We perform computational studies of static packings of a variety of nonspherical particles including circulo-lines, circulo-polygons, ellipses, asymmetric dimers, dumbbells, and others to determine which shapes form packings with fewer contacts than degrees of freedom (hypostatic packings) and which have equal numbers of contacts and degrees of freedom (isostatic packings), and to understand why hypostatic packings of nonspherical particles can be mechanically stable despite having fewer contacts than that predicted from naive constraint counting. To generate highly accurate force- and torque-balanced packings of circulo-lines and cir-polygons, we developed an interparticle potential that gives continuous forces and torques as a function of the particle coordinates. We show that the packing fraction and coordination number at jamming onset obey a masterlike form for all of the nonspherical particle packings we studied when plotted versus the particle asphericity A, which is proportional to the ratio of the squared perimeter to the area of the particle. Further, the eigenvalue spectra of the dynamical matrix for packings of different particle shapes collapse when plotted at the same A. For hypostatic packings of nonspherical particles, we verify that the number of “quartic” modes along which the potential energy increases as the fourth power of the perturbation amplitude matches the number of missing contacts relative to the isostatic value. We show that the fourth derivatives of the total potential energy in the directions of the quartic modes remain nonzero as the pressure of the packings is decreased to zero. In addition, we calculate the principal curvatures of the inequality constraints for each contact in circulo-line packings and identify specific types of contacts with inequality constraints that possess convex curvature. These contacts can constrain multiple degrees of freedom and allow hypostatic packings of nonspherical particles to be mechanically stable.

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:√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)] 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.

The critical behaviors of a granular system at the jamming transition have been extensively studied from both mechanical and thermodynamic perspectives. In this work, we numerically investigate the jamming behaviors of a variety of frictionless non-spherical particles, including spherocylinder, ellipsoid, spherotetrahedron and spherocube. In particular, for a given particle shape, a series of random configurations at different fixed densities are generated and relaxed to minimize interparticle overlaps using the relaxation algorithm. We find that as the jamming point (i.e., point J">J) is approached, the number of iteration steps (defined as the “time-scale” for our systems) required to completely relax the interparticle overlaps exhibits a clear power-law divergence. The dependence of the detailed mathematical form of the power-law divergence on particle shapes is systematically investigated and elucidated, which suggests that the shape effects can be generally categorized as elongation and roundness. Importantly, we show the jamming transition density can be accurately determined from the analysis of time-scale divergence for different non-spherical shapes, and the obtained values agree very well with corresponding ones reported in literature. Moreover, we study the plastic behaviors of over-jammed packings of different particles under a compression–expansion procedure and find that the jamming of ellipsoid is much more robust than other non-spherical particles. This work offers an alternative approximate procedure besides conventional packing algorithms for studying athermal jamming transition in granular system of frictionless non-spherical particles.

We generate a large number of monodisperse disk packings in two dimensions via geometric-based packing algorithms including the relaxation algorithm and the Torquato–Jiao algorithm. Using the geometric-structure approach, a clear boundary of the geometrical feasible region in the order map is found which quite differs from that of the jammed region. For a certain packing density higher than 0.83, the crystalline degree varies in different packing samples. We find that the local hexatic order may increase in two fairly different ways as the system densifies. Therefore, two typical non-equilibrium jammed patterns, termed polycrystal and distorted crystal, are defined at high packing densities. Furthermore, their responses to isotropic compression are investigated using a compression–relaxation molecular dynamic protocol. The distorted crystal pattern is more stable than the polycrystal one with smaller displacements despite its low occurrence frequency. The results are helpful in understanding the structure and phase transition of disk packings.

We apply the ideal octahedron model and the relaxation algorithm in generating octahedron packings. The cubatic order parameter [P4]1">[P4]1, bond-orientational order metric Q6">Q6, and local cubatic order parameter P4local">P4local of the packings are calculated and their correlations with the packing density are investigated in the order maps. The border curve of packing density separates the geometrically feasible and infeasible regions in the order maps. Observing the transition phenomenon on the border curve, we propose the concept of the maximally dense random packing (MDRP) as the densest packing in the random state in which the particle positions and orientations are randomly distributed and there is no nontrivial spatial correlations among particles. The MDRP characterizes the onset of nontrivial spatial correlations among particles. A special packing with a density about 0.7 is found in the order maps and considered to be the MDRP of octahedra. The P4local">P4local is proposed as a new order parameter for octahedron packings, which measures the average order degree in the neighborhoods of particles. The [P4]1">[P4]1, Q6">Q6 and P4local">P4local evaluate the order degree of orientation, bond orientation and local structures, respectively and are applied simultaneously to measure the order degree of the octahedron packings. Their thresholds in the random state are determined by Monte Carlo simulations.

