105 research outputs found
Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra
Systems of hard nonspherical particles exhibit a variety of stable phases
with different degrees of translational and orientational order, including
isotropic liquid, solid crystal, rotator and a variety of liquid crystal
phases. In this paper, we employ a Monte Carlo implementation of the
adaptive-shrinking-cell (ASC) numerical scheme and free-energy calculations to
ascertain with high precision the equilibrium phase behavior of systems of
congruent Archimedean truncated tetrahedra over the entire range of possible
densities up to the maximal nearly space-filling density. In particular, we
find that the system undergoes two first-order phase transitions as the density
increases: first a liquid-solid transition and then a solid-solid transition.
The isotropic liquid phase coexists with the Conway-Torquato (CT) crystal phase
at intermediate densities. At higher densities, we find that the CT phase
undergoes another first-order phase transition to one associated with the
densest-known crystal. We find no evidence for stable rotator (or plastic) or
nematic phases. We also generate the maximally random jammed (MRJ) packings of
truncated tetrahedra, which may be regarded to be the glassy end state of a
rapid compression of the liquid. We find that such MRJ packings are
hyperuniform with an average packing fraction of 0.770, which is considerably
larger than the corresponding value for identical spheres (about 0.64). We
conclude with some simple observations concerning what types of phase
transitions might be expected in general hard-particle systems based on the
particle shape and which would be good glass formers
A method for dense packing discovery
The problem of packing a system of particles as densely as possible is
foundational in the field of discrete geometry and is a powerful model in the
material and biological sciences. As packing problems retreat from the reach of
solution by analytic constructions, the importance of an efficient numerical
method for conducting \textit{de novo} (from-scratch) searches for dense
packings becomes crucial. In this paper, we use the \textit{divide and concur}
framework to develop a general search method for the solution of periodic
constraint problems, and we apply it to the discovery of dense periodic
packings. An important feature of the method is the integration of the unit
cell parameters with the other packing variables in the definition of the
configuration space. The method we present led to improvements in the
densest-known tetrahedron packing which are reported in [arXiv:0910.5226].
Here, we use the method to reproduce the densest known lattice sphere packings
and the best known lattice kissing arrangements in up to 14 and 11 dimensions
respectively (the first such numerical evidence for their optimality in some of
these dimensions). For non-spherical particles, we report a new dense packing
of regular four-dimensional simplices with density
and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 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
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