Superdense Crystal Packings of Ellipsoids

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Abstract

Particle packing problems have fascinated people since the dawn of civilization, and continue to intrigue mathematicians and scientists. Resurgent interest has been spurred by the recent proof of Kepler's conjecture: the face-centered cubic lattice provides the densest packing of equal spheres with a packing fraction \(\phi\approx0.7405\) \cite{Kepler_Hales}. Here we report on the densest known packings of congruent ellipsoids. The family of new packings are crystal (periodic) arrangements of nearly spherically-shaped ellipsoids, and always surpass the densest lattice packing. A remarkable maximum density of \(\phi\approx0.7707\) is achieved for both prolate and oblate ellipsoids with aspect ratios of \(\sqrt{3}\) and \(1/\sqrt{3}\), respectively, and each ellipsoid has 14 touching neighbors. Present results do not exclude the possibility that even denser crystal packings of ellipsoids could be found, and that a corresponding Kepler-like conjecture could be formulated for ellipsoids.

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Improving the density of jammed disordered packings using ellipsoids.

A. Donev (2004)

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 phi=pi/18 approximately 0.74. It is also well known that certain random (amorphous) jammed packings have phi approximately 0.64. Here, we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely-up to phi= 0.68 to 0.71 for spheroids with an aspect ratio close to that of M&M's Candies-and even approach phi approximately 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 approximately 10 for our spheroids, as compared to Z approximately 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.

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Is Random Close Packing of Spheres Well Defined?

Pablo Debenedetti, Salvatore Torquato, Thomas M. Truskett (2000)

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.