455 research outputs found
Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra
The determination of the densest packings of regular tetrahedra (one of the
five Platonic solids) is attracting great attention as evidenced by the rapid
pace at which packing records are being broken and the fascinating packing
structures that have emerged. Here we provide the most general analytical
formulation to date to construct dense periodic packings of tetrahedra with
four particles per fundamental cell. This analysis results in six-parameter
family of dense tetrahedron packings that includes as special cases recently
discovered "dimer" packings of tetrahedra, including the densest known packings
with density . This study strongly suggests that
the latter set of packings are the densest among all packings with a
four-particle basis. Whether they are the densest packings of tetrahedra among
all packings is an open question, but we offer remarks about this issue.
Moreover, we describe a procedure that provides estimates of upper bounds on
the maximal density of tetrahedron packings, which could aid in assessing the
packing efficiency of candidate dense packings.Comment: It contains 25 pages, 5 figures
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
Density of States for a Specified Correlation Function and the Energy Landscape
The degeneracy of two-phase disordered microstructures consistent with a
specified correlation function is analyzed by mapping it to a ground-state
degeneracy. We determine for the first time the associated density of states
via a Monte Carlo algorithm. Our results are described in terms of the
roughness of the energy landscape, defined on a hypercubic configuration space.
The use of a Hamming distance in this space enables us to define a roughness
metric, which is calculated from the correlation function alone and related
quantitatively to the structural degeneracy. This relation is validated for a
wide variety of disordered systems.Comment: Accepted for publication in Physical Review Letter
Stochastic reconstruction of sandstones
A simulated annealing algorithm is employed to generate a stochastic model
for a Berea and a Fontainebleau sandstone with prescribed two-point probability
function, lineal path function, and ``pore size'' distribution function,
respectively. We find that the temperature decrease of the annealing has to be
rather quick to yield isotropic and percolating configurations. A comparison of
simple morphological quantities indicates good agreement between the
reconstructions and the original sandstones. Also, the mean survival time of a
random walker in the pore space is reproduced with good accuracy. However, a
more detailed investigation by means of local porosity theory shows that there
may be significant differences of the geometrical connectivity between the
reconstructed and the experimental samples.Comment: 12 pages, 5 figure
Geometrical Ambiguity of Pair Statistics. I. Point Configurations
Point configurations have been widely used as model systems in condensed
matter physics, materials science and biology. Statistical descriptors such as
the -body distribution function is usually employed to characterize
the point configurations, among which the most extensively used is the pair
distribution function . An intriguing inverse problem of practical
importance that has been receiving considerable attention is the degree to
which a point configuration can be reconstructed from the pair distribution
function of a target configuration. Although it is known that the pair-distance
information contained in is in general insufficient to uniquely determine
a point configuration, this concept does not seem to be widely appreciated and
general claims of uniqueness of the reconstructions using pair information have
been made based on numerical studies. In this paper, we introduce the idea of
the distance space, called the space. The pair distances of a
specific point configuration are then represented by a single point in the
space. We derive the conditions on the pair distances that can be
associated with a point configuration, which are equivalent to the
realizability conditions of the pair distribution function . Moreover, we
derive the conditions on the pair distances that can be assembled into distinct
configurations. These conditions define a bounded region in the
space. By explicitly constructing a variety of degenerate point configurations
using the space, we show that pair information is indeed
insufficient to uniquely determine the configuration in general. We also
discuss several important problems in statistical physics based on the
space.Comment: 28 pages, 8 figure
Dense Packings of Superdisks and the Role of Symmetry
We construct the densest known two-dimensional packings of superdisks in the
plane whose shapes are defined by |x^(2p) + y^(2p)| <= 1, which contains both
convex-shaped particles (p > 0.5, with the circular-disk case p = 1) and
concave-shaped particles (0 < p < 0.5). The packings of the convex cases with p
