Dense packings of polyhedra: Platonic and Archimedean solids

Random close packing of semi-flexible polymers in two dimensions: Emergence of local and global order

Through extensive Monte Carlo simulations, we systematically study the effect of chain stiffness on the packing ability of linear polymers composed of hard spheres in extremely confined monolayers, corresponding effectively to 2D films. First, we explore the limit of random close packing as a function of the equilibrium bending angle and then quantify the local and global order by the degree of crystallinity and the nematic or tetratic orientational order parameter, respectively. A multi-scale wealth of structural behavior is observed, which is inherently absent in the case of athermal individual monomers and is surprisingly richer than its 3D counterpart under bulk conditions. As a general trend, an isotropic to nematic transition is observed at sufficiently high surface coverages, which is followed by the establishment of the tetratic state, which in turn marks the onset of the random close packing. For chains with right-angle bonds, the incompatibility of the imposed bending angle with the neighbor geometry of the triangular crystal leads to a singular intra- and inter-polymer tiling pattern made of squares and triangles with optimal local filling at high surface concentrations. The present study could serve as a first step toward the design of hard colloidal polymers with a tunable structural behavior for 2D applications.

Optimal three-dimensional particle shapes for maximally dense saturated packing

Saturated packing is a random packing state of particles widely applied in investigating the physicochemical properties of granular materials. Optimizing particle shape to maximize packing density is a crucial challenge in saturated packing research. The known optimal three-dimensional shape is an ellipsoid with a saturated packing density of 0.437 72(51). In this work, we generate saturated packings of three-dimensional asymmetric shapes, including spherocylinders, cones, and tetrahedra, via the random sequential adsorption algorithm and investigate their packing properties. Results show that the optimal shape of asymmetric spherocylinders gives the maximum density of 0.4338(9), while cones achieve a higher value of 0.4398(10). Interestingly, tetrahedra exhibit two distinct optimal shapes with significantly high densities of 0.4789(19) and 0.4769(18), which surpass all previous results in saturated packing. The study of adsorption kinetics reveals that the two optimal shapes of tetrahedra demonstrate notably higher degrees of freedom and faster growth rates of the particle number. The analysis of packing structures via the density pair-correlation function shows that the two optimal shapes of tetrahedra possess faster transitions from local to global packing densities.

Dense Disordered Jammed Packings of Hard Spherocylinders with a Low Aspect Ratio: A Characterization of Their Structure

This work numerically investigates dense disordered (maximally random) jammed packings of hard spherocylinders of cylinder length L and diameter D by focusing on L/D ∈ [0,2]. It is within this interval that one expects that the packing fraction of these dense disordered jammed packings ϕMRJ hsc attains a maximum. This work confirms the form of the graph ϕMRJ hsc versus L/D: here, comparably to certain previous investigations, it is found that the maximal ϕMRJ hsc = 0.721 ± 0.001 occurs at L/D = 0.45 ± 0.05. Furthermore, this work meticulously characterizes the structure of these dense disordered jammed packings via the special pair-correlation function of the interparticle distance scaled by the contact distance and the ensuing analysis of the statistics of the hard spherocylinders in contact: here, distinctly from all previous investigations, it is found that the dense disordered jammed packings of hard spherocylinders with 0.45 ≲ L/D ≤ 2 are isostatic.

Finding the grain size distribution that produces the densest arrangement in frictional sphere packings: Revisiting and rediscovering the century-old Fuller and Thompson distribution

By means of discrete-element methods, we investigate the joint effects of the grain size distribution (GSD) and contact friction on the structure of three-dimensional samples composed of spherical grains. Specifically, we compress these systems isotropically until jamming and then analyze their structure in terms of density, connectivity, coefficients of uniformity and curvature, and parameters of grading entropy. Our study focuses on power-law GSDs and particularly on the Fuller and Thompson distribution, proposed over a century ago. First, we show that, among the set of GSDs investigated, this particular distribution produces the densest and best-connected systems, falsifying a conjecture recently posed in the literature. Second, we find that the jamming packing fraction can be accurately predicted as a function of simple descriptors of the GSD, but among these descriptors the granular entropy concept proves to be the most useful. This allows for an alternative interpretation of both jamming and grading entropy concepts. Finally, we compare the Fuller and Thompson distribution with two well-known GSDs: that of the Apollonian sphere packing and that towards which granular systems evolve after intensive grain fracturing. Surprisingly, we find that these three GSDs are practically coincident in the limit of large size spans, despite having been introduced or discovered in different scientific contexts (i.e., engineering, mathematics, and earth sciences, respectively).

