646 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
Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape
We extend the results from the first part of this series of two papers by
examining hyperuniformity in heterogeneous media composed of impenetrable
anisotropic inclusions. Specifically, we consider maximally random jammed
packings of hard ellipses and superdisks and show that these systems both
possess vanishing infinite-wavelength local-volume-fraction fluctuations and
quasi-long-range pair correlations. Our results suggest a strong generalization
of a conjecture by Torquato and Stillinger [Phys. Rev. E. 68, 041113 (2003)],
namely that all strictly jammed saturated packings of hard particles, including
those with size- and shape-distributions, are hyperuniform with signature
quasi-long-range correlations. We show that our arguments concerning the
constrained distribution of the void space in MRJ packings directly extend to
hard ellipse and superdisk packings, thereby providing a direct structural
explanation for the appearance of hyperuniformity and quasi-long-range
correlations in these systems. Additionally, we examine general heterogeneous
media with anisotropic inclusions and show for the first time that one can
decorate a periodic point pattern to obtain a hard-particle system that is not
hyperuniform with respect to local-volume-fraction fluctuations. This apparent
discrepancy can also be rationalized by appealing to the irregular distribution
of the void space arising from the anisotropic shapes of the particles. Our
work suggests the intriguing possibility that the MRJ states of hard particles
share certain universal features independent of the local properties of the
packings, including the packing fraction and average contact number per
particle.Comment: 29 pages, 9 figure
Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings
We show that quasi-long-range (QLR) pair correlations that decay
asymptotically with scaling in -dimensional Euclidean space
, trademarks of certain quantum systems and cosmological
structures, are a universal signature of maximally random jammed (MRJ)
hard-particle packings. We introduce a novel hyperuniformity descriptor in MRJ
packings by studying local-volume-fraction fluctuations and show that
infinite-wavelength fluctuations vanish even for packings with size- and
shape-distributions. Special void statistics induce hyperuniformity and QLR
pair correlations.Comment: 10 pages, 3 figures; changes to figures and text based on review
process; accepted for publication at Phys. Rev. Let
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
Do Binary Hard Disks Exhibit an Ideal Glass Transition?
We demonstrate that there is no ideal glass transition in a binary hard-disk
mixture by explicitly constructing an exponential number of jammed packings
with densities spanning the spectrum from the accepted ``amorphous'' glassy
state to the phase-separated crystal. Thus the configurational entropy cannot
be zero for an ideal amorphous glass, presumed distinct from the crystal in
numerous theoretical and numerical estimates in the literature. This objection
parallels our previous critique of the idea that there is a most-dense random
(close) packing for hard spheres [Torquato et al, Phys. Rev. Lett., 84, 2064
(2000)].Comment: Submitted for publicatio
Collective Coordinate Control of Density Distributions
Real collective density variables [c.f.
Eq.\ref{Equation3})] in many-particle systems arise from non-linear
transformations of particle positions, and determine the structure factor
, where denotes the wave vector. Our objective is to
prescribe and then to find many-particle configurations
that correspond to such a target using a numerical optimization
technique. Numerical results reported here extend earlier one- and
two-dimensional studies to include three dimensions. In addition, they
demonstrate the capacity to control in the neighborhood of
0. The optimization method employed generates
multi-particle configurations for which , , and 1, 2, 4,
6, 8, and 10. The case 1 is relevant for the Harrison-Zeldovich
model of the early universe, for superfluid , and for jammed
amorphous sphere packings. The analysis also provides specific examples of
interaction potentials whose classical ground state are configurationally
degenerate and disordered.Comment: 26 pages, 8 figure
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
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
Nonequilibrium static growing length scales in supercooled liquids on approaching the glass transition
The small wavenumber behavior of the structure factor of
overcompressed amorphous hard-sphere configurations was previously studied for
a wide range of densities up to the maximally random jammed state, which can be
viewed as a prototypical glassy state [A. Hopkins, F. H. Stillinger and S.
Torquato, Phys. Rev. E, 86, 021505 (2012)]. It was found that a precursor to
the glassy jammed state was evident long before the jamming density was reached
as measured by a growing nonequilibrium length scale extracted from the volume
integral of the direct correlation function , which becomes long-ranged
as the critical jammed state is reached. The present study extends that work by
investigating via computer simulations two different atomic models: the
single-component Z2 Dzugutov potential in three dimensions and the
binary-mixture Kob-Andersen potential in two dimensions. Consistent with the
aforementioned hard-sphere study, we demonstrate that for both models a
signature of the glass transition is apparent well before the transition
temperature is reached as measured by the length scale determined from from the
volume integral of the direct correlation function in the single-component case
and a generalized direct correlation function in the binary-mixture case. The
latter quantity is obtained from a generalized Orstein-Zernike integral
equation for a certain decoration of the atomic point configuration. We also
show that these growing length scales, which are a consequence of the
long-range nature of the direct correlation functions, are intrinsically
nonequilibrium in nature as determined by an index that is a measure of
deviation from thermal equilibrium. It is also demonstrated that this
nonequilibrium index, which increases upon supercooling, is correlated with a
characteristic relaxation time scale.Comment: 26 pages, 14 figure
- …