Improving the Density of Jammed Disordered Packings Using
Sachs,2 Evan A.
Variano,2,6 Frank H.
P. M. Chaikin2,4
Packing problems, such as how densely objects can fill a
volume, are among the most ancient and persistent
problems in mathematics and science. For equal spheres,
it has only recently been proved that the face-centered
cubic lattice has the highest possible packing fraction
. It is also well known that
certain random (amorphous) jammed packings have 0.64. Here, we show
experimentally and with a new simulation algorithm that
ellipsoids can randomly pack more densely—up to = 0.68 to 0.71for
spheroids with an aspect ratio close to that of M&M's
Candies—and even approach 0.74 for ellipsoids
with other aspect ratios. We suggest that the higher
density is directly related to the higher number of
degrees of freedom per particle and thus the larger
number of particle contacts required to mechanically
stabilize the packing. We measured the number of contacts
per particle Z 10 for our spheroids, as compared to Z
6 for spheres.
Our results have implications for a broad range of
scientific disciplines, including the properties of
granular media and ceramics, glass formation, and discrete
1 Program in Applied and Computational Mathematics,
Princeton University, Princeton, NJ 08544, USA.
Department of Physics, Princeton University, Princeton, NJ 08544,
3 Department of Chemistry, Princeton University,
Princeton, NJ 08544, USA.
4 Princeton Materials
Institute, Princeton, NJ 08544, USA.
5 North Carolina
Central University, Durham, NC 27707, USA.
Department of Civil and Environmental Engineering, Cornell
University, Ithaca, NY 14853, USA.
7 Department of
Mathematics, Cornell University, Ithaca, NY 14853, USA.
* To whom correspondence
should be addressed. E-mail: email@example.com
The structure of liquids, crystals, and glasses is intimately
related to volume fractions of ordered and disordered
(random) hard-sphere packings, as are the transitions
between these phases (1).
Packing problems (2)
are of current interest in dimensions higher than three
for insulating stored data from noise (3),
and in two and three dimensions in relation to flow and
jamming of granular materials (4–6)
and glasses (7).
Of particular interest is random packing, which relates
to the ancient (economically important) problem of how
much grain a barrel can hold.
Many experimental and computational algorithms produce a
relatively robust packing fraction (relative density)
randomly packed monodisperse spheres as they proceed to
their limiting density (8).
This number, widely designated as the random close
packing (RCP) density, is not universal but generally
depends on the packing protocol (9).
RCP is an ill-defined concept because higher packing
fractions are obtained as the system becomes ordered, and
a definition for randomness has been lacking. A more
recent concept is that of the maximally random jammed (MRJ)
state, corresponding to the least ordered among all jammed
For a variety of order metrics, it appears that the MRJ
state has a density of 0.637 and is consistent with what
has traditionally been thought of as RCP (10).
Henceforth, we refer to this random form of packing as
the MRJ state.
We report on the density of the MRJ state of ellipsoid
packings as asphericity is introduced. For both oblate
and prolate spheroids, and Z (the average
number of touching neighbors per particle) increase
rapidly, in a cusp-like manner, as the particles deviate
from perfect spheres. Both reach high densities such as 0.71, and general
ellipsoids pack randomly to a remarkable 0.735, approaching the
density of the crystal with the highest possible density
for spheres (11)
. The rapid increases are
unrelated to any observable increase in order in these systems
that develop neither crystalline (periodic) nor liquid
crystalline (nematic or orientational) order.
Our experiments used two varieties of M&M's Milk
Chocolate Candies: regular and baking ("mini") candies
Both are oblate spheroids with small deviations from true
ellipsoids, r/r < 0.01. Additionally, M&M's
Candies have a very low degree of polydispersity
(principal axes 2a = 1.34 ± 0.02 cm, 2b =
0.693 ± 0.018 cm, a/b = 1.93 ± 0.05 for
regular; 2a = 0.925 ± 0.011 cm, 2b = 0.493 ±
0.018 cm, a/b = 1.88 ± 0.06 for minis). Several
sets of experiments were performed to determine the
packing fraction. A square box, 8.8 cm by 8.8 cm, was
filled to a height of 2.5 cm while shaking and tapping
the container. The actual measurements were performed by
adding 9.0 cm to the height and excluding the
contribution from the possibly layered bottom. After measuring
the average mass, density, and volume of the individual
candies, the number of candies in the container and their
volume fraction could be simply determined by weighing.