1 generated by a recently developed event-driven molecular dynamics (MD)
simulation algorithm [Donev, Torquato and Stillinger, J. Comput. Phys. 202
(2005) 737] suggest exact constructions of the densest known packings. We find
that the packing density (covering fraction of the particles) increases
dramatically as the particle shape moves away from the "circular-disk" point (p
= 1). In particular, we find that the maximal packing densities of superdisks
for certain p 6 = 1 are achieved by one of the two families of Bravais lattice
packings, which provides additional numerical evidence for Minkowski's
conjecture concerning the critical determinant of the region occupied by a
superdisk. Moreover, our analysis on the generated packings reveals that the
broken rotational symmetry of superdisks influences the packing characteristics
in a non-trivial way. We also propose an analytical method to construct dense
packings of concave superdisks based on our observations of the structural
properties of packings of convex superdisks.Comment: 15 pages, 8 figure
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
Reply to "Comment on `Jamming at zero temperature and zero applied stress: The epitome of disorder' "
We answer the questions raised by Donev, Torquato, Stillinger, and Connelly
in their "Comment on "Jamming at zero temperature and zero applied stress: The
epitome of disorder.' " We emphasize that we follow a fundamentally different
approach than they have done to reinterpret random close packing in terms of
the "maximally random jammed" framework. We define the "maximally random jammed
packing fraction" to be where the largest number of initial states, chosen
completely randomly, have relaxed final states at the jamming threshold in the
thermodynamic limit. Thus, we focus on an ensemble of states at the jamming
threshold, while DTSC are interested in determining the amount of order and
degree of jamming for a particular configuration. We also argue that
soft-particle systems are as "clean" as those using hard spheres for studying
jammed packings and point out the benefits of using soft potentials
Dense sphere packings from optimized correlation functions
Elementary smooth functions (beyond contact) are employed to construct pair
correlation functions that mimic jammed disordered sphere packings. Using the
g2-invariant optimization method of Torquato and Stillinger [J. Phys. Chem. B
106, 8354, 2002], parameters in these functions are optimized under necessary
realizability conditions to maximize the packing fraction phi and average
number of contacts per sphere Z. A pair correlation function that incorporates
the salient features of a disordered packing and that is smooth beyond contact
is shown to permit a phi of 0.6850: this value represents a 45% reduction in
the difference between the maximum for congruent hard spheres in three
dimensions, pi/sqrt{18} ~ 0.7405, and 0.64, the approximate fraction associated
with maximally random jammed (MRJ) packings in three dimensions. We show that,
surprisingly, the continued addition of elementary functions consisting of
smooth sinusoids decaying as r^{-4} permits packing fractions approaching
pi/sqrt{18}. A translational order metric is used to discriminate between
degrees of order in the packings presented. We find that to achieve higher
packing fractions, the degree of order must increase, which is consistent with
the results of a previous study [Torquato et al., Phys. Rev. Lett. 84, 2064,
2000].Comment: 26 pages, 9 figures, 1 table; added references, fixed typos,
simplified argument and discussion in Section IV
Modeling Heterogeneous Materials via Two-Point Correlation Functions: I. Basic Principles
Heterogeneous materials abound in nature and man-made situations. Examples
include porous media, biological materials, and composite materials. Diverse
and interesting properties exhibited by these materials result from their
complex microstructures, which also make it difficult to model the materials.
In this first part of a series of two papers, we collect the known necessary
conditions on the standard two-point correlation function S2(r) and formulate a
new conjecture. In particular, we argue that given a complete two-point
correlation function space, S2(r) of any statistically homogeneous material can
be expressed through a map on a selected set of bases of the function space. We
provide new examples of realizable two-point correlation functions and suggest
a set of analytical basis functions. Moreover, we devise an efficient and
isotropy- preserving construction algorithm, namely, the Lattice-Point
algorithm to generate realizations of materials from their two- point
correlation functions based on the Yeong-Torquato technique. Subsequent
analysis can be performed on the generated images to obtain desired macroscopic
properties. These developments are integrated here into a general scheme that
enables one to model and categorize heterogeneous materials via two-point
correlation functions.Comment: 37 pages, 26 figure
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