Descriptor-free unsupervised learning method for local structure identification in particle packings

Local structure identification is of great importance in many scientific and engineering fields. However, mathematical and supervised learning methods mostly rely on specific descriptors of local structure and can only be applied to particular packing configurations. In this work, we propose an improved unsupervised learning method, which is descriptor-free, for local structure identification in particle packing. The point cloud is used as the input of the improved method, which directly comes from spatial positions of particles and does not rely on specific descriptors. The improved method constructs an autoencoder based on the point cloud network combined with Gaussian mixture models for dimension reduction and clustering. Numerical examples show that the improved method performs well in local structure identification of quasicrystal disk and sphere packings, achieving comparable accuracy with previous methods. For disordered packings which have been considered nearly having no local structures, the improved method identifies a nontrivial 7-neighbor motif in the maximally dense random packing of disks, and finds acentric structural motifs in the random close packing of spheres, which demonstrate the ability on identification of new and unknown local structures. The improved unsupervised learning method would help mining information from massive simulation and experimental results, as well as devising new order parameters for particle packings.

Kinetic Frustration Effects on Dense Two-Dimensional Packings of Convex Particles and Their Structural Characteristics

The study of hard-particle packings is of fundamental importance in physics, chemistry, cell biology, and discrete geometry. Much of the previous work on hard-particle packings concerns their densest possible arrangements. By contrast, we examine kinetic effects inevitably present in both numerical and experimental packing protocols. Specifically, we determine how changing the compression/shear rate of a two-dimensional packing of noncircular particles causes it to deviate from its densest possible configuration, which is always periodic. The adaptive shrinking cell (ASC) optimization scheme maximizes the packing fraction of a hard-particle packing by first applying random translations and rotations to the particles and then isotropically compressing and shearing the simulation box repeatedly until a possibly jammed state is reached. We use a stochastic implementation of the ASC optimization scheme to mimic different effective time scales by varying the number of particle moves between compressions/shears. We generate dense, effectively jammed, monodisperse, two-dimensional packings of obtuse scalene triangle, rhombus, curved triangle, lens, and "ice cream cone" (a semicircle grafted onto an isosceles triangle) shaped particles, with a wide range of packing fractions and degrees of order. To quantify these kinetic effects, we introduce the kinetic frustration index K, which measures the deviation of a packing from its maximum possible packing fraction. To investigate how kinetics affect short- and long-range ordering in these packings, we compute their spectral densities χ̃_V(k) and characterize their contact networks. We find that kinetic effects are most significant when the particles have greater asphericity, less curvature, and less rotational symmetry. This work may be relevant to the design of laboratory packing protocols.

Shape or friction? Which of these characteristics drives the shear strength in granular systems?

The shape of the particles and local friction, separately, are known to strongly affect the macroscopic properties of an assembly of grains. But the combined effects of these two parameters still remain poorly described. By means of extensive two dimensional contact dynamics simulations, we perform a systematic analysis of the interplay between friction and shape on strength properties of granular systems. The shape of the particles is varied from disks to triangles, while the friction is varied from 0 to 0.7. We find that the macroscopic friction first increases with angularity, but it may decline (for low friction values), saturate (for intermediates friction values), or continue to increase (for large friction values) for the most angular shapes. In other words, the effect of the particle’s angularity on the shear strength depends on the level of sliding friction. In contrast, the effect of local friction on the shear strength does not depend on the specific properties of shape. The results presented here highlight the subtle coupling existing between shape and friction effects.

Classifying Crystals of Rounded Tetrahedra and Determining Their Order Parameters Using Dimensionality Reduction

Using simulations we study the phase behavior of a family of hard spherotetrahedra, a shape that interpolates between tetrahedra and spheres. We identify 13 close-packed structures, some with packings that are significantly denser than previously reported. Twelve of these are crystals with unit cells of N = 2 or N = 4 particles, but in the shape regime of slightly rounded tetrahedra we find that the densest structure is a quasicrystal approximant with a unit cell of N = 82 particles. All 13 structures are also stable below close packing, together with an additional 14th plastic crystal phase at the sphere side of the phase diagram, and upon sufficient dilution to packing fractions below 50-60% all structures melt. Interestingly, however, upon compressing the fluid phase, self-assembly takes place spontaneously only at the tetrahedron and the sphere side of the family but not in an intermediate regime of tetrahedra with rounded edges. We describe the local environment of each particle by a set of l-fold bond orientational order parameters q̅l, which we use in an extensive principal component analysis. We find that the total packing fraction as well as several particular linear combinations of q̅l rather than individual q̅l's are optimally distinctive, specifically the differences q̅4 - q̅6 for separating tetragonal from hexagonal structures and q̅4-q̅8 for distinguishing tetragonal structures. We argue that these characteristic combinations are also useful as reliable order parameters in nucleation studies, enhanced sampling techniques, or inverse-design methods involving odd-shaped particles in general.