These experiments yielded = 0.665 ± 0.01 for regulars and
= 0.695 ±
0.01 for minis. The same technique was used for 3.175 =
mm ball bearings (spheres) and yielded = 0.625 ± 0.01. A second set
of experiments was performed by filling 0.5-, 1-, and
5-liter round flasks (to minimize ordering due to wall
effects) with candies by pouring them into the flasks
while tapping (5 liters corresponds to about 23,000 minis
or 7500 regulars) (Fig.
1A). The volume fractions found in these more
reliable studies were = 0.685 ± 0.01 for both the minis and regulars
The same procedure for 30,000 ball bearings in the
0.5-liter flask yielded = 0.635 ± 0.01, which is close
to the accepted MRJ density.
| Fig. 1. (A) An
experimental packing of the regular candies. (B)
Computer-generated packing of 1000 oblate ellipsoids with =
A 5-liter sample of regular candies similar to that shown in
1A was scanned in a medical magnetic resonance imaging
device at Princeton Hospital. For several planar slices,
the direction (with respect to an arbitrary
axis) of the major elliptical axis was manually measured
and the two-dimensional nematic order parameter
S2 = 2 cos2 – 1 was computed,
yielding S2 0.05. This is consistent with the
absence of orientational order in the packing (14).
Our simulation technique generalizes the
Lubachevsky-Stillinger (LS) sphere-packing algorithm (15,
to the case of ellipsoids. The method is a hard-particle
molecular dynamics (MD) algorithm for producing dense
disordered packings. Initially, small ellipsoids are
randomly distributed and randomly oriented in a box with
periodic boundary conditions and without any overlap. The
ellipsoids are given velocities and their motion followed
as they collide elastically and also expand uniformly.
After some time a jammed state with a diverging collision
rate is reached and the density reaches a maximal value.
A novel event-driven MD algorithm (17)
was used to implement this process efficiently, based on
the algorithm used in (15)
for spheres and similar to the algorithm used for needles
A typical configuration of 1000 oblate ellipsoids (aspect
b/a = 1.9–1 0.526) is shown in Fig.
1B, with density of 0.70
and nematic order parameter S 0.02 to 0.05.
We have verified that the sphere packings produced by the LS
algorithm are jammed according to the rigorous
hierarchical definitions of local, collective, and strict
Roughly speaking, these definitions are based on
mechanical stability conditions that require that there
be no feasible local or collective particle displacements
and/or boundary deformations. On the basis of our
experience with spheres (10),
we believe that our algorithm (with rapid particle
expansion) produces final states that represent the MRJ
state well. The algorithm closely reproduces the packing
fraction measured experimentally.
The density of simulated packings of 1000 particles is shown
2A. Note the two clear maxima with 0.71, already close to
the 0.74 for the ordered face-centered cubic (fcc)/hexagonal
close-packed (hcp) packing, and the cusp-like minimum near
1 (spheres). Previous simulations for random sequential
addition (RSA) (21),
as well as gravitational deposition (22),
produce a similarly shaped curve, with a maximum at
nearly the same aspect ratios 1.5 (prolate) or 0.67 (oblate), but with
substantially lower volume fractions (such as 0.48 for RSA).
| Fig. 2. (A) Density versus
aspect ratio from simulations, for both prolate (circles) and
oblate (squares) ellipsoids as well as fully aspherical
(diamonds) ellipsoids. The most reliable experimental result
for the regular candies (error bar) is also shown; this likely
underpredicts the true density (38).
(B) Mean contact number Z versus aspect ratio
simulations [same symbols as in (A)], along with the
experimental result for the regular candies (cross). Inset:
Introducing asphericity makes a locally jammed particle free
to rotate and escape the cage of neighbors.
Why does the packing fraction initially increase as we
deviate from spheres? The rapid increase in packing
fraction is attributable to the expected increase in the
number of contacts resulting from the additional
rotational degrees of freedom of the ellipsoids. More
contacts per particle are needed to eliminate all local
and collective degrees of freedom and ensure jamming, and
forming more contacts requires a denser packing of the
particles. In the inset in Fig.
2B, the central circle is locally jammed. A uniform
vertical compression preserves , but the central ellipsoid
can rotate and free itself and the packing can densify.
The decrease in the density for very aspherical particles
could be explained by strong exclusion-volume effects in
orientationally disordered packings (23).
Results resembling those shown in Fig.
2A are also obtained for isotropic random packings of
but an argument based on "caging" (not jamming) of the
particles was given to explain the increase in density as
asphericity is introduced. Spherocylinders have a very
different behavior for ordered packings from ellipsoids
(the conjectured maximal density is , which is significantly higher
than for ellipsoids), and also cannot be oblate and are
always axisymmetric. The similar positioning of the
maximal density peak for different packing algorithms and
particle shapes indicates the relevance of a simple
By introducing orientational and translational order, it is
expected that the density of the packings can be further
increased, at least up to 0.74. As shown in Fig.
