Granular solids , liquids , and gases

Victor Hugo suggested the possibility that patterns created by the movement of grains of sand are in no small part responsible for the shape and feel of the natural world in which we live. 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, and construction. They clearly are also important for geological processes where landslides, erosion, and, on a related but much larger scale, plate tectonics determine much of the morphology of 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 chance perturbation will create an avalanche onto the floor. Moreover, Hugo hinted at the extreme sensitivity of the macroscopic world to the precise motion or packing of the individual grains. We may nevertheless think that he has overstepped the bounds of common sense when he related the creation of worlds to the movement of simple grains of sand. By the end of this article, we hope to have shown such an enormous richness and complexity to granular motion that Hugo’s metaphor might no longer appear farfetched and could have a literal meaning: what happens to a pile of sand on a table top is relevant to processes taking place on an astrophysical scale. Granular materials are simple: they are large conglomerations of discrete macroscopic particles. If they are noncohesive, then the forces between them are only repulsive so that the shape of the material is determined by external boundaries and gravity. If the grains are dry, any interstitial fluid, such as air, can often be neglected in determining many, but not all, of the flow and static properties of the system. Yet despite this seeming simplicity, a granular material behaves differently from any of the other familiar forms of matter—solids, liquids, or gases—and should therefore be considered an additional state of matter in its own right. In this article, we shall examine in turn the unusual behavior that granular material displays when it is considered to be a solid, liquid, or gas. For example, a sand pile at rest with a slope lower than the angle of repose, as in Fig. 1(a), behaves like a solid: 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, as seen in Fig. 1(b). However, this flow is clearly not that of an ordinary fluid because it only exists in a boundary layer at the pile’s surface with no movement in the bulk at all. (Slurries, where grains are mixed with a liquid, have a phenomenology equally complex as the dry powders we shall describe in this article.) There are two particularly important aspects that contribute to the unique properties of granular materials: ordinary temperature plays no role, and the interactions between grains are dissipative because of static friction and the inelasticity of collisions. We might at first be tempted to view any granular flow as that of a dense gas since gases, too, consist of discrete particles with negligible cohesive forces between them. In contrast to ordinary gases, however, the energy scale kBT is insignificant here. The relevant energy scale is the potential energy mgd of a grain of mass m raised by its own diameter d in the Earth’s gravity g . For typical sand, this energy is at least 1012 times kBT at room temperature. Because kBT is irrelevant, ordinary thermodynamic arguments become useless. For example, many studies have shown (Williams, 1976; Rosato et al., 1987; Fan et al., 1990; Jullien et al., 1992; Duran et al., 1993; Knight et al., 1993; Savage, 1993; Zik et al., 1994; Hill and Kakalios, 1994; Metcalfe et al., 1995) 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