Dense packings of hard circular arcs

This work investigates dense packings of congruent hard infinitesimally thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle θ∈(π,2π]. Differently than those denotable as minor whose subtended angle θ∈[0,π], it is impossible for two hard infinitesimally thin circular arcs with θ∈(π,2π] to arbitrarily closely approach once they are arranged in a configuration, e.g., on top of one another, replicable ad infinitum without introducing any overlap. This makes these hard concave particles, in spite of being infinitesimally thin, most densely pack with a finite number density. This raises the question as to what are these densest packings and what is the number density that they achieve. Supported by Monte Carlo numerical simulations, this work shows that one can analytically construct compact closed circular groups of hard major circular arcs in which a specific, θ-dependent, number of them (counter) clockwise intertwine. These compact closed circular groups then arrange on a triangular lattice. These analytically constructed densest-known packings are compared to corresponding results of Monte Carlo numerical simulations to assess whether they can spontaneously turn up.

Supramolecular Self-Assembly of Perylene Bisimide-Based Rigid Giant Tetrahedra

Recently, ordered structures constructed from rigid three-dimensional (3D) shaped polyhedra have been drawing general interest, with the tetrahedron being the simplest one but showing complicated assembly behaviors. Rigid tetrahedron building blocks have been shown to form quasicrystalline and crystalline phases with high packing fractions by both simulation and experiments. Nevertheless, the study of 3D tetrahedral building blocks is limited, especially in the field of supramolecular self-assembly. Here, we present an experimental study of rigid giant tetrahedral molecules constructed by attaching four bulky polyhedral oligomeric silsesquioxane (POSS) cages to a tetrahedral perylene bisimide (PBI) scaffold. Self-assembly of these giant tetrahedra is mediated by π-π interaction between the tetrahedral PBI-based scaffolds and their overall tetrahedral symmetry. A monolithic nearly centimeter-sized hexagonal supramolecular structure was observed in the giant tetrahedron with short flexible linkers between PBI and POSS cages, while a micrometer-sized crystalline helical structure formed in that with completely rigid aromatic linkers. Their significant difference in electrical conductivity could be explained by two completely different packing models of the giant tetrahedra.

Self-assembly of anisotropic nanoparticles into functional superstructures

Self-assembly of colloidal nanoparticles (NPs) into superstructures offers a flexible and promising pathway to manipulate the nanometer-sized particles and thus make full use of their unique properties. This bottom-up strategy builds a bridge between the NP regime and a new class of transformative materials across multiple length scales for technological applications. In this field, anisotropic NPs with size- and shape-dependent physical properties as self-assembly building blocks have long fascinated scientists. Self-assembly of anisotropic NPs not only opens up exciting opportunities to engineer a variety of intriguing and complex superlattice architectures, but also provides access to discover emergent collective properties that stem from their ordered arrangement. Thus, this has stimulated enormous research interests in both fundamental science and technological applications. This present review comprehensively summarizes the latest advances in this area, and highlights their rich packing behaviors from the viewpoint of NP shape. We provide the basics of the experimental techniques to produce NP superstructures and structural characterization tools, and detail the delicate assembled structures. Then the current understanding of the assembly dynamics is discussed with the assistance of in situ studies, followed by emergent collective properties from these NP assemblies. Finally, we end this article with the remaining challenges and outlook, hoping to encourage further research in this field.

Microstructural analysis of sheared polydisperse polyhedral grains

This article presents an analysis of the shear strength of numerical samples composed of polyhedra presenting a grain size dispersion. Previous numerical studies using, for instance, disks, polygons, and spheres, have consistently shown that microstructural properties linked to the fabric and force transmission allow granular media to exhibit a constant shear resistance although packing fraction can dramatically change as a broader grain-size distribution is considered. To have a complete picture of such behavior, we developed a set of numerical experiments in the frame of the discrete element method to test the shear strength of polydisperse samples composed of polyhedral grains. Although the contact networks and force transmission are quite more complex for such generalized grain shape, we can verify that the shear strength independence still holds up for 3D regular polyhedra. We make a particular focus upon the role of different contact types in the assemblies and their relative contributions to the granular connectivity and sample strength. The invariance of shear strength at the macroscopic scale results deeply linked to fine compensations at the microstructural level involving geometrical and force anisotropies of the assembly.