Particle shape plays a crucial role in determining packing characteristics. Real particles in nature usually have rounded corners. In this work, we systematically investigate the rounded corner effect on the dense packings of spherotetrahedral particles. The evolution of dense packing structure as the particle shape continuously deforms from a regular tetrahedron to a sphere is investigated, starting both from the regular tetrahedron and the sphere packings. The dimer crystal and the quasicrystal approximant are used as initial configurations, as well as the two densest sphere packing structures. We characterize the evolution of spherotetrahedron packings from the ideal tetrahedron (s = 0) to the sphere (s = 1) via a single roundness parameter s. The evolution can be partitioned into seven regions according to the shape variation of the packing unit cell. Interestingly, a peak of the packing density Φ is first observed at s ≈ 0.16 in the Φ-s curves where the tetrahedra have small rounded corners. The maximum density of the deformed quasicrystal approximant family (Φ ≈ 0.8763) is slightly larger than that of the deformed dimer crystal family (Φ ≈ 0.8704), and both of them exceed the densest known packing of ideal tetrahedra (Φ ≈ 0.8563).

The disordered packings of tetrahedra often show no obvious macroscopic orientational or positional order for a wide range of packing densities, and it has been found that the local order in particle clusters is the main order form of tetrahedron packings. Therefore, a cluster analysis is carried out to investigate the local structures and properties of tetrahedron packings in this work. We obtain a cluster distribution of differently sized clusters, and peaks are observed at two special clusters, i.e., dimer and wagon wheel. We then calculate the amounts of dimers and wagon wheels, which are observed to have linear or approximate linear correlations with packing density. Following our previous work, the amount of particles participating in dimers is used as an order metric to evaluate the order degree of the hierarchical packing structure of tetrahedra, and an order map is consequently depicted. Furthermore, a constraint analysis is performed to determine the isostatic or hyperstatic region in the order map. We employ a Monte Carlo algorithm to test jamming and then suggest a new maximally random jammed packing of hard tetrahedra from the order map with a packing density of 0.6337.

We investigate the excluded volumes of clusters in tetrahedral particle packing using an ideal tetrahedron model and Monte Carlo simulation. Both the influences of the size and topology of clusters on the excluded volume are studied. We find that the excluded volumes of the dimer composed of two tetrahedra and the wagon wheel composed of five tetrahedra are relatively lower than other cluster forms. For large clusters, the excluded volume decreases when the topology of a cluster approaches the wagon-wheel geometry. The results give an explanation to the cluster distribution which demonstrates that the dimer and wagon wheel are the dominative cluster forms in the packing structure of tetrahedra.

We present a new order metric for tetrahedral particle packing, which is observed to have a strong linear correlation with the packing density. We propose the concept of quasi-random packing to represent the hierarchical random packing structure of clusters. We also find that the nematic order of clusters can be used to classify the ordered and disordered packing of tetrahedra, which is also an indicator for quasi-random packing.

Studies on the macroscopic and microscopic packing properties of nonconvex particles are scarce. As a common concave form, the curved spherocylinder is used in the simulations, and its bending and elongation effects on the random packings are investigated numerically with sphere assembly models and a relaxation algorithm. The aspect ratio is demonstrated to be the main factor regarding the packing density. However, at certain aspect ratios of low densities around 0.3–0.4, the density of curved spherocylinders may increase by 15% more than that of the straight ones, indicating that bending is also a contributor to the packing density. The excluded volume of the curved spherocylinder decreases with the increase of the bending angle, indicating that the excluded volume is applicable in explaining the bending effect on the packing density variation of nonconvex particles. The packings are verified to be randomly distributed in orientation with no significant layering or in-plane order. The local arrangements are further analyzed from the radial distribution function and contact results. The results show that the random packings of nonconvex particles have significant differences and richer characteristics on both the macroscopic and microscopic properties compared with convex objects.

Regular tetrahedra have been demonstrated recently giving high packing density in random configurations. However, it is unknown whether the random-packing density of tetrahedral particles with other shapes can reach an even higher value. A numerical investigation on the random packing of regular and irregular tetrahedral particles is carried out. Shape effects of rounded corner, eccentricity, and height on the packing density of tetrahedral particles are studied. Results show that altering the shape of tetrahedral particles by rounding corners and edges, by altering the height of one vertex, or by lateral displacement of one vertex above its opposite face, all individually have the effect of reducing the random-packing density. In general, the random-packing densities of irregular tetrahedral particles are lower than that of regular tetrahedra. The ideal regular tetrahedron should be the shape which has the highest random-packing density in the family of tetrahedra, or even among convex bodies. An empirical formula is proposed to describe the rounded corner effect on the packing density, and well explains the density deviation of tetrahedral particles with different roundness ratios. The particles in the simulations are verified to be randomly packed by studying the pair correlation functions, which are consistent with previous results. The spherotetrahedral particle model with the relaxation algorithm is effectively applied in the simulations.