain inhomogeneities which arise throughout the bed.

g

q Figure 9.1: Sketch of a fluidized bed. The fluid superficial velocity (or velocity of the fluid– particles mixture) is q, and g is the acceleration due to gravity.

214

9 Fluidized Beds: From Waves to Bubbles

Fluidized beds are intermediate systems between suspensions in a viscous fluid and dry granular materials, with no fluid between the grains. They consist of a dispersion of particles with a low-viscosity fluid between the grains [1]. Particle–particle and fluid–particle interactions are thus very important. Direct contact between the solid grains is dominant when the interstitial fluid is a gas, or at very high concentrations, with a liquid medium. The interactions via the fluid dominate when the interstitial fluid is a liquid and when the concentration is not too high. Another important feature of fluidized beds is the significant relative motion between the particle and fluid phases arising from the difference in body force due to the difference in density between the fluid and the particles. The behavior of fluidized beds is therefor very complicated and far from being completely understood. Fluidized beds are not novel engineering devices and written records of their use dates back to the 16th century as a technique which was used to purify iron ore [2]. They were used more extensively at the beginning of the 20th century with the growing importance of fossil energy. They are still used very extensively in oil and coal industrial processes such as fluidized catalytic cracking and fluidized coal combustion, because of their excellent heat and mass transfer characteristics and ease of removal and rejuvenation of the fluid–solid mixture. But they can be found in other fields such as pharmaceutical processes and bio-reactors. Empirical correlations are still basic features of the practical design of fluidized beds, but since the mid-20th century there has been a growing interest in a more fundamental approach such as the modeling of the motion of the fluid and particle phases [3]. The scope of the present review is focused on the physical understanding of the dynamical processes leading to the evolution of a fluidized bed from uniform fluidization (when it exists) to voidage instabilities and large-amplitude structures which can propagate along the bed.

9.2 Flow Regimes and Instabilities Uniform and homogeneous fluidized beds are not common in practice. Fluidized beds usually present a variety of complex flow regimes above minimum fluidization that are depicted in Figure 9.2. Gas-fluidized beds are very unstable and rapidly attain a bubbling regime. In this regime, bubbles, which are regions almost free of particles, appear at the bottom of the bed and rise, causing the top surface of the bed to take the appearance of a boiling liquid. For large-diameter beds, bubbles adopt spherical cap forms while, for tall narrow beds, they grow in size by coalescence and become slugs which fill the pipe as shown in Figure 9.2. Most of what has been learned about these bubbles concerns the behavior of a single gas bubble injected into the bed. A bubble in a fluidized bed behaves like a large bubble in a low-viscosity liquid with low surface tension. Figure 9.3 shows a photograph by Rowe [4] of a bubble rising in a two-dimensional gas fluidized bed. The bubble is nearly devoid of particles and one can see the motion of the particles around the bubble. Some typical features were also found in the interior of a three-dimensional fluidized bed using X-ray imaging. The essential understanding of bubble motion came from a model by Davidson [5] and further developments can be viewed as an improvement of this (see, for instance, Chapter 5 of [3]). Experimental measurements support the main predictions of this model and, in particular, one recovers the classic Davies– Taylor expression relating the rising velocity of the bubble to the square root of its size.

9.2

Flow Regimes and Instabilities

Homogeneous

215

Waves

Bubbles

Slugs

Figure 9.2: The different flow regimes of fluidization.

The presence of bubbles in technological applications may or may not be welcome. On one hand, bubbles can stir the particles efficiently. On the other hand, if the reacting gas passes through the bubbles, there is a loss in chemical conversion via the catalytic particles and the bed is not efficient. Good reviews related to fully formed bubbles in fluidized beds can be found in [3, 6–8]. These approaches do not address the issue of the dynamical processes leading to the formation of bubbles, which will be tackled in Section 9.7. liquid-fluidized beds are less used in industry but their behavior is no less interesting. They are less unstable than gas fluidized beds and exhibit voidage-wave instability as shown in Figure 9.2. However, stable uniform fluidization can be observed just above minimum fluidization for viscous liquids or light and small particles [9]. If inertia is small enough, this state of uniform fluidization can even be observed over the full range of flow rate before the regime of pneumatic transport of particles. As mentioned in Section 9.1, the fluidized bed can then be considered as a stationary sedimenting suspension and has been studied as such [10, 11]. Note, however, that the boundary conditions are not the same in sedimentation and fluidization. In most experiments, inertia is not small and the bed becomes directly unstable above minimum fluidization [12–16]. This instability remains one-dimensional only in narrow beds, whereas in wider beds, transverse secondary instabilities can occur [13, 14, 17]. There exists also a turbulent regime at high flow rate. We will go into further details on the primary voidage-wave instability observed in narrow beds in Section 9.5. Most of the gas fluidized beds are bubbling while most of the liquid fluidized beds are not. Quite early Wilhelm and Kwauk [18] attempted to delineate the transition between bubbling and non-bubbling behavior (or aggregative and particulate, in their terminology) by using an empirical criterion based on the Froude number, suggesting that the bed is bubbling if

216


Figure 9.3: A bubble in a two-dimensional fluidized bed viewed from a reference frame moving up with the bubble. (From [4].)

F rm = u2m /gdp > 1 where um is the minimum fluidization velocity, dp the particle diameter, and g the acceleration due to gravity. Although this simple classification became very popular, the distinction between bubbling and non-bubbling is, in fact, more complicated. Early experimental studies reported of bubbling liquid-fluidized beds of high-density particles [6], showing that the transition between the two behaviors is not as sharp as expected. The question of the origin of the bubbles and of the distinction between these two types of behavior is one of the fundamental problems that has attracted much interest lately and that we will address in Section 9.7.

9.3 Instability Mechanism In Section 9.2, we have seen that uniform fluidized beds can be unstable. Moreover liquidfluidized beds are just as unstable as gas-fluidized ones, although bubbling is seen usually in the latter and less commonly in the former. The same destabilizing mechanism is at work in both case and small disturbances can grow exponentially in certain conditions. The following physical picture of the instability mechanism comes from Batchelor in 1988 [19] and also from a later review by Hinch [1]. Following the analysis of Batchelor [19], let us consider an homogeneous one-dimensional fluidized bed and take the usual reference frame of sedimentation for which the local mean

9.3

Instability Mechanism

217

material volume flux is zero. The particle mass conservation equation can be written as: ∂φ ∂(φV ) + = 0, ∂t ∂z

(9.1)

where φ is the local volume fraction of particles and V the particle local “sedimentation” velocity. Consider a small perturbation φ from the homogeneous base state φ0 . If V depend only on the volume fraction V = U (φ) (there is a good empirical relation, the Richardson– Zaki law [20], which gives U (φ) = U0 (1 − φ)n where U0 = U (0) is the sedimentation velocity of an isolated particle and the exponent n varies with the Reynolds number), the Eq. (9.1) can be linearized as: ∂φ ∂φ + c0 = 0, ∂t ∂z

(9.2)

with c0 = d(φU )/dφ. A small disturbance of the particle volume fraction propagates without change of shape with the kinematic wave speed c0 [21]. Up to now we have neglected inertia and the particle velocity V is simply a function of the local concentration and this function determines the wave speed. When included, inertia produces some phase-lag in the adjustment of the velocity to a change in the local concentration and this can cause the growth of the wave. Particle inertia was recognized early as the destabilizing mechanism and this was clearly formulated by Jackson in 1963 [22]. Jackson’s 1963 linear analysis, adopting a simple particle momentum equation together with the particle conservation equation, showed that all fluidized beds should be unstable and that shorter waves should grow faster. A short-wave cutoff was obtained later by introducing viscous dissipation, i.e., viscous stresses in the momentum equation [23–25]. The physical mechanism leading to this type of term in the equation is the momentum transport associated with fluctuations in particle velocity or particle contact interactions. When these terms are included, one obtained a maximum growth rate for a bounded value of the wavelength but the bed always remains unstable. Identifying a physical process which opposes the instability has proved to be more difficult and the issue is not entirely resolved. Momentum transport by particles associated with fluctuations in particle velocity also generates normal stresses. This was recognized by Anderson and Jackson in 1967 [24] as they identified an effective pressure associated with the particle phase but it was estimated to be a small effect. Later, it was found that the bed could be stabilized for a sufficiently large value of normal stress gradient [19, 26]. Note that Batchelor [19] related this gradient to the bulk modulus of elasticity of the particle configuration and proposed an additional source of bulk elasticity coming from hydrodynamic dispersion. In any case, when a normal stress gradient is included in the particle momentum equation, it is possible to find a specific criterion for the growth of small-amplitude concentration waves. This instability criterion takes a very simple form noticed in 1969 by Wallis [27] for a gas fluidized bed for which virtual mass effects are negligible compared to the drag force: the speed of the kinematic wave must exceed that of a dynamic wave associated with this bulk elasticity. For liquid fluidization, there is a similar instability criterion but the physical processes linked to the dynamic waves are less clear. A detailed review of the linear stability analysis of uniform fluidized bed can be found in Chapter 4 of [3]. The major difference between gas and liquid fluidization is that the growth rate of the linear wave is faster in the former case because of

218


the higher inertia of the particles relative to the fluid. As a consequence, detailed experimental observation of the growth of one-dimensional disturbances has been only practicable in liquid-fluidized bed as will be discussed in the Section 9.5.

9.4 Governing Equations While discussing the instability mechanism, we have identified some of the ingredients necessary for the modeling of fluidized suspensions. Since there is a significant relative motion between the particle and fluid phases, a basic description of a fluidized bed should be as two coexisting continua: a fluid phase and a solid phase. Such two-phase modeling has been used extensively over the past three decades and a complete report on the way the equations of motion can be derived can be found in the Chapter 2 of [3]. In principle, a complete modeling of the motion of the solid particles in a fluid is given by solving the Newtonian equations of motion for the translation and the rotation of each particle and the Navier–Stokes equation for the fluid with no-slip conditions on the surface of each particle and on the walls. Such types of direct calculation can now be done numerically but with a small number of particles compared to that found in a real fluidized bed. This is why one has employed two-phase governing equations which describe the system in an average sense. However, the formal process of averaging generates averaged quantities more numerous than the available equations and therefore there is a closure problem which is a central issue in fluidization theory. The minimal model consists in writing conservation equations for the mass and the momentum for the two phases. In order to close the equations, it is then necessary to postulate expressions for the fluid–particle interaction forces, i.e. the drag force and the virtual mass force, and the stress tensors associated with the fluid and particle phases. Note that there are more complicated models, such as those introducing a granular temperature, but with more complex closure relationships. The empirical Richardson–Zaki law [20] can be used for the drag force in terms of the relative motion between the phases. The drag force is proportional to the relative velocity of the two phases with a coefficient of proportionality varying with the volume fraction of the particles. The virtual mass force is proportional to the relative acceleration with a concentrationdependent virtual mass coefficient. There are some suggestions for this coefficient at moderate concentration, but it is not known at large concentration. In most of the studies, either it has been neglected (which is valid compared to the drag force for gas-fluidization) or it has been given its value for an isolated sphere (half the displaced mass of fluid). The biggest problem comes with the expression for the stresses of the fluid and solid phases. A Newtonian fluid form is usually adopted for the stress tensors. It is not clear, however, how to describe such important quantities as the pressure, ps , and the viscosity, µs , of the solid phase and many speculative expressions have been used in the literature as presented in Table 9.1. Most of the ad hoc expressions used in these theoretical or numerical studies are monotonic functions of the particle volume fraction φ which diverge at the volume fraction for random close-packing φcp . As we shall see in Section 9.6, recent measurements provide some insight into the rheology of the particle phase.

9.5

Primary Instability

219

Table 9.1: Various expressions for the particle pressure, ps , and viscosity, µs , proposed in the literature. (From [16].)

Authors

ps (φ)

Murray [28]

0

Fanucci et al. [29]

not given

Needham and Merkin [30]

Pφ

constant

Harris and Crighton [31]

P φ/(φcp − φ) P φ3 exp φcprφ−φ   C1 φ3 exp φcprφ−φ     C2 φ     C2 (φcpφ−φ)2 

constant

µs (φ)

Anderson et al. [32]

Glasser et al. [33]

M exp

Mφ µf φcp −φ

φcp −φ (1−φcp )(1−φ)

Mφ 1−(φ/φcp )1/3

Mφ 1−(φ/φcp )1/3

9.5 Primary Instability

Figure 9.4: Visualization of the voidage-wave instability in a 47% mean volume fraction suspension of glass spheres of diameter ds = 685 ± 28 µm and density ρs = 4.0 ± 0.1 g cm−3 fluidized by water in a cylindrical tube with an inner diameter of 0.70 ± 0.02 cm. The vertical scale is 1 cm. (From [34].)

The first systematic study of voidage-wave instability in water fluidized bed was published in 1969 by Anderson and Jackson [12]. Visual observations of the wave were easily made by backlighting the bed in this earlier work and in the following studies [9, 13–16]. A typical example of such visualization is shown in Figure 9.4. There is a propagating vertical stratifi-

220


a)

b)

Figure 9.5: Spatio-temporal plots of the one-dimensional wave forced at 0.5 Hz, a) and 1.5 Hz, b). The mean particle volume fraction is 0.500 ± 0.001 and the particles and fluid are the same as in Figure 9.4. The vertical scale (space) is 1 cm and the horizontal scale (time) is 2 s. The inclined white lines in these plots correspond to low concentration regions of the suspension moving upward with nearly constant phase velocity. The sinusoidal motion of the porous piston can also be seen. (From [15].)

cation of the particle volume fraction with regions of lower concentration which transmit more light appearing as propagating light bands. A more quantitative technique used by Anderson and Jackson [12] and the subsequent researchers [9, 13–16] was to measure the attenuation of the light through the suspension. The power spectra of the waves were found to be very broad. There was clear evidence, however, of a dominant low-frequency mode ∼ 1 − 2 Hz. The instability amplitude, which was small at the bottom of the bed, was found to increase along the bed and eventually to saturate [13, 15]. The spatial evolution of the wave was noted in the earlier studies but it was only later that the nature of the instability was formulated by Nicolas et al. [15, 35] within the absolute/convective framework [36]. A fluidized bed is an “open” flow where disturbances are convected along the bed and not reflected by the downstream boundary. Instabilities evolving in such “open” flows can be classified in two categories. If the instability is sensitive to external perturbations (noise), it is said to be convective. Perturbations are then amplified while convected by the mean flow, and the system can be described as a noise amplifier. On the contrary, if the instability has a local origin, it is said to be absolute. The flow then behaves as an unstable oscillator. Nicolas et al. [15] applied a sinusoidal displacement to the porous plate (called the distributor) which holds the suspension, and found that the waves were periodic and followed the frequency of the applied force as shown in Figure 9.5. This demonstrated that the instability is convective in nature. No experimental evidence of a transition towards an absolute instability was observed. A fluidized bed is a noise amplifier and any perturbation created at the bottom

9.5

Primary Instability

221

of the bed, the always present “natural” noise of the distributor or a controlled perturbation, propagates and grows along the bed. Away from the distributor, non-linearity was found to dominate quickly the wave evolution and both the “natural” noise and the controlled perturbation (if applied) participated in this non-linear evolution. The knowledge of the convective nature of the instability was an important step in understanding that each frequency mode needed to be investigated independently. The separation of the forcing mode from other modes was done by Duru et al. [16] with a synchronized average method using the forcing as a reference signal. Duru et al. [16] observed three types of behavior depending upon the forcing frequency. For high frequencies above a cutoff frequency (a shortwave cutoff), stable modes with spatially-decaying amplitude were found. At the cutoff frequency, a neutral mode propagated without change of amplitude or shape. For low frequencies, the modes were unstable and their amplitude and shape were observed to evolve along the bed. These observations reveal features which seemed in qualitative agreement with linear stability analyses. However, because of the large amplitude of the “natural” noise, it was very difficult to work with the small amplitude of forcing required to keep in the linear regime and the wave evolution was then rapidly dominated by non-linearities. The non-linear evolution and saturation of the unstable modes were analyzed in detail by Duru et al. [16]. These saturated waves were found to have a well-defined shape, with flat peaks of large particle concentration and narrow troughs of low concentration as can be seen in Figure 9.6. Note the slight asymmetry of this one dimensional saturated wave with a higher slope of concentrations where increasing in time compared with where decreasing. Numerous theoretical studies have been devoted to the effect of non-linearities on onedimensional waves and a complete record of them can be found in Chapter 5 of [3]. The predicted developed waves consist also of a dense plateau separated by narrow trough of lower density with a marked asymmetry of the profile for a gas-fluidized bed. However, this asymmetry is opposite to that of the experimental wave (see for example one of the most recent and extensive studies by Anderson et al. [32]). This discrepancy comes from the fact that these studies assume expressions for the particle phase viscosity and pressure in solving the equations of motion to find the shape of the steady wave (in fact it is due to the assumed expression for the pressure, as will be seen in Section 9.6). This shows the needs to search for empirical expressions for these quantities which will be discussed in Section 9.6. 0.65 0.6

φ

0.55 0.5 0.45 0

1

2

3

4

Time (s) Figure 9.6: Typical shape of a saturated wave with a mean volume fraction of 57% and a frequency of 1.4 Hz. (From [16].)

222


9.6 Rheology of the Particle Phase Rough estimates for the unknown solid viscosity and pressure were obtained by studying the general growth and propagation of the primary instability occurring in narrow liquidfluidized beds [12, 13]. But these measurements did not give any insight into the variations of these quantities with the volume fraction as they only provided values for a few particle concentrations. More recently, Duru et al. [16] used the experimental observation of the shape of the nonlinear saturated wave, such as that shown in Figure 9.6 to test governing equations of the type used by Anderson et al. [32]. The first result of this study is that the rheology of the particle phase can be considered as Newtonian, at least over the narrow range of parameters of the experiment. The second result is that it provided scaling laws and particle-concentration dependence for the particle-phase viscosity and pressure. They were also able to estimate the different terms of the momentum equation for the particle phase: in their experimental conditions, the drag on the particles nearly balances their weight, corrected for buoyancy, the small imbalance being due mostly to the viscous term with a much smaller contribution from the pressure (which made it more difficult to measure). They found that the variation of the solid-phase viscosity on the density ρs , diameter ds and terminal velocity vt of the particles and on the local particle concentration, could be approximated by the expression: µs (φ) ≈ 0.18

ρs ds vt , φrlp − φ

(9.3)

where φrlp denotes the particle volume fraction at random loose packing. The compressibility of the solid phase, i.e., the derivative of the particle pressure with respect to concentration, seems to be described by the expression: dps /dφ ≈ 0.7ρf vt2 , or dps /dφ ≈ 0.2ρs vt2 ,

(9.4)

over a range of most of the concentrations, with large negative values at the highest concentrations (with ρf being the density of the fluid). Because of the scatter of the experimental data, it was not possible to discriminate between these two scalings. However, earlier work by Ham et al. [9] on the study of the stability of the bed suggested a strong dependence of the stabilizing term with ρs . In his recent review, Sundaresan [37] compared this latter result with the measurements of the collisional pressure in a water fluidized bed by Zenit et al. [38] and the theoretical predictions by Koch and Sangani [39] for a fluidized suspension with large Stokes number and small Reynolds number. From these results, he concluded that the normalized particle pressure, ps /(0.5ρs vt2 ), should first increase with increasing volume fraction, reach a maximum (the value of which is still uncertain), and then decrease for larger volume fraction. Most of the previously proposed expressions in Table 9.1 have their viscosity and pressure increasing with concentration. This proposed concentration dependence seems correct for the viscosity, but not for the pressure. This has consequences on the shape of the non-linear saturated wave, such as that in Figure 9.6. While the viscosity is responsible for the amplitude of the wave and the characteristic shape, with a plateau of high concentration and trough of

9.7

Secondary Instability and the Formation of Bubbles

223

low concentration, the pressure gives a small asymmetry. Consequently, theoretical studies assuming an increase in the pressure with concentration predicted an opposite asymmetry, as discussed at the end of Section 9.5. A recent numerical study [40] using the more complex formalism of the granular temperature predicts a variation in the granular pressure similar to that of Zenit et al. [38] and produces a saturated wave with the correct asymmetry. The traditional justification for these increasing expressions is that the fluidized bed should be more stable at large concentrations. Duru et al. [16] explains the stability of a fluidized bed with their nearly-constant compressibility by noting that the destabilizing inertial terms decrease with concentration at high concentrations.

Figure 9.7: Sketch of the overturning instability. (From [44].)

9.7 Secondary Instability and the Formation of Bubbles It seems clear now that bubbling is a large-amplitude phenomenon which cannot be explained by linear stability analyses [3, 37, 41]. Batchelor and Nitsche proposed four successive stages for the formation of bubbles [19, 42–45]. First, the primary one-dimensional voidage-wave instability develops and creates a stratification of the suspension with consecutive dilute and dense layers. Secondly, the one-dimensional wave-train becomes unstable and produces a two-dimensional structure because of a gravitational overturning instability, which tends to tilt “heavy” layers of high particle concentration and “light” layers of low particle concentration and causes the accumulation of a dense heavy region in the troughs of the disturbance and a light region in the peaks, as depicted in Figure 9.7. Thirdly, this secondary instability ultimately creates a bubble-like void where an internal fluid circulation develops. Particles are

224


finally expelled by centrifugal forces from these buoyant blobs and this leads to bubbles of clear fluid rising steadily through an otherwise uniform bed. f) 17

12 c)10

14

z (cm)

z (cm)

16

8 10 68

15

12

14 13

10

0.585 < φ < 0.600

12

46 00

222

444

666

888

10 10 10

11

12 12

b) 11

0

e)

22

44

66

88

10 10

12

14 12

z (cm)

z (cm)

10 8 9

68 7

0.540 < φ < 0.555

12 10

0.525 < φ < 0.540

10 8

0.510 < φ < 0.525

46 5

0

22

44

66

88

10 10

68 0

12

d)13

7

10 12

z (cm)

a) 68 z (cm)

0.570 < φ < 0.585 0.555 < φ < 0.570

46 5

24

44

66

88

10 10

12

0.495 < φ < 0.510 0.480 < φ < 0.495 0.430 < φ < 0.480

11

8 10 9

68

3

0 00

22

222

444

666

x (cm)

888

10 10 10

12 12

7

0

22

4

66

88

10

12

x (cm)

Figure 9.8: Concentration map showing the two-dimensional destabilization of a plane wave resulting in short-lived buoyant blobs for the moderate-density particles with a mean concentration of 0.55. a) t = 0 s, b) t = 0.16 s, c) t = 0.28 s, d) t = 0.32 s, e) t = 0.40 s, f) t = 0.56 s. The z-direction is the vertical direction while the x-direction is the horizontal direction. The “zero” z-position is arbitrary and does not correspond to the bed bottom. (From [17].)

The first stage regarding the one-dimensional primary instability is well documented as discussed in Section 9.5. Most of the studies have been devoted to voidage-waves in liquidfluidized beds because of their slower growth rate. The voidage-wave instability stays onedimensional in constrained geometry, i.e., in narrow beds. The second stage, i.e., the twodimensional destabilization of the plane wave train, was observed in wider containers. It was first described experimentally by El-Kaissy and Homsy [13] in a two-dimensional liquidfluidized bed. They observed the break-up of the one-dimensional wave leading to the appearance of short-lived bubble-like voidage pockets. A later work by Didwania and Homsy [14], in a wider bed, characterized the different regimes of flow including wavy, transverse, turbulent, and bubbly states. More recently, Duru and Guazzelli [17] conducted experiments in a similar-sized water-fluidized bed and gave conclusive evidence for the complete succession of instabilities leading to the formation of bubbles. The most striking results are obtained with particles of approximately the same size (ds ≈ 1 mm) but different densities. For moderate-density particles (glass beads with ρs ≈ 4 g cm−3 ), the two-dimensional destabilization of the one-dimensional voidage wave results in the formation of short-lived buoyant blobs as can be seen in Figure 9.8. First, the plane wave reaches a certain amplitude (but is not saturated), see Figure 9.8a Secondly, the wave buckles and higher voidage pockets

9.7


225

20 20 18 18 16 16

0.585 < φ < 0.600 0.565 < φ < 0.585 0.555 < φ < 0.565 0.545 < φ < 0.555 0.535 < φ < 0.545 0.520 < φ < 0.535 0.505 < φ < 0.520 0.455 < φ < 0.505

z (cm)

14 14 12 12 10 10 88 66 44 22 0

0

22

44

66

8

10 10

12

x (cm) Figure 9.9: Concentration map above the break-up zone showing a complex two-dimensional regime for the moderate-density particles with a mean concentration of 0.536. The “zero” zposition is arbitrary and does not correspond to the bed bottom. (From [17].

appear, see Figure 9.8b and c. Thirdly, these pockets accelerate, see Figure 9.8c. Finally, the voidage pockets disappear, see Figure 9.8e) and f). Above this break-up, the bed reaches a complex two-dimensional regime with oblique waves which are remnant of the original plane-waves distorted by the break-up as can be seen in Figure 9.9. For high-density particles (stainless steel beads with ρs ≈ 8 g cm−3 ), the two-dimensional destabilization leads to the formation of real bubbles as can be seen in Figure 9.10. In this bubbling regime, bubbles appear continually within the bed, increase in size, split, and may even start to follow each other forming “trains” of bubbles, all patterns which are also observed in gas fluidization (for more details see [17]). The difference between moderate- and high-density particles comes from the evolution of the dilute pockets. The rate of filling of the pocket by particles from the top exceeds the rate of leaving it at the bottom, for moderate-density beads. Conversely, the pockets persist and grow for high-density beads. For smaller particles of low density (glass beads with ρs ≈ 2.5 g cm−3 and ds ≈ 685 µm), a transverse destabilization of the onedimensional voidage wave is still observed but it evolves smoothly and does not give rise to voidage pockets as can be seen in Figure 9.11. In his recent review, Sundaresan [37] summarized these results by using the Froude number at minimum fluidization as a control parameter. If F rm ∼ O(10−3 ), two-dimensional

226


e) t = 0.40 s

d) t = 0.32 s

c)

t = 0.2 s

b)

t = 0.12 s

a) t = 0s

1 cm Figure 9.10: Transverse destabilization of plane voidage wave leading to the formation of a real bubbles for the high-density particles. The sketches on the right show the position of the voidage perturbation. (From [17].)

destabilization occurs but there is a smooth evolution of the two-dimensional structure. If F rm ∼ O(10−2 ) bubble-like voids appear briefly and disappear, leading to a final complex two-dimensional regime made of oblique waves. If F rm ∼ O(10−1 ), the voids become real bubbles and their continuous appearance leads to a bubbling regime. The existence of the overturning instability was theoretically probed through linear stability analysis of the one-dimensional wave against two-dimensional perturbations for long wavelengths [32, 46–49]. The evolution of the structure following two-dimensional destabi-

9.7


227

14 14

12 12

0.560 < φ < 0.570 0.550 < φ < 0.560 0.540 < φ < 0.550 0.535 < φ < 0.540 0.530 < φ < 0.535 0.525 < φ < 0.530 0.520 < φ < 0.525 0.510 < φ < 0.520

z (cm)

10 10

88

66

44

22

0 0

22

44

66

88

10 10

12

x (cm) Figure 9.11: Concentration map showing the smooth transverse modulation of the wave-train for the smaller low-density particles with a mean concentration of 0.538. The “zero” z-position is arbitrary and does not correspond to the bed bottom. (From [17].)

lization was performed by Anderson et al. and Glasser et al. [32, 48, 49] by a series of direct computational studies of solutions of the full governing equations. They simulated both the growth of a two-dimensional perturbation from the uniformly stable bed and the transverse destabilization of a fully-saturated wave train. These simulations seem to capture well the general succession of destabilization events observed in non-bubbly beds, as can be seen in Figure 9.12. We can see the successive buckling of the one-dimensional wave and the appearance of the bubble-like void in Figure 9.12a and b, and then the acceleration, filling in, and fading of the void in Figure 9.12c and d. The manner in which the void disappears by being filled up from above is also well predicted. In bubbling beds, there is again a similarity between the succession of events leading to realbubble formation, from the initial buckling of the plane wave in Figure 9.13a and b to the final bubble structure in Figure 9.13d. However, quantitative predictions are still lacking. This may be related to the assumed expressions for the particle viscosity and pressure, in theoretical studies (see Table 9.1). Let us now return to the four-stage scenario of Batchelor and Nitsche [19, 42–45]. The buckling of the primary instability followed by the formation of regions of low concentration that develop into buoyant voidage pockets, demonstrate a similarity with the physical mechanism proposed by Batchelor and Nitsche [43]. The bubble-like voids of the third stage have been observed and we can expect fluid circulation in these voids. The difference between bubbling and non-bubbling beds appears at the late stages and is only seen in the expulsion

228


Figure 9.12: Concentration map showing the simulation of the growth of a perturbation of a fully-developed one-dimensional wave in a non-bubbling water-fluidized bed. (From [32].)

of particles from the low concentration region of the voidage pockets created by the overturning instability. In bubbling beds, the voidage within the pocket increases because particles leave the voidage pocket more quickly than they enter while, in non-bubbling beds, the pocket fades because it is filled in from above. This has been related to the role of inertia in deepening the low concentration region. Anderson et al. [32] argued that the larger growth rate of the primary instability for bubbling beds with higher inertia, produced sufficiently deep voidage stratification before the two-dimensional overturning instability occurred. Batchelor and Nitsche [45] proposed a different idea and attributed the expulsion of heavy particles from the buoyant blob to the centrifugal force.

