Granular and jammed materials

Andrea J. Liu

Sidney R. Nagel

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Although the word has been around for nearly four centuries, the realization that jamming is an important physical phenomenon that warrants scientific study in its own right is relatively new. This may be somewhat surprising considering that jamming is ubiquitous and is evident in many of the mundane yet beautiful and important phenomena taking place in our daily lives: sand forming graceful structures in a child's sandbox, powder refusing to flow out of a container, or foams standing rigid after being formed on a carbonated drink. This does not even include the related effects of jamming caused by collective social activities—cars trying to navigate traffic jams during rush hour, or crowds trying to evacuate a room during an emergency. Jamming thus causes problems for industry and problems for transport. But jamming can also be an asset in construction and in novel robot design to make objects whose rigidity can be controlled by slight changes in pressure.¹ The structures formed by jamming are remarkably robust yet reversible.

Slightly over a decade ago, physicists, chemists and engineers began to ask if there was any merit to thinking of specific disordered systems that develop rigidity as examples of a more general class of organization with a shared set of behaviors. This more comprehensive view of jamming attempted to link the behavior of supercooled glass-forming liquids, colloids, foams and sand.² Although this idea has had some traction, it has not been universally accepted.

Nevertheless, it is clear that there is a jamming transition that occurs at zero temperature that can be cleanly simulated and studied in the case of spheres (discs in two dimensions) interacting via finite-range repulsive forces in the absence of friction. This jamming transition is different from most other transitions, in that it has a discontinuous jump in one variable (the number of overlapping particles) but continuous variation in others (the shear modulus, for example). The solid formed from the jamming of such frictionless spheres is also unusual. Instead of having the ubiquitous Debye spectrum of normal modes for its low-frequency vibrations, it has a constant density of states all the way down to zero frequency. This dwarfs the contribution from the ordinary plane-wave modes. Moreover, the shear modulus can go to zero at the transition even as the bulk modulus remains large.

This model of a jamming transition is highly idealized. It contains no temperature, no friction, no three-body forces, no long-range attractions and no aspherical particles. Does it nevertheless have enough of the essential ingredients to describe any of the important features of real systems in the laboratory? For example, the presence of frictional interactions creates a much broader range of stability for granular matter than can be observed when friction is absent. Non-spherical interactions or three-body forces, change the way we must count degrees of freedom and constraints; particles without spherical symmetry have important rotational modes. It is not clear whether the glass transition, which occurs when a liquid solidifies into a glass by lowering the temperature, is related to the jamming of particulate solids where temperature is largely irrelevant. The introduction of thermal motion allows an exploration of different basins of stability that is strictly forbidden at zero temperature. These are just some of the issues that are being probed in ongoing work.

Meanwhile, experiments on jamming thrive—independent of any theoretical bias. Studies of flow in granular matter, foams, suspensions and colloids have long focused on understanding the rheology of these amorphous systems. Studies of supercooled liquids, for which the viscosity becomes increasingly large until the intrinsic relaxation times in the material become longer than those probed by the experiment, have long sought to define the phenomenology of the glass transition. These, too, must be considered when contemplating the broad scope of jamming research.

Over the last decade, there has been a lot of progress in understanding the nature of jamming transitions in its many different manifestations. Recent reviews have provided an overview of what has been done up to now.^3,4 In contrast, the present issue is devoted not to an overview of the entire field, but rather is meant to give a snapshot that captures some of the range of current important open questions in this area of research. In this volume, we loosely divide the work into three broad areas: structure, dynamics and thermodynamics.

In the first of these areas, we find several groups involved in understanding how the static structure of a jammed solid depends on various factors. Several papers deal with the effects of friction. These include the papers by Silbert, by Farrell et al., by Cheng and the two papers by Henkes et al. Several other papers deal with how the structure depends on polydispersity and the shape of the constituent objects. These include the papers by Xu and Ching, by Corwin et al. and by Schreck et al. The paper by Jacquin and Berthier deals with how the pair distribution function evolves with temperature and the paper by Pica Ciamarra et al. discusses over what range of densities a packing will support a load.

The section on dynamics contains papers that address three main concerns: elementary excitations, dynamical heterogeneities and flow. These topics have a considerable amount of overlap. (1) The papers by Vitelli and by Brito et al. discuss the elementary excitations of a packing above the jamming threshold while that by Kima and Pak studies the microscopic dynamics in experimental granular systems. (2) The papers by Mehta and by Katsuragi et al. discuss the nature of dynamical heterogeneities in systems near jamming and the paper by Sessoms et al. analyze the rearrangements in foams as they coarsen over time. (3) Finally several papers discuss the nature of flow when the packing is pushed beyond its limits of stability. These include the papers by Dijksman and van Hecke on flow in Couette geometries and by Brzinski and Durian on drag in an air fluidized granular system. The papers by Lörincz and Schall, by Heussinger et al. and by Lechenault et al. pay close attention to the fluctuations and heterogeneities that occur in flow and study diffusion of particles near the transition. The paper by Ciamarra et al. looks at the general problem of how the different control parameters govern the overall phase diagram for flow.

The third section on thermodynamics concerns nonequilibrium phenomena that can be understood in terms of a powerful generalized thermodynamics characterized by an effective temperature (see paper by Bouchbinder and Langer), a statistical approach to understanding stress transmission in static packs (see papers by Tighe et al., and by Chakraborty) and experiments testing the effective temperature concept (see paper by Lechenault and Daniels).

Currently, a new generation of experiments has been reported, is under way or is being planned that will confront the theoretical proposals, models and conjectures that have had their input largely from simulation. In particular, experimental methods have been developed⁵ to test predictions of the vibrational behavior near the jamming transition in colloidal systems.^5,6

This is an exciting time to be in this field. The study of jammed systems, be they granular, colloidal or molecular, has the potential to help us understand much about the general and robust behavior of amorphous solids and high-viscosity liquids. The current issue, we hope, conveys some of the rich potential of the field.

References

1. E. Steltz, A. Mozeika, N. Rodenberg, E. Brown and H. M. Jaeger, JSEL: Jamming Skin Enabled Locomotion, IEEE/RSJ International Conference on Intelligent Robots and Systems October 11–15, 2009, 2009, 5672–5677; E. Brown, N. Rodenberg, J. Amend, A. Mozeika, E. Steltz, M. R. Zakin, H. Lipson and H. M. Jaeger, Universal Robotic Gripper based on the Jamming of Granular Material, submitted, 2010.
2. A workshop at the Institute for Theoretical Physics at Santa Barbara was held in 1997. A book of relevant reprints and reviews of the some of the work in specific areas was modeled after this workshop: “ Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales,” Edited by A. J. Liu and S. R. Nagel, Taylor and Francis, London, 2001.
3. M. van Hecke, Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J. Phys.: Condens. Matter, 2010, 22, 033101.
4. A. J. Liu and S. R. Nagel, The Jamming Transition and the Marginally Jammed Solid, Annual Reviews of Condensed Matter Physics, 2010, 1, 14.1.
5. A. Ghosh, V. K. Chikkadi, P. Schall, J. Kurchan and D. Bonn, http://arxiv.org/abs/1003.1529v1, 2010.
6. K. Chen, W. G. Ellenbroek, Z. Zhang, D. T. N. Chen, P. Yunker, S. Henkes, C. Brito, O. Dauchot, W. van Saarloos, A. J. Liu, A. G. Yodh, http://arxiv.org/abs/1003.3065, 2010.

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Footnote

† This paper is part of a Soft Matter themed issue on Granular and jammed materials. Guest editors: Andrea Liu and Sidney Nagel.

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