3 for two dimensions, an affine deformation (stretch)
of the densest disk packing produces an ellipse packing
with the same volume fraction. However, this packing,
although the densest possible, is not strictly jammed
(i.e., it is not rigid under shear transformations). The
figure shows through a sequence of frames how one can distort
this collectively jammed packing (20),
traversing a whole family of densest configurations. This
mechanical instability of the ellipse packing as well as
the three-dimensional ellipsoid packing arises from the
additional rotational degrees of freedom and does not
exist for the disk or sphere packing.
| Fig. 3. Shearing the densest
packing of ellipses.
There have been conjectures (25,
that frictionless random packings have just enough
constraints to completely statically define the system
Z = 2f (i.e., that the system is isostatic),
where f is the number of degrees of freedom per
particle (f = 3 for spheres, f = 5 for
spheroids, and f = 6 for general ellipsoids) (28).
If friction is strong, then fewer contacts are needed,
Z = f + 1 (29).
Experimentally, Z for spheres was determined by
Bernal and Mason by coating a system of ball bearings
with paint, draining the paint, letting it dry, and
counting the number of paint spots per particle when the
system was disassembled (30).
Their results gave Z 6.4, surprisingly close to
isostaticity for frictionless spheres (31).
We performed the same experiments with the M&M's,
counting the number of true contacts between the
A histogram of the number of touching neighbors per
particle for the regular candies is shown in Fig.
4. The average number is Z = 9.82. In
simulations a contact is typically defined by a cutoff on
the gap between the particles. Fortunately, over a wide
range (10–9 to 10–4) of contact
tolerances, Z is reasonably constant. Superposed
4 is the histogram of contact numbers obtained for
simulated packings of oblate ellipsoids for = 0.526, from which
we found Z 9.80. In Fig.
2B we show Z as a function of aspect ratio
As with the volume fraction, the contact number appears
singular at the sphere value and rises sharply for small
deviations. Unlike , however, Z does not decrease for large
aspect ratios, but rather appears to remain
| Fig. 4. Comparison of
experimental (black bars, from 489 regular candies) and
simulated (white bars, from 1000 particles) distribution of
particle contact numbers.
We expect that fully aspherical ellipsoids, which have f =
6, will require even more contacts for jamming (Z
= 12 according to the isostatic conjecture) and larger
. Results from
simulations of ellipsoids with axes a = –1,
b = 1, and c = (where measures the
asphericity) are included in Fig.
we obtain a surprisingly high density of 0.735, with no significant
orientational ordering. The maximum contact number
observed in Fig.
2B is Z 11.4. It is interesting that for both spheroids
and general ellipsoids, Z reaches a constant value at
approximately the aspect ratio for which the density has
a maximum. This supports the claim that the decrease in
density for large is due to exclusion volume effects.
The putative nonanalytic behavior of Z and at = 1 is
striking and is evidently related to the randomness of
the jammed state. Crystal close packings of spheres and
ellipsoids show no such singular behavior, and in fact
are independent of for small deviations from unity. On the other
hand, for random packings, the behavior is not
discontinuous, whereas the number of degrees of freedom
jumps from three to five (or six) as soon as deviates from 1. In
several industrial processes such as sintering and
ceramic formation, interest exists in increasing the
density and number of contacts of powder particles to be
fused. If ellipsoidal instead of spherical particles are
used, we may increase the density of a randomly poured
and compacted powder to a value approaching that of the
densest (fcc) lattice packing.
October 2003; accepted 9 December
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||Wall effects yield lower measured
densities. Continued "tapping" of the candies may further
densify the system, as happens for granular material (36).
Furthermore, the somewhat higher density of the
computer-generated packings can be explained by taking into
account the influence of gravity and friction, which are not
included in the simulation. Gravitation-dominated packings
always have much lower packing fractions, as low as 0.4, and have
significant orientational ordering (22,
||Supported by American Chemical
Society PRF grant 36967-AC9 (S.T., A.D., F.H.S.), NASA grant
NAG3-1762 (P.M.C.), and NSF grants DMR-0213706 (S.T., P.M.C.,
A.D., F.H.S.), DMS-0312067 (S.T., A.D., F.H.S.), and
Include this information when
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Volume 303, Number 5660, Issue of 13 Feb 2004,
Copyright © 2004 by The American Association for the
Advancement of Science. All rights reserved.