9.8 Conclusions The present review has been focused on understanding voidage instabilities and explaining the origin of regions devoid of particles called “bubbles” that develop in fluidized beds. The evolution of these voidage structures evolves very quickly in a gas-fluidized bed. It is much slower in liquid-fluidized beds and thus more amenable to detailed experimental investigations. These experimental studies have established that one-dimensional voidage-wave instability occurred at first. A non-linear saturation of the voidage-wave was observed in constrained geometry, i.e., in narrow beds. In wider beds, there is a transverse destabilization of the one-dimensional wave and a succession of instabilities leading to bubble formation has

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229

Figure 9.13: Concentration map showing the simulation of the growth of a perturbation of a one-dimensional wave leading to bubble formation for water-fluidized lead-shot particles. (From [49].)

been identified. Theoretical and computational studies using two-phase modeling seem to have succeeded in accounting for most of these features, at least qualitatively. The nature of the difference between bubbling and non-bubbling behavior seems also to have been resolved. However, despite the success of this recent research, there is a lack of quantitative predictions. One of the possible causes may lie in the uncertainties of the closures adopted for the particle-phase rheology in the governing equations. Recent experimental work has provided scaling laws and concentration-dependence for the particle viscosity and pressure in a limited range of parameters, but the two-phase governing equations remain to be tested in different conditions.

Acknowledgment I would like to thank P. Duru, E. J. Hinch, and M. Nicolas for discussions and critical reading.

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[46] M. F. Göz, Transverse instability of plane voidage wavetrains in gas-fluidized beds, J. Fluid Mech. 303 (1995), 55–81. [47] M. F. Göz and S. Sundaresan The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds, J. Fluid Mech. 362 (1998), 83–119. [48] B. J. Glasser, I. G. Kevredikis, and S. Sundaresan, One- and two-dimensional travelling wave solutions in gas-fluidized beds, J. Fluid Mech. 306 (1996), 183–221. [49] B. J. Glasser, I. G. Kevredikis, and S. Sundaresan, Fully developed travelling wave solutions and bubble formation in fluidized beds, J. Fluid Mech. 334 (1997), 157–188.

10 Wind-blown Sand Hans J. Herrmann

10.1 Introduction If granular materials are submerged into a fluid such as air or water, which is in motion, this fluid will exert forces on the grains and in that way create a particle flux. This transport of granular material is among others responsible for the formation of dunes and beaches. The first to systematically study airborne sand transport was the British brigadier R. Bagnold who, during the time of World War II did experiments in wind channels and field measurements in the Sahara. He presented the first empiric expression for the saturated sand flux. Since then more refined expressions have been proposed. Bagnold also described for the first time the two basic mechanisms of sand transport: saltation and creep, and wrote the classic book on the subject which still is an excellent reference [1]. The sand flux on the surface modifies the shape of the landscape and spontaneously creates patterns on different scales: ripples in the range of ten to twenty centimeters and dunes in the range of two to two hundred meters. One example are the Barchan dunes shown in Figure 10.1 which arise when the wind always blows from the same direction and not much sand is available. This mobile topography can be described by a set of coupled equations of motion which contain as variable fields the wind shear stress and the sand flux. The resulting theoretical understanding allows to explain among others the different dune morphologies, their velocity and their minimum size.

Figure 10.1: Barchan dunes near Laâyoune, Morocco. The dune in the front, on the left side was measured in detail during our field trip in May 1999.

234

10 Wind-blown Sand

In the present review we will first introduce the properties of the turbulent wind field, then present the mechanisms of sand transport and then we will discuss the resulting patterns.

10.2 The Wind Field Air is a Newtonian fluid of density ρ = 1, 225 kg m−3 and a dynamic viscosity µ = 1.78 × 10−5 kg m−1 s−1 which is defined as τ =µ

dv dz

where τ is a small applied shear stress and dv dz the resulting velocity gradient. Its state is fully described by the velocity field v(r) and the pressure field p(r) when we assume constant temperature and density. Its time evolution is given by the Navier–Stokes equations and the incompressibility condition. The solution of this equation is mainly characterized by the dimensionless Reynolds number defined by Re =

Lv ν

where ν = µρ is the kinematic viscosity. L and v are a characteristic length and velocity of the problem (boundary conditions of the equation). Re represents the ratio of inertial forces to viscous forces. For low Reynolds numbers, flow is laminar, which means that it attains a final state which does not change in time and has the smallest spatial gradients. We encounter this situation for small wind velocities, in narrow channels (for instance between grains) and at a close distance to a surface. For high Reynolds numbers the flow is turbulent, which means that there are strong spatial and temporal fluctuations all the time. This situation is typical outdoors even at moderate wind velocities, due to the enormous size of the atmosphere. This complex behavior arises from the fact that, for large Re, the Navier–Stokes equation is dominated by the non-linear inertia term. Since the turbulent fluctuations in the velocity and pressure fields are random, one can only make predictions for the temporal averages. The critical Reynolds number at which the atmospheric boundary layer becomes turbulent is in the order of 6000 [2]. Hence, even small wind speeds create turbulent flows. The wind for which aeolian sand transport|( occurs is always turbulent. The air can sustain more shear stress due to turbulent fluctuations and eddies, compared to the laminar case. To model this effect, a turbulent viscosity η or a turbulent shear stress τT can be introduced, τ = τl + τT = (µ + η)

dv . dz

(10.1)

where we assume that the flow is in the x direction. The turbulent shear stress τT , equal to the transfer of momentum per unit time and unit area, is given by [3–5], τT = −ρu w = η

dv . dz

(10.2)

10.2 The Wind Field

235

Wind speed v in m s−1 height z in m 1

1

0.9 0.8

0.1

height

0.7 0.6

0.01

0.5 0.4

0.001 u∗ u∗ u∗ u∗ u∗

0.3 0.2

0.0001

0.1 0

1e-05 0

5

10

15 20 velocity

25

30

0

5

= 0.1 = 0.3 = 0.5 = 0.7 = 0.9

10 15 20 velocity

25

30

Figure 10.2: Logarithmic wind velocity profile of the atmospheric boundary layer above a surface with a roughness length z0 = 1.7 10−5 m.

The bar denotes the time average and u and w are velocity fluctuations in the x and z di¯ and rection, respectively. The velocity fluctuations are defined according to u = u − u ¯ where u, w denote the velocities and u ¯, w ¯ the time-averaged velocities. Usw = w − w, ing the mixing length theory [4] for the absorption of vertical momentum, the turbulent shear stress can be expressed by, 2 dv 2 , (10.3) τT = ρl dz where l is the vertical mixing length. Prandtl [4] assumed that the mixing length l = κz linearly increases with distance from the surface, where κ ≈ 0.4 is the von Kármán universal constant for turbulent flow. In a fully turbulent flow, where fluctuations of all length scales have been developed and the Reynolds number is clearly above the critical one, the dynamic viscosity µ is much smaller than the turbulent viscosity η and can be neglected. Therefore, the turbulent shear stress τT is identified with the overall shear stress τ . By integrating Eq. (10.3) from z0 to z, the well known logarithmic profile of the atmospheric boundary layer, illustrated in Figure 10.2, is obtained, v(z) =

z u∗ ln , κ z0

(10.4)

where z0 denotes the roughness length of the surface and u∗ = τ /ρ the shear velocity. The shear velocity u∗ characterizes the flow and has the dimensions of a velocity, although it is actually a measure of the shear stress. The roughness length z0 is either defined by the

236

10 Wind-blown Sand A)

B)

Figure 10.3: a) Small grains are immersed in the laminar sublayer which creates an aerodynamically smooth surface. b) Grains larger than the laminar sublayer are isolated objects and create an aerodynamically rough surface.

thickness of the laminar sublayer for aerodynamically smooth surfaces or by the size of the surface perturbations for aerodynamically rough surfaces, cf. Figure 10.3. Since these fluctuations can be of small scale and high frequency it is computationally too expensive to simulate them directly in practical applications. Instead, the Navier–Stokes equations can be time-averaged, ensemble-averaged, or otherwise manipulated to remove the small-scale dynamics, which results in a modified set of equations that are computationally less expensive to solve. The simplest models of turbulence are two-equation models in which the solution of two separate transport equations allows the turbulent velocity and length scales to be independently determined. The standard k- model [6] is a semi-empirical approach, based on transport equations for the turbulent kinetic energy k and its dissipation rate . In the derivation of the k- model it is assumed that the flow is fully turbulent, and the effects of molecular viscosity are negligible. The standard k- model is therefore valid only for fully turbulent flows. Many programs, packages, and libraries have been developed in order to solve the Navier Stokes equation in different geometries and boundary conditions using for instance the k model. However, three-dimensional turbulent flow is still difficult to treat and is limited by the performance of processors and memory. We have chosen FLUENT V5.0 [7] for the following calculations. In Figures 10.4 and 10.5 we show the velocity field of the wind over a crescent-shape obstacle which in fact corresponds to the measured topography of a Barchan dune (see Figure 10.1). We see from the cut in wind direction (Figure 10.4) that, behind the dune, an eddy of comparatively low velocity is formed while the strong wind seems to follow above an imaginary continuation of the initial hill following the line s(x) that delimits the eddy (separation line). The projection of Figure 10.5 shows that, in fact, there are two such eddies next to each other. The problem of the above type of calculation is that it is too time consuming from a computational point of view in order to be useful for an iterative calculation, such as the evolution of a dune, where the surface and thus the boundary evolves in time. Furthermore, the theoretical understanding is limited by using such a “black-box” model.

10.2 The Wind Field

237

Figure 10.4: Vertical cut along the symmetry plane, the central slice of the dune. The depicted velocity vectors clearly show the separation of flow at the brink and a large eddy that forms in the wake of the dune.

A dune or a smooth hill can be considered as a perturbation of the surface that causes a perturbation of the air flow. An analytical calculation of the shear stress perturbation due to a two-dimensional hill has been performed first by Jackson and Hunt [8]. Later, the work has been extended to three-dimensional hills and further refined [9–12]. The following discussion is mainly based on the work of Ref. [13]. They obtain, after a lengthy calculation, for the Fourier transformation of the shear stress perturbation τˆx in the wind direction, h(kx , ky )kx2 2 τˆx (kx , ky ) = |k| v 2 (l)

2 ln L|kx | + 4γ + 1 + i sign(kx )π 1+ ln l/z0

,

(10.5)

and the shear stress perturbation τˆy in the lateral direction, τˆy (kx , ky ) =

h(kx , ky )kx ky 2 , |k| v 2 (l)

(10.6)

where h(kx , ky )|k| = kx2 + ky2 and γ = 0.577216 (Euler’s constant). Furthermore, v(l) denotes the dimensionless velocity at the height l normalized by the velocity v0 , v(l) =

l u∗ ln , v0 κ z0

(10.7)

238

10 Wind-blown Sand

Figure 10.5: The bird’s eye view of the dune and the velocity vectors reveal the three-dimensional structure of the large wake eddy.

where κ ≈ 0.4 is the von Kármán’s constant and v0 the velocity of the undisturbed upwind profile at the intermediate height. We further simplify Eq. (10.5) by approximating the logarithmic term ln L|k| by the constant value that corresponds to the wavelength 4L of the hill and obtain, h(kx , ky )kx (kx + iB|kx |) , τˆx (kx , ky ) = A kx2 + ky2

(10.8)

where A and B depend logarithmically on ln L/z0 . The first term is a non-local term that is a direct consequence of the pressure perturbation over the hill. The second term is a correction that comes from the non-linearity of the NavierStokes equation and represents the effect of inertia. Both terms are depicted in Figure 10.6 for a Gaussian hill. The first term determines mainly the speed-up and is symmetric, whereas the second term is asymmetric. The superposition of both terms leads to an upwind shift of the maximum of the shear stress perturbation with respect to the maximum of the hill. The air shear stress onto a smooth surface perturbation can be calculated from Eq. (10.8) which is computationally very efficient. The drawback of this analytical formula is that it can only be used for smooth surfaces. However, for a dune with slip face flow, separation occurs at the sharp brink as seen in Figure 10.4 which represents a discontinuity of the surface. One can treat the complicated problem of flow separation in an heuristic way. The flow is divided into two parts by the separating streamline s(x) that reaches from the point of flow separation to the bottom. Inside the area that is enclosed by the separating streamline and the

10.3 Aeolian Sand Transport

239

1.5

h(x) τˆ(x) τÂ (x) τˆB (x)

τˆ and h/H

1

0.5

0

−0.5 −5

−4

−3

−2

−1

0

1

2

3

4

5

x/σ Figure 10.6: Calculated shear stress at the surface. The first part τA , which is non-local and symmetric, causes most of the speed-up and the depressions before and behind the hill. The second part τB , which is local and asymmetric, leads to a symmetry breaking of the overall shear stress τ (x) and thus to an upwind shift of the maximum of the shear stress with respect to the maximum of the profile h(x).

surface, called the separation bubble, a re-circulating flow develops,whereas the (averaged) flow outside is laminar as shown in Figure 10.4. The general idea, suggested by Ref. [10], is that the air shear stress τ (x) on the windward side can be calculated using the envelope that ˜ comprises the dune and the separation bubble, Figure 10.7. The envelope h(x) is defined as, ˜ h(x) = max (h(x), s(x)) .

(10.9)

The minimal model for a “smooth” separating streamline s(x) is a third-order polynomial.

10.3 Aeolian Sand Transport The most obvious property of sand is the grain diameter d which ranges from d ≈ 2 mm for very coarse sand to d ≈ 0.05 mm for very fine sand. The sand itself consists mostly of quartz (Si02 ) which has a density ρquartz of 2650 kg m−3 . This density is more than 2000 times larger than that of air. A further classification can be made with respect to the shape of the grains [14]. Dune sand has a sharply peaked distribution with an average diameter of approximately 0.2–0.25 mm. An agitating medium such as air exerts two types of force on grains when blowing over a bed of sand. The first is called drag force Fd and acts horizontally in the direction of the flow. For turbulent flow it scales quadratically with the velocity and is called Newton’s turbulent drag, Fd = βρu2∗

πd2 , 4

(10.10)

240

10 Wind-blown Sand

2.5

h(x) s(x) τ (x)

h/H and τ /τ0

2

1.5

1

0.5

0 −3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x/L ˜ Figure 10.7: The envelope h(x) of the windward profile of a dune h(x) and the separating streamline s(x) together form a smooth object which is used to calculate the air shear stress τ (x) on the windward side using Eq. (10.8). In the region of re-circulation, the air shear stress τ is set to zero.

where β is a phenomenological parameter that includes some characteristics of the bed such as its packing. The second force is called lift force Fl and arises from the static pressure difference ∆p between the bottom and the top of a grain, caused by the strong velocity gradient of the air near the ground, Fl = ∆p

πd2 πd2 = CL ρv 2 , 4 2

(10.11)

where CL = 0.0624 [15]. v denotes the air velocity at a height of 0.35d with respect to the zero level at z0 . In a turbulent flow, where high velocity and pressure fluctuations occur, the short-term lift forces are sufficiently large to eject grains. Chepil [15] showed, further, that the ratio c = 0.85 between the forces of drag and lift is constant within the relevant range of Reynolds numbers, F l = c Fd .

(10.12)

Inertia and gravity oppose the aerodynamic forces. Most important is the grains’ weight Fg that directly counteracts the lift force, F g = ρ g

πd3 6

(10.13)

where ρ = ρquartz − ρair is the reduced density of the sand grains in the air. In addition, cohesive and adhesive forces would have to be taken into account for small grains and wet


241

Fl

Fd φ p

Fg

Figure 10.8: The grain starts to role when the drag and lift forces exceed the gravitational force. This can be expressed by a momentum balance with respect to the pivot point p.

sand beds, but are neglected here. The uppermost layer of grains in the bed is free to move when the aerodynamic forces overcome the gravitational force. The balance of momentum for a grain that is about to rotate around its pivot point, as depicted in Figure 10.8, can be expressed as, d d Fd cos φ = (Fg − Fl ) sin φ 2 2

(10.14)

By inserting Eqs. (10.10), (10.12), and (10.13), Eq. (10.14) defines the minimal shear stress for grain movement which is called fluid threshold or aerodynamic entrainment threshold τta = ρair u2∗ta , 2 sin φ τta = . (10.15) ρ gd 3 β cos φ + c sin φ The fluid threshold shear stress τta on a flat surface is directly proportional to the immersed density ρ and the diameter d of the grains. The packing of the grains is reflected by the angle φ which can be understood in a similar way to the angle of internal friction of a sand pile. However, its value may be different. The shape and sorting of the grains is determined by β. Shields [16] introduced a dimensionless coefficient Θ that expresses the ratio of the applied tangential force to the resisting grain movement, Θ(Re∗ ) ≡

τta . ρ gd

(10.16)

The Shields parameter Θ depends on the Reynolds friction number Re∗ = u∗ dν −1 . For Re∗ > 1, Θ is constant with a value that ranges from 0.01 to 0.014. Ref. [1] used the dimen-

242

10 Wind-blown Sand

sionless Shields parameter Θ to define the fluid threshold shear velocity u∗ta , ρ gd . u∗ta = Θ ρair

(10.17)

Yet, this expression is only valid as long as cohesive and adhesive forces can be neglected and thus for grain diameters larger than 0.2 mm. The typical value for the fluid threshold shear velocity u∗ta = 0.25 m s−1 is obtained for d = 250 µm using Θ = 0.012. When sediment transport has started and sand grains are present in the air, they impact onto the bed. The momentum transfer from an impacting grain to a resting grain on the bed changes the mass balance and lowers the threshold for entrainment. This has already been observed by Bagnold [17] who called this lowered threshold the impact threshold u∗t . The impact threshold shear velocity u∗t can be calculated in an analogous way and expressed by Eq. (10.17) with an effective Shields parameter Θ = 0.0064. The concept of a single threshold becomes more and more difficult if sediments are poorly sorted or if the effect of moisture and cementing agents become important. Moreover, inclined beds, and thereby gravity, alters the threshold velocity. A detailed discussion of these effects can be found in the book of Pye and Tsoar [14]. Conventionally, different mechanisms of aeolian sand transport such as suspension and bed-load are distinguished according to the degree of detachment of the grains from the ground. Bed-load transport can further be divided into saltation, reptation, and creep. However, the different subclassifications of bed-load are subtle and depend often on the author’s nomenclature [14]. Grains are suspended in air if they travel long distances on irregular trajectories before reaching the ground again. A turbulent air flow can keep grains in suspension when the vertical component of the turbulent fluctuations w , Eq. (10.2), exceeds the settling velocity wf of the grains. In typical sand storms, when shear velocities are in the range of 0.18–0.6 m s−1 [14], particles with a maximum diameter of 0.04–0.06 mm can be transported in suspension, as seen in Figure 10.9. The grains of typical dune sand have a diameter of the order of 0.25 mm and are transported via bed-load and mainly saltation. For this reason, suspension is neglected in the following discussion and the next section focuses on the saltation process. Sediment transport by saltation is the most relevant bed-load mechanism. To initiate saltation some grains have to be entrained directly by the air. This is called direct aerodynamic entrainment. However, if there is already a sufficiently large number of grains in the air, the direct aerodynamic entrainment is negligible and grains are mainly ejected by collisions of impacting grains. The entrained grains are accelerated by the wind along their trajectory, mainly by the drag force, before they impact onto the bed again. The interaction between an impacting grain and the bed is called the splash process and can produce a jet of grains that are ejected into the air. It is currently the subject of theoretical and experimental investigations [18, 19]. Finally, the momentum transferred from the air to the grains gives rise to a deceleration of the air. Through this negative feedback mechanism, after a transient time, saltation reaches a constant transport rate. Direct aerodynamic entrainment occurs if the air shear stress τ exceeds τta from Eq. (10.15). A simple way to model the rate of aerodynamically entrained grains has been


243

n lta

m

tio n

od

0.1

sa

0.05

0.02

wf u∗

0.01 0.01

= 0.1

0.02

wf u∗

typical wind speeds and dune sand

en sp su d ifie

0.2

s io

su sp en sio n

shear velocity

0.5

=1

0.05

0.1

0.2

0.5

grain diameter Figure 10.9: The mechanism of transport depends on the shear velocity of the air and the grain diameter. For typical dune sand (0.2 mm < d < 0.3 mm) and wind velocities (0.2 m s−1 < u∗ < 0.6 m s−1 ) on Earth, aeolian sand transport occurs by saltation (area inside the rectangle).

proposed by Anderson [20]. He takes the number of entrained grains proportional to the excess shear stress, Na = ζ(τ − τta ),

(10.18)

where Na is the number of entrained grains per time and ζ a proportionality constant, chosen by Ref. [20] to be of the order of 105 grains N−1 s−1 . Once launched into the airstream, aerodynamic forces (lift and drag) and gravity act on the grain and determine its trajectory [1, 21–23]. In a saltation trajectory, the vertical motion is mainly determined by the gravitational force Fg , Fg = mg,

(10.19)

where m denotes the mass of the grain and g the gravitational acceleration. The acceleration in the horizontal direction is essentially caused by the drag force Fd , Fd =

1 πd2 ρair Cd (v(z) − u) |v(z) − u| , 2 4

(10.20)

where d is the grain diameter, v(z) the velocity of the air, u the velocity of the grain, and Cd the drag coefficient that depends on the local Reynolds number Re = |v − u|d/ν. Hence, the trajectory is close to that of a simple ballistic trajectory, of flight time T and height h T =

2uz0 ; g

h=

u2z0 , 2g

(10.21)

244

10 Wind-blown Sand

where uz0 is the vertical component of the initial velocity. More elaborate calculations [22,25] have shown that the simple approximation using the ballistic formula gives values which are 10–20% too high. Wind tunnel measurements [18] lead to the values: T (u∗ ) ≈ 1.7 u∗ /g, h(u∗ ) ≈ 1.5 u∗ /g, and the saltation length l(u∗ ) ≈ 18 u2∗ /g. For u∗ = 0.5 m s−1 a flight time T ≈ 0.08 s, hop height h ≈ 3.8 cm and hop length l ≈ 45 cm are obtained. The momentum transfer from the air to the grains, lowers the wind speed near the ground. This is called a negative feedback mechanism that finally limits the number of grains in the air and drives the system into a steady state. A possibility of including this effect is to add a body force f that models the average momentum transfer from the air to the grains to the right of the Navier–Stokes equation. ρair ∂t v + ρair (v∇)v = −∇p + ∇τ + f

(10.22)

The microscopic picture of saltation as discussed before is too detailed for many macroscopic applications concerning sand filling, desertification, and dune formation. They have to be linked to macroscopic variables such as the sand flux q per unit width and time. The sand flux q depends on the shear velocity u∗ , the threshold u∗t , the grain diameter d, and many other properties. It may even depend on the history if non-equilibrium conditions and transients are important. However, most of the known sand flux relations are restricted to the saturated case where the sand flux is an expression of the form q(u∗ , u∗t , d, . . .). Sand flux measurements performed in wind tunnels [26, 27] show that sand flux starts at a threshold u∗t and scales with the cube of the shear velocity (q ∝ u3∗ ) for high shear velocities (u∗ u∗t ). In the vicinity of the threshold the functional dependence is not well understood and empirical and theoretical flux relations differ considerably. The simplest flux relation that predicts a cubic relation between sand flux q and shear velocity u∗ was proposed by Bagnold [1], ρair d 3 u , (10.23) qB = CB g D ∗ where d is the grain diameter and D = 250 µm a reference grain diameter. This simple relation is still of theoretical interest at high wind speeds, far from the threshold, where most of the sand is transported. Later a threshold has been incorporated into the sand flux relations in order to account for the fact that sand transport cannot be maintained below a certain shear velocity. Many phenomenological sand flux relations have been proposed and have been summarized, for instance in Ref. [14]. A sand flux relation that is widely used is the one by Lettau and Lettau [28], qL = C L

ρair 2 u (u∗ − u∗t ) g ∗

(10.24)

where CL is a fit parameter. Analytical calculations that predict the sand flux by averaging over the microscopic processes have greatly increased the understanding of aeolian sediment transport [21,29–31]. A result of such a theoretical calculation was obtained by Sørensen [22], qS = CS

ρair u∗ (u∗ − u∗t )(u∗ + 7.6 ∗ u∗t + 2.05 m s−1 ), g

(10.25)


245

0.09

u∗ u∗ u∗ u∗ u∗

0.08

q in kg m−1 s−1

0.07

= = = = =

0.3 0.4 0.5 0.6 0.7

0.06 0.05 0.04 0.03 0.02 0.01 0 0

1

2

3

4

5

6

7

8

9

10

x in m Figure 10.10: Numerical solution of the sand flux equation (10.26) for different shear velocities u∗ . The model parameter γ = 0.2 of Eq. (10.27) that defines the length and time of the saturation transients was chosen here so that saturation is reached between 1 s and 2 s.

where CS is a parameter that has analytically been determined. Fitting Eq. (10.25) to wind tunnel data, revealed that the analytical value of CS is about four times too small. However, the functional structure of the relation reproduces the data very well [27]. All the preceding relations of the form q(u∗ , . . .) assume that the sediment transport is in a steady state, i.e., the sand flux is saturated. In order to overcome this limitation and to obtain information about the dynamics of the aeolian sand transport|), numerical simulations based on the grain scale have been performed [20, 23, 32]. They showed that, on a flat surface, the typical time to reach the equilibrium state in saltation is approximately two seconds, which was later confirmed by wind tunnel measurements [26]. If one assumes that each splash event produces on average the same number of ejected new particles, the increase of saltating grains would be exponential in time. After a saturation time Ts , however, the flux saturates to qs . From this microscopic picture Sauermann et al. [33–35] have derived the equation for the evolution of the flux 1 q ∂q = q 1− , (10.26) ∂x ls qs where ls is the saturation length. Solutions of Eq. (10.26) are shown in Figure 10.10. We want to emphasize that Ts (τ, u) and ls (τ, u) = Ts u are not constant, but depend on the external shear stress τ of the wind and the mean grain velocity u. We can relate the characteristic time Ts and length ls of the saturation transients to the saltation time T and the saltation length l of the average grain trajectory, τt τt , ls = l (10.27) Ts = T γ(τ − τt ) γ(τ − τt )

246

10 Wind-blown Sand

where τt is the entrainment threshold shear stress and γ is a constant. For typical wind speeds, the time to reach saturation is of the order of 2 s [20, 23, 32]. Assuming a grain velocity of 3–5 m s−1 [24] we obtain a length scale of the order of 10 m for saturation. This length scale is large enough to play an important role in dune formation. The solution of Eq. (10.26), together with the saturated sand flux qs for two cosine shaped heaps with the same aspect ratio, is depicted in Figure 10.11. We can see that the solution of the large hill is close to the saturated flux, apart from the region near the foot of the heap. The situation at the foot is different, there is no deposition and the surface velocity decreases, which predicts a steepening of the foot area. For small hills the shear stress gradient, ∂τ /∂x ∝ 1/L, becomes large and the sand flux lags a certain distance behind the saturated sand flux. We want to emphasize that this lag compensates the upwind shift of the shear stress to a certain degree, or may even overcompensate it, as for the small hill in Figure 10.11. The effect of this lag can be seen in the surface velocity of the windward side (left side). The large hill shows a tendency to steepen due to a decreasing surface velocity, whereas the small hill will flatten due to an increasing surface velocity. Finally, a further consequence of the saturation transients is an absolute minimal dune size.