Hard convex lens-shaped particles: Characterization of dense disordered packings

Among the family of hard convex lens-shaped particles (lenses), the one with aspect ratio equal to 2/3 is "optimal" in the sense that the maximally random jammed (MRJ) packings of such lenses achieve the highest packing fraction ϕ_{MRJ}≃0.73 [G. Cinacchi and S. Torquato, Soft Matter 14, 8205 (2018)1744-683X10.1039/C8SM01519H]. This value is only a few percent lower than ϕ_{DKP}=0.76210⋯, the packing fraction of the corresponding densest-known crystalline (degenerate) packings [G. Cinacchi and S. Torquato, J. Chem. Phys. 143, 224506 (2015)JCPSA60021-960610.1063/1.4936938]. By exploiting the appreciably reduced propensity that a system of such optimal lenses has to positionally and orientationally order, disordered packings of them are progressively generated by a Monte Carlo method-based procedure from the dilute equilibrium isotropic fluid phase to the dense nonequilibrium MRJ state. This allows us to closely monitor how the (micro)structure of these packings changes in the process of formation of the MRJ state. The gradual changes undergone by the many structural descriptors calculated here can coherently and consistently be traced back to the gradual increase in contacts between the hard particles until the isostatic mean value of ten contact neighbors per lens is reached at the effectively hyperuniform MRJ state. Compared to the MRJ state of hard spheres, the MRJ state of such optimal lenses is denser (less porous), more disordered, and rattler-free. This set of characteristics makes them good glass formers. It is possible that this conclusion may also hold for other hard convex uniaxial particles with a correspondingly similar aspect ratio, be they oblate or prolate, and that, by using suitable biaxial variants of them, that set of characteristics might further improve.

Predicting maximally random jammed packing density of non-spherical hard particles via analytical continuation of fluid equation of state

We developed a formalism for accurately predicting the density of MRJ packing state of a wide spectrum of congruent non-spherical hard particles in 3D via analytical fluid EOS.

Phase diagram of a system of hard cubes on the cubic lattice

We study the phase diagram of a system of 2×2×2 hard cubes on a three-dimensional cubic lattice. Using Monte Carlo simulations, we show that the system exhibits four different phases as the density of cubes is increased: disordered, layered, sublattice ordered, and columnar ordered. In the layered phase, the system spontaneously breaks up into parallel slabs of size 2×L×L where only a very small fraction cubes do not lie wholly within a slab. Within each slab, the cubes are disordered; translation symmetry is thus broken along exactly one principal axis. In the solidlike sublattice-ordered phase, the hard cubes preferentially occupy one of eight sublattices of the cubic lattice, breaking translational symmetry along all three principal directions. In the columnar phase, the system spontaneously breaks up into weakly interacting parallel columns of size 2×2×L, where only a very small fraction cubes do not lie wholly within a column. Within each column, the system is disordered, and thus translational symmetry is broken only along two principal directions. Using finite-size scaling, we show that the disordered-layered phase transition is continuous, while the layered-sublattice and sublattice-columnar transitions are discontinuous. We construct a Landau theory written in terms of the layering and columnar order parameters which is able to describe the different phases that are observed in the simulations and the order of the transitions. Additionally, our results near the disordered-layered transition are consistent with the O(3) universality class perturbed by cubic anisotropy as predicted by the Landau theory.

Revealing a Vacancy Analog of the Crowdion Interstitial in Simple Cubic Crystals

Vacancies in simple cubic crystals of hard cubes are known to delocalize over one-dimensional chains of several lattice sites. Here, we use computer simulations to examine the structure and dynamics of vacancies in simple cubic crystals formed by hard cubes, right rhombic prisms (slanted cubes), truncated cubes, and particles interacting via a soft isotropic pair potential. We show that these vacancies form a vacancy analog of the crowdion interstitial, generating a strain field which follows a soliton solution of the sine-Gordon equation, and diffusing via a persistent random walk. Surprisingly, we find that the structure of these "voidions" is not significantly affected by changes in density, vacancy concentration, and even particle interaction. We explain this structure quantitatively using a one-dimensional model that includes the free-energy barrier particles have to overcome to slide between lattice sites and the effective pair interaction along this line. We argue that voidions are a robust phenomenon in systems of repulsive particles forming simple cubic crystals.

