An Introduction to Granular Physics

What are granular materials and why are they interesting?

Granular materials are very simple: they are large conglomerations of discrete macroscopic particles. If they are non-cohesive, then the forces between them are essentially only repulsive so that the shape of the material is determined by external boundaries and gravity. If they are dry then any interstitial fluid, such as air, can often be neglected in determining many, but as we will see below, not all of the flow and static properties of the system.

Yet despite this seeming simplicity, granular materials behave differently from any of the other standard and familiar forms of matter : solids, liquids or gases, and should therefore be considered an additional state of matter in its own right. We will see that at the root of this unique status are three important aspects: the existence of static friction, the fact that temperature is effectively zero and, for moving grains, the inelastic nature of their collisions.

No one can seriously doubt that granular materials, of which sand is but one example, are ubiquitous in our daily lives. They play an important role in many of our industries, such as mining, agriculture, civil engineering and pharmaceutical manufacturing. They clearly are also important for geological processes where landslides and erosion and, on a larger scale, plate tectonics determine much of the morphology of the Earth. Practically everything that we eat started out in a granular form and all the clutter on our desks is often so close to the angle of repose that a small perturbation will create an avalanche onto the floor. We may still think that Hugo has overstepped the bounds of common sense when he likens the creation of worlds to the movement of simple grains of sand. However, by the end of our recent review article, we hope to have shown that there is enormous richness and complexity to granular motion. Even the possibility that Victor Hugo's metaphor (quoted at the very beginning of this web site) could have a literal meaning might no longer appear far fetched: first connections are emerging between granular dynamics and processes taking place on an astrophysical scale.

In some cases, such as a sandpile at rest with a slope less than the angle of repose, static friction produces solid-like behavior: the material remains at rest even though gravitational forces create macroscopic stresses on its surface. If the pile is tilted several degrees above the angle of repose grains start to flow, like in a fluid. However, this flow is clearly not that of an ordinary fluid because it only exists in a boundary layer at the pile's surface. We might view this flow, or any granular flow, as that of a dense gas since gases, too, are made up of discrete particles with negligible cohesive forces between them. Unlike in an ordinary gas, however, k T plays no role in a granular material. Instead, the relevant energy scale is the potential energy, mgd, of a grain of mass m raised by its own diameter, d , in the gravity of the Earth, g. For typical sand this potential energy is at least 10¹² times kT at room temperature. Because kTis effectively zero, ordinary thermodynamic arguments become useless. For example, many studies have shown that vibrations or rotations of a granular material will induce particles of different sizes to separate into different regions of the container. Since there are no attractive forces between the particles, this separation would at first appear to violate the increase of entropy principle, which normally favors mixing. In a granular material, on the other hand, kT = 0 implies that entropy considerations can easily be outweighed by dynamical effects that now become of paramount importance.

Perhaps the most important role of temperature is to allow a system to explore phase space. With kT = 0 this does not occur in a granular material. Unless perturbed by external disturbances, each metastable configuration of the material will last indefinitely, and no thermal averaging over nearby configurations will take place. Because each configuration has its unique properties, the reproducibility of granular behavior, even on large scales and certainly near the static limit, can only be defined in terms of ensemble averages. A second role of temperature in ordinary gases or fluids is to provide a microscopic velocity scale. Again, in granular materials this role is completely suppressed, and the only velocity scale is the one imposed by any macroscopic flow itself. It is possible to formulate an effective, "granular temperature" in terms of velocity fluctuations around the mean flow velocity. Yet, as we describe in more detail in the full version of this review article, such approaches in general cannot recover thermo- or hydrodynamics because of the inelastic nature of each individual, granular collision.

The science of granular media has a long history. Much of the engineering literature has been devoted to understanding how to deal with these materials. In the literature, there are many notable names such as Coulomb, who proposed the ideas of static friction, Faraday, who discovered the convective instability in a vibrated container filled with powder, and Reynolds, who introduced the notion of dilatancy, which implies that a compacted granular material must expand in order for it to undergo any shear. There has been a resurgence of interest in this field in recent years within physics. Sand has become a fruitful metaphor for describing many other, and often more microscopic, dissipative dynamical systems. De Gennes originally used sandpile avalanches as a macroscopic picture for the motion of flux lines in a type-II superconductor. A recent, intriguing use of this metaphor is based on the idea of Self-Organized Criticality, originally described in terms of the avalanches in a sandpile close to its angle of repose. The self-organization paradigm was postulated to have a wide realm of applicability to a variety of natural phenomena spanning from the microscopic to the astrophysical scale. The physics that has been uncovered in the past few years in this field has clear relevance to what is being done in other areas of condensed matter physics. Slow relaxations are found in vibrated sandpiles which bear close similarity to the slow relaxation found in glasses, spin glasses and flux lattices. Fluid-like behavior can be found in these materials which very much resemble similar phenomena exhibited by conventional liquids. Nonlinear dynamical phenomena are observed which are relevant to breakdown phenomena in semiconductors, stick-slip friction on a microscopic and earth-quake dynamics on a macroscopic scale.

Finally, there is one other, vitally important reason for the recent activity in this field. As mentioned above, many of our industries rely on transporting and storing granular materials. These include the pharmaceutical industry which relies on the processing of powders and pills, agriculture and the food processing industry where seeds, grains and foodstuffs are transported and manipulated, as well as all construction-based industries. Additional manufacturing processes, e.g. in the automotive industry, rely on casting large metal parts in carefully packed beds of sand. Yet the technology for handling and controlling granular materials is poorly developed. Estimates show that we waste 60% of the capacity of many of our industrial plants due to problems related to the transport of these materials from part of the factory floor to another. Hence even a small improvement in our understanding of how granular media behave should have an profound impact for industry.