10.4 Dunes We have seen that the lee side of a hill has the tendency to steepen. When the wind is blowing long enough from the same direction, the lee side reaches the angle of repose Θ ≈ 34◦ , which is the steepest stable angle of a free sand surface. If this angle is exceeded, avalanches start to 2.5

q/q0 and h/H

2

qs q(small) q(large) h

1.5 1 0.5 0

Figure 10.11: The figure shows the saturated flux qs , Eq. (10.25), and the flux q, Eq. (10.26), including saturation transients, calculated for two cosine shaped hills with an aspect ratio of H/L = 1/8. The hills have a height of H = 1 m and H = 10 m, the saturation length is ls = 0.8 m. The saturated flux qs is scale invariant and thus identical for both hills, but the flux q becomes completely different in the case of the small hill, L ≈ 10ls .

10.4 Dunes

247

slip down the hill until the surface has relaxed to a slope equal or below the angle of repose. Now gravity instead of the wind is responsible for the sand flux. In two dimensions, one could easily redistribute the sand in such a way that the slip face is a straight line at the angle of repose. However, in three dimensions this process is not obvious. Bouchaud et al. (BCRE) [36] proposed a model to study avalanches on inclined surfaces. The advantage of this model is that it can easily be extended to three dimensions. The full three-dimensional model is defined by the three variable fields h(x, y), q(x, y) and τ (x, y). τ is calculated from h through the Fourier transformation of Eq. (10.8). Then q is obtained from τ through Eq. (10.26) using qs from Eq. (10.24) and ls from Eq. (10.26). Then the new topography h is obtained from q, using mass conservation: 1 ∂h = ∇s q ∂t ρsand

(10.28)

If ∇s h > tan Θ the, just mentioned, BCRE equations are applied. ∇s denotes the spatial derivative in direction of the strongest gradient in the wind direction. Once h is obtained one goes back to calculate τ again, etc. With this loop one iteratively solves the time evolution of the three fields. The above system of iteratively solved coupled equations describes fully the motion of the free granular surface under the action of wind and can be used to calculate the formation, evolution and shape of dunes. If a dune changes shape invariantly with velocity vd one has ∂h = vd ∇s h ∂t

(10.29)

which, inserted into Eq. (10.28) gives vd =

1 dq ρ dh

(10.30)

or if qb is the flux over the maximum (brink) of the dune of height H vd =

qb ρH

(10.31)

This result, that dune velocity is inversely proportional to its height, is also known from Bagnold [1]. Depending on the amount of sand and on the variability of the wind direction, one finds different dune morphologies. If the wind always comes from the same direction, one obtains transverse dunes if there is much sand and Barchan dunes (see Figure 10.1) if little sand is available. As shown schematically in Figure 10.12, Barchans are crescent-shaped dunes moving with a velocity given by Eq. (10.31). They exist in large fields in Marocco, Peru, Namibia, etc., typically parallel to the coast. In Figure 10.13 we see a longitudinal cut through the highest point of Barchans of different size normalized in such a way that they all have the same maximum [37]. On the windward side all curves fall on top of each other, while the crest moves inwards for increasing dune height. In Figure 10.14 we show the corresponding results from the numerical solution of the equations. Only the wind comes from the opposite side. We see the same behavior as a

248

10 Wind-blown Sand

wind

crest

luv

lee

horn Figure 10.12: Schematic diagram of a Barchan dune.

1 0.9

Normalized height z˜

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.5

0

0.5

1

1.5

Normalized length x ˜ Figure 10.13: Profile along the symmetry plane of the dunes (thin lines) using normalized variables (from ref. [37]).

function of dune height. The dashed curve gives a reasonable fit to the windward side. Also the transverse cuts scale with height and the numerical calculation agrees with observation. A consequence of the above scaling relations is a well-known linear dependence between dune height, length and width as has been already reported by Bagnold [1]. Viewing the dune from the top, the brink has the shape of a parabola [37]. Due to the competition between the saturation length and the size of the separation bubble behind the dune, one can calculate the

10.5 Conclusion

249

1

normalized height z

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

normalized length x

Figure 10.14: Normalized longitudinal profiles of the dunes. A fit with cosh4 (x) (dashed line) reproduces the windward side quite well. The shear velocity is u∗ = 0.5 ms−1 (from Ref. ( [38]).

minimal height of a stable dune to be about 1.5 meters. The shear stress of the wind and the sand flux on a Barchan dune have also been measured and have been very favorably compared to the numerical results of the equations [39]. Using the system of equations of motion for dunes, one can also calculate entire systems of dunes and create virtual landscapes. One example is shown in Figure 10.15. With these computer dunes it has lately been shown [40] that when a small Barchan bumps into a larger one it can either be swallowed (if it is too small), or it can coalesce but produce at each horn a new baby Barchan, or it can, if the two initial dunes are of similar size, show solitary behavior. In this last case, the dunes seem to transverse each other unaltered except for an eventual change in relative size.

10.5 Conclusion We have shown that, using our knowledge on turbulence and exploiting the mechanism of saltation, it is possible to set up equations of motion for a wind-driven free granular surface. This set of three coupled equations containing, as variable fields; the shear stress of the wind, the sand flux at the surface and the profile of the landscape, can be solved numerically. The resulting patterns agree very well with phenomenological knowledge on dunes and with quantitative field measurements. The advantage of solving dune motion on the computer consists of being able to make predictions over long timescales since, in the real world, the motion is extremely slow. One

250

10 Wind-blown Sand

Figure 10.15: Complex dune pattern, calculated with the full three-dimensional model. Wind is blowing from the left to the right. When Barchan dunes are too close they interact, are eventually connected, and form complex dune structures. The large dunes are shielding the small dunes from the arriving sand flux which then constantly lose volume.

can predict the effect of protective measures like the BOFIX technique of Meunier [41] and one can also calculate the dunes on Mars.

References [1] Bagnold, R. A. (1941). The Physics of Blown Sand and Desert Dunes. London: Methuen. [2] Houghton, J. T. (1986). The Physics of Atmospheres, Volume 2nd edn. Cambridge: Cambridge Univ. Press. [3] Kármán, T. (1935). Some aspects of the turbulence problem. Proc. 4th Int. Congr. Appl. Mech. Cambridge, 54–91. [4] Prandtl, L. (1935). The mechanics of viscous fluids. In W. F. Durand (Ed.), Aerodynamic Theory, Volume Vol. III, pp. 34–208. Berlin: Springer. [5] Sutton, O. G. (1953). Micrometeorology. New York: McGraw–Hill. [6] Launder, B. E. and Spalding, D. B. (1972). Lectures in Mathematical Models of Turbulence. London, England: Academic Press. [7] Fluent Inc. (1999). Fluent 5. Finite Volume Solver. [8] Jackson, P. S. and Hunt, J. C. R. (1975). Turbulent wind flow over a low hill. Q. J. R. Meteorol. Soc. 101, 929. [9] Sykes, R. I. (1980). An asymptotic theory of incompressible turbulent boundary layer flow over a small hump. J. Fluid Mech. 101, 647–670.

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[10] Zeman, O. and Jensen, N. O. (1988). Progress report on modeling permanent form sand dunes. Risø National Laboratory M-2738. [11] Carruthers, D. J. and Hunt J. C. R. (1990). Atmospheric Processes over Complex Terrain, Volume 23, Chapter Fluid Mechanics of Airflow over Hills: Turbulence, Fluxes, and Waves in the Boundary Layer. Am. Meteorological. Soc. [12] Weng, W. S., Hunt, J. C. R., Carruthers, D. J., Warren, A., Wiggs, G. F. S., Livingstone, I. and Castro, I. (1991). Air flow and sand transport over sand–dunes. Acta Mechanica (Suppl.) 2, 1–22. [13] Hunt, J. C. R., Leibovich, S. and Richards, K. J. (1988). Turbulent wind flow over smooth hills. Q. J. R. Meteorol. Soc. 114, 1435–1470. [14] Pye, K. and Tsoar, H. (1990). Aeolian Sand and Sand Dunes. London: Unwin Hyman. [15] Chepil, W. S. (1958). The use of evenly spaced hemispheres to evaluate aerodynamic forces on a soil surface. Trans. Am. Geophys. Union 39(397–403). [16] Shields, A. (1936). Applications of similarity principles and turbulence research to bedload movement. Technical Report Publ. No. 167, California Inst. Technol. Hydrodynamics Lab. Translation of: Mitteilungen der preussischen Versuchsanstalt für Wasserbau und Schiffsbau. W. P. Ott and J. C. van Wehelen (translators). [17] Bagnold, R. A. (1937). The size-grading of sand by wind. Proc. R. Soc. London 163 (Ser. A), 250–264. [18] Nalpanis, P., Hunt, J. C. R. and Barrett, C. F. (1993). Saltating particles over flat beds. J. Fluid Mech. 251, 661–685. [19] Rioual, F., Valance, A. and Bideau, C. (2000). Experimental study of the collision process of a grain on a two-dimensional granular bed. Phys. Rev. E 62, 2450–2459. [20] Anderson, R. S. (1991). Wind modification and bed response during saltation of sand in air. Acta Mechanica (Suppl.) 1, 21–51. [21] Owen, P. R. (1964). Saltation of uniformed sand grains in air. J. Fluid. Mech. 20, 225– 242. [22] Sørensen, M. (1991). An analytic model of wind-blown sand transport. Acta Mechanica (Suppl.) 1, 67–81. [23] McEwan, I. K. and Willetts, B. B. (1991). Numerical model of the saltation cloud. Acta Mechanica (Suppl.) 1, 53–66. [24] Willetts, B. B. and Rice, M. A. (1985). Inter-saltation collisions. In O. E. BarndorffNielsen (Ed.), Proceedings of International Workshop on Physics of Blown Sand, Volume 8, pp. 83–100. Memoirs. [25] Anderson, R. S. and Hallet, B. (1986). Sediment transport by wind: toward a general model. Geol. Soc. Am. Bull. 97, 523–535. [26] Butterfield, G. R. (1993). Sand transport response to fluctuating wind velocity. In N. J. Clifford, J. R. French, and J. Hardisty (Eds.), Turbulence: Perspectives on Flow and Sediment Transport, Chapter 13, pp. 305–335. John Wiley & Sons Ltd. [27] Rasmussen, K. R. and Mikkelsen, H. E. (1991). Wind tunnel observations of aeolian transport rates. Acta Mechanica Suppl 1, 135–144. [28] Lettau, K. and Lettau, H. (1978). Experimental and micrometeorological field studies of dune migration. In H. H. Lettau and K. Lettau (Eds.), Exploring the World’s Driest Climate. Center for Climatic Research, Univ. Wisconsin: Madison.

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[29] Ungar, J. E. and Haff, P. K. (1987). Steady state saltation in air. Sedimentology 34, 289–299. [30] Sørensen, M. (1985). Estimation of some eolian saltation transport parameters from transport rate profiles. In O. E. B.-N. et al. (Ed.), Proc. Int. Wkshp. Physics of Blown Sand., Volume 1, Denmark, pp. 141–190. University of Aarhus. [31] Werner, B. T. (1990). A steady-state model of wind blown sand transport. J. Geol. 98, 1–17. [32] Anderson, R. S. and Haff, P. K. (1988). Simulation of eolian saltation. Science 241, 820. [33] Sauermann, G., Kroy K. and Herrmann H. (2001), A continuum saltation model for sand dunes. Phys. Rev. E 64, 31305. [34] Kroy, K., Sauermann G. and Herrmann H. J. (2002), A minimal model for sand dunes. Phys. Rev. Lett. 88, 054301. [35] Kroy K., Sauermann G. and Herrmann H. J. (2002), Aeolian sand dunes preprint condmat/0203040 [36] Bouchaud, J. P., Cates, M. E., Ravi Prakash J., and Edwards S. F. (1994). Hysteresis and metastability in a continuum sandpile model. J. Phys. France I 4, 1383. [37] Sauermann, G., Poliakov, A., Rognon, P. and Herrmann, H. J. (2000), The shape of the Barchan dunes of southern Morocco, Geomorphology 36, 47–62. [38] Schwämmle, V. and Herrmann, H. J. (2003). A model of Barchan dunes including lateral shear stress, submitted to EPJE, cond-mat/0304695 [39] Sauermann G., Andrade J. S., Maia L. P. Costa U. M. S., Araújo A. D. and Herrmann H. J. (2003), Wind velocity and sand transport on a Barchan dune, Geomorphology 1325, 1–11. [40] Schwämmle V. and Herrmann H. J. (2003), Budding and solitary wave behaviour of dunes, submitted to Nature. [41] Meunier J. and Rognon P. (2000), Une méthode écologique pour détruire les dunes mobiles, Secheresse 11, 309–316.

Part V

Charged and Magnetic Granular Matter

11 Electrostatically Charged Granular Matter Stephan M. Dammer, Jochen Werth, and Haye Hinrichsen

11.1 Introduction In many industrial applications and experimental situations, the handling of granular matter involves friction and is therefore accompanied by triboelectric charging of the grains. This applies in particular to dry granulates made of insulating material, but also to more complex systems such as powders suspended in a non-polar liquid. In fact, whenever a granular material is processed, stirred or shaken, such that contacts between different grains are continuously established and broken, surface charges are separated to some extent, leading to mutual electrostatic forces. The process of charge separation is a complex phenomenon and depends strongly on the material as well as on the surface properties of the grains. Even the sign of the mean charge may depend on the choice of the material. Therefore, interesting effects are expected if grains of different materials are mixed or if the grains interact with surfaces made of a different material. The study of electrically charged granular matter is particularly fascinating because of the long-range nature of Coulomb interactions. In fact, various electrostatic properties such as the integral charge and the macroscopic electric field can be detected and quantified easily. Sometimes the influence of electrostatic forces can even be seen with the naked eye. For example, simple plastic spheres poured between two plexiglass plates and shaken, become equally charged and arrange themselves in a beautiful pseudo-crystalline configuration because of their mutual Coulomb repulsion (see Figure 11.1). Apart from the material properties, the influence of electrostatic interactions depends significantly on the size of the grains: Since in the case of insulators the charges are located on the surface of the particle, the typical charge per grain will scale with its surface area. Hence the average charge per unit mass will decrease with increasing particle size. Therefore, the influence of Coulomb forces is expected to be particularly strong for very small particles, especially for fine powders. On the other hand, approaching scales of 1 micron or below, the grains are so small that the number of elementary charges per grain is limited or even equal to one. This means that for granular matter on the nanometer scale, the quantized nature of electric charge is expected to play a significant role as well. The purpose of this article is to summarize some interesting recent developments in the theoretical understanding and modeling of charged granular matter. Regarding the theoretical modeling, the simplest system of interest is a granular gas of equally charged particles, which will be discussed in Section 11.2. Another important physical situation, on which we will focus in the present article, is electrically charged granular matter suspended in a non-polar liquid (see Section 11.3). If the suspended particles are very small, the influence of the van

256

11 Electrostatically Charged Granular Matter

Figure 11.1: A simple experiment for didactic purposes: Charged plastic spheres poured between two plexiglass plates arrange themselves in a pseudo-crystalline pattern.

der Waals interaction has to be considered as well. As will be explained in Section 11.4, it is in most cases justified to regard the van der Waals interaction as an attractive short-range force, leading to irreversible aggregation. The agglomeration process continues until the flakes reach a certain size with a self-focussing size distribution. As an application, which makes use of Coulomb forces, we finally discuss an electrically supported coating process, in which charged particles on the micrometer scale are coated by oppositely charged nanoparticles in order to improve the flowability of the powder.

11.2 Charged Granular Matter in Vacuum The simplest situation to be studied is the dynamics of charged granular matter in vacuum. A comprehensive introduction to this topic was given by Scheffler in Ref. [1]. In this section we summarize some of the most important results. A granular gas in vacuum is a system of ballistically moving and colliding granular particles (see e.g. [2–4] for recent reviews and Chapter 8 in this book). Like molecules in a real gas the particles are scattered by mutual collisions in a chaotic manner, making it possible to apply concepts of ordinary statistical physics, such as entropy and temperature on a mesoscopic level. For example, one may introduce a granular temperature [5, 6] Tg ≡

1 (v − v )2 , 3

(11.1)

which is defined as the mean square velocity of the particles with respect to their average velocity. Using this analogy the Boltzmann constant of ordinary thermodynamics has to be replaced by the average mass of the particles m so that the average kinetic energy of the particles measured in a co-moving frame is proportional to mT . However, in contrast to molecular gases, where collisions are fully elastic so that energy is strictly conserved, the contact forces between granular particles are dissipative because of friction, converting some of the kinetic energy into thermal energy. This means that a granular gas seen on the mesoscopic

11.2 Charged Granular Matter in Vacuum

257

scale of its individual grains is not an independent statistical realm, instead it couples to the microscopic thermal degrees of freedom. Therefore, the granular temperature is not a genuine thermodynamic quantity in the usual sense, defined in terms of how the entropy changes with respect to a conserved quantity, rather it may be considered as an approximate concept with temperature-like features. In particular, a non-trivial stationary state can only be maintained if the dissipative loss of energy is compensated by an external driving force, e.g., by gravitation or mechanical vibrations of the container. In this sense, granular gases are generally out of equilibrium, although certain concepts of equilibrium statistical physics may prove to be useful in an approximate sense. Without an external driving force, the kinetic energy of a granular gas is continuously converted into thermal energy until an equilibrium is reached, where Tg is practically zero. This process, called granular cooling, is unusual in so far as Tg decreases with increasing particle density, leading eventually to the formation of dense clusters provided that the average density is high enough (see also Chapter 5 in this book). The loss of energy is a complex process described by phenomenological concepts such as sliding and rolling friction. Here an important quantity is the dimensionless restitution coefficient en = −vn /vn ,

(11.2)

where vn and vn are the relative normal velocities of the particle before and after the collision. The loss of energy per collision depends quadratically on the restitution coefficient. Usually the restitution coefficient is considered to be velocity-independent, although in experiments a weak dependence is observed. As in molecular gases, where the dilute limit is well approximated by the ideal gas, a good starting point for the theoretical analysis of granular gases is the limit of low densities, where the dynamics is dominated by ballistic propagation and binary collisions. In this limit it can be shown that collisional cooling without external driving leads to a loss of granular temperature as dT ∝ −n2 T 3/2 , dt

(11.3)

where n is the density of the particles. Hence Tg scales as t−2 and approaches zero for t → ∞. Turning to higher densities, a dynamic instability occurs, leading to the formation of clusters, in which case Tg decreases even more rapidly [7–9]. As outlined in the introduction, electrically neutral granular gases are an exception rather than a rule since most materials are inevitably charged as soon as the grains collide. The influence of electrostatic forces is particularly pronounced if the particles are made of insulating materials, where charges can be accumulated on the surface. Although triboelectrical charging is also observed in the case of conducting materials (see e.g. [10]), let us in the following restrict ourselves to the, technologically more important, case of insulating granular matter. The charging process of insulators is a complex phenomenon and depends crucially on surface structure, defects, and adsorbates [11]. Since so far the predictive power of theoretical studies has been limited, the properties of different materials are empirically measured and ranked in so-called triboelectric series [12] according to the sign and the efficiency of the charging process. Since triboelectric charging is of enormous importance in many industrial

258


applications such as solvent-free varnishing and selective sorting of recycled materials, empirical triboelectric series are well established and can be used as a reliable starting point for choosing appropriate granular materials. One of the simplest models of charged granular matter is a monodisperse powder of spheres of mass m and diameter d, which are monopolar charged and interacting with a Coulomb potential Φi,j = qi qj /rij . Ignoring rotational degrees of freedom the typical kinetic energy of the spheres is Ekin = 32 N mTg , where n = N/V is the number density of particles, which is often expressed in terms of the so-called packing density π (11.4) ν = nd3 , 6 defined as the volume of the sphere divided by the volume per particle in the container and ranging between 0 (dilute limit) and approximately 0.74 (dense packing of spheres). The kinetic energy has to be compared with the average electrostatic energy Eq = q 2 /a, where 3V 1/3 is the typical distance between neighboring particles. The quotient of these a = 4πN energies defines a dimensionless coupling parameter Γ=

2q 2 /a . 3mTg

(11.5)

For example, a very large Γ would describe a gas dominated by Coulomb forces such as in Figure 11.1, while a small value of Γ would indicate that the dynamics is governed by kinetic energy due to random motion of the grains. Although a simple molecular dynamics simulation of such a model seems to be straight forward, some points have to be considered with care. For example, in order to avoid the cloud of particles expanding in the simulation volume, due to mutual repulsive forces, let us assume periodic boundary conditions, whose implementation will be explained below. Another difficulty is the long-range nature of the Coulomb force, requiring computation of the interaction of all pairs of particles, slowing down the computation when the number of particles is large. Another crucial point concerns the location of the charges. In the limit of dilute granular gases, it is reasonable to assume that the charges are located in the centers of the spheres. For higher densities, however, the actual position and distribution on the particle’s surface is expected to play a role. Assuming periodic boundary conditions, each charge induces infinitely many charges in the periodically continued volume, leading to a diverging Coulomb field U=

N q2 1 → ∞, 2 |rij + n| n

(11.6)

i=j

where n runs over all image displacements. There are several ways to circumvent this problem. One of them is the so-called minimum image method, in which the Coulomb interaction is calculated only between the particles in the simulation volume and their nearest images. Using this method the Coulomb energy is given by Umi =

N q 2 fmi (rij + n) , 2 |rij + n| n

i=j

(11.7)

11.2 Charged Granular Matter in Vacuum

259

where fmi (r) is equal to 1 if the maximum norm of r is less than half of the lateral box size and zero otherwise. Another method would be to work directly with the forces Fiel = q 2

N fmi (rij + n) |rij + n|3 n

(11.8)

j=i

instead of the potential, taking advantage of the fact that the forces are still finite even in the presence of infinitely many image charges. The Coulomb energy can also be regularized by using the well-known standard technique of Ewald summation. Here every charge is compensated by a Gaussian countercharge distribution of the form ρ− (r) = −

κ3 exp (−κ2 (r − ri )2 ) , π 3/2

(11.9)

which effectively leads to a screening and thus makes Coulomb interaction short-ranged [13– 15]. A central problem concerning molecular dynamics simulations of charged granular matter is the so-called braking failure, i.e., numerical errors due to the rapidly varying forces during a collision of hard spheres. In order to circumvent this problem, it is useful to treat collisions and free ballistic motion separately [16]. This can be done by using so-called event-driven simulation techniques which work as follows [14]. For a given configuration of particles the deterministic trajectories and the time of the next collision are computed analytically. After advancing all particles up to this collision time, the actual collision process is carried out separately, taking the restitution coefficient and other physically relevant parameters into account. In the presence of long-range Coulomb forces, however, event-driven simulations are not feasible since it is difficult to compute the particle trajectories analytically. As a compromise a hybrid simulation technique, the so-called Verlet algorithm [14], has been devised which combines the advantages of ordinary molecular dynamics [17] and event-driven simulations. The Verlet algorithm works as follows. Instead of calculating the trajectories between collisions analytically, they are iterated numerically according to the discrete evolution equation 1 ri (t + ∆t) = ri (t) + ∆t vi (t) + ∆t2ai (t) 2 1 vi (t + ∆t) = vi (t) + ∆t[ai (t) + ai (t + ∆t)] . 2

(11.10)

Whenever an overlap between two particles is detected, indicating a collision that was supposed to happen shortly before, the simulation evolves backwards in time to the moment of collision in order to update the velocity vectors according to the laws of hard-sphere collisions [18–20]. Performing numerical simulations of charged granular matter leads to the following main result. Granular cooling is indeed observed and leads at sufficiently high densities to the onset of cluster formation. According to Ref. [8] the size of these clusters is of the order of d . L0 ∝ ν 1 − e2n

(11.11)

260


However, as expected, these clusters do not persist. Instead, the electrostatic repulsion makes them unstable, i.e., they lose their integrity after some time [21]. Therefore, the cooling process is much slower and the system eventually reaches a gas-like state, when the kinetic energy of the particles becomes smaller than the Coulomb barrier needed to be overcome in order for collision to take place.

11.3 Charged Granular Matter in Suspension Let us now turn to a different kind of system, namely, electrically charged powders suspended in a non-polar liquid. In this section we will focus on two different aspects. First, we discuss the influence of the surrounding liquid on the particles. Second, we summarize some facts about van der Waals forces, which play an essential role in aggregation processes. Electrically charged granular matter suspended in a non-polar liquid differs significantly from colloidal suspensions, which have been investigated intensively in recent years (see, e.g., Refs. [22–24, 27]). While in the latter case the electrostatic forces are screened by layers of counterions surrounding the particles, the Coulomb interaction in a non-polar liquid is longranged. In the following we are especially interested in very fine powders (with particle sizes of a few micrometers or below) suspended in a non-polar medium, such as in liquid nitrogen [25]. Because of frequent interactions with the surrounding molecules of the liquid the particles no longer move ballistically, instead they are almost completely thermalized and thus perform an effective random walk driven by Brownian forces. This means that the concept of a granular temperature Tg , as discussed in the previous section in the context of granular gases, is no longer valid, instead it has to be replaced by the ordinary temperature T of the fluid which serves as a thermal reservoir. The assumption of random-walk-like motion can be justified as follows. For a spherical particle with radius a and mass m, the timescale on which the particle moves ballistically is of the order of m , (11.12) trelax = 6πηa where η is the viscosity of the fluid. This timescale has to be compared with the typical time needed for the particle to diffuse on average by its own diameter, tdiff =

12πηa3 . kB T

(11.13)

As shown in Ref. [26], under typical experimental conditions, the relaxation time trelax is always by at least three orders of magnitude smaller than the diffusion time tdiff , even when considering particles as small as one nanometer in diameter. In modeling the dynamics this comparison justifies the use of the so-called overdamped limit, ignoring inertia and considering the trajectories of the particles effectively as random walks. In the previous section, discussing electrically charged granular gases, we assumed that, on collision, the particles change their velocity vectors according to classical collision laws and separate again. However, fine powders may exhibit cohesive properties, leading to aggregation

11.3 Charged Granular Matter in Suspension

261

and agglomeration of the grains. There are different sources of cohesion. One of them is caused by surface adsorbates, e.g., by humidity. Another important source, which is especially relevant for very fine powders is the van der Waals interaction. In fact, for small particles of 1 micron or below, van der Waals forces are the dominating source of stickiness. Van der Waals forces are of quantum-mechanical origin and are caused by fluctuations of virtual dipoles. This means that a molecule, having a vanishing dipole moment, may aquire a non-zero dipole moment for a short time ∆t as long as the corresponding change of energy ∆E is smaller than /2δt. This temporary dipole then induces another dipole in neighboring particle, leading to a mutual attractive force. The strength of the van der Waals force between two molecules decays quickly as r −6 . At distances larger than 100 nm it decays even more rapidly since the time for the induced dipole to be established (given by the velocity of light traveling between the molecules) exceeds ∆t. For this reason the van der Waals force can be considered effectively as a force with a finite range of about 100 nm. However, for small r, the van der Waals force becomes very strong, especially in the case of very small particles. More specifically, the force between two spherical particles of radii a1 , a2 , whose surfaces are separated by a distance h, is given by [27] F =−

A a 1 a2 f (p) , 6h2 (a1 + a2 )

(11.14)

where p is related to the London wavelength λL ≈ 100nm by p = 2πh/λL , and f (p) approximates the intensity of the forces at short distances as a series by   1+3.54p for p < 1 1+1.77p (11.15) f (p) = 0.067  0.98 − 0.434 for p ≥ 1 . p p2 + p3 Here A is the so-called Hamaker constant, which varies typically in the range between 10−21 J and 10−18 J [28, 29]. In many experimental situations it is reasonable to consider the van der Waals interaction as an irreversible short-range sticky force, i.e., once the particles touch each other they stick together irreversibly. Consequently, the granular material aggregates, forming clusters of many primary particles which often look like flakes (see Figure 11.2). Although the rigidity of such flakes is finite so that they may break under mechanical stress, it is often a good approximation to consider them as being rigid. An important quantity that characterizes the aggregation of charged particles in a liquid is the so-called Bjerrum length B =

q2 , 4π 0 r kB T

(11.16)

where q is the charge of the particles and r is the relative dielectric constant of the medium in which the experiment is performed. The Bjerrum length defines the typical distance at which the Coulomb barrier between two equally charged particles is of the same order as the thermal energy kB T . Thus, in practical terms, a suspension can be regarded as stable

262


Figure 11.2: TEM picture of Aerosil flakes (courtesy of K.-E. Wirth and M. Linsenbühler).

if the particle diameter d is small compared to the Bjerrum length. Similarly, starting with an unstable suspension of primary particles, the aggregation is expected to continue until the Bjerrum length, which grows quadratically with the accumulated charge, exceeds the size of the aggregate.

