Session P53: Disordered Systems: Packing

Analytical Construction of A Dense Packing of Truncated Tetrahedra and Its Melting Properties

Dense polyhedron packings are useful models of a variety of condensed matter and biological systems and have intrigued scientists and mathematicians for centuries. Here, we analytically construct the densest known packing of truncated tetrahedra with a remarkably high packing fraction 207/208=0.995 192, which is amazingly close to unity and strongly implies its optimality. This construction is based on a generalized organizing principle for polyhedra that lack central symmetry that we introduce here. The packing characteristics and equilibrium melting properties of the putative optimal packing as the system undergoes decompression are discussed.   

Packing fraction of dimers and anisotropic objects

We present a statistical theory and computer simulations for the calculation of the average volume in jammed assemblies of dimer shaped objects and other anisotropic particles like spherocylinders. The theory predicts the volume fraction as a function of the coordination number of the particles.   

Statistical Mechanics of Athermal Packings: Incorporating Basin Volumes

We present a first principles formalism for the statistical mechanics of athermal packings subject to driving beyond the weak limit. Edwards hypothesized a statistical mechanics of flat measure associated with packings explored at fixed density. This ensemble has been found to work well in the limit of very weak (but non-zero) driving. Beyond the weak driving limit, the probability measures associated with jammed states become proportional to the volume of basins of attractions associated with the packings on the density landscape. We propose here, a statistical mechanics which takes into consideration the volume of basins of attraction under certain approximations. Further, the statistical mechanics takes into account the protocol by writing the partition function in terms of an integral over protocol dependent generalized coordinates. This will allow an extremum principle to determine states, in these out of equilibrium systems.   

Polydisperse sphere packing in high dimensions, a search for an upper critical dimension

The recently introduced granocentric model for polydisperse sphere packings has been shown to be in good agreement with experimental and simulational data in two and three dimensions. This model relies on two effective parameters that have to be estimated from experimental/simulational results. The non-trivial values obtained allow the model to take into account the essential effects of correlations in the packing. Once these parameters are set, the model provides a full statistical description of a sphere packing for a given polydispersity. We investigate the evolution of these effective parameters with the spatial dimension to see if, in analogy with the upper critical dimension in critical phenomena, there exists a dimension above which correlations become irrelevant and the model parameters can be fixed \textit{a priori} as a function of polydispersity. This would turn the model into a proper theory of polydisperse sphere packings at that upper critical dimension. We perform infinite temperature quench simulations of frictionless polydisperse sphere packings in dimensions 2-8 using a parallel algorithm implemented on a GPGPU. We analyze the resulting packings by implementing an algorithm to calculate the additively weighted Voronoi diagram in arbitrary dimension.   

Constraint percolation on hyperbolic lattices

Constraint percolation models include constraints on the occupation of sites to, for example, better understand the onset of glassiness in glass-forming liquids. The dynamical glass transition in the Fredrickson-Andersen model simplifies to the study of the percolation transition in $k$-core percolation where every occupied site must have at least $k$ occupied neighbors. Other constraint percolation models, such as force-balance percolation, have been introduced to begin to account for mechanical equilibrium on each particle arising during the onset of jamming. To study a mean-field-like version of force-balance percolation in which the directionality of forces becomes important, we consider clusters with occupied particles satisfying the $k=3$-core condition and lying inside a triangle determined by three of its occupied neighbors. The model is constructed on a tessellation of the Poincar\'e disk, thus, bearing a hyperbolic structure. Models on such spaces exhibit mean-field-like behavior and also play an important role in generating geometric frustration in glassy systems. We analytically investigate the conditions under which there exists a transition as well as the underlying nature of the transition. We also present numerical results to compare with our analytical results.   

Cavity method for Edwards ensemble of jammed matter

Two theoretical frameworks have emerged to investigate the problem of random close packings: the Edwards statistical mechanics and the cavity method for glassy systems. Here we propose a model that combines both approaches into a single Hamiltonian imposing force balance constraints and minimization of the volume of the system. The formalism can be put into the framework of constraint optimization problems as recently proposed. The cavity method then solves the problem of force balance providing a prediction of the coordination number of the jammed packing. The model can be applied to spherical frictionless and frictional particles as well as non-spherical particles providing a prediction of the coordination number as a function of the aspect ratio of the particles.   

Glass Transition and Random Close Packing above Three Dimensions

Motivated by a recently identified severe discrepancy between a static and a dynamic theory of glasses, we numerically investigate the behavior of dense hard spheres in spatial dimensions 3 to 12. Our results are consistent with the static replica theory, but disagree with the dynamic mode-coupling theory, indicating that key ingredients of high-dimensional physics are missing from the latter. We also obtain numerical estimates of the random close packing density, which provides new insights into the mathematical problem of packing spheres in large dimension.   

Packing Squares in a Torus

We study the densest packings of N unit squares in a torus (i.e., using periodic, square boundary conditions in 2D) using both analytical methods and simulated annealing. We find a rich array of dense packing solutions: density-one packings when N is the sum of two square integers, a family of ``gapped bricklayer'' Bravais lattice solutions with density N/(N+1), and some surprising non-Bravais lattice configurations -- including lattices of holes, as well as a configuration for N=23 in which not all squares share the same orientation. We assess the entropy of some of these configurations, as well as the frequency and orientation of density-one solutions as N goes to infinity.   

