Anyone who has tried to pack as many nonoverlapping pennies as possible on a table top has quickly learned to arrange them very regularly; the densest packings of spheres in 2 (or 3, and maybe other) dimensions are crystalline. Similar results are expected for "most" shapes, not just spheres, and even if several different shapes are allowed. The study of the symmetries of the densest possible packings of space, by congruent copies of one or more basic shapes, is one of the deepest areas of research in geometry [R1].

Such regularity easily fails for packings that have less than maximum density; it is easy to construct examples of this. But this is misleading. The regularity phenomenon of optimally dense packings does indeed extend to packings which are less than optimally dense - if one takes into account the statistics of various types of packings, a profound generalization of the phenomenon which has been explored by physicists rather than mathematicians for the past 50 years. Simply put, the overwhelming majority, of those packings of spheres whose volume fraction is close to maximum, are not only approximately crystalline in a naive geometric sense but have global features that distinguish them intrinsically from packings at low volume fraction. Such phenomena are important in science for many reasons; here we concentrate on their use in modelling materials, especially colloids and granular matter. We will consider some interesting phenomena that appear in models of these two types of material, and their mathematical significance for the geometry of sphere packings.

By "granular" matter we mean (static) bulk matter composed of many macroscopic noncohesive parts, typically sedimented in a fluid, which is often air. The prototypical example is a pile of sand. (Noncohesive) colloids are also composed of many macroscopic noncohesive parts immersed in a fluid (not usually air), but the colloidal elements are typically small enough, on the order of 1 micron diameter, to form a suspension in the surrounding fluid rather than a sediment. Typical examples are paint and milk. It is common to model either type of material, to first approximation, as a large collection of congruent, impenetrable spheres. In colloids the spheres undergo Brownian motion because of interaction with the surrounding fluid. Granular systems on the other hand are significantly affected by gravity, and by interparticle friction, and the elements are in mechanically stable, static configurations.

For certain colloids a useful model is that of "hard spheres" [Lo], from classical statistical mechanics. In that model many physical properties can be computed from the (Helmholtz) free energy density, which, for a system with a fixed number N of spheres in a fixed box of volume V, is (T/N) log(P), where T is the "temperature", an independent parameter, and P is the volume of "phase space", the Euclidean space of all possible configurations of all the particles. One of the basic advantages of such a probabilistic theory is the formalism that has been developed called the "infinite volume (or thermodynamic) limit", which is an approach to understand how some bulk quantities, of systems with many degrees of freedom, have extraordinarily sharp values - the probabilistic aspect gets surpressed. For instance, statistical mechanics is the only known fundamental theory for the "phases" and "phase transitions" in the physics of bulk matter in thermal equilibrium; see [FR] for a short, simple introduction. In the infinite volume limit, as N goes to infinity with phi=N/V held fixed (creating another independent parameter, phi), the free energy density of the hard sphere model turns out, in this model, to be a function of only one parameter, T/phi. We will therefore fix T=1 and discuss dependence on phi alone.

The hard sphere model can be studied for spheres of any dimension. It is not very interesting for dimension 1, but in both dimensions 2 and 3 it is very interesting indeed; see [Lo] for a good review. Not much can be proven about the model (see [BL] for a recent attempt), but in early computer simulations the model was shown to exhibit a solid/fluid phase transition. In 3 dimensions the following is very generally accepted from the simulation evidence: there is a high density phase in the range 0.55 < phi < 0.74, a low density phase in the range 0 < phi < 0.49, and a mixed phase in between, 0.49 < phi < 0.55; and there are sharp phase transitions at the edges of the mixed phase, namely a freezing transition at phi=0.49 at which the chaotic packings in the low density phase begin to freeze, and a melting transition at 0.55 at which the crystalline packings in the high density phase start to melt. Although I have used the common physics terminology to describe these phenomena, it is straightforward to interpret these statements in terms of the one parameter family of uniform probability distributions, on the space of all possible sphere packings, which are concentrated on the subspace of packings of fixed volume fraction phi (the parameter). The phase transitions make precise how "almost all" high density packings abruptly change character, at volume fraction phi=0.55, and "almost all" low density packings abruptly change character at phi=0.49. As noted above these facts have been understood by physicists for 50 years. We will now turn to granular media (see [dG] for a review), and their associated sphere packings, which have other restrictions besides volume fraction. Unlike the case for colloids, the sphere packings associated with granular matter exhibit phenomena that are not well understood - and for that reason are perhaps more interesting.

One of the incentives for studying granular matter is the unusual properties of these substances, the best known being: dilatancy, random close packing, and random loose packing.

Dilatancy was popularized by Reynolds around 1895 [Re] to denote the unusual response of sand to shear: dense sand expands when sheared. Loosely packed sand collapses when sheared, which is less surprising. The volume fraction (about 0.6) between these two regimes is called the critical state density in soil science. A common modern example of Reynolds' dilatancy is vacuum packed coffee. There is strong resistance to deforming such a package because to do so requires expanding the contents against atmospheric pressure; if the vacuum is eliminated by a puncture the package immediately loses its rigidity. Intuitively, if one tries to shear a dense collection of spheres they need to get out of each others way, thereby expanding the collection.

