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 `ordered'. 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 which have less than maximum density. But this is misleading. The regularity phenomenon of optimally dense packings can indeed extend to packings which are less than optimally dense, if one takes into account the relative number 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. As we will see in the `hard sphere model', 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 the overwhelming majority of packings at low volume fraction. Such phenomena are important in science for many reasons; here we concentrate on their use in modeling materials, especially colloidal, granular and crumpled materials. We will consider some interesting phenomena that appear in these types of `soft' matter, 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 - we're mostly interested in roundish particles - to form a suspension in the surrounding fluid rather than a sediment. Typical examples are paint and milk. And by `crumpled matter' we mean a material such as a sheet of stiff paper that has been compacted into a wad of small diameter. To fit sheets into a similar framework as colloids and granular matter we'll think of a sheet as composed of many small planar elements joined together.

Colloidal, granular and crumpled materials are all called `soft' because they can be macroscopically deformed with much less force than is typical of solids. It is common to model any of these materials, in a crude approximation, as a large collection of congruent, impenetrable spheres, perhaps subject to certain further constraints. In colloids the spheres undergo Brownian motion because of interaction with the surrounding fluid, instead of the ballistic motion of molecules, though this has little to do with equilibrium behavior. This is the only difference between colloids and molecular systems, as far as the modeling is concerned, and they are consequently the best understood of the soft materials we consider. Granular systems on the other hand are significantly affected by gravity, and by interparticle friction, and the elements are in mechanically stable, static configurations. The elements of (crumpled) sheets can be modeled as spheres held together in a well defined but flexible network - as if they were the knots in a fisherman's net.

For noncohesive colloids a useful model is that of `hard spheres' [Lo]. In that model the basic quantity, from which many physical properties can be computed, is the (reduced) `free energy density' F, which, for a system with a fixed number N of spheres in a fixed box of volume V, is (1/N)log(P), where P is the volume of `phase space', the Euclidean space of all possible configurations in the box of all the particles. (We are ignoring the velocity degrees of freedom since they can be easily integrated out in this model.) Clearly all configurations in the box with the same number of spheres are treated equally, so it is relevant to know, for given V and N, what most such configurations are like. To quantify this one uses the family of uniform probability distributions parameterized by N and V. Moreover, for the quantities in which we are primarily interested it is useful to take the `infinite volume (or thermodynamic) limit', in which V and N go to infinity with phi=N/V held fixed, thereby replacing N and V by a single parameter, the volume fraction phi, which takes values between 0 and 0.74 (the highest possible volume fraction for a packing of congruent spheres). One then studies F(phi) in this limit. An important feature of the model is that the concave function F has a flat portion in its graph, between phi=0.49 and phi=0.54. The significance of this facet is the following. For phi below 0.49 almost all packings are quite random, defining the `fluid phase' of packings. For phi above 0.54 almost all packings are crystalline, defining the `solid phase' of packings. And for phi between 0.49 and 0.54 the probability distribution represents appropriate mixtures of the packings at 0.49 and at 0.54, the `mixed phase', which in a large finite system consist of part of space filled at one density and the complement filled at the other density. One way to understand the presence of the mixed phase is that the number of homogeneous sphere packings at, say, volume fraction 1/2, is very much smaller than the number of inhomogeneous `mixed' packings of average volume fraction 1/2, as described above. Therefore if we tried to smoothly adjust the volume fraction of a `typical' sphere packing between the high and low extremes, starting at one end, we would find a bottleneck with too few paths to take, and be forced to resort to the inhomogeneous intermediaries in the volume fraction interval (0.49, 0.54). This is presumably the geometric situation for packings of spheres (and probably most other shapes).

The hard sphere model is said to exhibit a first order phase transition because of the above features. Intuitively, there is a freezing transition at phi=0.49 at which fluid packings start to crystallize, producing a mixture of mostly fluid and a bit of crystal. And there is a melting transition at 0.54 at which crystalline packings start to become disordered, producing a mixture of mostly crystal and a bit of fluid. The 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 these phase transitions were shown to hold by early computer simulations (see [Kr]). And for us a key fact is that the model is an accurate portrayal of noncohesive colloidal materials [RD], which exhibit well-defined fluid and crystalline phases, separated by a mixed phase, just as in the model.

As we have seen, noncohesive colloids can be accurately modeled using the set of all sphere packings, partitioned by volume fraction. We will now turn to granular media (see [dG] for a review), and crumpled sheets (see [Wi] for a review), and their associated sphere packings, which require restrictions on the packings that are used. Unlike the case for colloids, these materials exhibit phenomena that are not well understood - and for that reason are perhaps more interesting to try to model.