11.4 Agglomeration of Monopolar Charged Suspensions In this section we consider the dynamical aspects of aggregation of equally charged granular particles suspended in a non-polar liquid. In particular we argue that the distribution of flake sizes is expected to self-focus towards a universal distribution [30], as will be demonstrated below. Irreversible aggregation of uncharged particles has been studied extensively both theoretically and experimentally in various contexts including, for example, aerosol coalescence, polymerization, gelation, and planetesimal accumulation [31–35]. In most cases a simple mean field theory, based on Smoluchowski’s rate equation, provides an accurate description. The reason why mean field theories are able to describe the asymptotic behavior correctly can be made plausible by studying the simple case of pairwise coagulation or annihilation of diffusing point-like particles [36]. Since the particles perform a simple random walk before they coagulate, the aggregation process is essentially governed by a random walk of primary particles and is therfore characterized by an upper critical dimension dc = 2. While the mean field approximation becomes asymptotically exact in d > dc dimensions, fluctuation effects are expected to change the scaling behavior in low dimensions d < dc . In fact, in onedimensional systems the particles become anticorrelated and their average density decays as ρ(t) ∼ t−1/2 , while a mean field rate equation predicts ρ(t) ∼ 1/t. Therefore, in one spatial dimension (anti)correlations play an important role so that the mean field results are no longer valid. In two spatial dimensions, however, these anticorrelations become marginal, generating an effective mean field behavior plus logarithmic corrections. Finally, in three dimensions anticorrelations between particles are irrelevant and the mean field approximation becomes exact in the asymptotic limit t → ∞. A similar scenario is expected to hold in the case of non-point-like particles in three dimensions, justifying the use of mean field rate equations. In

11.4 Agglomeration of Monopolar Charged Suspensions

263

fact, as will be shown below, the mean field predictions are in excellent agreement with the numerical results obtained in three dimensions.

11.4.1 Mean Field Rate Equation In order to set up appropriate mean field equations, we consider Smoluchowski’s coagulation equation for monodisperse initial conditions, i.e., all particles have initially the same mass m∗ , the same radius a∗ , and the same charge q ∗ . As the particles collide they form clusters with increasing mass and charge. Since the mass and the charge of the clusters are proportional to the number of primary particles, it is sufficient to describe them by a single index i, leading to the equation ∞ 1 dni (t) = Rjk nj (t)nk (t) − ni (t) Rij nj (t). dt 2 j=1

(11.17)

j+k=i

Here ni (t) denotes the number density of clusters with mass mi =im∗ at time t. Moreover, it is assumed that clusters of primary particles may also be considered as spherical with their total charge qi =iq ∗ located in the center. The size of the clusters is given by an effective radius ai =iα a∗ ,

(11.18)

where 1/α denotes the fractal dimension of the aggregates (e.g., α=1/3 for spherical droplets). Starting with primary particles the initial condition reads n1 (t=0)=1 ,

ni (t=0)=0 for i>1 .

(11.19)

The matrix Rij in the rate equation is referred to as the coagulation kernel and describes at which rate clusters of size i and j merge into a single one. The form of the coagulation kernel depends on the particular physical situation and takes all interactions including electrostatic and hydrodynamic forces into account. Based on computer simulations, we observed that the long-range part of the hydrodynamic forces seen from the perspective of a primary particle basically amounts to fluctuations which can be accounted for by the modification of the diffusion constant [26]. Similarly, the short-range part of hydrodynamic forces, in particular the so-called lubrication force, will certainly modify the effective coagulation rate but it is not expected to change the overall physical situation. Therefore, to lowest order, we assume that the hydrodynamic forces can be neglected by absorbing their influence in the effective rates for diffusion and coagulation. The Coulomb force, however, indeed changes the physical situation. For this reason let us consider the uncharged and the charged cases separately. 11.4.1.1 Uncharged Case Let us assume that the clusters of mass mi diffuse in the suspension with an effective diffusion constant Di . With these assumptions two clusters of masses mi and mj will touch each other and aggregate as soon as the distance between their centers equals ai +aj . In this case Rij can be calculated by solving the diffusion equation in the presence of an absorbing sphere [34,35], leading to Rij = 4π(ai + aj )(Di + Dj ) ≡ rij .

(11.20)

264


Assuming Einstein’s formula to be valid, the diffusion constants are given by (11.21)

Di =kB T /6πηai ,

where η is the viscosity of the carrying fluid and ai is the effective radius defined in Eq. (11.18). With these assumptions, one is led to the coagulation kernel rij =

2kB T α (i + j α )(i−α + j −α ) . 3η

(11.22)

Because of the property rbi bj =bλ rij

(11.23)

(b > 0)

these rates are homogenous functions of degree λ = 0. We note that for most uncharged coagulating systems discussed in the literature, the coagulation kernels are homogeneous functions of degree λ with 0≤λ≤2 [31]. For example, coagulation of spherical particles in the kinetic gas regime the homogeneity parameter is found to be λ=1/6 [34]. In fact, as shown in [31], λ is the most important parameter for the uncharged case. Depending on its value, the coagulation equation shows two types of qualitatively different solutions: • For λ≤1 the cluster size distribution decays exponentially for large sizes at all times. In the scaling limit the solutions are self-similar and (for λ < 1) obey the scaling relation 1

ni (t) = s(t)−2 φ (i/s(t)) with s(t) ∼ t 1−λ ,

(11.24)

where s(t) is the average number of primary particles per cluster and φ is a scaling function that depends on the details of the reactions. • For 1 the final position of the surface charges is random, while in the case of a being much smaller than B the charges always find each other exactly. This simple criterion tells us under which conditions the Coulomb interaction will influence the morphology and the electrical properties of the deposited layer.

11.6 Summary Triboelectric charging is a very common phenomenon whenever granular materials evolve dynamically in such a way that the grains collide. Therefore, the influence of Coulomb interactions is expected to play an important role in very diverse situations especially if the particles are very small. This applies for example to small ice particles in thunderclouds, where triboelectric charging eventually leads to lightning, as well as to enormous electrical fields generated during the eruption of volcanos. Generally any handling of fine granular matter is to some extent accompanied by electrostatic phenomena, even when powders are suspended into non-polar liquids. In the present article we have discussed basic physical concepts in order to estimate under which conditions Coulomb forces have an impact concerning the structure of

278


granular matter or the morphology of a deposited layer. In most cases these criteria are based on a comparison of the typical energy scale of the dynamics with the characteristic Coulomb energy barrier. Coulomb interactions are not always a disturbing side effect but may be used as a tool in order to achieve certain technological goals. As an example, we have discussed a coating process, where micrometer particles are coated by small nanoparticles with the aim of improving the flow properties of the powder – an application which plays an enormous role in the pharmaceutical industry. Here it is possible to use Coulomb forces in a constructive manner to support the coating process, increasing the efficiency of the task and probably improving the quality of the deposited layers.

Acknowledgments We would like to thank Z. Farkas, H. Knudsen, M. Linsenbühler, K.-E. Wirth, and D. E. Wolf. This work was partly supported by the Deutsche Forschungsgemeinschaft (DFG), grant Hi/744.

References [1] T. Scheffler, Kollisionskühlung in elektrisch geladener granularer Materie, PhD thesis, University of Duisburg (2000). (electronic download from http://www.ub.uni-duisburgessen.de/recherch/eltexte/eltexte.shtml) [2] T. Poeschel and N.V. Brilliantov (Eds.) Granular Gas Dynamics, Lecture Notes in Physics, vol. 624, Springer (2003). [3] N.V. Brilliantov and T. Poeschel, Kinetic Theory of Granular Gases, Oxford University Press, (2004) in press [4] T. Poeschel and S. Luding, Granular Gases, Lecture Notes in Physics, vol. 564, Springer (2002). [5] S. Ogawa, Multitemperature theory of granular materials, in: Proceedings of the USJapan Seminar on Continuum-Mechanical and Statistical Approaches in the Mechanics of Granular Materials, ed. by S.C. Cowin and M. Satke, Tokyo (1978). [6] I. Ippolito, C. Annic, J. Lemaitre, L. Oger, and D. Bideau, Granular temperature: Experimental analysis, Phys. Rev. E 52, 2072 (1995). [7] I. Goldhirsch and G. Zanetti, Clustering instability in dissipative gases, Phys. Rev. Lett. 70, 1619 (1993). [8] I. Goldhirsch, Microstructures and Kinetics in Rapid Granular Flows, in: Traffic and Granular Flow, ed. by D. E. Wolf, M. Schreckenberg, and A. Bachem, p. 251, World Scientific (1996). [9] S. McNamara and W. R. Young, Inelastic collapse and clumping in a one-dimensional granular medium, Phys. Fluids A 4, 496 (1992). [10] J. Lowell and A. C. Rose-Innes, Contact electrification, Adv. Phys. 29, 947 (1980). [11] U. Malaske, C. Tegenkamp, M. Henzler, and H. Pfnür, Defect-induced band gap states and the contact charging effect in wide band gap insulators, Surf. Sci. 408, 1998 (1998).

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[12] D. M. Taylor and P. E. Secker, in: Industrial Electrostatics: Fundamentals and Measurements, Wiley & Sons, New York, p. 96 (1994). [13] P. P. Ewald, Annalen der Physik 64, 253 (1921). [14] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, (1987). [15] R. S. Haberlandt, S. Fritzsche, G. Peinel, and K. Heinzinger, Molekulardynamik, Vieweg, Braunschweig (1995). [16] J. Schäfer and D. E. Wolf, Bistability in granular flow along corrugated walls, Phys. Rev. E 51, 6154 (1995). [17] P. A. Cundall and O. D. L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29, 47 (1979). [18] W. J. McNeil and W. G. Madden, A new method for the molecular dynamics simulation of hard core molecules, J. Chem. Phys. 76, 6221 (1982). [19] M. Y. Louge, Computer simulations of rapid granular flows of spheres interacting with a flat, frictional boundary, Phys. Fluids A 6, 2253 (1994). [20] M. A. Hopkins and M. Y. Louge, Inelastic microstructures in rapid granular flows of smooth disks, Phys. Fluids A 3, 47 (1991). [21] T. Scheffler, D.E. Wolf, Granular Matter 4 103 (2002). [22] G. Nägele and P. Baur, Long-time dynamics of charged colloidal suspensions: hydrodynamic interaction effects, Physica A 245, 297 (1997). [23] R. Messina, C. Holm, and K. Kremer, Effect of colloidal charge discretization in the primitive model, Eur. Phys. J. E 4, 363–370 (2001). [24] R. B. Jones, Stability of colloidal clusters in shear flow near a wall: Stokesian dynamics simulation studies, J. Chem. Phys. 115, 5319 (2001). [25] G. Huber and K.-E. Wirth, Electrostatically supported surface coating of solid particles using liquid nitrogen, Proceeding PARTEC 2001. [26] J. H. Werth, M. Linsenbühler, S. M. Dammer, Z. Farkas, H. Hinrichsen, K.-E. Wirth, and D. E. Wolf, Agglomeration of charged nanopowders in suspensions, Powder Technology 133, 106 (2003). [27] T. van den Ven, Colloidal Hydrodynamics, Academic Press, London (1989). [28] H. C. Hamaker, The London-van der Waals attraction between spherical particles Physica 4, 1058 (1937). [29] L. Bergstrom, Hamaker constants of inorganic materials, Adv. in Coll. and Interface Sci. 70, 125 (1997). [30] S. M. Dammer and D. E. Wolf, Self-Focussing dynamics in monopolarly charged suspensions, eprint cond-mat/0404546, accepted by Phys. Rev. Lett. [31] see e.g. M. H. Ernst, in Fractals in Physics, p. 289, ed. by L. Pietronero and E. Tosatti, North-Holland, Amsterdam (1986), and references therein. [32] M. H. Lee, J. Phys. A 34, 10219 (2002). [33] A.V. Ivlev, G.E. Morfill and U. Konopka, Phys. Rev. Lett. 89, 195502 (2002) [34] S.K. Friedlander, Smoke, Dust and Haze, Wiley-Interscience, New York (1972) [35] N.A. Fuchs The Mechanics of Aerosols, Elmsford, New York, Pergamon Press (1964)

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[36] L. Peliti, Renormalization of fluctuation effects in the A + A → A reaction, J. Phys. A: Math. Gen. 19, L365 (1986). [37] J.E. Barnes and P. Hut, A hierarchical O(N Log N) force calculation algorithm, Nature 324, 446, (1986). [38] I. Zimmermann, M. Eber, and K. Meyer, Zeitschrift für physikalische Chemie 218, 51 (2004). [39] J.H. Werth, M. Linsenbuhler, S.M. Dammer, Z. Farkas, H. Hinrichsen, K.E. Wirth, and D.E. Wolf, Powder Technology 133 (1-3), 106 (2003) [40] M. Linsenbühler, K.-E. Wirth, Properties of tailor-made functionalized micro-particles, Proceedings PARTEC 2004 [41] J.H. Werth, H. Knudsen, and H. Hinrichsen, Agglomeration of oppositely charged particles in nonpolar liquids, 2004, submitted to Phys. Rev. E.

12 Magnetized Granular Materials Daniel L. Blair and Arshad Kudrolli

12.1 Introduction Granular materials in their simplest form interact only during contact. In addition to the hardcore repulsion, the coefficients of normal and tangential inelasticity, and sliding friction determine the dissipative interaction between particles. Complex properties ranging from solid to liquid-like transitions may occur solely due to these interactions [1]. There exist numerous contexts where granular materials have relevance, and a large number of those involve interactions between the particles that go beyond those just outlined. For example, the presence of humidity introduces capillary bridges between particles resulting in cohesive forces which affect packing, flow, and segregation properties [2–4]. Capillary interaction is relatively short range and is important when particles are nearly in or leaving contact. An example where particles interact at long range occurs when they become magnetized, where the interaction potential is anisotropic. Magnetic interactions are important in applications ranging from micro-sized toners to the processing of centimeter-sized mineral ores. Therefore, it is surprising that very few fundamental studies have been conducted on the impact of magnetic fields on the properties of granular materials. Recent exceptions being the work of Forsyth, et al. [5], who examined the effect of an applied magnetic field on the packing and cohesivity of a pile composed of steel beads, and the work of Fazekas, et al. [6], who simulated the patterns formed as magnetized particles are poured into a silo. Although a complete theory describing the observed phenomena in granular materials that includes even the basic interactions is not yet available, interesting progress is being made in the regime of low dissipation and rapid motion. In this limit, the kinetic theory of gases has been modified to include dissipation [7,8] and experiments have become available to guide the development of the models [9–12]. Therefore, we chose such a “granular gas” as the starting point to understand the effects of added magnetic interactions. In the limit where the dissipative interaction can be reduced to zero, the system under consideration can be mapped to the dipolar hard-sphere model [13, 14]. Considerable theoretical work has been accomplished using the tools of equilibrium statistical mechanics which can serve as a reference point in our understanding of the impact of magnetization on weakly dissipative granular system. In this chapter, we discuss an experimental study of a model magnetized granular system first introduced by us in Ref. [15]. The system consists of a vibrated container with a submonolayer of uniformly magnetized steel spheres. The experiments are of particular interest due to the direct visualization of particles. A granular gas is observed at high vibration amplitudes, and magnetization of the particles appears to be insignificant. In contrast, when the vibration is lowered below a critical value, simple chain and ring structures are observed to

282

12 Magnetized Granular Materials

initially self-assemble because of the anisotropic interactions between the particles. These structures grow rapidly to form either compact clusters or networks of chains, depending on the depth of the quench. We compare and contrast our results with the dipolar hard-sphere model which neglects dissipative interactions. The model predicts a network of chains to form below a critical temperature, based on the propensity of the particles to align along the magnetic poles, energy costs associated with free ends, and entropic considerations. Network structures are observed in our experiments, but are metastable. The stable clusters are generally more compact than anticipated in the models. We measure the velocity of the particles and thus the granular temperature in order to understand the source of the discrepancy. Equipartition is not observed as the temperature of the particles in the cluster is significantly lower than the temperature of the isolated particles. We discuss the effect of dissipation on the observed phases, and compare our results with those for the equilibrium dipolar hard-sphere model.

12.2 Background: Dipolar Hard Spheres We begin by discussing the dipolar hard-sphere model (DHSM) which serves as an extremum model for particles with anisotropic interactions and its implications. Developed to understand the kinetic behavior associated with ferro-fluids [13,14], DHSM has seen a resurgence of simulations and theoretical treatments that have mainly focused on the question of the existence of a critical liquid–gas transition [16]. The idealized interaction between two dipolar hard spheres separated by distance r is defined as, Udhs = Uhs +

1 3 (µi · µj ) − 5 (µi · rij )(µj · rij ), r3 r

(12.1)

where Uhs corresponds to the hard-core repulsion interaction, µ is the dipole moment, and r is the inter-particle vector connecting the centers of dipoles i, j. Clearly from an energy point of view, neighboring particles like to align head to tail which lowers their energy by 2µ2 /σ 3 . Once a chain of magnetized particles forms, energy is required to bend the chain which may be supplied by kinetic energy acquired from collisions with neighboring particles. When a chain bends so much so that the ends close to form a circle, then energy is lowered below that for a chain when the number of particles in the ring exceeds or equals four. Thus depending on the thermal energy, the particles can be found in a number of metastable configurations. When the potential in Eq. (12.1) is averaged over all particle positions, then an r−6 potential similar to the van der Waals interaction for isotropic liquids is obtained [14, 16]. Thus, DHSM is expected to have a well-defined liquid–gas transition just like a van der Waals liquid. However, early simulations of the DHSM did not observe phase coexistence between the gas and liquid phases but instead found ferromagnetically oriented chains of dipoles that spanned the system [17]. It was therefore postulated that the weakly interacting chains preclude phase coexistence [18, 19]. More recent large-scale Monte Carlo simulations revealed the existence of a liquid phase, where chaining is greatly reduced and particles have high coordination number [20]. Building on these studies, Tlusty and Safran [21] have developed a topological model

12.3 Experimental Technique

283

to investigate the critical liquid–gas transition. They have found phase coexistence as well as critical behavior by considering the concentration of three-fold junctions and free ends. Although direct and indirect observations in colloidal systems exist [22–24], these experiments have not directly visualized the nature of the phases under conditions that satisfy those stipulated by the model. Therefore, the models and simulations have not been thoroughly tested, even for equilibrium hard spheres.

12.3 Experimental Technique The apparatus consists of a circular, flat, anodized aluminum cell, with a diameter D = 30.0 cm, and side walls of height h = 1.0 cm (see Figure 12.1). The system is leveled to within 0.01 cm to ensure that the plate is uniformly accelerated. The measured acceleration of the plate Γ = Aω 2 /g (where A, ω are the amplitude and angular frequency and g is the acceleration due to gravity) is varied between Γ = 0 − 3.0 g, at ω = 377 rad s−1 . The particles used are chrome steel spheres with a diameter of σ = 0.3 cm (with a high degree of sphericity δσ/σ ∼ 10−4 ) and mass m = 0.12 g. Each sphere has been placed in a ramped field of 1 × 104 G to embed a permanent moment of µ ∼ 10−2 emu per particle. The surface fraction of magnetic particles, φ, defined as the ratio of the area of the particles to that of a close-packed mono-layer, is varied from φ = 0.01 → 0.15. We also place glass particles of equivalent mass, with a fixed surface fraction of φp = 0.15. The glass particles act as a constant thermal bath.

CCD 30 cm Accelerometer

10cm

Lock in Amplifier Shaker

Amplifier Computer

Figure 12.1: A schematic diagram of the experimental apparatus. The plate has a diameter of D = 30 cm with side walls of height h = 1.0 cm. The plate is leveled to within 0.01 cm to ensure that the acceleration is uniform. The shaker is driven by a power amplifier and the driving signal originates from an arbitrary wave-form generator. A lock-in amplifier filters the signal of a 10 mV/g accelerometer that is mounted to the bottom of the driving plate. Image data is acquired from overhead by a high-speed digital camera. Each device is interfaced via a microcomputer workstation.

284

12 Magnetized Granular Materials 10−1

φ = 0.01 φ = 0.05 φ = 0.09

10−3

10−4

φ = 0.01 φ = 0.05 φ = 0.09

8.0 T [g (cm s−1 )2 ]

P (v)

10−2

9.0

(a)

7.0 6.0 5.0 4.0

Γ = 2.0 3.0

10

−5

-40

-30

-20

-10 0 10 v (cms−1 )

20

30

40

2.0 1.2

(b) 1.4

1.6

1.8 Γ/g

2.0

2.2

2.4

Figure 12.2: a) The distribution of particle velocities of the P (vx ) versus vx , component of the velocity in the horizontal direction for Γ = 2.0, at φ = 0.01, 0.05, 0.09. The distributions have not been rescaled, thus demonstrating that over a broad range of surface fraction the distribution is unchanged. The solid line is a Gaussian fit for φ = 0.09. b) The granular temperature T , versus Γ, the dimensionless acceleration. The data is essentially independent of φ.

Image data is acquired through a high-speed Kodak SR-1000 digital camera with a spatial (temporal) resolution of 512 × 480 pixels (250 frames s−1 ). By utilizing the Hoshen– Kopelmen algorithm [25], individual clusters that form are identified. Having identified particles as members of clusters, the number of particles, the centroid, and the radius of gyration can be measured as a function of time. The frame rate is sufficiently rapid that instantaneous velocities of the particles can be measured. In any discussion of phases in hard-sphere systems, the two most important parameters are the volume fraction (or area fraction in a quasi-two-dimensional system) and the temperature. Driven granular systems are non-equilibrium, making the thermal energy scale irrelevant to the properties at the macroscopic level. Therefore, a kinetic quantity called the “granular temperature” has been postulated as equivalent to the thermodynamic temperature in equilibrium systems [26]. The granular temperature is given by the width of the velocity distribution of the grains. In our experimental system, the parameter we can directly control is the vibration strength Γ. We use the glass particles to define a system temperature by performing calibration experiments as a function of Γ. Energy equipartition is not observed in systems out of equilibrium, therefore care must be taken while interpreting the granular temperature. We will hopefully clarify these distinctions at appropriate points within the discussion. The distribution of particle velocity components in the horizontal direction is shown in Figure 12.2a for three values of φ at a fixed ω and Γ = 2.0. The data corresponding to both glass and steel particles in the gas-like state. Because the masses of the particles were chosen to be similar, their distributions are found to be identical over a broad range of surface fraction. The distributions are non-Gaussian, consistent with previous observations [9], and have a kurtosis of 3.73 (a Gaussian distribution has a kurtosis of 3.0). Due to the self-consistency of

12.4 The Phase Diagram

285

the distributions, we can define the granular temperature as follows: T =

3 mvi2 , 2

(12.2)

where i represents the individual components of the velocity vector. Here we have assumed that the velocity distributions in the horizontal and vertical directions are the same, although in reality they may be different by a factor given by the coefficient of restitution [12]. Using this definition, we have obtained the granular temperature as a function of Γ (see Figure 12.2b). We observe that the points (to within experimental accuracy) fall onto one master curve irrespective of φ. The resulting calibration allows us to simply utilize a local fitting window on the points in Figure 12.2b to obtain the temperature of the bath T by knowing only Γ. We will see in a later section that the temperature of the magnetized particles is, in fact, not the same as the bath temperature but is state dependent.

12.4 The Phase Diagram To study the transition from a gas to a clustered phase, experiments with the following protocol are performed. The container with the a fixed number of particles is first vibrated at high amplitude (Γ = 3) so that a gas-like state is observed and then the amplitude is lowered. Below a critical amplitude, clusters nucleate and grow. The order of magnitude where the transition occurs can be determined by considering the temperature of the system and the dipole energy 2µ2 /σ 3 . We denote the temperature at which clusters are first observed to nucleate as the transition temperature Ts . As Ts is approached from above, (i.e., by lowering T from the gas temperature), prior to actual nucleation, the system begins to support the existence of short-lived di-mers and tri-mers that act as the initial seeds for the nucleating clusters. In Figure 12.3a–c typical initial structures are shown. The formation and growth of these initial structures depends on the bath temperature to which the system is lowered. When the vibration amplitude is lowered to Ts or slightly below, we observe that long chains, like those found in Figure 12.3a are unstable, and in time will give way to more compact structures. Chains of particles will either form ring configurations, or more compact configurations (see Figure 12.3b,c). There are at least two possible scenarios for the change from the chain configuration: (1) Chains are highly mobile. Mobility, due to the rotational and translational degrees of freedom, allow chains to align head-to-tail and grow through a coarsening process. (2) Due to the flexibility of the chains, and the long-range attractive interactions of their free ends, they will eventually attach themselves either to another chain or will form a loop or a closed Y (see Figure 12.3b). The process described as scenario (1) above is displayed in Figure 12.4a–c for a particular chain at T = Ts . Initially, the chain (Figure 12.4a) consists of the beginnings of a Y structure at the top and a ring of four particles in the center. Figure 12.4b shows an intermediate configuration, and Figure 12.4c shows the cluster after 1080 s, that coexists with individual particles. To characterize the gas and clustered phases described above, and to describe the nature of the transition, we plot the phase diagram (see Figure 12.5). The connected points reflect the measured transition temperature Ts , that separates the gas phase (Figure 12.6a) from the

286


(a)

(b)

(c)

Figure 12.3: Initial structures that nucleate from the gas phase at and below Ts . a) A chain of dipoles that rapidly evolves into a more stable and energetically favorable configuration. b) A ring of 13 dipoles c) A close packed cluster. The scale bar denotes 1 cm. T = 4.0 erg, φ = 0.09.

(a)

(b)

(c)

Figure 12.4: The evolution of a chain at φ = 0.05, T = Ts over t = 1080 s. The chain begins as an extended structure, a) that has formed from multiple short chains joining together and in time has become more compact. The white scale bar denotes 1 cm.

clustered phase (Figure 12.6b), and demonstrates a monotonic increase with φ. If clusters are nucleated and T is then increased above Ts , cluster evaporation is hysteretic. The apparent hysteresis depends on the ramping rate of T , and is not observed to disappear over laboratory timescales. Therefore, it appears that the observed phase transition is of first order. Within the cluster phase, there exists a large network phase (see Figure 12.5) that is produced by rapidly quenching the system directly from the gas phase far below Ts . The result of such a rapid thermal quench is shown in Figure 12.6c. The qualitative features of the network phase can be understood as follows. When the system is quenched, the kinetic energy of the particles is dramatically decreased, and the effects of the long-range dipole potential becomes important. Energetically, the local arrangement is dominated by head-to-tail alignment due to the inherent anisotropy in the embedded potential (see Eq. (12.1)). As discussed earlier, from the energy standpoint it is far more favorable to form ring and compact states when the number of particles in a chain reaches four. However, at low temperatures, the probability that a chain receives significant kinetic energy to over come the bending potential barrier, decreases. Thus the metastable chain-like structures can be observed over long times. Consequently, the further we lower the temperature in the quench, the more the structure formed is ramified and survives as such for a longer time. Over very long times, the configuration evolves and the eventual configuration is a more compact cluster with very few free ends.

12.4 The Phase Diagram

287

6.0

4.0

2.0

2.5 111111111111111111111111 000000000000000000000000 Gas 000000000000000000000000 111111111111111111111111 Hysteresis 000000000000000000000000 111111111111111111111111 000000000000000000000000 2.0 111111111111111111111111 000000000000000000000000 111111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 1.5 11111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 11111111111111111111111 Clustered 00000000000000000000000 11111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 1.0 11111111111111111111111 00000000000000000000000 11111111111111111111111 000000000000000000000000 111111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111

0.0 0.00

Γ

Tg [ g (cm s-1 )-2 ]

8.0

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 Network 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111

0.05

φ

0.10

0.5 0.0

0.15

Figure 12.5: The phase diagram of temperature T versus the surface fraction of the particles φ. The driving acceleration Γ, is also shown for clarity. A gas phase consisting of single particles and shortlived di-mers and tri-mers are observed above a transition temperature Ts that depends on φ, shown by the solid points. To evaporate a cluster in the gas phase one must go past Ts denoted by the hysteresis region. At and below Ts , di-mers and tri-mers act as seeds to the formation of compact clusters that co-exist with single particles. If T is rapidly quenched from the gas region to very low T highly ramified networks of particles form (Figure 12.6c).

