Method for dense packing discovery

Dense crystalline packings of ellipsoids

An ellipsoid, the simplest nonspherical shape, has been extensively used as a model for elongated building blocks for a wide spectrum of molecular, colloidal, and granular systems. Yet the densest packing of congruent hard ellipsoids, which is intimately related to the high-density phase of many condensed matter systems, is still an open problem. We discover an unusual family of dense crystalline packings of self-dual ellipsoids (ratios of the semiaxes α:sqrt[α]:1), containing 24 particles with a quasi-square-triangular (SQ-TR) tiling arrangement in the fundamental cell. The associated packing density ϕ exceeds that of the densest known SM2 crystal [ A. Donev et al., Phys. Rev. Lett. 92, 255506 (2004)10.1103/PhysRevLett.92.255506] for aspect ratios α in (1.365, 1.5625), attaining a maximal ϕ≈0.75806... at α=93/64. We show that the SQ-TR phase derived from these dense packings is thermodynamically stable at high densities over the aforementioned α range and report a phase diagram for self-dual ellipsoids. The discovery of the SQ-TR crystal suggests organizing principles for nonspherical particles and self-assembly of colloidal systems.

Statistical mechanics of the lattice sphere packing problem

We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond previous methods, not only in exploring higher dimensions but also in shedding light on the statistical mechanics underlying the problem in question. We observe evidence of a phase transition in the thermodynamic limit d→∞. In the dimensions explored in the present work, the results are consistent with a first-order crystallization transition but leave open the possibility that a glass transition is manifested in higher dimensions.

Efficient linear programming algorithm to generate the densest lattice sphere packings

Finding the densest sphere packing in d-dimensional Euclidean space R(d) is an outstanding fundamental problem with relevance in many fields, including the ground states of molecular systems, colloidal crystal structures, coding theory, discrete geometry, number theory, and biological systems. Numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations, even for the restricted problem of finding the densest lattice sphere packings. In this paper we apply the Torquato-Jiao packing algorithm, which is a method based on solving a sequence of linear programs, to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19. We show that the TJ algorithm is appreciably more efficient at solving these problems than previously published methods. Indeed, in some dimensions, the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones. We also study the suboptimal local density-maxima solutions (inherent structures or "extreme" lattices) to gain insight about the nature of the topography of the "density" landscape.

Random perfect lattices and the sphere packing problem

Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d=19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily. Their number, however, grows superexponentially with the dimension, so to get an idea of their properties we propose to study a randomized version of the generating algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Monte Carlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best known packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A(d) and D(d)), and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve the Minkowsky bound φ~2(-(0.84±0.06)d), and a competitor in which their packing fraction decreases superexponentially, namely, φ~d(-ad) but with a very small coefficient a=0.06±0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6±0.1.

Dense-packing crystal structures of physical tetrahedra

We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal mathematical tetrahedron into a less ideal, physical, tetrahedron and all the way to the sphere. Thus, we also connect the two well-studied problems of sphere packing and tetrahedron packing on a single axis. Our numerical results uncover a rich optimal-packing behavior, compared to that of other continuous families of particles previously studied. We present four structures as candidates for the optimal packing at different values of asphericity, providing an atlas of crystal structures that might be observed in systems of nanoparticles with tetrahedral symmetry.

Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming

We have formulated the problem of generating dense packings of nonoverlapping, nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive-shrinking cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d-dimensional Euclidean space R(d), 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 R(d) for d=2, 3, 4, 5, and 6 with a diversity of disorder and densities up to the respective maximal densities. A novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures (locally maximally dense and mechanically stable packings), besides the usual disordered ones (such as the maximally random jammed state), with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere. For example, 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 [0.6, 0.7408...], where π/√18 = 0.7408... corresponds to the density of the densest packing. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d=2, 4, 5, and 6. Our jammed sphere packings are characterized and compared to the corresponding packings generated by the well-known Lubachevsky-Stillinger (LS) molecular-dynamics packing algorithm. 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.