Hard convex lens-shaped particles: metastable, glassy and jammed states

We generate and study dense positionally and/or orientationally disordered, including jammed, monodisperse packings of hard convex lens-shaped particles (lenses). Relatively dense isotropic fluid configurations of lenses of various aspect ratios are slowly compressed via a Monte Carlo method based procedure. Under this compression protocol, while 'flat' lenses form a nematic fluid phase (where particles are positionally disordered but orientationally ordered) and 'globular' lenses form a plastic solid phase (where particles are positionally ordered but orientationally disordered), 'intermediate', neither 'flat' nor 'globular', lenses do not form either mesophase. In general, a crystal solid phase (where particles are both positionally and orientationally ordered) does not spontaneously form during lengthy numerical simulation runs. In correspondence to those volume fractions at which a transition to the crystal solid phase would occur in equilibrium, a 'downturn' is observed in the inverse compressibility factor versus volume fraction curve beyond which this curve behaves essentially linearly. This allows us to estimate the volume fraction at jamming of the dense non-crystalline packings so generated. These packings are nematic for 'flat' lenses and plastic for 'globular' lenses, while they are robustly isotropic for 'intermediate' lenses, as confirmed by the calculation of the τ order metric, among other quantities. The structure factors S(k) of the corresponding jammed states tend to zero as the wavenumber k goes to zero, indicating they are effectively hyperuniform (i.e., their infinite-wavelength density fluctuations are anomalously suppressed). Among all possible lens shapes, 'intermediate' lenses with aspect ratio around 2/3 are special because they are those that reach the highest volume fractions at jamming while being positionally and orientationally disordered and these volume fractions are as high as those reached by nematic jammed states of 'flat' lenses and plastic jammed states of 'globular' lenses. All of their attributes, taken together, make such 'intermediate' lens packings particularly good glass-forming materials.

Evolutions of packing properties of perfect cylinders under densification and crystallization

Cylindrical particles are ubiquitous in nature and industry, and a cylinder is a representative shape of rod-like particles. However, the disordered packing results of cylinders in previous studies are quite inconsistent with each other. In this work, we obtain the MRJ (maximally random jammed) packings and the MDRPs (maximally dense random packings) of perfect cylinders with the aspect ratio (height/diameter) 0.2 ≤ w ≤ 6.0 using the ASC (adaptive shrinking cell) algorithm and the IMC (inverse Monte Carlo) method, respectively. The optimal aspect ratio corresponding to the maximal packing density is w = 0.9 in the MRJ state, while the value is w = 1.2 in the MDRP state. Then we investigate the evolutions of packing properties of perfect cylinders under densification and crystallization. We compare the different final packing states generated via the two methods with different compression rates and order constraints. In the densification procedure, we generate jammed and random packings of cylinders with various compression rates via the ASC and IMC method, respectively. When decreasing the compression rate, we find that the packing density increases but the optimal w remains the same in both methods. In the crystallization procedure, the order constraint in the IMC method is gradually released which means the degree of order in the packings is allowed to increase, and we find that the optimal w shifts from 1.2 to 0.9 while the packing density increases as well. Meanwhile, the random packings evolve into the jammed packings in the crystallization procedure which reflects the competition mechanism between the randomness and jamming. These results also indicate that the optimal w is solely related to the degree of order in the cylinder packings but not determined by the protocol or packing density. Furthermore, a uniform shape elongation effect on the random-packing densities of various shaped particles is found via a new proposed definition of the scaled aspect ratio. Finally, a rough linear relationship between the mean and standard deviation of the reduced Voronoi cell volumes is obtained only for the random packings. Our findings should lead to a better understanding toward the jammed and random packings and are helpful in guiding the granular material design.

Reconfigurable Polymer Shells on Shape-Anisotropic Gold Nanoparticle Cores

Reconfigurable hybrid nanoparticles made by decorating flexible polymer shells on rigid inorganic nanoparticle cores can provide a unique means to build stimuli-responsive functional materials. The polymer shell reconfiguration has been expected to depend on the local core shape details, but limited systematic investigations have been undertaken. Here, two literature methods are adapted to coat either thiol-terminated polystyrene (PS) or polystyrene-poly(acrylic acid) (PS-b-PAA) shells onto a series of anisotropic gold nanoparticles of shapes not studied previously, including octahedron, concave cube, and bipyramid. These core shapes are complex, rendering shell contours with nanoscale details (e.g., local surface curvature, shell thickness) that are imaged and analyzed quantitatively using the authors' customized analysis codes. It is found that the hybrid nanoparticles based on the chosen core shapes, when coated with the above two polymer shells, exhibit distinct shell segregations upon a variation in solvent polarity or temperature. It is demonstrated for the PS-b-PAA-coated hybrid nanoparticles, the shell segregation is maintained even after a further decoration of the shell periphery with gold seeds; these seeds can potentially facilitate subsequent deposition of other nanostructures to enrich structural and functional diversity. These synthesis, imaging, and analysis methods for the hybrid nanoparticles of anisotropically shaped cores can potentially aid in their predictive design for materials reconfigurable from the bottom up.

