593 research outputs found
Optimal Packings of Superballs
Dense hard-particle packings are intimately related to the structure of
low-temperature phases of matter and are useful models of heterogeneous
materials and granular media. Most studies of the densest packings in three
dimensions have considered spherical shapes, and it is only more recently that
nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs
(whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a
versatile family of convex particles (p >= 0.5) with both cubic- and
octahedral-like shapes as well as concave particles (0 < p < 0.5) with
octahedral-like shapes. In this paper, we provide analytical constructions for
the densest known superball packings for all convex and concave cases. The
candidate maximally dense packings are certain families of Bravais lattice
packings. The maximal packing density as a function of p is nonanalytic at the
sphere-point (p = 1) and increases dramatically as p moves away from unity. The
packing characteristics determined by the broken rotational symmetry of
superballs are similar to but richer than their two-dimensional "superdisk"
counterparts, and are distinctly different from that of ellipsoid packings. Our
candidate optimal superball packings provide a starting point to quantify the
equilibrium phase behavior of superball systems, which should deepen our
understanding of the statistical thermodynamics of nonspherical-particle
systems.Comment: 28 pages, 16 figure
Non-universal Voronoi cell shapes in amorphous ellipsoid packings
In particulate systems with short-range interactions, such as granular matter
or simple fluids, local structure plays a pivotal role in determining the
macroscopic physical properties. Here, we analyse local structure metrics
derived from the Voronoi diagram of configurations of oblate ellipsoids, for
various aspect ratios and global volume fractions . We focus
on jammed static configurations of frictional ellipsoids, obtained by
tomographic imaging and by discrete element method simulations. In particular,
we consider the local packing fraction , defined as the particle's
volume divided by its Voronoi cell volume. We find that the probability
for a Voronoi cell to have a given local packing fraction shows the
same scaling behaviour as function of as observed for random sphere
packs. Surprisingly, this scaling behaviour is further found to be independent
of the particle aspect ratio. By contrast, the typical Voronoi cell shape,
quantified by the Minkowski tensor anisotropy index ,
points towards a significant difference between random packings of spheres and
those of oblate ellipsoids. While the average cell shape of all cells
with a given value of is very similar in dense and loose jammed sphere
packings, the structure of dense and loose ellipsoid packings differs
substantially such that this does not hold true. This non-universality has
implications for our understanding of jamming of aspherical particles.Comment: 6 pages, 5 figure
Novel Features Arising in the Maximally Random Jammed Packings of Superballs
Dense random packings of hard particles are useful models of granular media
and are closely related to the structure of nonequilibrium low-temperature
amorphous phases of matter. Most work has been done for random jammed packings
of spheres, and it is only recently that corresponding packings of nonspherical
particles (e.g., ellipsoids) have received attention. Here we report a study of
the maximally random jammed (MRJ) packings of binary superdisks and
monodispersed superballs whose shapes are defined by |x_1|^2p+...+|x_2|^2p<=1
with d = 2 and 3, respectively, where p is the deformation parameter with
values in the interval (0, infinity). We find that the MRJ densities of such
packings increase dramatically and nonanalytically as one moves away from the
circular-disk and sphere point. Moreover, the disordered packings are
hypostatic and the local arrangements of particles are necessarily nontrivially
correlated to achieve jamming. We term such correlated structures "nongeneric".
The degree of "nongenericity" of the packings is quantitatively characterized
by determining the fraction of local coordination structures in which the
central particles have fewer contacting neighbors than average. We also show
that such seemingly special packing configurations are counterintuitively not
rare. As the anisotropy of the particles increases, the fraction of rattlers
decreases while the minimal orientational order increases. These novel
characteristics result from the unique rotational symmetry breaking manner of
the particles.Comment: 20 pages, 8 figure
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
Hypoconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids
Continuing on recent computational and experimental work on jammed packings
of hard ellipsoids [Donev et al., Science, vol. 303, 990-993] we consider
jamming in packings of smooth strictly convex nonspherical hard particles. We
explain why the isocounting conjecture, which states that for large disordered
jammed packings the average contact number per particle is twice the number of
degrees of freedom per particle (\bar{Z}=2d_{f}), does not apply to
nonspherical particles. We develop first- and second-order conditions for
jamming, and demonstrate that packings of nonspherical particles can be jammed
even though they are hypoconstrained (\bar{Z}<2d_{f}). We apply an algorithm
using these conditions to computer-generated hypoconstrained ellipsoid and
ellipse packings and demonstrate that our algorithm does produce jammed
packings, even close to the sphere point. We also consider packings that are
nearly jammed and draw connections to packings of deformable (but stiff)
particles. Finally, we consider the jamming conditions for nearly spherical
particles and explain quantitatively the behavior we observe in the vicinity of
the sphere point.Comment: 33 pages, third revisio
Dense periodic packings of tori
Dense packings of nonoverlapping bodies in three-dimensional Euclidean space
are useful models of the structure of a variety of many-particle systems that
arise in the physical and biological sciences. Here we investigate the packing
behavior of congruent ring tori, which are multiply connected nonconvex bodies
of genus 1, as well as horn and spindle tori. We analytically construct a
family of dense periodic packings of unlinked tori guided by the organizing
principles originally devised for simply connected solid bodies [Torquato and
Jiao, PRE 86, 011102 (2012)]. We find that the horn tori as well as certain
spindle and ring tori can achieve a packing density higher than the densest
known packing of both sphere and ellipsoids. In addition, we study dense
packings of cluster of pair-linked ring tori (i.e., Hopf links).Comment: 15 pages, 7 figure
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