Particle Packings in Low Dimensions

We have published a paper in Physical Review Letters on random sphere packings.

Abstract:

Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). We argue that the current picture of RCP cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm. We suggest that this impasse can be broken by introducing the new concept of a maximally random jammed state, which can be made precise.

Here is a link to the full paper (Physical Review Letters, 84, 2064 (2000).)

See also, Random Heterogeneous Materials: Microstructure and Macroscopic Properties by S. Torquato.

Here are some links to discussions of this work in the popular press:

- Science

- BBC

We have recently published a paper in Science on random packing of ellipsoids.

Abstract:

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 ~0.74. 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 0.71 for 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 geometry.

Here is a link to the full paper (Science, 303, 990 (2004).)

- New York Times

- CNN.com

- Science News

- We have recently discovered the densest known packings of congruent ellipsoids. See the paper entitled Unusually Dense Crystal Ellipsoid Packings.

References for a Colloquium Talk entitled Optimal Packings: Problems for the Ages presented by Professor Torquato on June 29, 2006 at the Aspen Center for Physics Workshop Physical and Mathematical Aspects of Packing.

Executables (produced from Fortran 90 codes) to generate and analyze monodisperse or polydisperse jammed sphere packings of hard spheres in 2 and 3 dimensions can be found at http://cherrypit.princeton.edu/Packing/Fortran. The programs can also handle ellipsoids in 2 and 3 dimensions, please contact us if you'd like to use this feature.

A C++ program to generate jammed packings of hyperspheres in d dimensions can be found at http://cherrypit.princeton.edu/Packing/C++.

Note that the output depends sensitively on the choice of expansion rate, the initial conditions, and also the velocity thermostat (cooling), and ensuring that the packings are truly jammed is nontrivial. The interested person should read the papers:

1. A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, Jamming in Hard Sphere and Disk Packings, Journal of Applied Physics, 95, 989-999 (2004).
2. A. Donev, S. Torquato, and F. H. Stillinger, Neighbor List Collision-Driven Molecular Dynamics for Nonspherical Hard Particles: I. Algorithmic Details, Journal of Computational Physics, 202, 737 (2005).
3. A. Donev, S. Torquato, and F. H. Stillinger, Neighbor List Collision-Driven Molecular Dynamics for Nonspherical Hard Particles: II. Applications to Ellipses and Ellipsoids, Journal of Computational Physics, 202, 765 (2005).

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