Frustrated packing in a granular system under geometrical confinement

Optimal packings of uniform spheres are solved problems in two and three dimensions. The main difference between them is that the two-dimensional ground state can be easily achieved by simple dynamical processes while in three dimensions, this is impossible due to the difference in the local and global optimal packings. In this paper we show experimentally and numerically that in 2 + ε dimensions, realized by a container which is in one dimension slightly wider than the spheres, the particles organize themselves in a triangular lattice, while touching either the front or rear side of the container. If these positions are denoted by up and down the packing problem can be mapped to a 1/2 spin system. At first it looks frustrated with spin-glass like configurations, but the system has a well defined ground state built up from isosceles triangles. When the system is agitated, it evolves very slowly towards the potential energy minimum through metastable states. We show that the dynamics is local and is driven by the optimization of the volumes of 7-particle configurations and by the vertical interaction between touching spheres.

Inertial shear flow of assemblies of frictionless polygons: Rheology and microstructure

Motivated by the understanding of shape effects in granular materials, we numerically investigate the macroscopic and microstructural properties of anisotropic dense assemblies of frictionless polydisperse rigid pentagons in shear flow, and compare them with similar systems of disks. Once subjected to large cumulative shear strains their rheology and microstructure are investigated in uniform steady states, depending on inertial number I, which ranges from the quasistatic limit ([Formula: see text]) to 0.2. In the quasistatic limit both systems are devoid of Reynolds dilatancy, i.e., flow at their random close packing density. Both macroscopic friction angle [Formula: see text], an increasing function of I , and solid fraction [Formula: see text], a decreasing function of I, are larger with pentagons than with disks at small I, but the differences decline for larger I and, remarkably, nearly vanish for [Formula: see text]. Under growing I , the depletion of contact networks is considerably slower with pentagons, in which increasingly anisotropic, but still well-connected force-transmitting structures are maintained throughout the studied range. Whereas contact anisotropy and force anisotropy contribute nearly equally to the shear strength in disk assemblies, the latter effect dominates with pentagons at small I, while the former takes over for I of the order of 10^-2. The size of clusters of grains in side-to-side contact, typically comprising more than 10 pentagons in the quasistatic limit, very gradually decreases for growing I.

Statistical characterization of microstructure of packings of polydisperse hard cubes

Polydisperse packings of cubic particles arise in several important problems. Examples include zeolite microcubes that represent catalytic materials, fluidization of such microcubes in catalytic reactors, fabrication of new classes of porous materials with precise control of their morphology, and several others. We present the results of detailed and extensive simulation and microstructural characterization of packings of nonoverlapping polydisperse cubic particles. The packings are generated via a modified random sequential-addition algorithm. Two probability density functions (PDFs) for the particle-size distribution, the Schulz and log-normal PDFs, are used. The packings are analyzed, and their random close-packing density is computed as a function of the parameters of the two PDFs. The maximum packing fraction for the highest degree of polydispersivity is estimated to be about 0.81, much higher than 0.57 for the monodisperse packings. In addition, a variety of microstructural descriptors have been calculated and analyzed. In particular, we show that (i) an approximate analytical expression for the structure factor of Percus-Yevick fluids of polydisperse hard spheres with the Schulz PDF also predicts all the qualitative features of the structure factor of the packings that we study; (ii) as the packings become more polydisperse, their behavior resembles increasingly that of an ideal system-"ideal gas"-with little or no correlations; and (iii) the mean survival time and mean relaxation time of a diffusing species in the packings increase with increasing degrees of polydispersivity.