(a)

(b)

(c)

Figure 12.6: Images of the observed phases. a) The gas phase for T = 7.5 erg, φ = 0.09. b) The cluster phase T = Ts , φ = 0.09. c) The network phase after a rapid quench from T = 7.5 → 3.3 erg, φ = 0.15.


g(r)

288 16 14 12 10 8 6 4 2 0

(a)

(b)

12 g(r)

10 8 6 4 2 0

0

1

2

3

4

5

r/σ

Figure 12.7: The radial distribution function g(r) vs r/σ for the a) clustered and b) network phases. In the cluster phase, (see Figure 12.6a) a splitting of the second and third peaks which indicates the existence of short range structure. The network phase demonstrates characteristics associated with liquids. The parameters for the plots are (a) φ = 0.09, T = 5.8 erg and (b) φ = 0.15, T = 2.9 erg.

The reason why the network phase may be interpreted as a metastable liquid can be shown as follows. We plot the radial distribution function of the particle positions, g(r) =

Nc N N 1 δ(r − rij ) 1 , Nc n=0 N 2 i πrδr

(12.3)

j=i

where rij is the inter-particle spacing and Nc is the number of separate configurations sampled for the particles in a cluster at T = Ts and for a network (see Figure 12.7a,b, respectively). Like the g(r) for a liquid, the plot shows peaks at unit intervals.

12.5 The Non-equipartition of Energy Before continuing the systematic exploration of the phase diagram, we first discuss the impact of the magnetic fields on the velocity distributions and consequently the granular temperature. By using the particle tracking and cluster identification methods utilized in Section 12.3 we measure the temperature of the particles with the clusters. In Figure 12.8 the distribution of particle velocities for various species is shown. The species of the particles are defined by the phase in which the particles exist. The three broad distributions correspond to the following species: Dipolar particles in a purely gas phase at T = Ts and φ = 0.09 just prior to a cluster nucleating, given by (); the glass particles at T = Ts and φ = 0.15 given by (); the

12.5 The Non-equipartition of Energy

289

P (v)

10−1

10−2

10−3

10−4 -40

-30

-20

-10 0 10 −1 v (cms )

20

30

40

Figure 12.8: The probability distribution functions P (v) of the velocity components in the gas and clustered phases on a log linear scale. () dipolar particles in a purely gas phase at T = Ts and φ = 0.09 just prior to a cluster nucleating, () glass particles at T = Ts and φ = 0.15, () magnetic particles that coexist in the gas phase with a cluster at φ = 0.09 T = Ts , (◦) particles within the cluster at T = Ts and φ = 0.09. The solid lines are Gaussian fits. Even though the phases co-exist, velocity distribution of particles in the gas phase are significantly greater than that for the clustered phases.

0.12

Tc [ g (cm s−1 )2 ]

0.10 0.08 0.06 0.04 0.02 0.00

0

5

10

15 r/σ

20

25

30

Figure 12.9: The cluster temperature Tc vs. r/σ for φ = 0.09. The temperature of the cluster decreases as a function of time and therefore as a function of cluster size.

290


magnetic particles that coexist in the gas phase with a cluster at φ = 0.09 T = Ts given by (); the particles within the cluster at T = Ts and φ = 0.09 given by (◦). The distributions are shown in absolute units, demonstrating that when particles are in a gas phase, prior to and after a cluster has nucleated, their granular temperatures are well defined and equipartition holds. However, once particles enter a cluster, their motion is highly limited, thus the reduction in their kinetic defined temperature. We also observe that clusters in fact cool as their size increases. In Figure 12.9 the temperature of a cluster at T = Ts and φ = 0.09 is shown, versus its radius. The ratio of the cluster temperature Tc to that of the system temperature T , is T /Tc ≈ 60. Therefore, both the mobility or diffusion of particles has been reduced, (not unlike the analog of colloidal gels) and the mean square velocity no longer reflects the mean square velocity of the system. Thus we have demonstrated the apparent breakdown of equipartition in a magnetized granular system.

12.6 Cluster Growth Rates We now discuss how the clusters grow once they are nucleated. It is observed that the closer the temperature is to Ts , the fewer clusters nucleate; conversely, more clusters nucleate the further the temperature is lowered below Ts . Therefore, we first discuss the growth when the temperature of the gas phase is lowered so that T = Ts , and then discuss the cluster growth at lower constant temperature. By carefully lowering the temperature to Ts we are able to nucleate a single cluster (if a single cluster does not form, the process is repeated). 1

1.2

(a)

(b)

1.0

0.8

rc /r∞

rc /r∞

0.8

0.6 0.4

0.6 φ = 0.01 φ = 0.02 φ = 0.03 φ = 0.04 φ = 0.05 φ = 0.06 φ = 0.07

0.4

φ = 0.03 φ = 0.05 φ = 0.07 φ = 0.09

0.2 0 0.0

0.5

1.0 t/τ

1.5

0.2

2.0

0.0

0

2

4

6 t/τ

8

10

12

Figure 12.10: a) Rescaled radius of gyration of the clusters, rc /r∞ vs. t/τ , at T = Ts for various φ. r∞ and τ are obtained by fitting the data to Eq. (12.4). Notice that the radius of the clusters never reaches the asymptotic value over the time of the experiment for T = Ts . b) Rescaled radial growth of clusters, rc /r∞ vs. t/τ , at T = 2.3 erg for various φ. The curves display a universal scaling. The clusters show a rapid approach and saturation that is markedly different from those at T = Ts where saturation did not occur over the lifetime of the experiment.

12.6 Cluster Growth Rates

291

1000 T = Ts T = 1.2

τ (s)

800 600 400 200 0

0

0.02

0.04

0.06

0.08

0.1

φ Figure 12.11: The characteristic cluster growth time τ vs. φ taken from the fits to Eq. (12.4). The open (closed) symbols are from the fits to the data in Figure 12.10a,b.

We observe that the change in the radius of gyration is well described by a simple exponential form, rc = r∞ (1 − e−t/τ ),

(12.4)

where r∞ is the asymptotic radius, and τ is the saturation time. By rescaling each data set shown in Figure 12.10a by the asymptotic radius and the saturation time, we demonstrate that the growth equation is universal. The values of τ are plotted versus φ in Fig 12.11. It also appears that the clusters formed at T = Ts never reach their asymptotic radius in the present experimental time window. The saturation of the radius of gyration occurs because the total number of magnetized particles decreases at a rate n(t) = (N − nc (t)1/d ), where N is the total number of particles and d is the fractal dimension of the cluster. We never observe that all of the free particles join the cluster. However, the overall surface fraction is lowered by the cluster growth, therefore leading to a reduction in the growth. We speculate that there may also be secondary effects that further act to arrest the growth, such as changes in the curvature of the outer edge of the cluster, and the shielding of the clusters by the side walls. In the previous paragraph and in Figure 12.10a, we demonstrated that the cluster radius never reached r∞ for all φ. To investigate the response of the system to temperatures below Ts , we have lowered the temperature of the system from the gas phase to the isotherm T = 2.3 erg for φ = 0.01 − 0.07. We observe that by lowering the temperature to slightly below Tc at that particular φ, cluster nucleation is quite rapid as compared to the observations made at Ts (see Figure 12.10b). Also, the number of clusters formed ranges from n = 1 − 5 depending on the surface fraction, but does not follow any systematic trend. The radial growth equation (12.4) is also fitted to the

292


data and demonstrates a rapid growth to a saturation value determined by φ. In Figure 12.10b the evolution of the radius of gyration rescaled for all φ, is plotted. The good collapse of the curves further indicates a universal form given by Eq. (12.4). The characteristic time τ is plotted as a function of φ (see Figure 12.11) and demonstrates a linear decrease over a broad range of surface fractions. We note, however, that at temperatures far below Ts , the saturation to r = r∞ does indeed occur, unlike the data presented at T = Ts where the saturation was limited by co-existence. We interpret this saturation as an effect of dissipation. Due to the lowered temperature, particles that remain in the gas phase are less likely to impact the clusters with large velocities. Thus, the thermal particles are less likely to remove particles from the clusters. This, coupled with the dissipation due to collisions and the attractive interactions between particles, leads to a rapid approach to r∞ for all φ.

12.7 Compactness of the Cluster In Figure 12.12, the average number of particles contained in a cluster nc = nc /n, where n is the total number of clusters formed, is plotted versus the dimensionless radius of gyration rc /σ. We propose a general definition for nc , as an average value, because the same quantity will be used when multiple clusters form (when T = Ts , n = 1). By using the scaling relation nc = αRgd ,

(12.5)

√ where α is determined by the dimensionality of the mass elements, (for disks α = π 2 /2 3) we calculate the fractal dimension of the clusters. 100 (a)

(b)

y = x1.4

100

nc

nc

y = x1.5

y = x2

10

10 φ = 0.01 φ = 0.02 φ = 0.03 φ = 0.04 φ = 0.05 φ = 0.06 φ = 0.07

φ = 0.03 φ = 0.05 φ = 0.07 φ = 0.09 1

1

10 rc /σ

1

1

10 rc /σ

Figure 12.12: a) The average number of particles in a cluster nc vs. rc /σ, the radius of gyration over a range of φ at T = Ts . The dashed lines indicate two unique scalings for the dimensionality of each cluster demonstrating a cross-over at rc /σ = 6. b) The average number of particles in a cluster nc vs. rc /σ, the radius of gyration over a range of φ at T = 2.3 erg. The dashed lines indicate a single scaling that captures the overall scaling of the clusters.

12.8 Migration of Clusters

293

Figure 12.12a clearly demonstrates a cross-over in the dimensionality of the clusters. Clusters with rc < 5.0 σ follow closely to the spatial dimension. As the number of particles per cluster increases with time, the trend for the fractal dimension, d, over all φ, is more consistent with d = 1.4±0.1. Thus we demonstrate that clusters are initially more compact and isotropic at early times, and become more extended for long times at T = Ts . The change in dimensionality may be understood by observing the growth of a cluster. As particles join the cluster they do so by forming highly mobile chains that are tethered by one end to the surface of the cluster. If the free end of the chain is able to connect back onto the cluster, an excluded area is established. During this process free particles may become trapped by the chains. The trapped particles can, in time, produce rearrangement and re-opening of these regions. The fractal-like character of the clusters can be seen in Figure 12.6b and 12.13. We also measure the fractal dimension of the clusters at T = 2.3 erg. In Figure 12.12b the radius of gyration rc , scaled by the particle diameter, of each cluster is plotted versus the number of particles within the cluster. We observe that the fractal dimension has a value of d ≈ 1.5, that does not depend on the surface fraction. The nature of the clusters is quite different from those measured at Ts . First, the clusters have a constant and somewhat larger fractal dimension. Second, due to the much lower system temperature, the particles are less likely to rearrange once they have joined into the clustered phase and, as a consequence, the motion of the clusters is very small compared to the example shown in Figure 12.13.

12.8 Migration of Clusters Next we comment on the interesting dynamics of the cluster as a whole. Soon after clusters form, they can nearly always be observed to migrate towards the container boundary. The trajectory of a cluster over time migration is shown in Figure 12.13; the positions of the particles within the cluster at time t = 1100 s are also plotted. Soon after the cluster forms, it diffuses until it reaches the boundary. The absence of “thermal” particles near the side wall results in an unbalanced pressure which pins the cluster to the side wall. Once in contact, the cluster demonstrates the existence of a depletion force as seen in colloids [28]. The depletant in this case is the “thermal” particles (magnetic and glass) impinging on the side of the cluster.

12.9 Summary We have introduced a new experimental system to study the impact of magnetic interactions on granular systems. The particles interact via their magnetic fields and inelastic collisions. Using a quasi-two-dimensional geometry, we are able to visualize the state of the system as a function of area fraction and external driving amplitude. Because the interactions are anisotropic, both simple and complex structures can be made to self-assemble by changing the system parameters. We have found that the nature and growth of the clusters is history dependent. Therefore, the formation of the clusters was explored using two protocols. When the vibration amplitude was lowered slightly below the critical amplitude, simple structures such as rings or chains

294


30 25

y (cm)

20 15 10 5 0

0

5

10

15 x (cm)

20

25

30

Figure 12.13: The trajectory of the center of mass of a cluster in the cell. The solid line represents each space-time point for the cluster center of mass. The dashed line denotes the inner boundary of the cell, while the individual points show the final spot of the particles within the cluster at the end of the experiment.

formed which continued to grow, adsorbing particles from the gas phase. The radial growth of the radius of gyration of the cluster follows a universal decaying exponential form. However, the asymptotic radius is never reached over laboratory timescales. The fractal dimension of the clusters undergoes a cross-over from a dimension that matches the spatial dimension to a lower value. The cross-over occurs as particles join the cluster through a chaining and looping process. If the temperature is lowered below Ts , the clusters that nucleate have a very different conformation. The radial growth still universally scales to a decaying exponential form, but the asymptotic radius is always reached. Below Ts , the cluster dimensionality does not demonstrate a cross-over but remains fixed at a value somewhat greater than that of clusters at Ts implying that the compactness of clusters is always less at lower temperatures. We also observe that, within the cluster region, in the phase diagram there exists a network phase that exhibits a metastable liquid character. The network is comprised of particles that are connected as chains. These chains demonstrate the inherent anisotropy in the interactions between particles. The metastability can be understood from the energy landscape that the network must traverse not only to overcome the energy penalty of bending the chains but also the dissipation and friction that is the hallmark of this non-equilibrium system. When clusters co-exist with single or gas-like particles, we find that the granular temperature of the particles in the clusters is significantly lower than that in the gas phase. Furthermore, the temperature of the particles in the clusters decreases as its size increases. Thus equipartition is not observed. The reason for the lower temperature in the clustered phase

References

295

appears to be friction and inelastic collision, which quickly suppress rapid relative motion between particles. Thus we have shown that a granular system with magnetic interactions shows a rich variety of phases and phenomena. Using the dipolar hard-sphere model, one can understand some of the observed structures. However, a deeper understanding of the dissipative interactions has to be reached to understand why the observed structures are more compact than those shown by simulations conducted on equilibrium systems. Our study may be helpful in developing micro-mechanical devices where the understanding of processes that lead to self-assembly of complex structures from components, is of vital importance.

Acknowledgment This work was supported by the National Science Foundation under Grant # DMR-9983659.

References [1] G. H. Ristow, Pattern formation in granular matter (Springer-Verlag, 2000). [2] D.J. Hornbaker, R. Albert, I. Albert, A.-L. Barabási, and P. Schiffer, What keeps sandcastles standing? Nature (London) 387, 765 (1997). [3] A. Samadani and A. Kudrolli, Segregation transitions in wet granular matter, Phys. Rev. Lett. 85, 5102 (2000). [4] P. Tegzes, T. Vicsek, and P. Schiffer, Development of correlations in the dynamics of wet granular avalanches, Phys. Rev. E 67, 051303 (2003) [5] A. J. Forsyth, S. R. Hutton, C. F. Osborne, and M. J. Rhodes, Effects of interparticle force on the packing of spherical granular material Phys. Rev. Lett. 87, 244301 (2001). [6] S. Fazekas, J. Kertész, and D. E. Wolf, Computer simulations of magnetic grains, in: Traffic and Granular Flow 03, eds. S. Hoogendoorn, S. Luding et al. (Springer, Berlin, 2004). [7] P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech. 134, 401 (1983). [8] J. T. Jenkins and S. B. Savage, A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles, J. Fluid Mech. 130, 187 (1983). [9] W. Losert, D. Copper, J. Delour, A. Kudrolli, and J. P. Gollub, Velocity statistics in vibrated granular media, Chaos 9, 682 (1999). [10] J. S. Olafsen and J. S. Urbach, Velocity distributions and density fluctuations in a granular gas, Phys. Rev. E 60, R2468 (1999). [11] F. Rouyer and N. Menon, Velocity fluctuations in a homogeneous 2D granular gas in steady state, Phys. Rev. Lett. 85, 3676 (2000) [12] D. L. Blair and A. Kudrolli, Collision statistics of driven granular materials, Phys. Rev. E 67, 041301 (2003) [13] R. E. Rosensweig, Ferrohydrodynamics (Cambridge, Cambridge University Press, 1985).

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[14] P. G. deGennes and P. A. Pincus, Pair correlations in a ferromagnetic colloid, Phys. Kondens. Mater. 11, 189 (1970). [15] D. L. Blair and A. Kudrolli, Clustering transitions in vibrofluidized magnetized granular materials, Phys. Rev. E 67, 021302 (2003). [16] Y. Levin, What happened to the gas–liquid transition in the system of dipolar hard spheres?, Phys. Rev. Lett. 83, 1159 (1999). [17] M. E. van Leeuwen and B. Smit, What makes a polar liquid a liquid?, Phys. Rev. Lett. 71, 3991 (1993). [18] R. P. Sear, Low-density fluid phase of dipolar hard spheres, Phys. Rev. Lett. 76, 2310 (1996). [19] R. van Roij, Theory of chain association versus liquid condensation, Phys. Rev. Lett. 76, 3348 (1996). [20] P. J. Camp, J. C. Shelley, and G. N. Patey, Isotropic fluid phases of dipolar hard spheres, Phys. Rev. Lett. 84, 115 (2002). [21] T. Tlusty and S. A. Safran, Defect-induced phase separation in dipolar fluids, Science 290, 1328 (2000). [22] H. Wang, Y. Zhua, C. Boyd, W. Luo, A. Cebers, and R. E. Rosensweig, Periodic branched structures in a phase-separated magnetic colloid, Phys. Rev. Lett. 72, 1929 (1994). [23] H. Mamiya, I. Nakatani, and T. Furubayashi, Phase transitions of iron-nitride magnetic fluids, Phys. Rev. Lett. 84, 6106 (2000). [24] K. Butter, P.H.H. Bomans, P.M. Frederik, G.J. Vroege and A.P. Philipse, Direct observation of dipolar chains in iron ferrofluids by cryogenic electron microscopy. Nature Materials 2, 88 (2003). [25] J. Hoshen and R. Kopelman, Percolation and cluster distribution. i. cluster multiple labeling technique and critical concentration algorithm, Phys. Rev. B 14, 3438 (1976). [26] S. Ogawa, Multitemperature theory of granular materials, in Proc. US–Japan Semin. Contin-Mech. and Stat. Approaches Mech. Granular Mater., page 208 (Gukujustu Bunken Fukyakai, Tokyo, 1978). [27] W. Wen, F. Kun, K. F. Pál, D. W. Zheng, and K. N. Tu, Aggregation kinetics and stability of structures formed by magnetic microspheres, Phys. Rev. E 59, R4758 (1999). [28] P. D. Kaplan, J. L. Rouke, A. G. Yodh, and D. J. Pine, Entropically driven surface phase separation in binary colloidal mixtures, Phys. Rev. Lett. 72, 582 (1994). [29] K. Fetosa and N. Menon, Breakdown of energy equipartition in a 2d binary vibrated granular gas, Phys. Rev. Lett. 88, 198301 (2002).

Part VI

Computational Aspects

13 Molecular Dynamics Simulations of Granular Materials Stefan Luding

Abstract One challenge of todays research is the realistic simulation of granular materials consisting of millions of particles. In this chapter, the molecular dynamics method for the simulation of many-particle systems is briefly introduced and some examples of applications are presented. There exist two basically different approaches, the so-called soft particle molecular dynamics and the hard sphere, event-driven method. The former is straightforward, easy to generalize, and has numerous applications, while the latter is optimized for rigid interactions and is mainly used for collisional, dissipative granular gases. The examples given include homogeneous and clustering granular gases, and bi-axial or cylindrical compression/shearing of dense packings.

13.1 Introduction A straightforward approach towards the understanding of macroscopic material behavior by simply modeling and simulating all atoms in a macroscopic system is not possible due to the huge number of degrees of freedom. Therefore, one could reduce the size of the system under consideration, so that a microscopic simulation of atoms is possible. However, the possible length of such a “probe” is in general too small in order to regard it as macroscopic. Therefore, methods and tools to perform a so-called micro–macro transition [1–3] are discussed in this study, starting from the molecular dynamics simulations. These “microscopic” simulations of a small sample (representative volume element) are used to derive macroscopic constitutive relations needed to describe the material within the framework of a macroscopic continuum theory. For granular materials, as an example, the particle properties and interaction laws are inserted into a discrete particle molecular dynamics (MD) and lead to the collective behavior of the dissipative many-particle system. From the particle simulation, one can extract, e.g., the pressure of the system as a function of density. This equation of state allows a macroscopic description of the material, which can be viewed as a compressible, non-Newtonian complex fluid [4], including a fluid–solid phase transition. In the following, two versions of the molecular dynamics simulation method are introduced. The first is the so-called soft sphere molecular dynamics (MD), as described in Section 13.2. It is a straightforward implementation to solve the equations of motion for a system of many interacting particles [5, 6]. For MD, both normal and tangential interactions, like friction, are discussed for spherical particles. The second method is the so-called event-driven

300

13 Molecular Dynamics Simulations of Granular Materials

(ED) simulation, as discussed in Section 13.3, which is conceptually different from MD, since collisions are dealt with via a collision matrix that determines the momentum change on physical grounds. For the sake of brevity, the ED method is only discussed for smooth spherical particles. A comparison and a way to switch between the soft and hard particle methods is provided in Section 13.4. As one ingredient of a micro–macro transition, the stress is defined for a dynamic system of hard spheres, in Section 13.5, by means of kinetic-theory arguments [2], and for a quasistatic system by means of volume averages [7]. Examples are presented in the following Sections 13.6 and 13.7, where the above-described methods are applied.

13.2 The Soft-particle Molecular Dynamics Method One possibility of obtaining information about the behavior of granular media is to perform experiments. An alternative would be to perform simulations with the molecular dynamics (MD) or discrete-element model (DEM) [1, 8–15]. Note that both methods are identical in spirit, but different groups of researchers use these (and also other) names. Conceptually, the MD or DEM method has to be separated from the hard-sphere eventdriven (ED) molecular dynamics, see Section 13.3, and also from the so-called contact dynamics (CD). The former will be discussed below, the latter (CD) will be introduced in Chapter 14 by Brendel, Unger, and Wolf. Alternative methods, such as cell models or lattice gas methods are not discussed here at all.

13.2.1 Discrete-particle Model The elementary units of granular materials are mesoscopic grains which deform under stress. Since the realistic modeling of the deformations of the particles is much too complicated, we relate the interaction force to the overlap δ of two particles, see Figure 13.1. Note that the evaluation of the inter-particle forces based on the overlap may not be sufficient to account for the inhomogeneous stress distribution inside the particles. Consequently, our results presented below are of the same quality as the simple assumptions about the force-overlap relation, see Figure 13.1.

13.2.2 Equations of Motion If all forces f i acting on the particle i, either from other particles, from boundaries or from external forces, are known, the problem is reduced to the integration of Newton’s equations of motion for the translational and rotational degrees of freedom: mi

d2 d2 r = f + m g , and I ϕ = ti i i i i dt2 dt2 i

(13.1)

with the mass mi of particle i, its position r i the total force f i = c f ci acting on it due to contacts with other particles or with the walls, the acceleration due to volume forces like gravity g, the spherical particles moment of inertia Ii , its angular velocity ω i = dϕi /dt and the total torque ti = c lci × f ci .

13.2 The Soft-particle Molecular Dynamics Method

301

f hys

k1δ

k2δ

ri δ

0

f0 rj

f b)

a)

min

δ min δ0

δ max

δ

−k c δ

Figure 13.1: a) Two particle contact with overlap δ. b) Schematic graph of the piecewise linear, hysteretic, cohesive force-displacement model used below.

The equations of motion are thus a system of D+D(D−1)/2 coupled ordinary differential equations to be solved in D dimensions. With tools from numerical integration, as nicely described in textbooks as [5,6], this is straightforward. The typically short-ranged interactions in granular media, allow for a further optimization by using linked-cell or alternative methods [5, 6] in order to make the neighborhood search more efficient. In the case of long-range interactions, (e.g., charged particles with Coulomb interaction, or objects in space with selfgravity) this is no longer possible, so that more advanced methods for optimization have to be applied – for the sake of brevity, we restrict ourselves to short-range interactions here.

13.2.3 Contact Force Laws 13.2.3.1 Linear Normal Contact Model Two spherical particles i and j, with radii ai and aj , respectively, interact only if they are in contact so that their overlap δ = (ai + aj ) − (ri − r j ) · n

(13.2)

is positive, δ > 0, with the unit vector n = nij = (ri − r j )/|r i − r j | pointing from j to i. The force on particle i, from particle j, at contact c, can be decomposed into a normal and a tangential part as f c := f ci = f n n + f t t, where f n is discussed first. The simplest normal contact force model, which takes into account excluded volume and dissipation, involves a linear repulsive and a linear dissipative force f n = kδ + γ0 vn ,

(13.3)

with a spring stiffness k, a viscous damping γ0 , and the relative velocity in the normal direction ˙ vn = −v ij · n = −(v i − v j ) · n = δ. This so-called linear spring dashpot model allows to view the particle contact as a damped harmonic oscillator, for which the half-period of a vibration around an equilibrium position,

302


see Figure 13.1, can be computed, and one obtains a typical response time on the contact level, π (13.4) tc = , with ω = (k/m12 ) − η02 , ω with the eigenfrequency of the contact ω, the rescaled damping coefficient η0 = γ0 /(2mij ), and the reduced mass mij = mi mj /(mi + mj ). From the solution of the equation of a half-period of the oscillation, one also obtains the coefficient of restitution r = vn /vn = exp (−πη0 /ω) = exp (−η0 tc ) ,

(13.5)

which quantifies the ratio of relative velocities after (primed) and before (unprimed) the collision. For a more detailed discussion of this and other, more realistic, non-linear contact models see e.g. [16] and Chapter 8 by Brilliantov and Pöschel and Chapter 14 by Brendel, Unger, and Wolf. The contact duration in Eq. (13.4) is also of practical technical importance, since the integration of the equations of motion is stable only if the integration time-step ∆tMD is much smaller than tc . Furthermore, it depends on the magnitude of dissipation. In the extreme case of an overdamped spring, tc can become very large. Therefore, the use of neither too weak nor too strong dissipation is recommended. 13.2.3.2 Cohesive Normal Contact Model Here we apply a variant of the linear hysteretic spring model [16–18], as an alternative to the frequently applied spring-dashpot models. This model is the simplest version of some more complicated nonlinear-hysteretic force laws [17, 19, 20], which reflect the fact that, at the contact point, plastic deformations may take place. The repulsive (hysteretic) force can be written as  for loading, if k2 (δ − δ0 ) ≥ k1 δ  k1 δ if k1 δ > k2 (δ − δ0 ) > −kc δ (13.6) k2 (δ − δ0 ) for un/reloading, f hys =  −kc δ for unloading, if − kc δ ≥ k2 (δ − δ0 ) with k1 ≤ k2 , see Figure 13.1. During the initial loading, the force increases linearly with the overlap δ, until the maximum overlap δmax is reached (which has to be kept in memory as a history parameter). The line with slope k1 thus defines the maximum force possible for a given δ. During unloading the force drops from its value at δmax down to zero at overlap δ0 = (1 − k1 /k2 )δmax , on the line with slope k2 . Reloading at any instant leads to an increase in the force along this line, until the maximum force is reached; for still increasing δ, the force again follows the line with slope k1 and δmax has to be adjusted accordingly. Unloading below δ0 leads to negative, attractive forces until the minimum force −kc δmin is reached at the overlap δmin = (k2 − k1 )δmax /(k2 + kc ). This minimum force, i.e., the maximum attractive force, is obtained as a function of the model parameters k1 , k2 , kc , and the history parameter δmax . Further unloading leads to attractive forces f hys = −kc δ on the cohesive branch with slope −kc . The highest possible attractive force, for given k1 and k2 , is hys = −(k2 −k1 )δmax . Since this would lead to a discontinuity reached for kc → ∞, so that fmax at δ = 0, it is avoided by using finite kc .