Among the unusual properties of granular matter, the best known are: 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 dilatancy onset in soil science. A common modern example of Reynolds' dilatancy is coffee vacuum packed in a flexible plastic bag. 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 first simple model analyzing dilatancy onset in a quantitative manner is [AR4].

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 appropriate/allowed types of 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].

There is a theory for granular matter (`static sand'), introduced by Edwards et al in 1989 [EO], which is a simple variant of the hard sphere model sketched above. The granular modification consists of including friction and gravity - or at least some of its effects, such as requiring that the particles touch. In the Edwards theory the basic object, the free energy as a function of volume fraction phi, is again the logarithm of the volume of the space of all possible configurations of spheres at fixed volume fraction, but now the sphere positions are restricted to those which are mechanically realizable, including friction and gravity. 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, are all conjectured, in the Edwards theory, to be interpretable in terms of the relative numbers of all possible piles of marbles for each volume fraction. A recent experiment [RRSS], [JORSS] suggests that a dynamical (nonequilibrium) theory would be more suitable [RS2].

Dilatancy onset has been experimentally associated with a phase transition in [SN, MS]. Monte Carlo simulations of Edwards-style models using packings of squares in the plane, and cubes in space, have been shown to exhibit random loose packing in [AR1], and a phase transition associated with dilatancy onset in [AR4]. But of most significance to this review is that random close packing has been associated with a different phase transition, closely analogous to the freezing transition of colloids noted above, on the theoretical side in [R2] and [AR2] and on the experimental side in [RRSS], [JORSS]. Further, these works can be used to base random close packing directly on the hard sphere model, rather than a subensemble [R4].

One reason crumpled sheets are interesting is the way its strength is developed. When we squeeze a stiff sheet of paper into a tight wad, we not only (reversibly) bend the sheet, but also create creases where the material has been irreversibly deformed. It is the bends and creases which give the wad its ability to withstand surprisingly large deforming forces (and therefore make it a useful, cheap packing material). We are concerned here not with energy distribution, which has been the main object of recent work on crumpled sheets, but in the geometry of the crumpled material. Think of a sheet as a fisherman's net, but with large spherical knots or nodes, held together by short flexible threads which keep neighboring sphere centers close together: say at separation no more than 11/10 of a sphere diameter. (Not convenient for fishing!) So the spheres can move relative to one another, but not a lot, the restricted motion being a crude model of a cost for bending. Now consider a sheet of such a network, initially flat and with large linear dimension, much larger than that of the diameter of the spherical nodes. To clarify the necessary dimensions we introduce a third linear scale: the diameter of a tight `wad' into which we want to deform the sheet. The diameter of the wad should be much bigger than that of the spherical nodes of the network.

If we wanted to create a really tight wad from the sheet, of high volume fraction, we would fold the sheet into parallel sections which are stacked together as in an accordion, using a minimal number of bends since bends create empty space. Note that such a folded sheet, or its parallel sections, has an orientation in space, a `broken symmetry'. This can be quantified in any sheet using the planes going through the nodes of each of the unit cells of the network. The sheet could be said to have an orientation whenever these planes have an average orientation (computed in terms of their normal lines for instance) of macroscopic size, i.e. comparable to the volume of the configuration. Optimally low volume fraction corresponds to the flat sheet, which also has an orientation. But even at very low volume fraction the material quickly loses its orientation and becomes symmetric. One may wonder whether the orientation, i.e. broken symmetry, described above at optimally high volume fraction, is also quickly lost away from the optimum.

In [AR3, AR5] there is a toy model of crumpled sheets and it shows that the symmetry of low volume fraction states is suddenly lost to an oriented state at a moderate volume fraction. And in fact this is again a first order `freezing' transition, with a flat portion in the graph of the analogous free energy, the facet representing mixed packings, partly random and partly ordered (folded).

As we noted above, the phenomenon of random close packing of granular matter, and the phenomenon of bulk folding in crumpled media, can both be modeled, within appropriate classes of sphere packings, as analogous to the freezing transition of colloids or the hard sphere model. For crumpled media the spheres in the packings are tethered together in a 2 dimensional network, and for granular media the spheres in the packings are appropriately constrained to `sand pile'-like configurations. In summary, the geometry of sphere packings can, if studied as a function of volume fraction phi, keeping track of the relative number of packings for each phi, model very interesting physical phenomena. For different materials one simply varies the set of allowed packings.

Finally, we note that there is a version of these ideas applicable to large, dense networks or graphs, in which the `structure' at high density is multipartite. Indeed this can be thought of as a crude mean-field approach to modeling molecular materials [RS]. There is an expository article on this use of network models in [R3], and the solid/fluid transition is discussed in some depth in this context in [RRS].

References

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