Phase behaviour and gravity-directed self assembly of hard convex spherical caps

We investigate the phase behaviour and self-assembly of convex spherical caps using Monte Carlo simulations. This model is used to represent the main features observed in experimental colloidal particles with mushroom-cap shape [Riley et al., Langmuir, 2010, 26, 1648]. The geometry of this non-centrosymmetric convex model is fully characterized by the aspect ratio χ* defined as the spherical cap height to diameter ratio. We use NPT Monte Carlo simulations combined with free energy calculations to determine the most stable crystal structures and the phase behaviour of convex spherical caps with different aspect ratios. We find a variety of crystal structures at each aspect ratio, including plastic and dimer-based crystals; small differences in chemical potential between the structures with similar morphology suggest that convex spherical caps have the tendency to form polycrystalline phases rather than crystallising into a single uniform structure. With the exception of plastic crystals observed at large aspect ratios (χ* > 0.75), crystallisation kinetics seem to be too slow, hindering the spontaneous formation of ordered structures. As an alternative, we also present a study of directing the self-assembly of convex spherical caps via sedimentation onto solid substrates. This study contributes to show how small changes to particle shape can significantly alter the self-assembly of crystal structures, and how a simple gravity field and a template can substantially enhance the process.

Maximally dense random packings of cubes and cuboids via a novel inverse packing method

The packings of cubes and cuboids (i.e., "elongated" or "compressed" cubes) are ubiquitous in nature. The high symmetry and space-tiling nature of such particles make them easily packable in dense configurations with a high degree of orientational and translational order. In this paper, we devise a novel inverse packing method that enables one to generate dense hard-particle packings with a controllable degree of disorder/order quantified by predefined order metrics via stochastic Monte Carlo optimizations. We employ the inverse packing method to generate and investigate the maximally dense random packings (MDRPs) of hard cubes and cuboids with aspect ratio α, in which a series of newly introduced normalized local cubatic order parameters sensitive to the onset of any spatial order in packings of cubes and cuboids is minimized. The density of the MDRP of cubes is φ ≈ 0.637, which increases as the shape deviates from the cube limit (α = 1) and reaches the maximal values for cuboids with aspect ratios α = 0.7 or 1.5. These special α values associated with local density extrema are almost identical for those associated with the random packings of spherocylinders, spheroids and superellipsoids, suggesting a universal influence of shape elongation on random packing density. Our inverse packing method can be readily utilized to study the MDRPs of other hard particles and the normalized local cubatic order parameter introduced here is applicable to other shaped particles characterized by three principal axes.

Orientationally Ordered Silicon Nanocrystal Cuboctahedra in Superlattices

Uniform silicon nanocrystals were synthesized with cuboctahedral shape and passivated with 1-dodecene capping ligands. Transmission electron microscopy, electron diffraction, and grazing incidence wide-angle and small-angle X-ray scattering show that these soft cuboctahedra assemble into face-centered cubic superlattices with orientational order. The preferred nanocrystal orientation was found to depend on the orientation of the superlattices on the substrate, indicating that the interactions with the substrate and assembly kinetics can influence the orientation of faceted nanocrystals in superlattices.

Packing of nonoverlapping cubic particles: Computational algorithms and microstructural characteristics

Packing of cubic particles arises in a variety of problems, ranging from biological materials to colloids and the fabrication of new types of porous materials with controlled morphology. The properties of such packings may also be relevant to problems involving suspensions of cubic zeolites, precipitation of salt crystals during CO_{2} sequestration in rock, and intrusion of fresh water in aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We present a detailed simulation and microstructural characterization of packings of nonoverlapping monodisperse cubic particles, following up on our preliminary results [H. Malmir et al., Sci. Rep. 6, 35024 (2016)2045-232210.1038/srep35024]. A modification of the random sequential addition (RSA) algorithm has been developed to generate such packings, and a variety of microstructural descriptors, including the radial distribution function, the face-normal correlation function, two-point probability and cluster functions, the lineal-path function, the pore-size distribution function, and surface-surface and surface-void correlation functions, have been computed, along with the specific surface and mean chord length of the packings. The results indicate the existence of both spatial and orientational long-range order as the the packing density increases. The maximum packing fraction achievable with the RSA method is about 0.57, which represents the limit for a structure similar to liquid crystals.

Binary mixtures of disks and elongated particles: Texture and mechanical properties

We analyze the shear strength and microstructure of binary granular mixtures consisting of disks and elongated particles by varying systematically both the mixture ratio and degree of homogeneity (from homogeneous to fully segregated). The contact dynamics method is used for numerical simulations with rigid particles interacting by frictional contacts. A counterintuitive finding of this work is that the shear strength, packing fraction, and, at the microscopic scale, the fabric, force, and friction anisotropies of the contact network are all nearly independent of the degree of homogeneity. In other words, homogeneous mixtures have the same strength properties as segregated packings of the two particle shapes. In contrast, the shear strength increases with the proportion of elongated particles correlatively with the increase of the corresponding force and fabric anisotropies. By a detailed analysis of the contact network topology, we show that various contact types contribute differently to force transmission and friction mobilization.

