The "most probable" sphere packings, and models of soft matter
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
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
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
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. 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. (Technically,
the set of all packings is mixing under translations with respect to
each distribution for phi less than 0.49). For phi above 0.54 almost
all packings are crystalline, defining the "solid phase" of
packings. (This is harder to define. However in particular the set of
all packings is nonmixing under translations with respect to any
distribution for phi larger than 0.54.) 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".
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
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
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 we 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 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 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
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, 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. 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.
Dilatancy onset has been experimentally associated with a phase transition
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
and a phase transition associated with dilatancy onset in
But of most significance to this review is that random close packing has been
with a different phase transition, closely analogous to the freezing
transition of colloids noted above,
In other words, using the subcollection of mechanically stable
packings, again partitioned by
volume fraction phi, there is a sharp phase transition, signalled by
a flat portion in the graph of the free energy, representing mixed
packings which are partly random and partly crystalline.
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 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
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
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