13.2 The Soft-particle Molecular Dynamics Method

303

The lines with slope k1 and −kc define the range of possible force values and departure from these lines takes place in the case of unloading and reloading, respectively. Between these two extremes, unloading and reloading follow the same line with slope k2 . Possible equilibrium states are indicated as circles in Figure 13.1, where the upper and lower circle correspond to a pre-stressed and stress-free state, respectively. Small perturbations lead, in general, to small deviations along the line with slope k2 as indicated by the arrows. A non-linear un/reloading behavior would be more realistic. However, due to a lack of detailed experimental information, we use the piece-wise linear model as a compromise. One refinement is a k2 value dependent on the maximum overlap that implies small and large plastic deformations for weak and strong contact forces, respectively. One model, as implemented ∗ , so that k2 (δmax ) is increasing recently [21], requires an additional model parameter, δmax ∗ from k1 to k2 (linear interpolation) with the maximum overlap, until δmax is reached1 : k2 (δmax ) =

k2 ∗ k1 + (k2 − k1 )δmax /δmax

∗ if δmax ≥ δmax ∗ if δmax < δmax

.

(13.7)

While in the case of collisions of particles with large deformations, dissipation takes place due to the hysteretic nature of the force-law, stronger dissipation of small-amplitude deformations is achieved by adding the viscous, velocity dependent dissipative force from Eq. (13.3) to the hysteretic force, such that f n = f hys + γ0 vn . The hysteretic model contains the linear contact model as the special case k1 = k2 = k. 13.2.3.3 Tangential Contact Model The force in the tangential direction is implemented in the spirit of Cundall and Strack [8], who introduced a tangential spring in order to account for static friction. Various authors have used this idea and numerous variants were implemented, see [22] for a summary and discussion. Since we use a formulation which is generalized to dimensions D = 2 and D = 3, and which also takes into account static and dynamic friction coefficients, it is necessary to repeat the model and define the details of the implementation. Note, however, that here we only discuss sliding/sticking friction and disregard rolling [23] and torsion friction [24]. The tangential force is coupled to the normal force via Coulomb’s law, i.e., f t ≤ µs f n , where, for the limiting, case one has dynamic friction with f t = µd f n . The dynamic and the static friction coefficients follow, in general, the relation µd ≤ µs . The static situation requires an elastic spring in order to allow for a restoring force, i.e., a non-zero remaining tangential force in static equilibrium due to activated Coulomb friction. If a repulsive contact is established, and thus one has f n > 0, the tangential force is active. In the presence of cohesion, Coulomb’s law has to be slightly modified in so far that f n is replaced by f n + kc δ. In other words, the reference criterion for a contact is no longer the zero force level, but is the cohesive, attractive force level along −kc δ. 1

A limit to the slope k2 is needed for practical reasons. If k2 is not limited, the contact duration could become very small so that the time-step would have to be reduced below reasonable values.

304


If a contact is active, we project the tangential spring into the actual tangential plane 2 ξ = ξ − n(n · ξ ) ,

(13.8)

where ξ is the old spring from the last iteration. This action is relevant only for an already existing spring; if the spring is new, the tangential spring-length is zero anyway. However, its change is well defined even for the first initiation step. The tangential velocity, as needed in the following, is v t = v ij − n(n · v ij ) ,

(13.9)

with the total relative velocity of the particle surfaces at the contact v ij = v i − v j + ai n × ω i + aj n × ω j .

(13.10)

In order to compute the changes in the tangential spring, we first compute the tangential testforce as the sum of the tangential spring and a tangential viscous force (in analogy to the normal viscous force) f to = −kt ξ − γt v t ,

(13.11)

with the tangential spring stiffness kt and the tangential dissipation parameter γt . As long as |f to | ≤ fCs , with fCs = µs (f n + kc δ), one has static friction and, on the other hand, if the limit |f to | > fCs is reached, sliding friction is active with magnitude fCd = µd (f n + kc δ). (As soon as |f to | becomes smaller than fCd , static friction is active again.) In the former static case, the tangential spring increases as ξ = ξ + v t ∆tMD ,

(13.12) t

f to

from Eq. (13.11) is used. to be used in the next iteration in Eq. (13.8), and the force f = In the latter, sliding case, the tangential spring is adjusted to a length which is consistent with Coulomb’s condition 1 (13.13) ξ = − fCd t + γt v t , kt with the tangential unit vector, t = f to /|f to |, defined by Eq. (13.11), and thus the magnitude of the Coulomb force is used. Inserting ξ from Eq. (13.13) into Eq. (13.11) leads to f to ≈ fCd t. Note that f to and v t are not necessarily parallel in three dimensions. However, the mapping in Eq. (13.13) always works, rotating the new spring such that the direction of the frictional force is unchanged and, at the same time, limiting the spring in length according to Coulomb’s law. In short notation the tangential contact law reads (13.14) f t = f t t = +min fC , |f to | t , where fC follows the static/dynamic selection rules described above. Note that the tangential force described above is identical to the classical Cundall–Strack spring only in the limits µ = µs = µd and γt = 0. The sequence of computations and the definitions and mappings into the tangential direction, however, are new (to our knowledge) insofar as they can be easily used in three dimensions as well as in two. 2

This is necessary, since the frame of reference of the contact may have rotated since the last time-step.

13.3 Hard-sphere Molecular Dynamics

305

13.2.3.4 Background Friction Note that the viscous dissipation takes place in a two-particle contact. In the bulk material, where many particles are in contact with each other, dissipation is very inefficient for longwavelength cooperative modes of motion [25, 26]. Therefore, an additional damping with the background can be introduced, so that the total force on particle i is f n n + f t t − γb v i , (13.15) fi = j

with the damping artificially enhanced in the spirit of a rapid relaxation and equilibration. The sum in Eq. (13.15) takes into account all contact partners j of particle i, but the background dissipation can be attributed to the medium between the particles. Note that the effect of γb should be checked for each simulation in order to exclude artificial effects. Results of the soft-particle method are presented in Sections 13.6 and 13.7.

13.3 Hard-sphere Molecular Dynamics In this section, the hard-sphere model is introduced, together with the event-driven algorithm. A generalized model takes into account the finite-contact duration of realistic particles and, besides providing a physical parameter, saves computing time because it avoids the “inelastic collapse”. In the framework of the hard-sphere model, particles are assumed to be perfectly rigid and they follow an undisturbed motion until a collision occurs, as described below. Due to the rigidity of the interaction, the collisions occur instantaneously, so that an event-driven simulation method [27–30] can be used. Note that the ED method was only recently implemented in parallel [29, 31]. However, we will not discuss this in detail. The instantaneous nature of hard-sphere collisions is artificial, although it is a valid limit in many circumstances. Even though details of the contact- or collision behavior of two particles are ignored, the hard-sphere model is valid when binary collisions dominate and multi-particle contacts are rare [32]. The lack of physical information in the model allows a much simpler treatment of collisions than is described in Section 13.2 by just using a collision matrix based on momentum conservation and energy-loss rules. For the sake of simplicity, we restrict ourselves to smooth hard spheres, here. Collision rules for rough spheres are extensively discussed elsewhere, see e.g. [33, 34], and references therein.

13.3.1 Smooth Hard-sphere Collision Model Between collisions, hard spheres travel independently of each other. A change in velocity – and thus a change in energy – can occur only at a collision. The standard interaction model for instantaneous collisions of identical particles with radius a, and mass m, is used in the following. The post-collisional velocities v of two collision partners in their center-of-mass reference frame are given, in terms of the pre-collisional velocities v, by v 1,2 = v 1,2 ∓ (1 + r)vn /2 ,

(13.16)

306


with v n ≡ [(v 1 − v 2 ) · n] n, the normal component of the relative velocity v 1 −v 2 , parallel to n, and the unit vector pointing along the line connecting the centers of the colliding particles. If two particles collide, their velocities are changed according to Eq. (13.16), with the change of the translational energy at a collision ∆E = −m12 (1 − r 2 )vn2 /2, with dissipation for restitution coefficients r < 1.

13.3.2 Event-driven Algorithm Since we are interested in the behavior of granular particles, possibly evolving over several decades in time, we use an event-driven (ED) method which discretizes the sequence of events with a variable time-step adapted to the problem. This is different from classical MD simulations, where the time-step is usually fixed. In the ED simulations, the particles follow an undisturbed translational motion until an event occurs. An event is either the collision of two particles or the collision of one particle with a boundary of a cell (in the linked-cell structure) [5]. The cells have no effect on the particle motion here; they were solely introduced to accelerate the search for future collision partners in the algorithm. Simple ED algorithms update the whole system after each event, a method which is straightforward, but inefficient for large numbers of particles. In Ref. [27] an ED algorithm was introduced which updates only those two particles involved in the last collision. Because this algorithm is “asynchronous” insofar that an event, i.e., the next event, can occur anywhere in the system, it is so complicated to parallelize it [29]. For the serial algorithm, a double buffering data structure is implemented, which contains the “old” status and the “new” status, each consisting of: time of event, positions, velocities, and event partners. When a collision occurs, the old and new status of the participating particles are exchanged. Thus, the former new status becomes the actual old one, while the former old status becomes the new one and is then free for the calculation and storage of possible future events. This seemingly complicated exchange of information is carried out extremely simply and fast by only exchanging the pointers to the new and old status respectively. Note that the old status of particle i has to be kept in memory, in order to update the time of the next contact, tij , of particle i with any other object j, if the latter independently changed its status due to a collision with yet another particle. During the simulation such updates may be necessary several times so that the predicted new status has to be modified. The minimum of all tij is stored in the new status of particle i, together with the corresponding partner j. Depending on the implementation, positions and velocities after the collision can also be calculated. This would be a waste of computer time, since before the time tij , the predicted partners i and j might be involved in several collisions with other particles, so that we apply a delayed update scheme [27]. The minimum times of event, i.e., the times which indicate the next event for a certain particle, are stored in an ordered heap tree, such that the next event is found at the top of the heap with a computational effort of O(1); changing the position of one particle in the tree from the top to a new position requires O(log N ) operations. The search for possible collision partners is accelerated by the use of a standard linked-cell data structure and consumes O(1) of numerical resources per particle. In total, this results in a numerical effort of O(N log N ) for N particles. For a detailed description of the algorithm see Ref. [27]. Using all these algorithmic tricks, we are able to

13.4 The Link between ED and MD via the TC Model

307

simulate about 105 particles within reasonable time on a low-end PC [35], where the particle number is more limited by memory than by CPU power. Parallelization, however, is a means pf overcoming the limits of one processor [29]. As a final remark concerning ED, one should note that the disadvantages connected with the assumptions made which allow us to use an event-driven algorithm, limit the applicability of this method. Within their range of applicability, ED simulations are typically much faster than MD simulations, since the former accounts for a collision in one basic operation (collision matrix), whereas the latter requires about one hundred basic steps (integration time-steps). Note that this statement is also true in the dense regime. In the dilute regime, both methods give equivalent results, because collisions are mostly binary [26]. When the system becomes denser, multi-particle collisions can occur and the rigidity assumption within the ED hardsphere approach becomes invalid. The most striking difference between hard and soft spheres is the fact that soft particles dissipate less energy when they are in contact with many others of their kind. In the following chapter, the so-called TC model is discussed as a means to account for the contact duration tc in the hard-sphere model.

13.4 The Link between ED and MD via the TC Model In the ED method the contact duration is implicitly zero, matching well the corresponding assumption of instantaneous contacts used for kinetic theory [36, 37]. Due to this artificial simplification (which disregards the fact that a real contact always takes finite time) ED algorithms run into problems when the time between events tn becomes too small. In dense systems with strong dissipation, tn may even tend towards zero. As a consequence the socalled “inelastic collapse” can occur, i.e., the divergence of the number of events per unit time. The problem of inelastic collapse [38] can be avoided using restitution coefficients dependent on the time elapsed since the last event [28, 32]. For the contact that occurs at time tij between particles i and j, one uses r = 1 if at least one of the partners involved had a collision with another particle later than tij − tc . The time tc can be seen as a typical duration of a contact, and allows for the definition of the dimensionless ratio τc = tc /tn .

(13.17)

The effect of tc on the simulation results is negligible for large r and small tc – for a more detailed discussion see [28, 32, 35]. In assemblies of soft particles, multi-particle contacts are possible and inelastic collapse is avoided. The TC model can be seen as a means to allow for multi-particle collisions in dense systems [28, 39, 40]. In the case of a homogeneous cooling system (HCS), one can explicitly compute the corrected cooling rate (r.h.s.) in the energy balance equation d E = −2I(E, tc ) , dτ

(13.18)

2 with the dimensionless time τ = (2/3)At/t

E (0) for 3D systems, scaled by A = (1 − r )/4, −1 and the collision rate tE = (12/a)νg(ν) T /(πm), with T = 2K/(3N ). In these units,

308


a)

b)

Figure 13.2: a) Deviation from the HCS, i.e., rescaled energy E/Eτ , where Eτ is the classical solution Eτ = (1 + τ )−2 . The data are plotted against τ for simulations with different τc (0) = tc /tE (0) as given in the inset, with r = 0.99, and N = 8000. Symbols are ED simulation results, the solid line results from the third-order correction. b) E/Eτ plotted against τ for simulations with r = 0.99, and N = 2197. Solid symbols are ED simulations, open symbols are MD (soft-particle simulations) with three different tc as given in the inset.

the energy dissipation rate I is a function of the dimensionless energy E = K/K(0) with the kinetic energy K, and the cut-off time tc . In this representation, the restitution coefficient is hidden in the rescaled time via A = A(r), so that inelastic hard-sphere simulations with different r, scale on the same master curve. When the classical dissipation rate E 3/2 [36] is extracted from I, so that I(E, tc ) = J(E, tc )E 3/2 , one has the correction-function J → 1 for tc → 0. The deviation from the classical HCS is [32]: J(E, tc ) = exp (Ψ(x)) ,

(13.19)

2 3 + O(x4 ) in the with the series expansion Ψ(x) −1.268x √ + 0.01682x √− 0.0005783x √ √ √ = −1 collision integral, with x =√ πtc tE (0) E = πτc (0) E = πτc [32]. This is close to the result ΨLM = −2x/ π, proposed by Luding and McNamara, based on probabilistic mean-field arguments [28] 3 . Given the differential equation (13.18) and the correction due to multi-particle contacts from Eq. (13.19), it is possible to obtain the solution numerically, and to compare it to the classical Eτ = (1 + τ )−2 solution. Simulation results are compared to the theory in Figure 13.2a. The agreement between simulation and theory is almost perfect in the examined range of tc values, only when deviations from homogeneity are evidenced does one expect disagreement between simulation and theory. The fixed cut-off time tc has no effect when the time between collisions is very large tE tc , but strongly reduces dissipation when the > −1 collisions occur with high frequency t−1 E ∼ tc . Thus, in the homogeneous cooling state, there is a strong effect initially, and if tc is large, but the long-time behavior tends towards the classical decay E → Eτ ∝ τ −2 . 3

ΨLM thus neglects non-linear terms and underestimates the linear part.

13.5 The Stress in Particle Simulations

309

The final check of whether the ED results obtained using the TC model are reasonable is to compare them to MD simulations, see Figure 13.2b. Open and solid symbols correspond to soft and hard-sphere simulations, respectively. The qualitative behavior (the deviation from the classical HCS solution) is identical. The energy decay is delayed due to multi-particle collisions, but later the classical solution is recovered. A quantitative comparison shows that the deviation of E from Eτ is larger for ED than for MD, given that the same tc is used. This weaker dissipation can be understood from the strict rule used for ED: namely that dissipation is inactive if any particle already had a contact. The disagreement between ED and MD is systematic and should disappear if a value of tc of about 30 percent smaller is used for ED. The disagreement is also plausible, since the TC model disregards all dissipation for multiparticle contacts, while the soft particles still dissipate energy – even though much less – in the case of multi-particle contacts. The above simulations show that the TC model is in fact a “trick” to make hard particles soft and therefore to form a connection between the two types of simulation models: soft and hard. The only change made to traditional ED involves a reduced dissipation for (rapid) multi-particle contacts.

13.5 The Stress in Particle Simulations The stress tensor is a macroscopic quantity that can be obtained by the measurement of forces per area, or via a so-called micro–macro homogenization procedure. Both methods will be discussed below. During derivation, it also turns out that stress has two contributions, the first is the “static stress” due to particle contacts, a potential energy density, the second is the “dynamics stress” due to momentum flux, as in the ideal gas, a kinetic energy density. For the sake of simplicity, we here restrict ourselves to the case of smooth spheres.

13.5.1 Dynamic Stress For dynamic systems, one has momentum transport via the flux of the particles. This simplest contribution to the stress tensor is the standard stress in an ideal gas, where the atoms (mass N points) move with a certain fluctuation velocity v i . The kinetic energy E = i=1 mvi2 /2 due to the fluctuation velocity vi can be used to define the temperature of the gas kB T = 2E/(DN ), with dimension D and particle number N . Given a number density n = N/V , the stress in the ideal gas is then isotropic and thus quantified by the pressure p = nkB T ; note that we will disregard kB in the following. In the general case, the dynamic stress is σ = (1/V ) i mi v i ⊗ v i , with the dyadic tensor product denoted by ⊗, and the pressure p = trσ/D = nT being the kinetic energy density. The additional contribution to the stress is due to collisions and contacts and will be derived from the principle of virtual displacement for soft interaction potentials, and then modified for hard-sphere systems.

310


13.5.2 Static Stress from Virtual Displacements From the centers of mass r 1 and r 2 of two particles, we define the so-called branch vector l = r 1 − r 2 , with the reference distance l = |l| = 2a at contact, and the corresponding unit vector n = l/l. The deformation in the normal direction, relative to the reference configuration, is defined as δ = 2an − l. A virtual change of the deformation is then ∂δ = δ − δ ≈ ∂l = ε · l ,

(13.20)

where the prime denotes the deformation after the virtual displacement described by the tensor ε. The corresponding potential energy density due to the contacts of one pair of particles is u = kδ 2 /(2V ), expanded to second order in δ, leading to the virtual change 1 k k 2 δ · ∂δ + (∂δ) ≈ δ · ∂ln , ∂u = (13.21) V 2 V where k is the spring stiffness (the prefactor of the quadratic term in the series expansion of the interaction potential), V is the averaging volume, and ∂ln = n(n · ε · l) is the normal component of ∂l. Note that ∂u depends only on the normal component of ∂δ due to the scalar product with δ, which is parallel to n. From the potential energy density, we obtain the stress from a virtual deformation by differentiation with respect to the deformation tensor components k 1 ∂u = δ⊗l = f ⊗l, (13.22) ∂ε V V where f = kδ is the force acting at the contact, and the dyadic product ⊗ of two vectors leads to a tensor of rank two. σ=

13.5.3 Stress for Soft and Hard Spheres Combining the dynamic and the static contributions to the stress tensor [41], one has for smooth, soft spheres:

1 σ= (13.23) mi v i ⊗ v i − f c ⊗ lc , V i c∈V

where the right sum runs over all contacts c in the averaging volume V . Replacing the force vector by momentum change per unit time, one obtains for hard spheres:   1  1 σ= mi v i ⊗ v i − pj ⊗ l j  , (13.24) V ∆t n i j where pj and lj are the momentum change and the center-contact vector of particle j at collision n, respectively. The sum in the left term runs over all particles i, the first sum in the right term runs over all collisions n occurring in the averaging time ∆t, and the second sum in the right term concerns the collision partners of collision n [28]. Exemplary stress computations from MD and ED simulations are presented in the following section.

13.6 2D Simulation Results

311

13.6 2D Simulation Results Stress computations from two-dimensional MD and ED simulations are presented in the following subsections. First, a global equation of state, valid for all densities, is proposed based on ED simulations, and second, the stress tensor from a slow, quasi-static deformation is computed from MD simulations with frictional particles.

13.6.1 The Equation of State from ED The mean pressure in two dimensions is p = (σ1 + σ2 )/2, with the eigenvalues σ1 and σ2 of the stress tensor [4, 41, 42]. The 2D dimensionless, reduced pressure P = p/(nT ) − 1 = pV /E − 1 contains only the collisional contribution and the simulations agree nicely with the theoretical prediction P2 = 2νg2 (ν) for elastic systems, with the pair-correlation function g2 (ν) = (1 − 7ν/16)/(1 − ν)2 , and the volume fraction ν = N πa2 /V , see Figure 13.3. A better pair-correlation function is g4 (ν) =

1 − 7ν/16 ν 3 /16 − , (1 − ν)2 8(1 − ν)4

(13.25)

which defines the non-dimensional collisional stress P4 = 2νg4 (ν). For a system with homogeneous temperature, we note that the collision rate is proportional to the dimensionless pressure t−1 n ∝ P.

Figure 13.3: The dashed lines are P4 and Pdense as functions of the volume fraction ν, and the symbols are simulation data, with standard deviations as given by the error bars in the inset. The thick solid line is Q, the corrected global equation of state from Eq. (13.26), and the thin solid line is Q0 without empirical corrections.

312


When plotting P against ν with a logarithmic vertical axis, in Figure 13.3, the simulation results can hardly be distinguished from P2 for ν < 0.65, but P4 leads to better agreement up to ν = 0.67. Crystallization is evidenced at the point of the liquid–solid transition νc ≈ 0.7, and the data clearly deviate from P4 . The pressure is strongly reduced due to the increase of free volume caused by ordering. Eventually, the data diverge at the maximum packing fraction √ νmax = π/(2 3) for a perfect triangular array. For high densities, one can compute from free-volume models, the reduced pressure Pfv = 2νmax /(νmax − ν). Slightly different functional forms do not lead to much better agreement [42]. Based on the numerical data, we propose the corrected high-density pressure Pdense = Pfv h(νmax − ν) − 1, with the empirical fit function h(x) = 1 + c1 x + c3 x3 , and c1 = −0.04 and c3 = 3.25, in perfect agreement with the simulation results for ν ≥ 0.73. Since, to our knowledge, there is no conclusive theory available to combine the disordered and the ordered regime [43], we propose a global equation of state Q = P4 + m(ν)[Pdense − P4 ] ,

(13.26)

with an empirical merging function m(ν) = [1 + exp (−(ν − νc )/m0 )]−1 , which selects P4 for ν νc and Pdense for ν νc , with the transition density νc and the width of the transition m0 . In Figure 13.3, the fit parameters νc = 0.702 and m0 ≈ 0.0062 lead to qualitative and quantitative agreement between Q (thick line) and the simulation results (symbols). However, a simpler version Q0 = P2 + m(ν)[Pfv − P2 ], (thin line) without empirical corrections leads already to reasonable agreement when νc = 0.698 and m0 = 0.0125 are used. In the transition region, this function Q0 has no negative slope but is continuous and differentiable, so that it allows for an easy and compact numerical integration of P . We selected the parameters for Q0 as a compromise between the quality of the fit on the one hand, and the simplicity and treatability of the function on the other. As an application of the global equation of state, the density profile of a dense granular gas in the gravitational field has been computed for mono-disperse [41] and bi-disperse situations [4,42]. In the latter case, however, segregation was observed and the mixture theory could not be applied. The equation of state and also other transport properties are extensively discussed in Refs. [44–47] for 2D, bi-disperse systems.

13.6.2 Quasi-static MD Simulations In contrast to the dynamic, collisional situation discussed in the previous section, a quasi-static situation, with all particles almost at rest most of the time, is discussed in the following. 13.6.2.1 Model Parameters The systems examined in the following contain N = 1950 particles with radii ai randomly drawn from a homogeneous distribution with minimum amin = 0.5 10−3 m and maximum amax = 1.5 10−3 m. The masses mi = (4/3)ρπa3i , with the density ρ = 2.0 103 kg m−3 , are computed as if the particles were spheres. This is an artificial choice and introduces some dispersity in mass in addition to the dispersity in size. Since we are mainly concerned about slow deformation and equilibrium situations, the choice for the calculation of mass should not


313

matter. The total mass of the particles in the system is thus M ≈ 0.02 kg with the typical reduced mass of a pair of particles with mean radius, m12 ≈ 0.42 10−5 kg. If not explicitly mentioned, the material parameters are k2 = 105 N m−1 and γ0 = 0.1 kg s−1 . The other spring-constants k1 and kc will be defined in units of k2 . In order to switch on cohesion, k1 < k2 and kc > 0 is used; if not mentioned explicitly, k1 = k2 /2 is used, and k2 is constant, independent of the maximum overlap previously achieved. Using the parameters k1 = k2 and kc = 0 in Eq. (13.4) leads to a typical contact duration (half-period): tc ≈ 2.03 10−5 s for γ0 = 0, tc ≈ 2.04 10−5 s for γ0 = 0.1 kg s−1 , and tc ≈ 2.21 10−5 s for γ0 = 0.5 kg s−1 for a collision. Accordingly, an integration time-step of tMD = 5 10−7 s is used, in order to allow for a “safe” integration of contacts involving smaller particles. Large values of kc lead to strong cohesive forces, so that more energy can also be dissipated in one collision. The typical response time of the particle pairs, however, is not affected so that the numerical integration works well from a stability and accuracy point of view. 13.6.2.2 Boundary Conditions The experiment chosen is the bi-axial box set-up, see Figure 13.4, where the left and bottom walls are fixed, and stress- or strain-controlled deformation is applied. In the first case a wall is subject to a predefined pressure, in the second case, the wall is subject to a pre-defined strain. In a typical “experiment”, the top wall is strain controlled and slowly shifted downwards while the right wall moves, stress-controlled, dependent on the forces exerted on it by the material in the box. The strain-controlled position of the top wall as a function of time t is here z(t) = zf +

z0 − zf z (1 + cos ωt) , with εzz = 1 − , 2 z0

(13.27)

where the initial and the final positions z0 and zf can be specified, together with the rate of deformation ω = 2πf so that after a half-period T /2 = 1/(2f ) the extremal deformation is reached. In other words, the cosine is active for 0 ≤ ωt ≤ π. For larger times, the top wall is fixed and the system can relax indefinitely. The cosine function is chosen in order to allow for a smooth start-up and finish of the motion so that shocks and inertia effects are reduced. However, the shape of the function is arbitrary as long as it is smooth. The stress-controlled motion of the side wall is described by ¨(t) = Fx (t) − px z(t) − γw x(t) ˙ , mw x

(13.28)

where mw is the mass of the right side wall. Large values of mw lead to slow adaptation, small values allow for a rapid adaptation to the actual situation. Three forces are active: (i) the force Fx (t) due to the bulk material, (ii) the force −px z(t) due to the external pressure, and (iii) a strong frictional force which damps the motion of the wall so that oscillations are reduced. 13.6.2.3 Initial Configuration Initially, the particles are randomly distributed in a huge box, with rather low overall density. Then the box is compressed, either by moving the walls to their desired position, or by defining

314


z

εzz

z( t ) z0

px

zf x

a)

b)

0

T /2

t

Figure 13.4: a) Schematic drawing of the model system. b) Position of the top wall as a function of time for the strain-controlled situation.

an external pressure p = px = pz , in order to achieve an isotropic initial condition. Starting from a relaxed, isotropic initial configuration, the strain is applied to the top wall and the response of the system is examined. In Figure 13.5, snapshots from a typical simulation are shown during compression. εzz = 0

εzz = 0.042

εzz = 0.091

Figure 13.5: Snapshots of the simulation at different εzz for constant side pressure p. The greyscale indicates the average forces on the particles; large and small forces correspond, respectively, to bright and dark, even black, particles.

In the following, simulations are presented with different side pressures p = 20, 40, 100, 200, 400, and 500. The behavior of the averaged scalar and tensor variables during the simulations is examined in more detail for situations with small and large confining pressure. The


315

averages are performed such that ten to twenty percent of the total volume is disregarded in the vicinity of each wall in order to avoid boundary effects. A particle contact is taken into account for the average if the contact point lies within the averaging volume V . 13.6.2.4 Compression and Dilation The first quantity of interest is the density (volume fraction) ν and, related to it, the volumetric strain εV = ∆V /V . From the averaged data, we find compression for small deformation and large side pressure. This initial regime follows strong dilation, for all pressures, until a quasisteady-state is reached, where the density is almost constant besides a weak tendency towards further dilation. An initially dilute granular medium (weak confining pressure) thus shows dilation from the beginning, whereas a denser granular material (strong confining pressure) can be compressed even further by the relatively strong external forces until dilation starts. The range of density change is about 0.02 in volume fraction and spans up to 3 % change in volumetric strain.

a)

b)

P Figure 13.6: a) Volume fraction ν = i πa2i /V for different confining pressure p. b) Volumetric strain – negative values mean compression, whereas positive values correspond to dilation.