Competing Interactions between Various Entropic Forces toward Assembly of Pt3Ni Octahedra into a Body-Centered Cubic Superlattice

Anisotropic nanocrystal assembled supercrystals with open superlattices (SLs) manifest novel and unique properties, but poor understanding of the nucleation/growth mechanisms limits their design and fabrication for practical applications. Using highly anisotropic Pt3Ni octahedral nanocrystals, we have grown large single supercrystals with an open body-centered cubic (bcc) superlattice that has a low filling factor of 26.8%. Synchrotron-based X-ray structural reconstruction fully revealed the coherence of translational and orientational orderings and determined that the constituent octahedra arrange themselves with the vertex-to-vertex and face-to-face configurations along the SL[100] and SL[111] directions, respectively. The large face-to-face separation and flexible vertex-to-vertex elastic contact provided the rattle space and supporting axis for local rotations of Pt3Ni octahedra within the bcc superlattice. Development of orientational disordering along with robust preservation of translational ordering during the heating process of a supercrystal in the oleic acid wetting environment confirmed the dominance of rotational entropy of hard octahedra in the formation of the open bcc superlattice. Ultimate achievement of dynamic equilibrium between the vertex-oriented elastic repulsions and the face-oriented attractions of surface-coating ligands governs the structural and mechanical stability of the supercrystal. This discovery provides significant insights into the design and control of geometrical shapes for the fabrication of highly anisotropic nanocrystals into desired open superlattices with tailored optical and electronic properties.

Convergence Properties of Crystal Structure Prediction by Quasi-Random Sampling

Generating sets of trial structures that sample the configurational space of crystal packing possibilities is an essential step in the process of ab initio crystal structure prediction (CSP). One effective methodology for performing such a search relies on low-discrepancy, quasi-random sampling, and our implementation of such a search for molecular crystals is described in this paper. Herein we restrict ourselves to rigid organic molecules and, by considering their geometric properties, build trial crystal packings as starting points for local lattice energy minimization. We also describe a method to match instances of the same structure, which we use to measure the convergence of our packing search toward completeness. The use of these tools is demonstrated for a set of molecules with diverse molecular characteristics and as representative of areas of application where CSP has been applied. An important finding is that the lowest energy crystal structures are typically located early and frequently during a quasi-random search of phase space. It is usually the complete sampling of higher energy structures that requires extended sampling. We show how the procedure can first be refined, through targetting the volume of the generated crystal structures, and then extended across a range of space groups to make a full CSP search and locate experimentally observed and lists of hypothetical polymorphs. As the described method has also been created to lie at the base of more involved approaches to CSP, which are being developed within the Global Lattice Energy Explorer (Glee) software, a few of these extensions are briefly discussed.

Confined disordered strictly jammed binary sphere packings

Disordered jammed packings under confinement have received considerably less attention than their bulk counterparts and yet arise in a variety of practical situations. In this work, we study binary sphere packings that are confined between two parallel hard planes and generalize the Torquato-Jiao (TJ) sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)] to obtain putative maximally random jammed (MRJ) packings that are exactly isostatic with high fidelity over a large range of plane separation distances H, small to large sphere radius ratio α, and small sphere relative concentration x. We find that packing characteristics can be substantially different from their bulk analogs, which is due to what we term "confinement frustration." Rattlers in confined packings are generally more prevalent than those in their bulk counterparts. We observe that packing fraction, rattler fraction, and degree of disorder of MRJ packings generally increase with H, though exceptions exist. Discontinuities in the packing characteristics as H varies in the vicinity of certain values of H are due to associated discontinuous transitions between different jammed states. When the plane separation distance is on the order of two large-sphere diameters or less, the packings exhibit salient two-dimensional features; when the plane separation distance exceeds about 30 large-sphere diameters, the packings approach three-dimensional bulk packings. As the size contrast increases (as α decreases), the rattler fraction dramatically increases due to what we call "size-disparity" frustration. We find that at intermediate α and when x is about 0.5 (50-50 mixture), the disorder of packings is maximized, as measured by an order metric ψ that is based on the number density fluctuations in the direction perpendicular to the hard walls. We also apply the local volume-fraction variance σ(τ)(2)(R) to characterize confined packings and find that these packings possess essentially the same level of hyperuniformity as their bulk counterparts. Our findings are generally relevant to confined packings that arise in biology (e.g., structural color in birds and insects) and may have implications for the creation of high-density powders and improved battery designs.

Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres

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).