3,341 research outputs found
Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings via Linear Programming
We have formulated the problem of generating periodic dense paritcle packings
as an optimization problem called the Adaptive Shrinking Cell (ASC) formulation
[S. Torquato and Y. Jiao, Phys. Rev. E {\bf 80}, 041104 (2009)]. Because the
objective function and impenetrability constraints can be exactly linearized
for sphere packings with a size distribution in -dimensional Euclidean space
, it is most suitable and natural to solve the corresponding ASC
optimization problem using sequential linear programming (SLP) techniques. We
implement an SLP solution to produce robustly a wide spectrum of jammed sphere
packings in for and with a diversity of disorder
and densities up to the maximally densities. This deterministic algorithm can
produce a broad range of inherent structures besides the usual disordered ones
with very small computational cost by tuning the radius of the {\it influence
sphere}. In three dimensions, we show that it can produce with high probability
a variety of strictly jammed packings with a packing density anywhere in the
wide range . We also apply the algorithm to generate various
disordered packings as well as the maximally dense packings for
and 6. Compared to the LS procedure, our SLP protocol is able to ensure that
the final packings are truly jammed, produces disordered jammed packings with
anomalously low densities, and is appreciably more robust and computationally
faster at generating maximally dense packings, especially as the space
dimension increases.Comment: 34 pages, 6 figure
Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. I. Polydisperse spheres
Hyperuniform many-particle distributions possess a local number variance that
grows more slowly than the volume of an observation window, implying that the
local density is effectively homogeneous beyond a few characteristic length
scales. Previous work on maximally random strictly jammed sphere packings in
three dimensions has shown that these systems are hyperuniform and possess
unusual quasi-long-range pair correlations, resulting in anomalous logarithmic
growth in the number variance. However, recent work on maximally random jammed
sphere packings with a size distribution has suggested that such
quasi-long-range correlations and hyperuniformity are not universal among
jammed hard-particle systems. In this paper we show that such systems are
indeed hyperuniform with signature quasi-long-range correlations by
characterizing the more general local-volume-fraction fluctuations. We argue
that the regularity of the void space induced by the constraints of saturation
and strict jamming overcomes the local inhomogeneity of the disk centers to
induce hyperuniformity in the medium with a linear small-wavenumber nonanalytic
behavior in the spectral density, resulting in quasi-long-range spatial
correlations. A numerical and analytical analysis of the pore-size distribution
for a binary MRJ system in addition to a local characterization of the
n-particle loops governing the void space surrounding the inclusions is
presented in support of our argument. This paper is the first part of a series
of two papers considering the relationships among hyperuniformity, jamming, and
regularity of the void space in hard-particle packings.Comment: 40 pages, 15 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
On the hard sphere model and sphere packings in high dimensions
We prove a lower bound on the entropy of sphere packings of of
density . The entropy measures how plentiful such
packings are, and our result is significantly stronger than the trivial lower
bound that can be obtained from the mere existence of a dense packing. Our
method also provides a new, statistical-physics-based proof of the lower bound on the maximum sphere packing density by showing
that the expected packing density of a random configuration from the hard
sphere model is at least when the
ratio of the fugacity parameter to the volume covered by a single sphere is at
least . Such a bound on the sphere packing density was first achieved
by Rogers, with subsequent improvements to the leading constant by Davenport
and Rogers, Ball, Vance, and Venkatesh
Densest local packing diversity. II. Application to three dimensions
The densest local packings of N three-dimensional identical nonoverlapping
spheres within a radius Rmin(N) of a fixed central sphere of the same size are
obtained for selected values of N up to N = 1054. In the predecessor to this
paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305
(2010)], we described our method for finding the putative densest packings of N
spheres in d-dimensional Euclidean space Rd and presented those packings in R2
for values of N up to N = 348. We analyze the properties and characteristics of
the densest local packings in R3 and employ knowledge of the Rmin(N), using
methods applicable in any d, to construct both a realizability condition for
pair correlation functions of sphere packings and an upper bound on the maximal
density of infinite sphere packings. In R3, we find wide variability in the
densest local packings, including a multitude of packing symmetries such as
perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest
local packings of N spheres near a central sphere to minimal-energy
configurations of N+1 points interacting with short-range repulsive and
long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that
they are in general completely different, a result that has possible
implications for nucleation theory. We also compare the densest local packings
to finite subsets of stacking variants of the densest infinite packings in R3
(the Barlow packings) and find that the densest local packings are almost
always most similar, as measured by a similarity metric, to the subsets of
Barlow packings with the smallest number of coordination shells measured about
a single central sphere, e.g., a subset of the FCC Barlow packing. We
additionally observe that the densest local packings are dominated by the
spheres arranged with centers at precisely distance Rmin(N) from the fixed
sphere's center.Comment: 45 pages, 18 figures, 2 table
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