From the initial slope, one can obtain the Poisson ratio of the bulk material, and from the slope in the dilatant regime, one obtains the so-called dilatancy angle, which is a measure of the magnitude of dilatancy required before shear is possible [48, 49]. 13.6.2.5 Stress Tensor The sums of the normal and the tangential stress contributions are displayed in Figure 13.7 for two side-pressures p = 20 and p = 200. The lines show the stress measured on the walls, and the symbols correspond to the stress measured via the micro–macro average in Eq. (13.23), proving the reasonable quality of the micro–macro transition as compared to the wall stress “measurement”.

316


There is also other macroscopic information hidden in the stress–strain curves in Figure 13.7. From the initial, rapid increase in stress, one can determine the compressibility of the bulk material. Later, the stress reaches a peak at approximately 2.6p and then saturates at about 2p. From both the peak and saturation stresses, one obtains the yield stresses at peak and in critical state flow, respectively [50].

a)

b)

Figure 13.7: Total stress tensor σ = σ n + σ t for a) small and b) high pressure – the agreement between the wall pressure and the averaged stress is almost perfect.

Note that for the parameters used here, both the dynamic stress and the tangential contributions to the stress tensor are more than one order of magnitude smaller than the normal contributions. As a cautionary note, we remark also that the artificial stress induced by the background viscous force is negligible here (about two percent), when γb = 10−3 kg s−1 and a compression frequency f = 0.1 s−1 are used. For faster compression with f = 0.5 s−1 , one obtains about ten percent contribution to stress from the artificial background force.

13.7 Large-scale Computational Examples In this section, several examples of rather large particle numbers simulated using MD and ED are presented. The ED algorithm is first used to simulate a freely cooling, dissipative gas in two and three dimensions [30, 35]. Then, a peculiar three-dimensional ring-shear experiment is modeled with soft sphere MD.

13.7.1 Cluster Growth (ED) In the following, a two-dimensional system of length L = l/d = 560 with N = 99856 dissipative particles of diameter d = 2a is examined [28, 35], with volume fraction ν = 0.25 and restitution coefficient r = 0.9. This 2D system is compared to a three-dimensional system

13.7 Large-scale Computational Examples

317

of length L = l/d = 129 with N = 512000 dissipative spheres of diameter d and volume fraction ν = 0.25 with r = 0.3 [30]. 13.7.1.1 Initial Configuration Initially the particles are arranged on a square lattice with random velocities drawn from an interval with constant probability for each coordinate. The mean total velocity, i.e., the random momentum due to the fluctuations, is eliminated in order to have a system with its center of mass at rest. The system is allowed to evolve for some time, until the arbitrary initial condition is forgotten, i.e., the density is homogeneous, and the velocity distribution is a Gaussian in each coordinate. Then dissipation is switched on and the evolution of the system is reported for the selected r. In order to avoid inelastic collapse, the TC model is used, which reduces dissipation if the time between collisions drops below a value of tc = 10−5 s.

a)

b)

Figure 13.8: a) Collision frequency of individual particles from a 2D simulation, after about 5200 collisions per particle. b) Cluster visualization from a 3D simulation. The greyscale indicates large (bright) and small (dark) collision rates; white corresponds to no particles at all.

13.7.1.2 System Evolution For the values of r used here, the system becomes inhomogeneous quite rapidly [30, 35]. Clusters, and thus also dilute regions, build up and have the tendency to grow. Since the system is finite, their extension will reach system size at a finite time. Thus we distinguish between three regimes of system evolution: (i) the initially (almost) homogeneous state, (ii) the cluster growth regime, and (iii) the system size dependent final stage where the clusters have reached system size. We note that a cluster does not behave like a solid body, but has internal motion and can eventually break into pieces after some time. These pieces (small clusters) collide and can merge to larger ones. In Figure 13.8, snapshots are presented and the collision rate is grey-scaled. The collision rate and the pressure are higher inside the clusters than at their surface. Note that most of the computational effort is spent in predicting collisions and to compute the velocities after

318


the collisions. Therefore, the regions with the largest collision frequencies require the major part of the computational resources. Due to the TC model, this effort stays limited and the simulations can easily continue for many thousands of collisions per particle.

13.7.2 3D Ring-shear Cell Simulation The simulation in this section models a ring-shear cell experiment, as recently proposed [51, 52]. The interesting observation in the experiment is a universal shear zone, initiated at the bottom of the cell and becoming wider and moving inwards while propagating upwards in the system. 13.7.2.1 Model System The numerical model chosen here is MD with smooth particles in three dimensions. In order to save computing time, only a quarter of the ring-shaped geometry is simulated. The walls are cylindrical, and are rough on the particle scale, due to some attached particles. The outer cylinder wall with radius Ro , and part of the bottom r > Rs are rotating around the symmetry axis, while the inner wall with radius Ri , and the attached bottom-disk r < Rs remain at rest. In order to resemble the experiment, the geometry data are Ri = 0.0147 m, Rs = 0.085 m, and Ro = 0.110 m. Note that the small Ri value is artificial, but it does not affect the results for small and intermediate filling heights. The slit in the bottom wall at r = Rs triggers a shear band. In order to examine the behavior of the shear band as a function of the filling height H, this system is filled with 6000 to 64000 spherical particles with mean radius 1.0 mm and radii range 0.5 mm < a < 1.5 mm, which interact here via repulsive and dissipative forces only. The particles are forced towards the bottom by the gravity force f g = mg here and are kept inside the system by the cylindrical walls. In order to provide some wall roughness, a fraction of the particles (about 3 percent) that are originally in contact with the walls are glued to the walls and move with them. 13.7.2.2 Material and System Parameters The material parameters for the particle–particle and particle–wall interactions are k = 102 N/m and γ0 = 2.10−3 kg/s. Assuming a collision of the largest and the smallest particle used, the reduced mass m12 = 2.94 10−6 kg, leads to a typical contact duration tc = 5.4 10−4 s and a restitution coefficient of r = 0.83. The integration time-step is tMD = 5.10−6 s, i.e., two orders of magnitude smaller than the contact duration. The simulations run for 25 s with a rotation rate fo = 0.01 s−1 of the outer cylinder, with angular velocity Ωo = 2πfo . For the average of the displacement, only times t > 10 s are taken into account. Within the averaging accuracy, the system seemingly has reached a quasisteady state after about 8 s. The empty cell is shown in Figure 13.9, while three realizations with different filling heights are displayed in Figure 13.10, from a top and front view. 13.7.2.3 Shear Deformation Results From the top view, it is evident that the shear band moves inwards with increasing filling height, and it also becomes wider. From the front view, the same information can be evi-

13.7 Large-scale Computational Examples

319

b)

a)

Figure 13.9: Snapshots from the quarter-cylinder geometry. Visible here are only those particles glued to the wall; the cylinder and slit positions are indicated by the lines. a) Top view and b) front view. Dark and bright particles are at rest or moving, respectively.

denced and, in addition, the shape of the shear band inside the bulk is visible. The inwards displacement happens deep in the bulk and the position of the shear band does not change a lot closer to the surface. In order to allow for a more quantitative analysis of the shear band, both on the top and as a function of depth, we perform fits with the universal shape function proposed in [51]: vϕ (r) r − Rc , = A 1 + erf rΩo W

(13.29)

where A is a dimensionless amplitude A = 0.50 ± 0.02, Rc is the center of the shear band, and W is its width. The fits to the simulations qualitatively confirm the experimental findings insofar as the center of the shear band, as observed on top of the material, see Figure 13.11, moves inwards with an Rc ∝ H 5/2 behavior, and that the width of the shear band increases almost linearly with H. For filling heights larger than H ≈ 0.05 m, deviations from this behavior are observed, because the inner cylinder is reached and thus sensed by the shear band. Slower shearing does not affect the center, but slightly reduces the width – as checked by one simulation. As in the experiments, the behavior of the shear band within the bulk, see Figure 13.12, deviates qualitatively from the behavior seen from the top. Instead of a slow motion of the shear band center inwards, the shear band rapidly moves inwards at small distances h, and reaches a saturation distance with small change closer to the surface. Again, a slower rotation does not affect the center but reduces the width.

320


N = 16467

N = 34518

N = 60977

Figure 13.10: Snapshots from simulations with different filling heights seen from the top and from the front; the particle number N is given in the inset. The dark-bright-dark pattern (from left-bottom to right-top) denotes particles with rdφ ≤ 0.5 mm, rdφ ≤ 2 mm, rdφ ≤ 4 mm, and rdφ > 4 mm, i.e., the displacement in tangential direction per second, respectively. The filling heights in these simulations are H = 0.018 m, 0.037 m, and 0.061 m (from left to right).

a)

b)

Figure 13.11: a) Distance of the top-layer shear band center from the slit, plotted against the filling height H. The open symbols are simulation results, the solid symbol is a simulation with slower rotation fo = 0.005 s−1 , and the line is a fit with constant cR = 30. b) Width of the shear band from the same simulations; the line is a fit with cW = 2/5.

13.8 Conclusion

a)

321

b)

Figure 13.12: (Left) Distance of the bulk shear band center from the slit and, (Right) width of the shear band, both plotted against the height h. The open symbols are simulation results obtained with fo = 0.01 s−1 , the solid symbols are obtained with slower rotation fo = 0.005 s−1 . Squares, circles and triangles correspond to the filling heights H = 0.037 m, 0.049 m, and 0.061 m, respectively. The curves are identical to those plotted in Figure 13.11.

13.7.2.4 Discussion In summary, the example of a ring shear cell simulation in 3D has shown that, even without the more complicated details of fancy interaction laws, experiments can be reproduced at least qualitatively. A more detailed study of quantitative agreement has been performed in 2D [15], and is in progress for the 3D case. A challenge for the future remains the micro– macro transition in 3D and the continuum theory formulation of the shear band problem.

13.8 Conclusion This chapter has given a summary of the most important details about soft-particle molecular dynamics (MD) and hard-particle event-driven (ED) simulations, together with an attempt to link the two approaches in the dense limit where multi-particle contacts become important. As an example for a micro–macro transition, the stress tensor was defined and computed for dynamic and quasi-static systems, using ED and MD, respectively. This led, for example, to a global equation of state, valid for all attainable densities, and also to the partial stresses due to normal and tangential (frictional) contacts. For the latter situation, the micro–macro average is compared to the macroscopic stress (=force/area) measurement, with reasonable agreement. In the last section, some examples of larger simulations were provided in order to illustrate applications at the front of ongoing research, these are cluster formation, or shear band formation. In conclusion, molecular dynamics methods have proven to be a helpful tool for the understanding of granular systems. The qualitative approach of the early years has now developed into the attempt of a quantitative predictive modeling of the diverse modes of complex behavior in granular media. The achievement of this goal will be a research challenge for the next

322


few decades, involving enhanced kinetic theories for dense collisional flows and elaborate constitutive models for quasi-static, dense systems with shear band localization.

Acknowledgments We acknowledge the financial support of several funding institutions that supported this research, and also helpful discussions with the many persons who contributed to these results.

References [1] P. A. Vermeer, S. Diebels, W. Ehlers, H. J. Herrmann, S. Luding, and E. Ramm, editors. Continuous and Discontinuous Modelling of Cohesive Frictional Materials, Berlin, 2001. Springer. Lecture Notes in Physics 568. [2] T. Pöschel and S. Luding, editors. Granular Gases, Berlin, 2001. Springer. Lecture Notes in Physics 564. [3] J. G. Kirkwood, F. P. Buff, and M. S. Green. The statistical mechanical theory of transport processes. J. Chem. Phys., 17(10):988, 1949. [4] S. Luding, M. Lätzel, and H. J. Herrmann. From discrete element simulations towards a continuum description of particulate solids. In A. Levy and H. Kalman, editors, Handbook of Conveying and Handling of Particulate Solids, pages 39–44, Amsterdam, The Netherlands, 2001. Elsevier. [5] M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, 1987. [6] D. C. Rapaport. The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge, 1995. [7] M. Lätzel, S. Luding, and H. J. Herrmann. Macroscopic material properties from quasi-static, microscopic simulations of a two-dimensional shear-cell. Granular Matter, 2(3):123–135, 2000. cond-mat/0003180. [8] P. A. Cundall and O. D. L. Strack. A discrete numerical model for granular assemblies. Géotechnique, 29(1):47–65, 1979. [9] Y. M. Bashir and J. D. Goddard. A novel simulation method for the quasi-static mechanics of granular assemblages. J. Rheol., 35(5):849–885, 1991. [10] S. van Baars. Discrete Element Analysis of Granular Materials. PhD thesis, Technische Universiteit Delft, Delft, Nederlands, 1996. [11] H. J. Herrmann and S. Luding. Modeling granular media with the computer. Continuum Mechanics and Thermodynamics, 10:189–231, 1998. [12] C. Thornton. Numerical simulations of deviatoric shear deformation of granular media. Géotechnique, 50(1):43–53, 2000. [13] C. Thornton and S. J. Antony. Quasi-static deformation of a soft particle system. Powder Technology, 109(1–3):179–191, 2000. [14] C. Thornton and L. Zhang. A DEM comparison of different shear testing devices. In Y. Kishino, editor, Powders & Grains 2001, pages 183–190, Rotterdam, 2001. Balkema.

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[33] S. Luding, M. Huthmann, S. McNamara, and A. Zippelius. Homogeneous cooling of rough dissipative particles: Theory and simulations. Phys. Rev. E, 58:3416–3425, 1998. [34] R. Cafiero O. Herbst and S. Luding. A mean field theory for a driven dissipative gas of frictional particles. preprint, 2004. [35] S. Luding and H. J. Herrmann. Cluster growth in freely cooling granular media. Chaos, 9(3):673–681, 1999. [36] P. K. Haff. Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech., 134:401–430, 1983. [37] J. T. Jenkins and M. W. Richman. Kinetic theory for plane shear flows of a dense gas of identical, rough, inelastic, circular disks. Phys. of Fluids, 28:3485–3494, 1985. [38] S. McNamara and W. R. Young. Inelastic collapse in two dimensions. Phys. Rev. E, 50(1):R28–R31, 1994. [39] S. Luding, E. Clément, J. Rajchenbach, and J. Duran. Simulations of pattern formation in vibrated granular media. Europhys. Lett., 36(4):247–252, 1996. [40] S. Luding. Surface waves and pattern formation in vibrated granular media. In Powders & Grains 97, pages 373–376, Amsterdam, 1997. Balkema. [41] S. Luding, M. Lätzel, W. Volk, S. Diebels, and H. J. Herrmann. From discrete element simulations to a continuum model. Comp. Meth. Appl. Mech. Engng., 191:21–28, 2001. [42] S. Luding. Liquid–solid transition in bi-disperse granulates. Advances in Complex Systems, 4(4):379–388, 2002. [43] H. Kawamura. A simple theory of hard disk transition. Prog. Theor. Physics, 61:1584– 1596, 1979. [44] M. Alam, J. T. Willits, B. O. Arnarson, and S. Luding. Kinetic theory of a binary mixture of nearly elastic disks with size and mass-disparity. Physics of Fluids, 14(11):4085– 4087, 2002. [45] M. Alam and S. Luding. How good is the equipartition assumption for transport properties of a granular mixture? Granular Matter, 4(3):139–142, 2002. [46] M. Alam and S. Luding. Rheology of bidisperse granular mixtures via event-driven simulations. J. Fluid Mech., 476:69–103, 2003. [47] M. Alam and S. Luding. First normal stress difference and crystallization in a dense sheared granular fluid. Phys. Fluids, 15(8):2298–2312, 2003. [48] S. Luding and H. J. Herrmann. Micro–macro transition for cohesive granular media. in: Bericht Nr. II-7, Inst. für Mechanik, Universität Stuttgart, S. Diebels (Ed.), 2001. [49] S. Luding. Micro–macro transition for anisotropic, periodic, elastic solids. Int. J. Sol. Struct. 41, 5821–5836, 2004. [50] J. Schwedes. Review on testers for measuring flow properties of bulk solids. Granular Matter, 5(1):1–45, 2003. [51] D. Fenistein and M. van Hecke. Kinematics – wide shear zones in granular bulk flow. Nature, 425(6955):256, 2003. [52] D. Fenistein, J. W. van de Meent, and M. van Hecke. Universal and wide shear zones in granular bulk flow. cond-mat/0310409, 2003.

14 Contact Dynamics for Beginners Lothar Brendel, Tamas Unger, and Dietrich E. Wolf

14.1 Introduction For the dynamical properties of dense granular media where lasting contacts dominate, steric hindrance and solid friction play a crucial role. Examples are the withdrawal of material from a silo, the compaction of powders, imprinting of one’s foot on a beach, or the stability of an ancient vault in an earthquake. For sufficiently rigid materials, elastic or plastic deformation of the particles can be so small in these processes that they can be safely neglected. What matters is the rearrangement of rigid particles. Contact dynamics is a simulation method that was developed to deal with rigid, frictional particles. The purpose of this article is threefold: It contains a description of the contact dynamics simulation method, it discusses when this method is more efficient than molecular dynamics, and finally it describes how the basic algorithm can be extended to simulate cohesive powders. In the first two parts, cohesion is largely considered to be negligible, but this is not true for fine powders (particle diameters of about 1 µm and smaller), nor for wet sand. Contact Dynamics (CD) is a discrete element method like Molecular Dynamics (MD), i.e., the equations of motion are integrated for each particle. However, by considering the particles as perfectly rigid, contact dynamics suppresses phenomena caused by particle deformation. It represents the deformation of the granular medium as a whole, in an idealized way, exclusively by particle rearrangements. Obviously the volume exclusion of perfectly rigid particles is a constraint that is formulated as an inequality: the distance between the particle surfaces (“gap”) must be larger than or equal to zero. Such constraints are called unilateral. They are only active if the gap is zero, and otherwise have no effect. Therefore, the number of degrees of freedom in the system depends on the number of contacts (more precisely: active constraints) and is itself a dynamical variable, which explains the name “contact dynamics”. By contrast, in soft-particle MD as well as in event-driven MD, the number of degrees of freedom does not change with time. Imposing constraints requires implicit forces (constraint forces) which cannot be calculated from the positions and velocities of the particles alone. The constraint forces are determined so as to compensate all forces that would cause constraint-violating accelerations. The volume exclusion constraint allows only complementary values of gap g and constraint force Rn , which is normal to the tangent surface at the contact point. Their product must be zero, gRn = 0. This is expressed by the Signorini graph, on the left of Figure 14.1. As long as g > 0 the constraint is not active, hence Rn = 0. If g = 0, the constraint force must prevent interpenetration of the particles. Hence it must be repulsive and can take whatever non-negative value is needed for this purpose.

326

14 Contact Dynamics for Beginners

t |/Rn 6 |R

6 Rn µs µd

-|Vt |

-g a)

b)

Figure 14.1: Volume exclusion constraint a): Gap g and constraint force R n are complementary to each other (Signorini graph). Non-sliding constraint b): The constraint force (static friction force) plus the sliding friction force constitute the Coulomb graph. In this paper the static and dynamic friction coefficients are assumed to be equal, µs = µd = µ.

In addition to volume exclusion we have to deal with a second type of constraint, the non t| sliding constraint of frictional contacts. It is only active if the tangential relative velocity |V is zero. In this case the static friction force can be non-zero and assumes whatever tangential t | ≤ µs Rn are needed to prevent sliding. If a constraint force direction and value 0 ≤ |R t = outside this Coulomb cone would be needed, sliding cannot be avoided, and one obtains V 0 and the well-defined sliding friction −µd Rn Vt /|Vt |. The absolute values of the tangential velocity and the friction force lie on the Coulomb graph (Figure 14.1b). Although both graphs have infinitely steep parts they can be implemented in the CD method without any change, in contrast to MD. The CD technique can handle rigid particles and static frictional contacts without regularizing the graphs, Figure 14.1. Hence it is able to overcome some difficulties that arise in soft-particle molecular dynamics (MD) (see Stefan Luding’s contribution in this book) or in event-driven simulations [11, 17]. Algorithms for contact dynamics were already developed in the 1980s [1, 16]. In the context of granular media they were made known to a wider physics community by Jean and Moreau [12, 13, 18]. Two recent reviews were given by Schwager and Pöschel [24] and by Unger and Kertész [26]. The following sections closely follow the presentation given in [25, 26].

14.2 Discrete Dynamical Equations Collisions of rigid particles give rise to discontinuous velocities during the time-evolution. In such non-smooth mechanics the use of second or higher-order schemes for the numerical integration of the motion is not beneficial and could even be problematic. Therefore first-order schemes are applied, e.g., an implicit Euler integration in our CD code: 1 Fi (t + ∆t) ∆t . mi ri (t + ∆t) = ri (t) + vi (t + ∆t) ∆t

vi (t + ∆t) = vi (t) +

(14.1) (14.2)

14.3 Volume Exclusion in a One-dimensional Example

327

The two equations describe the change of velocity and center of mass position during one timestep for the ith particle. The vector Fi denotes the sum of the forces acting on the particle and is calculated in each step such that the constraints remain fulfilled. The time-stepping is similar for the rotational degrees of freedom. The orientation of a i we particle is updated with the new angular velocity ωi (t + ∆t), while for the update of ω use the torque Ti (t + ∆t) exerted by the contact forces.

14.3 Volume Exclusion in a One-dimensional Example Before we describe the three-dimensional implementation of contact dynamics, the structure of the algorithm will be explained with the simplest possible example, the central collision of two non-rotating equal spheres, labeled i = 1 or 2,with zero restitution coefficient (see Figure 14.2). In this one-dimensional example, only the volume exclusion constraint occurs, and the constraint force has only one component, R. As the particles only interact if they are in contact, it is important to keep a list of existing and incipient contacts, i.e., contacts that may form during the next time-step. With each of these contacts one can associate a relative velocity V = dg/dt which is zero for closed contacts, negative for incipient contacts, and positive for particles that move away from each other. For the one-dimensional example it is trivial to connect the contact-related quantities, V and R, to the particle velocities v1 and v2 and the interaction forces R1 and R2 experienced by the particles: v1 , (14.3) V = v2 − v1 = (−1, 1) · v2 R1 −1 = R. (14.4) 1 R2 Equation (14.4) is simply the action–reaction principle. and the constraint force R have a normal as In three dimensions, the relative velocity V well as a tangential component for each contact. We shall see below that they are related to the velocities and angular velocities, and the interaction forces and torques, respectively, by a straightforward generalization of the linear relations (14.3) and (14.4). Newton’s equation of motion relates the particle acceleration to the sum of the interaction force Ri and a possible external force Fiext : d dt

v1 v2

1 = m

R1 R2

+

F1ext F2ext

.

(14.5)

The task is to calculate the interaction forces Ri such that the acceleration will not lead to a violation of the volume exclusion constraint. For example, if both particles are already in contact, their relative velocity must remain zero, i.e., Ri + Fiext must be the same for both particles. This is borne out most easily by transforming Newton’s equations (14.5) into an equation of motion “of the contact”, i.e., of the relative velocity, by using Eqs. (14.3) and

328


½

¾

Figure 14.2: Central collision between two non-rotating equal spheres: A simple example for an incipient contact.

(14.4): dV 1 = (−1, 1) · dt m

−1 1

R +

F1ext F2ext

=

1 dVfree R + . M dt

In this equation, M = m/2 is the reduced mass of the two particles, and ext 1 1 dVfree F1 = (−1, 1) · = (F2ext − F1ext ) ext F dt m m 2

(14.6)

(14.7)

would be the relative acceleration without any interaction of the particles. Solving Eq. (14.6) for R in the Euler scheme (14.1) gives the constraint force for the new time-step, Rnew = M

Vnew − Vfree,new , ∆t

(14.8)

as a linear function of the relative velocity for the new time-step, Vnew . Both are unknown and will be determined simultaneously from the constraint conditions. Here and in the following the superscript “new” refers to the value at time t + ∆t, while values of g, V and R without this superscript are taken at time t. Note that in the one-contact case worked out here Vfree,new = V +

1 ext F2 − F1ext ∆t m

(14.9)

is known. In addition to Eq. (14.8) one needs the constraint in order to determine the two unknowns, Vnew and Rnew . Three conditions must be fulfilled: • volume exclusion, g new = g + Vnew ∆t ≥ 0, • contact condition, g new Rnew = 0, • non-cohesiveness (constraint forces purely repulsive), Rnew ≥ 0. This means that the set of allowed pairs (Vnew , Rnew ) is the Signorini graph shown in Figure 14.3. Its intersection with the linear relation (14.8) determines both values simultaneously. Obviously one gets g (14.10) Vnew = max − , Vfree,new , ∆t

14.4 The Three-dimensional Single Contact Case Without Cohesion

329

6 Rnew n M ∆t

−g ∆t

-

Vnew

Vfree,new

Figure 14.3: The constraint force R new and the relative velocity V new for the new time-step are related by a Signorini graph (bold line) and by the linear equation of motion (14.8) (dashed line). The intersection of the two graphs determines both values simultaneously.

and new

R

M g free,new +V = max 0, − ∆t ∆t M g 1 + V + (F2ext − F1ext )∆t . = max 0, − ∆t ∆t m

(14.11)

This solves the task of calculating the interaction forces Ri , Eq. (14.4), for Newton’s equations of motion, Eq. (14.5). The next section contains the generalization of this to three dimensional space as an easy reference for those who want to write a CD-program. It can be skipped, if one is not interested in the practical algorithmic questions.

14.4 The Three-dimensional Single Contact Case Without Cohesion We consider a pair of rigid particles already in contact or with a small gap between them. They are numbered 1 and 2 and are subject to constant external forces F1ext , F2ext acting on the centers of mass (Figure 14.4). Their restitution coefficient is assumed to be zero. Volume at this contact, where we use exclusion and Coulomb friction may require a constraint force R acts on particle 1. In this the convention that R acts on particle 2 while its reaction force −R section we will show how R is calculated. Each particle has six degrees of freedom, three translational and three rotational. Accordingly the equations of motion for particle i involve two three-component vectors, the center i . The conof mass velocity, vi , and the angular velocity with respect to the center of mass, ω enters the equations of motion for the particle degrees of freedom in terms of straint force R i and interaction torques Ti , interaction forces R , 1 = −R R

, 2 = R R

, T1 = −l1 × R

, T2 = l2 × R

(14.12)

330


Figure 14.4: Two rigid particles with an incipient contact.

where the vectors l1 and l2 point from the centers of mass to the expected contact point. (For general particle shapes there may be more than one expected contact point.) It is useful to introduce generalized velocity and force vectors:      ext  1 R F1 v1     0   ω T   1 1 ext   , F = V= (14.13)  v2  , R =  F ext  , 2  R 2 ω2 0 T2 where R contains the interaction forces and torques, while Fext contains the external forces (external torques are not taken into account here). and As in Eqs. (14.4) and (14.3) the contact quantities R = v2 + ω (14.14) 2 × l2 − v1 + ω1 × l1 , V are linearly related to the corresponding generalized vectors: R = HR T = H V, V

(14.15) (14.16)

where HT is the transpose of the matrix H. These two matrices (defined by Eqs. (14.14) and (14.12)) describe the geometry and allow to transform contact quantities into particle quantities and vice versa. The equations of motion for the two particles read:   m1 1 0 0 0  0 dV 0 0 I1  . = M−1 (R + Fext ) , M= (14.17)  1 0 0 0 m dt 2 0 0 0 I2 M−1 is the inverse of the generalized 12 × 12 mass matrix M, which contains the masses and the matrices of the moments of inertia of the particles (1 denotes the 3 × 3 unit matrix). In Eq. (14.17), we neglected a term including the inverse of dM/dt, which takes care of the change of the Ii due to rotation (and therefore is absent for spheres). For a slowly deforming

14.4 The Three-dimensional Single Contact Case Without Cohesion

331

granular system, this contribution of higher order in ω can be neglected, though. (We will make a similar approximation again in the next paragraph.) and hence the particle interaction R, In order to determine the constraint force R by apEq. (14.15), one transforms Eq. (14.17) into an equation for the relative velocity V T T plying H (cf. Eq. (14.6)) (note that the term (dH /dt) V describing the geometrical change is neglected here, which is typically a good approximation) : free dV ˆ −1 R + dV =M , dt dt free dV ˆ −1 = HT M−1 H . = HT M−1 Fext , and M dt

(14.18) (14.19)

free /dt has the meaning of the acceleration without any interaction between the particles, dV ˆ denotes the reduced-mass matrix of the contact, which replaces the reduced mass in and M ˆ −1 acts in a simple the special case considered in the previous section. It can be shown that M way for contacting spheres and can be characterized by two parameters mn and mt (normal and tangential mass respectively): = ˆ −1 R M 1 mn

=

1 1 Rt , Rnn + mn mt 1 1 + , m1 m2

l 2 l 2 1 1 = + 1 + 2 . mt mn I1 I2

(14.20) (14.21)

Here the moments of inertia (I1 and I2 ) are numbers and n denotes the normal unit vector (perpendicular to the tangent plane), which points from particle 1 towards particle 2. Eq. (14.20) shows that normal and tangential components are not coupled for spheres, which is not true in general. As in Eq. (14.8) we solve Eq. (14.18) for the contact force and obtain new free,new ˆ V −V new = M , R ∆t

(14.22)

where according to Eq. (14.19) free,new = V + HT M−1 Fext ∆t V

(14.23)

has the meaning of the new velocity if there was no interaction. Now the volume exclusion new , which completes the calculation of the and non-sliding constraints are used to determine V constraint forces (14.22). This is a bit more complicated than in the one-dimensional case and is done in three steps in the algorithm. 1. First we check whether the gap g remains positive after the time-step ∆t, if the interaction between the particles is not taken into account, i.e., whether g + Vnfree,new ∆t > 0 .