The classic experiments on random loose packing and random close packing were performed by Scott et al in the 1960's [SK], using samples of many thousands of congruent ball bearings. They found that by carefully pouring the spheres into a container they could achieve a volume fraction down to about 0.61. On the other hand they could, by vertical shaking, raise the volume fraction up to about 0.64. In other words there seemed to be rather well defined limits on the volume fraction (to within one percent) that could be achieved by certain types of bulk manipulation. Conversely, by individual manipulation of the spheres - placing each one where one wanted it - one could achieve close to 0.74, and alternatively, if their surfaces were rough enough, one could achieve volume fractions much lower than 0.61. So somehow these intriguing limits - a "random loose packing" lower limit of about 0.61 and a "random close packing" upper limit of about 0.64 - required certain constraints on manipulation. In fact Scott et al showed [SC] that by another type of manipulation, cyclic shearing, one could easily achieve volume fractions above 0.64, up to 0.66; and it is noteworthy that when the density passed 0.64 there seemed to always be crystal-like clusters of spheres in the material. This was confirmed and extended by Pouliquen et al [ND, PN]. The results on random loose packing have also been significantly extended by Schröter et al [JS].

A probabilistic theory of granular matter was put forward by Edwards et al in 1984 [EO], which should be able to clarify the above phenomena among others. The theory is a relatively small modification of the hard sphere model of classical statistical mechanics sketched above. The granular modification consists of ignoring the momentum variables of the spheres, and therefore the temperature, but including friction, and gravity - or at least some of its effects, such as forcing the particles to touch. In the Edwards theory the basic object is again the logarithm of the volume of the space of all possible configurations of spheres, but now the sphere positions are restricted to those which are mechanically realizable. In geometric terms we are simply looking at subensembles of the original uniform ensembles of all possible packings at fixed volume fraction; in the new ensembles we only consider packings which are also mechanically stable, like a pile of marbles. So the physical phenomena of dilatancy, random close packing, and random loose packing, should all be interpretable in terms of the statistics of all possible piles of marbles.

Dilatancy has been experimentally associated with a phase transition in [SN], and random close packing has been associated theoretically with a different phase transition, in [R2] and [AR2]. Monte Carlo simulations using the Edwards theory have been applied to random loose packing in [AR1].

References

[AR1] D. Aristoff and C. Radin, Random loose packing in granular matter, J. Stat. Phys. 135(2009) 1-23.

[AR2] D. Aristoff and C. Radin, Random close packing in a granular model, preprint.

[BL] L. Bowen, R. Lyons, C. Radin and P. Winkler, A solidification phenomenon in random packings, SIAM J. Math. Anal. 38(2006), 1075-1089.

[dG] P.G. de Gennes, Granular matter: a tentative view, Rev. Mod. Phys. 71 (1999) S374-S382.

[EO] S.F. Edwards and R.B.S. Oakeshott, Theory of powders, Physica A 157 (1989) 1080-1090.

[FR] M.E. Fisher and C. Radin, Definitions of thermodynamic phases and phase transitions, AIM workshop report.

[JS] M. Jerkins, M. Schröter, H.L. Swinney, T.J. Senden, M. Saadatfar and T. Aste, Onset of mechanical stability in random packings of frictional particles, Phys. Rev. Lett. 101 (2008) 018301.

[Lo] H. Löwen, Fun with hard spheres, pages 295-331 in Statistical physics and spatial statistics : the art of analyzing and modeling spatial structures and pattern formation, ed. K.R. Mecke and D. Stoyen, Lecture notes in physics No. 554, Springer-Verlag, Berlin, 2000.

[ND] M. Nicolas, P. Duru and O. Pouliquen, Compaction of a granular material under cyclic shear, Eur. Phys. J. E 3 (2000) 309-314.

[PN] O. Pouliquen, M. Nicolas and P.D. Weidman, Crystallization of non-brownian spheres under horizontal shaking, Phys. Rev. Lett. 79 (1997) 3640-3643.

[R1] C. Radin, Orbits of orbs: sphere packing meets Penrose tilings, Amer. Math. Monthly 111 (2004) 137-149.

[R2] C. Radin, Random close packing of granular matter, J. Stat. Phys. 131 (2008) 567-573.

[Re] O. Reynolds, On the dilatancy of media composed of rigid particles in contact, with experimental illustrations, Phil. Mag. Series 5 20 (1885) 469-481.

[SC] G.D. Scott, A.M. Charlesworth and M.K. Mak, On the random packing of spheres, J. Chem. Phys. 40 (1964) 611-612.

[SK] G.D. Scott and D.M. Kilgour, The density of random close packing of spheres, Brit. J. Appl. Phys. (J. Phys. D) 2 (1969) 863-866.

[SN] M. Schröter, S. Nägle, C. Radin and H.L. Swinney, Phase transition in a static granular system, Europhys. Lett. 78 (2007) 44004.