Characterization of Basin Volumes in Mechanically Stable Packings

There are a finite number of distinct mechanically stable (MS) packings in granular systems composed of frictionless particles. For typical packing-generation protocols employed in experimental and numerical studies, the probabilities with which the MS packings occur are highly nonuniform and depend strongly on preparation protocol. Despite intense work, it is extremely difficult to predict {\it a priori} the MS packing probabilities. We describe a novel computational method for calculating the MS packing probabilities by directly measuring the volume of the MS packing `basin of attraction', which we define as the collection of initial points in configuration space at {\it zero packing fraction} that map to a given MS packing by following a particular dynamics in the density landscape. We show that there is a small core region with size $l_c$ surrounding each MS packing in configuration space in which all initial conditions map to a given MS packing. However, we find that the MS packing probabilities are not strongly correlated with $l_c$ and thus they are determined by complex geometric features of the landscape that are distant from the MS packing.   

Hierarchical freezing in a lattice model with a nonperiodic ground state

A recent result in tiling theory provides a two-dimensional lattice model with nearest and next-nearest neighbor interactions that has a limit-periodic ground state. During a slow quench from the high temperature, disordered phase, the ground state emerges through an infinite sequence of phase transitions, all related by renormalizations of the temperature scale with the sequence of critical temperatures approaching zero. As the temperature is decreased, sublattices with increasingly large lattice constants become ordered. Quenching at any finite rate eventually results in glass-like state due to kinetic barriers created by simultaneous freezing on sublattices with different lattice constants.   

Degenerate Quasicrystal of Hard Triangular Bipyramids Stabilized by Entropic Forces

The assembly of hard polyhedra into novel ordered structures has recently received much attention. Here we focus on triangular bipyramids (TBPs)- i.e.~dimers of hard tetrahedra- which pack densely in a simple triclinic crystal with two particles per unit cell [1]. This packing is referred to as the TBP crystal. We show that hard TBPs do not form this densest packing in simulation. Instead, they assemble into a different, far more complicated structure, a dodecagonal quasicrystal, which, in the level of monomers, is identical to the quasicrystal recently discovered in the hard tetrahedron system [2], but the way that tetrahedra pair into TBPs in the nearest neighbor network is random, making it the first degenerate quasicrystal reported in the literature [3]. This notion of degeneracy is in the level of decorating individual tiles and is different from the degeneracy of a quasiperiodic random tiling arising from phason flips [4]. The $(3.4.3^2.4)$ approximant of the quasicrystal is shown to be more stable than the TBP crystal at densities below $79.7\%$.\\[4pt] [1] Chen ER, Engel M, Sharon SC, Disc. Comp. Geom. 44:253 (2010).\\[0pt] [2] Haji-Akbari A, Engel M, et al.~Nature 462:773 (2009).\\[0pt] [3] Haji-Akbari A, Engel M, Glotzer SC, arXiv:1106.5561 [PRL, in press].\\[0pt] [4] Elser V, PRL 54: 1730 (1985)   

A Novel Decomposition of the Structure of Jammed Packings

We use simulations of 2D bidisperse disks to determine the properties of jammed packings and investigate the statistical mechanics of these systems. We have created a novel method for classifying structural subunits of a packing, using the structures to calculate relevant physical quantities. The classification scheme is based on a 20 type decomposition of the Delaunay triangles extracted from the centers of the particles in the packing. We find that the distribution of each type has a universal form, independent of total number of particles N in the packing for N=8-10,000, and that the parameters describing this form saturate as N is increased beyond N=20. We measure the distribution of the particle connections, the area distributions of the different structures, and nearest neighbor distributions. We explore the extent to which the nearest-neighbor distributions can predict the properties of the entire packing.   

8x8 and 10x10 Hyperspace Representations of SU(3) and 10-fold Point-Symmetry Group of Quasicrystals

In order to further elucidate the unexpected 10-fold point-symmetry group structure of quasi-crystals for which the 2011 Nobel Prize in chemistry was awarded to Daniel Shechtman, we explore a correspondence principle between the number of (projective) geometric elements (points[vertices] + lines[edges] + planes[faces]) of primitive cells of periodic or quasi-periodic arrangement of hard or deformable spheres in 3-dimensional space of crystallography and elements of quantum field theory of particle physics [points ( particles, lines ( particles, planes ( currents] and hence construct 8x8 =64 = 28+36 = 26 + 38, and 10x10 =100= 64 + 36 = 74 + 26 hyperspace representations of the SU(3) symmetry of elementary particle physics and quasicrystals of condensed matter (solid state) physics respectively, As a result, we predict the Cabibbo-like angles in leptonic decay of hadrons in elementary-particle physics and the observed 10-fold symmetric diffraction pattern of quasi-crystals.   

Competition of Bergman-type approximants with other packing motifs in the Cu-Zr system

Knowledge about the topological and chemical ordering in metallic liquids and glasses is essential in predicting phase selection and understanding glass formation dynamics. Taking the Cu-Zr system as an example, previous studies have established Bergman-type medium-range ordering (MRO) from a structural analysis with cluster alignment methods [1]. In this study, we examine the thermodynamic stability of a crystalline approximant of Bergman-type quasicrystals [2] against packing geometries existing in other intermetallic compounds for a wide range of Cu compositions. The most stable structures for each structural motif at each Cu composition are obtained using an efficient genetic-algorithm search. Our results show that the Bergman-type approximant structure is thermodynamically favored over other packing geometries at the glass-forming region with Cu compositions around 65{\%}, reaffirming the Bergman-type MRO is the lowest energy in Cu-Zr glasses.\\[4pt] [1] X. W. Fang, C. Z. Wang, Y. X. Yao, Z. J. Ding, and K. M. Ho, Phys. Rev. B 82, 184204 (2010).\\[0pt] [2] G. Bergman J. L. T. Waugh, and L. Pauling, Acta Cryst. 10, 254 (1957).