(14.24)

332


free,new . The normal component of the relative velocity is given by Vnfree,new = nT · V The normal vector n is parallel to the shortest connection between the surfaces of the two particles.1 If condition (14.24) is fulfilled, the incipient contact did not close during new = V free,new . If the new = 0, and V the time step so that the contact force is zero, R left-hand side of Eq. (14.24) is zero or negative, the algorithm continues with the second step. 2. In this step the algorithm makes an attempt to establish a non-sliding contact, i.e., we require, on one hand, that the gap closes: g + Vnew n ∆t = 0 ,

(14.25)

on the other hand, that no slip occurs: new = 0 . V t

(14.26)

new = −(g/∆t)n. This determines the contact force Therefore, the new velocity is V Eq. (14.22): ˆ g n + V free,new . new = − 1 M (14.27) R ∆t ∆t However, this contact force can only be accepted if it lies within the Coulomb cone t | ≤ µRn . If this does not hold, we have to give up the assumption of a non-sliding |R new is recalculated in the third step. contact. Then the contact will be a sliding one and R 3. For a sliding contact the condition (14.25) remains valid, but Eq. (14.26) does not. Then new from the following condition: the tangenVnew must be determined together with R t t tial part of ˆ g n − V new + V new = − 1 M free,new R (14.28) t ∆t ∆t must be equal to sliding friction, i.e., new new = −µRnew Vt R . t n new | |V

(14.29)

t

new and the tangential comThere are only three unknowns: the normal component of R new . The three equations (14.28) (one for each component) determine these ponents of V unknowns. These three points form a contact law that in general provides the contact force in every time-step. It can be applied for colliding particles, but also for pre-existing contacts. In this 1

This is unique for convex particles. In special cases, e.g., for polygons in two dimensions, a planar contact may form which is modeled by two or more point contacts. Then one should not only consider the shortest connection, but at the same time all other contact candidates.

14.5 Iterative Determination of Constraint Forces in a Multi-contact System

333

sense no distinction has to be made. Note that this contact law corresponds to a completely inelastic collision, i.e., to zero value of the normal restitution coefficient. To accomplish such a collision, two time-steps are needed by this scheme. In the first time step the normal relative velocity is only reduced, but it is not set to zero, in order to let the gap close and then in the following time-step the relative normal velocity vanishes completely. For practical reasons, a slight change is recommended in the contact law [12], that is the application of g pos = max (g, 0) instead of g in Eqs. (14.27) and (14.28). This, in principle, makes no difference because g should be non-negative. However, due to inaccurate calculations some small overlaps can be created between neighboring particles. These overlaps would be immediately eliminated by the first version of the inelastic contact law by applying larger repulsive force in order to satisfy Eq. (14.25). This self-correcting mechanism, nonetheless, has the non-negligible drawback that it pumps kinetic energy into the system, when thrusting the overlapping particles away from each other. With the application of g pos one avoids this. Moreover an already existing overlap is not eliminated, only its further growth is inhibited. This can be used to monitor the numerical inaccuracies of a CD-simulation. For spherical particles the inelastic contact law simplifies, because the reduced-mass matrix M is diagonal for spheres. The three steps are then: if Vfree n ∆t + g pos > 0 new = 0 then R

if

pos  1 g  free  Rnew m + V = − n n n ∆t ∆t else 1   R free new = − mt V t t ∆t new new Rt > µRn   new new = µRnew Rt then R n  t new R t

(no contact)

(sticking contact)

(sliding contact) (14.30)

Note that, for a sliding contact, the recalculation of Rn is not necessary in this special case. Simulations may also involve certain confining objects (e.g., container, fixed wall, moving piston, rotating drum). Therefore the algorithm has to be able to handle not only spheresphere contacts, but also sphere–plane and sphere–cylinder contacts. One can easily verify that if planes and cylinders with infinite moments of inertia are used (I2 = ∞), the same simple contact law can be applied as the one derived here for spheres.

14.5 Iterative Determination of Constraint Forces in a Multi-contact System So far we have only discussed how to treat a single incipient or existing contact in the framework of contact dynamics. However, the most interesting applications of CD involve dense

334


11 00 1 0 11 00

00 ½ 11 1 ·½0 0 1 · ¾ 00 11 110 00 1 Ê ½ Ê Ê ·½

Figure 14.5: A one-dimensional array of spheres in contact.

granular media, where many particles interact simultaneously within a contact network that may span a substantial part of the whole system. A simple one-dimensional example is given in Figure 14.5. Let us assume, that none of the contacts has newly formed in the last time-step, so that all gaps gi and all relative velocities Vi are zero, but that the whole array will be accelerated or perhaps disrupted by some external forces acting only on some far away particles of the chain which are not depicted in Figure 14.5. Eq. (14.11) can be used for the calculation of the constraint force at the ith contact, but the role of the external forces is now played by the constraint forces of the ext by Rnew neighboring contacts. Replacing F2ext by −Rnew i+1 and F1 i−1 in Eq. (14.11) one obtains = Rnew i

1 new Ri−1 + Rnew i+1 , 2

(14.31)

where we used the fact that the reduced mass is M = m/2 in this simple case and that the right-hand side of Eq. (14.31) is ≥ 0. This is a discretized Laplace equation which couples all constraint forces in the contact network. This example shows that using constraint forces has a serious consequence. A contact force depends also on adjacent contact forces that press the two particles together. Thus, for a compressed cluster of rigid particles, the contact forces cannot be computed locally. This is a natural consequence of perfect rigidity. As the velocity of sound is infinite, a collision can immediately affect forces even very far away. Whereas, in the simple one-dimensional example of Eq. (14.31) the exact calculation of globally consistent constraint forces is feasible, it becomes exceedingly cumbersome for large, complex, three-dimensional contact networks. There may even be more than one solution satisfying all constraints [21, 27]. Different algorithms have been used to determine globally consistent constraint forces (e.g. [21, 24]), but in general one uses an iterative scheme (called the iterative solver). It is applied in every timestep before the implicit Euler integration can proceed one step further with the newly provided forces. This method works as follows. At each iteration step we update every contact independently in the sense that, for one existing or incipient contact, a “new” contact force is calculated based on the contact law for the one-contact case, presuming that the current forces of the neighboring contacts were already the correct ones. The resulting force is stored for the given contact and a new contact is chosen for the next update. In that way all the contact forces are updated one by one, sequentially. Of course, one update per contact (i.e., one iteration step) does not yet provide a global solution. Such iteration steps have to be repeated many times letting the forces relax, according to their neighborhood, towards a globally consistent state.

14.5 Iterative Determination of Constraint Forces in a Multi-contact System

335

After satisfactory convergence is reached, the iteration loop can be stopped. By convergence, we mean that further update of the contact forces gives only negligible changes, thus the constraint conditions are practically fulfilled for the whole system. The applied number of iterations NI within one time-step depends on the accuracy of the convergence criterion [2,26]. Higher NI provide more accurate forces but require more computational effort. As an example let us return to the one-dimensional case, Eq. (14.31). If one associates a virtual time-step ∆t∗ = ∆t/NI with each of the NI iteration steps performed within a single real time-step ∆t of the simulation, the forces relax towards a consistent solution of the Eqs (14.31) according to NI Ri (t∗ + ∆t∗ ) − Ri (t∗ ) (Ri+1 (t∗ ) − 2Ri (t∗ ) + Ri−1 (t∗ )) . = ∆t∗ 2∆t

(14.32)

The change of Ri per iteration step is equal to the difference between the left and right hand side of the consistency equation (14.31). The virtual time evolution (14.32) is simply a discretized one-dimensional diffusion equation [25] with diffusion constant D ∝ NI

r2 , ∆t

(14.33)

where r is the particle radius. Also in three dimensions, the force consistency with the constraints spreads diffusively during the iteration. For a system of linear size L convergence requires D∆t > L2 ∼ (N 1/d r)2 , where N is the number of particles in the system, which is assumed to be connected throughout, and d is the space dimension. This implies NI > N 2/d .

(14.34)

The number of iterations needed to reach convergence of the constraint forces for a single time-step, grows with the number of particles in the system. When applying the inelastic contact law in three dimensions and replacing Fiext by the contact forces from neighboring particles, one should not forget that they also exert torques T1ext and T2ext . They have to be included in the generalized vector Fext in Eq. (14.13), where the two torques originally were set to zero. Regarding the order of the update sequence within the list of (existing and incipient) contacts, it is preferably random and different for each sweep. In this way one avoids any bias in the information spreading. If the update order was from top to bottom, information would pass faster through the contact network downwards than upwards. It has to be mentioned that the random sweep described here differs from the well known random sequential update. While, in the latter, the choice of a contact is independent of the previous choices (the same contact could be selected twice), the random sweep selects each contact exactly once within one iteration step. We note that, in contrast to this sequential process, a simple parallel update would be unstable.

336


14.6 Computational Effort: Comparison Between CD and MD In this section we estimate the computing time Tcomp needed for the simulation of a dense N particle system in d dimensions for a certain real time Treal . This gives a guidance, for which problems it will be advantageous to use Contact Dynamics rather than Molecular Dynamics. In the derivation of the inelastic contact law (14.30) changes of the matrix H were neglected. This is only justified if the relative displacement of adjacent particles during one time-step is small compared to the particle size and to the radius of curvature of the surfaces in contact. This means that the time-step in contact dynamics must be a fraction of r/v, where v is a typical relative velocity and r a typical radius of curvature. Each time-step requires NI ∼ N 2/d force iterations, each of which takes order N computations. Hence the computational effort for a contact dynamics simulation is (CD) ∼ N 1+2/d Treal v/r. Tcomp

(14.35)

In molecular dynamics with elastic interactions modeled by a linear spring of stiffness K, each collision must be time resolved, so that a much shorter time-step than in CD is needed. It must be a fraction of the duration of a collision, m/K, where m is the particle mass. The computational effort per time-step, however, scales only with the particle number N . Hence (MD) Tcomp ∼ N Treal

K/m.

(14.36)

Putting this together we expect

(CD)

Tcomp

(MD) Tcomp

∼N

2/d

mv 2 . Kr 2

(14.37)

Systems where this is smaller than 1 can, in principle, be simulated with CD more efficiently than with MD, see Figure 14.6. mv 2 Kr 2

is the ratio between a typical kinetic energy per particle and the elastic energy cost to deform a particle substantially, i.e., on the scale of its radius. In most physical situations this factor should be small, because in general the kinetic energy does not suffice to deform collision partners substantially. In particular, it is small for quasi-static systems of rigid particles. For such systems it is advantageous to take the limit of infinite rigidity and to use CD instead of MD, provided that the particle number is not too large. The factor N 2/d ∝ NI is the price required for simulating perfectly rigid particles. For large systems with finite rigidity of the particles, MD costs less computing time than CD. However, if one is willing to use CD with incomplete force relaxation, i.e., with fixed NI N 2/d , the CD-algorithm leads to pseudo-elastic behavior, analogous to soft-particle MD simulations [25]. This involves sound propagation with finite speed and can be described by a (CD) (MD) damped wave equation. Then Tcomp ∼ Tcomp .

14.7 Rolling and Torsion Friction

Kr 2 6 mv 2

337

CD

MD N Figure 14.6: Domains where either CD or MD simulations are more efficient are separated by a power law N 4/d .

14.7 Rolling and Torsion Friction So far we have characterized the relative motion of two particles by the relative velocity V only. However, their relative orientation can also change, if the relative angular velocity = Ω ω2 − ω1

(14.38)

=0 is non-zero. A rigid rotation or translation of two arbitrary particles in contact requires V and Ω = 0. The first condition means that the particles stay in contact (Vn = 0) and that t = 0). The second condition means that there is no torsion there is no slip at the contact (V (Ωn = 0) nor rolling motion (Ωt = 0) at the contact. Torsion and rolling friction are torques T counteracting relative angular velocity. They are explained microscopically by forces of different sign acting on opposite sides of a contact region as illustrated in Figure 14.7. Strictly speaking, they are not possible for perfectly rigid particles with a single point contact. However, real particles are not perfectly rigid. Therefore, one wants to allow torsion and rolling friction also in the idealized limit considered in contact dynamics. Figure 14.7 shows that the contact torque acts with opposite sign on the angular velocities ωi of the particles. Therefore one has to replace Ti in Eq. (14.12) by − T, T1 = −l1 × R

+ T. T2 = l2 × R 6 6 ? a 6 ? ?

Figure 14.7: An example of forces (vertical vectors) causing rolling friction.

(14.39)

338


A common heuristic contact law for these rotational degrees of freedom is analogous to Coulomb’s friction law, with the relative tangential velocity replaced by the normal, respec and the friction force replaced by the corresponding comtively tangential, components of Ω ponents of the contact torque T (Figure 14.8). Such an ansatz is capable of stabilizing a static heap of spheres on a flat plane [29]. Its implementation in contact dynamics is a straightforward generalization of the implementation of Coulomb friction described above (leading to a ˆ 6 × 6 reduced mass matrix M). 6 |Tt |/Rn

6 |Tn |/Rn

µt

µn -|Ωt |

a)

-|Ωn | b)

Figure 14.8: Graphs describing rolling friction a) and torsion friction b) in contact dynamics.

Recently a rolling friction law was derived based on linear viscoelasticity of the particle t → 0 in this case. material [5, 20]. In contrast to Figure 14.8 rolling friction vanishes for Ω As microscopic justification for the heuristic ansatz, Figure 14.8, one can imagine surface roughness, sinter necks or plastic deformation as the origin of rolling friction, instead of viscoelasticity. In general, one has to expect that the different types of friction are coupled. For sliding and torsion friction this has been worked out [8, 9, 28]. Ignoring the coupling as we do in this paper, overestimates friction. The additional parameters introduced by these contact laws are the coefficients of rolling friction, µt , and of torsion friction, µn . Unlike their companion µ, they relate torques to a force by |Tt | ≤ µt Rn ,

|Tn | ≤ µn Rn

,

(14.40)

therefore having the dimension of length. In the literature of applied physics dealing with rolling friction, this is sometimes obfuscated by considering a wheel of radius R being pushed by a force acting perpendicular to the contact normal Fpush = |T|/R. Consequently, approximate values for the dimensionless coefficient µt /R are provided and turn out to be small compared to µ (confirming that a wheel is indeed a very good idea compared to a sledge). It is very tempting to presume that the length scale contained in µt is proportional to the particle radius r and hence to regard µ ≡ µt /r as a mere material parameter. Assuming furthermore that with two particles of different radii, the geometry enters only via the sum of the particles’ curvatures, the corresponding length scale becomes the reduced radius, i.e., µt = r ∗ µ

(14.41)

14.8 Attractive Contact Forces

339

with r∗ =

r1 r 2 r 1 + r2

(14.42)

.

Another classical approach [10] to rolling friction employs rate independent (i.e., nonviscous) hysteretic losses (expressed as a fraction α of the elastic energy put in). In this case |T|max ∼ αaRn

(14.43)

,

but the contact diameter a (stemming from the particles’ deformation) itself depends on Rn and (nonlinearly) on r ∗ , namely, according to the Hertz law, (14.44) a ∝ r ∗ Rn for discs (or a cylinder on a plane) and a ∝ (r ∗ Rn )1/3

(14.45)

for spheres. An essentially fixed a could also be justified, though, in the case of surfaces √ which exhibit a micro-roughness with an amplitude of order ξ r ∗ , providing an a ∼ ξr ∗ . Otherwise, incorporating (14.43) as a contact law renders the equations of contact dynamics non-linear.

14.8 Attractive Contact Forces Up to now we have not taken any kind of attractive interaction between the particles into account. For sufficiently coarse dry granular materials, adhesive forces are indeed so weak compared to other forces, that this is a good approximation. However, for wet granular media and for fine powders, adhesion is important. Here we explain how one can include it in contact dynamics simulations. 6Rn

6Rn

-g

-g

−FC

−FC dC

a)

b)

Figure 14.9: Extensions of Signorini’s graph to include adhesion. Maximal attractive force FC at zero distance only, a) and within finite range dC , b).

While the force/distance-relationship differs for adhesion forces of different origin (van der Waals forces, fluid menisci, . . . ), a common characteristic quantity is FC , the maximal

340


tensile force that the contact can bear. The simplest extension of Signorini’s graph is therefore a part of length FC on the negative force axis as shown in Figure 14.9a. Such a zero-range attraction is in accordance with the general concept of contact dynamics and, at first sight, seems perfectly reasonable. However, this would lead to an unphysical behavior in the limit of vanishing time-step ∆t. Because of the rigidity of the particles, a finite momentum ∆p can be transmitted instantaneously, if the connected cluster of particles, to which the contact belongs, collides with some other particle or cluster. This corresponds to a force ∆p/∆t which becomes arbitrarily large if ∆t → 0. All cohesive contacts with a geometry such that this force acts as a tensile load, would open for the slightest shock, if the time step chosen is small enough. In other words, the principally technical parameter ∆t picks up a physical meaning which is highly undesired in numerical simulations. The missing second ingredient is EC , the energy needed to separate the two particles. This binding energy is zero in Figure 14.9a. The simplest contact law containing nothing else but a cohesion force and a cohesion energy, is a constant force FC up to a distance dC ≡ EC /FC as depicted in Figure 14.9b. A contact can only open, if an external pulling force exceeds the threshold FC and performs work E larger than EC so that the particles separate with a kinetic energy E − EC . The opening of a contact usually needs several time-steps, in which the pulling force exceeds FC . In our implementation, a contact which started to open, but is not yet as wide as dC , is not pulled back by the cohesive force, if the tensile load again becomes smaller than FC . Such a weakened, but not yet broken, contact can only be strengthened again (closing of the gap), if the particles are pushed together. This simplifies the algorithm and is the reason why, in the graph, all pairs of values (Rn , g) within the rectangle with 0 ≤ g ≤ dC and −FC ≤ Rn ≤ 0 are permitted. Another question arising with the presence of adhesion is its influence on the friction laws. While various surface effects can be brought into play, the most basic approach is that along the lines of the DMT model [7] where the attractive force can be considered as an additional external “pushing”, i.e., the normal force Rn in the friction laws has to be replaced by Rn +FC .

14.9 Conclusion We have tried to give a didactic introduction into the simulation method of contact dynamics, also pointing out its strengths and limitations. The algorithms presented in this article have been applied to investigate the physics of dense granular media by more and more scientists over the last decade, but still molecular dynamics is much wider known and often regarded as easier. This does not mean that contact dynamics is less powerful. On the contrary, the two techniques have complementary strengths. We have described how to extend the basic algorithm in order to simulate the effects of rolling and torsion friction and of cohesion. Animated examples of these simulations [14] can be found on the CD included with this book. We restricted ourselves to the case where all particles have zero restitution coefficient. As dense granular media provide an enormous number of collective dissipation mechanisms due to rearrangements, frustrated rotations etc., a grain hitting such a packing will hardly bounce back: the effective restitution coefficient

References

341

is close to zero, which justifies our assumption. The way in which non-zero restitution coefficients may be implemented is described in [19]. Most simulations were done with round particles in two dimensions, but a few simulation results for polygonal particles can be found (e.g., in [15]). We are not aware of any contact dynamics simulation of polyhedra in three dimensions, although this is certainly feasible, but many cases of incipient contact configurations (corner-face, edge-face, edge-edge etc.) have to be distinguished. Three-dimensional simulations of cohesive spheres were done in order to investigate the influence of rolling and torsion friction on the compactibility of porous powders [3]. A nice quantitative validation of the basic contact dynamics algorithm can be found in [6], where the experimental shear bands of a packing of parallel rods (a quasi two-dimensional system) could be reproduced in great detail starting the simulation with exactly the same initial configuration. Another stimulating comparison between contact dynamics simulations and experiments is presented in [4]. There the uniaxial compaction of porous powders was studied. The key result is a power-law relationship between compacting stress and obtained porosity. Another active research area, where contact dynamics has been successfully applied, are the statistical properties of contact forces in a granular packing under load and their relation to jamming. For example, it was shown in [23], that the anisotropic load-bearing network of strong force lines is stabilized by the weak forces, which contribute nearly isotropically to the pressure. This work was extended in [22] where the role of tensile contact forces between cohesive grains (without rolling friction) was investigated. Finally, a topic which is currently intensively studied is the non-uniqueness of realizations of force equilibrium in a dense frictional packing of rigid particles. Contact dynamics is ideally suited to address this question [27].

Acknowledgments Many friends and colleagues were involved in the development of the Contact Dynamics code and later in its application. We thank them for the stimulating collaboration: G. Bartels, Z. Farkas, H. Hinrichsen, K. Johnson, D. Kadau, J. Kertész, M. Morgeneyer, F. Radjai, M. Sasvári, J. Schwedes. This work was supported by the German Science Foundation within SFB 445 and by the grant WO 577/3, by the BMBF through grant HUN 02/011, and by Federal Mogul GmbH.

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Index

adhesion 196 adhesive particles 189–209 aeolian sand transport 234–245 algebraic tails 89, 92 anisotropic interactions 282 anomalous velocity statistics 89 Barchan dune 247 bi-disperse granular gas 135 bifurcation 123–133 Bjerrum length 261 Boltzmann approach, to granular matter 61 Boltzmann equation, classical 59 boundary conditions 6, 13, 20 Boussinesq solution 13, 20 bubbles 213–215, 223 bulk elasticity 217 centrifugal force 224, 228 chains 281–294 charged granular gas 260 clustering 117–136 clusters 282–294 coagulation 262 coarse-graining (averaging) 8 coarsening 127 coating process 256, 272 coefficient of restitution 199 adhesive particles 205 Padé approximation 204 cohesion 325, 340 cohesive forces 281 collision of particles 199 collision processes 89, 91, 106, 109 compaction 325, 341 compactivity 56 compartmentalized system 117–135 constitutive relations 4, 5

constraint forces 325, 331, 334 contact area 191, 195 contact dynamics 165–185, 325–341 contact ellipse 195 contact gain 182 contact loss 179 contact network 24, 334 contact problem 189 continuum description 165, 168, 169, 186 continuum mechanics 4, 8, 9 convective instability 220 convex bodies 192 coordination number 170 Coulomb friction 165, 168, 329, 338 Coulomb interaction 255, 258 critical exponents 123, 128 critical states 171 dilatancy 176 dilation angle 168, 169, 176 dipolar hard-sphere model (DHSM) 282 directed transport 135 displacement field 4, 9 dissipative force 199 dissipative interaction 281 dissipative system 122 drag force 213, 217, 218 dune simulation 247 dunes 246 dynamic wave 217 effective temperature 47 Ehrenfest urn model 118, 127 elastic interaction force 193 elasticity 9, 11, 23–25, 39 electrostatic force 255–272 elliptic PDE 23 energy dissipation 89, 96, 98, 99

346 energy equipartition 89, 107 entrainment 242 entropy 56 escape time 134 event driven simulation 259, 299–321 fabric tensor 171 structural anisotropy 173 flow rule 168, 177 fluctuations 117, 130, 172–174, 184 fluid–particle interactions 214 fluidization 213–218, 225 flux model 121–135 force chains 11 force distributions, in a jammed emulsion 67 force lines 341 force models 10 force statistics 19 force-displacement laws 301 friction 11, 23, 24 friction models 299, 303, 305 frictionless packings 23, 24 Froude number 215, 225 gas-fluidized bed 214, 221 Ginzburg–Landau equation 146 granular avalanches 147–152 granular gas 281 granular gases 89, 117–121 granular materials 281, 299, 300 granular piles 6 granular response 15 granular slabs 6 granular suspensions 260 granular temperature 117, 282, 284 Green functions 24, 28 hardening 169, 174 Hertz contact law 193 Hertz law 339 homogeneous cooling 307, 308 hydrodynamic description 117, 118 of a vibrated granular gas 133 hydrodynamic dispersion 217 hydrodynamics 112 hyperbolic models 3, 15 hyperbolic PDE 23 hyperstatic networks 26 hypostatic networks 25

Index hysteresis 126, 131 inelastic collisions 117, 120, 134, 293 inelastic gases 89, 101, 109, 111, 112 internal angle of friction 168, 178 internal variables 165, 170, 171, 175, 176, 181, 183 isostatic jamming 76 isostatic networks 24–40 isostaticity 24–41 iterative solver 334 jamming 45 condition 49 in compressed emulsions 74 in glassy systems 46 in granular matter 48 JKR theory 196 kinetic theory 89, 91, 92 Kramers escape problem 118 lattice gas 109 Lennard–Jones interaction 197 liquid-fluidized bed 215, 216, 222, 224 Maxwell Demon effect 117–136 Maxwell model 89 micro–macro transition 299, 300, 315, 321 microstructure 165, 175, 176 minimum image method 258 mixtures 89, 90, 102, 108, 109, 112 molecular dynamics 73, 299, 300, 321, 325, 326, 336 multi-particle contacts 305, 307–309, 321 multiplicative noise 29, 30, 32–34, 40 multiplicative process 24, 32, 33, 35 non-equilibrium gases 89 non-equilibrium system 120 non-smooth mechanics 326 non-uniqueness of force equilibrium 341 nonsmooth interactions 166 normal stress 217 order parameter 144–148, 152–160 out of equilibrium systems 46, 47, 50, 78 overdamped limit 260 overpopulated tails 92 overturning instability 223, 226

Index parabolic PDE 23 particle pressure 219, 222 particle tracking 288 particle viscosity 227 particle–particle interactions 214 phase transition 123, 128, 130 plastic behaviour 168 plasticity 5 Poisson ratio 190 polygonal particles 341 positional disorder 33, 34, 36 Potts model 124 power-law distribution 23, 32, 37, 40 pseudo-elastic behavior 336 quasi-static approximation 204 quasi-static deformation 194 quasi-static loading 166, 183 reduced mass 328, 331, 334, 338 response functions 23, 24 restitution coefficient 327, 329, 333, 340, see coefficient of restitution Reynolds number 234 rheology of the particle phase 218, 222 rigid rotation 176 rigidity 24–26 ring structures 281 rolling friction 337–339, 341 roughness length 235 saltation 242 sand flux 244 saturation length 245 saturation time 245 sedimentation 215–217 self-assembly 282 self-focussing 265–267 separation bubble 239 separation occurs 238 shear band 143 shear deformation 318

347 shear stress 234 shear stress perturbation 237 shear velocity 235 Shields parameter 241 Signorini graph 325, 328 slugs 214 Smoluchowski equation 262 solid fraction 168–181 stick-slip oscillation 143, 147, 159 sticking collision 206 strain field 8–10 stress field 12, 14, 16 stress tensor 190 dissipative component 194 elastic component 190 sudden collapse of a cluster 127, 131 superposition 17 suspensions 214, 218 systems far from equilibrium 120 Tabor parameter 198 TC model 307 topological disorder 33, 35, 36 torsion friction 338, 340, 341 traffic jam formation 136 triboelectric charging 255, 257, 277 turbulent 234 two-phase governing equations 218 unilateral constraints 325 universal size distribution 262 velocity distributions 89–112, 284, 288 Verlet algorithm 259 virtual mass force 218 virtual work principle or theorem 27, 28 viscous interaction force 193 voidage wave 224, 225 yield 5 Young module 190

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