Packings Of Non-spherical Particles Research Articles

Disordered packings of binary mixtures of dimer particles

Disordered packings of non-spherical particles and their mixtures are abundant in nature, but have so far attracted only few systematic studies. Previous investigations of binary mixtures of specific convex shapes have established two generic properties: (i) the existence of a unique density maximum when shape or mixture composition of the two species are varied; (ii) the validity of an ideal mixing law indicating that the packing density is independent of the segregation state. These findings were so far only observed for mixtures of convex particles such as spherocylinders, ellipsoids, and spheres. Here, we investigate the packing properties of binary mixtures of frictionless dimer particles simulated by a gravitational pouring protocol in LAMMPS. Our results demonstrate the validity of (i, ii) also for such packings of non-convex particles. Moreover, we investigate the contact statistics of these packings to elucidate the microstructural features that underlie (i, ii). Our results show that the contact number per species also satisfies a simple mixing law and that similar microscopic rearrangements of contacts as in monodisperse dimer packings accompany the formation of the density peak in binary mixtures largely independent of the mixture composition.

Characterization of void space, large-scale structure, and transport properties of maximally random jammed packings of superballs

Dense, disordered packings of particles are useful models of low-temperature amorphous phases of matter, biological systems, granular media, and colloidal systems. The study of dense packings of nonspherical particles enables one to ascertain how rotational degrees of freedom affect packing behavior. Here, we study superballs, a large family of deformations of the sphere, defined in three dimensions by $|{x}_{1}{|}^{2p}+|{x}_{2}{|}^{2p}+{|{x}_{3}|}^{2p}\ensuremath{\le}1$, where $p\ensuremath{\in}(0,\ensuremath{\infty})$ is a deformation parameter indicating to what extent the shape deviates from a sphere. As $p$ increases from the sphere point ($p=1$), the superball tends to a cuboidal shape and approaches a cube in the $p\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. As $p\ensuremath{\rightarrow}0.5$, it approaches an octahedron, becomes a concave body with octahedral symmetry for $p<0.5$, and approaches a three-dimensional cross in the limit $p\ensuremath{\rightarrow}0$. Previous characterization of superball packings has shown that they have a maximally random jammed (MRJ) state, whose properties (e.g., packing fraction $\ensuremath{\phi}$, average contact number $\overline{Z}$) vary nonanalytically as $p$ diverges from unity. Here, we use an event-driven molecular dynamics algorithm to produce MRJ superball packings with $0.85\ensuremath{\le}p\ensuremath{\le}1.50$. To supplement the previous work on such packings, we characterize their large-scale structure by examining the behaviors of their structure factors $S(Q)$ and spectral densities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(Q)$, as the wave number $Q$ tends to zero, and find that these packings are effectively hyperuniform for all values of $p$ examined. We show that the mean width $\overline{w}$ is a useful length scale to make distances dimensionless in order to compare systematically superballs of different shape. Moreover, we compute the complementary cumulative pore-size distribution $F(\ensuremath{\delta})$ and find that the pore sizes tend to decrease as $|1\ensuremath{-}p|$ increases. From $F(\ensuremath{\delta})$, we estimate how the fluid permeability, mean survival time, and principal diffusion relaxation time vary as a function of $p$. Additionally, we compute the diffusion ``spreadability'' $\mathcal{S}(t)$ [Torquato, Phys. Rev. E 104, 054102 (2021)] of these packings and find that the long-time power-law scaling indicates these packings are hyperuniform with a small-$Q$ power law scaling of the spectral density ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(Q)\ensuremath{\sim}{Q}^{\ensuremath{\alpha}}$ with an exponent $\ensuremath{\alpha}$ that ranges from 0.64 at the sphere point to 0.32 at $p=1.50$, and decreases as $|1\ensuremath{-}p|$ increases. Each of the structural characteristics computed here exhibits an extremum at the sphere point and varies nonanalytically as $p$ departs the sphere point. We find the nonanalytic behavior in $\ensuremath{\phi}$ on either side of the sphere point is nearly linear, and determine that the rattler fraction ${\ensuremath{\phi}}_{R}$ decreases rapidly as $|1\ensuremath{-}p|$ increases. Our results can be used to help inform the design of colloidal or granular materials with targeted densities and transport properties.

Mechanical response of packings of nonspherical particles: A case study of two-dimensional packings of circulo-lines.

We investigate the mechanical response of jammed packings of circulo-lines in two spatial dimensions, interacting via purely repulsive, linear spring forces, as a function of pressure P during athermal, quasistatic isotropic compression. The surface of a circulo-line is defined as the collection of points that is equidistant to a line; circulo-lines are composed of a rectangular central shaft with two semicircular end caps. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power law, 〈G(P)〉∼P^{β}, with β∼0.5, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of 〈G(P)〉 over the same range of pressure for aspect ratios R≳1.2. However, the power-law scaling exponent β∼0.8-0.9 is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose 〈G〉 into separate contributions from geometrical families, G_{f}, and from changes in the interparticle contact network, G_{r}, such that 〈G〉=〈G_{f}〉+〈G_{r}〉. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase and decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution 〈G_{f}〉 is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent for 〈G(P)〉.

Experimental study of particle shape dependence of avalanches inside a rotating drum

We investigate the avalanches of spherical and non-spherical granular particles inside half-filled rotating drums. The time series of the center of gravity of the particle assemblies are obtained via image analysis and their single-sided amplitude (SSA) spectra are analyzed. The spectra features of this new indicator turn out to be characteristic for the avalanches, in terms of the existence of peaks in the low-frequency range and the decay rate of high frequency components. The SSA spectrum has a peak for the packings of non-spherical particles but not for the spherical particles. The high frequency part is characterized by a power law decay 1/ f a (a > 0) . A 1/ f -decay is found only for the spherical particles. For the packings of cornered particles, the exponents significantly deviate from a = 1. As 1/ f spectra are often associated with self-organized criticality and therefore a scale invariance of the dynamics, we may conclude that there is no scale-invariant structure for granular avalanches. Considering the small number of particles and the regularity of convex particle shapes being used, the spectral features revealed in this study could be utilized for validating particle simulations.

Shape effects on packing properties of bi-axial superellipsoids

We generate the maximally random jammed (MRJ) packings and maximally dense random packings (MDRPs) of bi-axially elongated superballs with different shape parameters. On the MRJ packing state, the packings of ellipsoids, general superellipsoids and cuboids are on different random degrees and the shape elongation effects on the disordered-packing densities are not uniform. The effects of order degree and particle shape are compounded. However, we find uniform shape elongation effects on the MDRP state. Bi-axial elongation will improve the random-packing densities. The packing densities reach the maximum when the aspect ratio is about 1.5 for all the surface shape parameters and oblateness. The influence of order degree is eliminated on the MDRP state, which demonstrates the advantages of the MDRP state as a platform for comparing the packing density of disordered packings. Our work helps to explore the shape effects on disordered packings of non-spherical particles.

Explicit contact force model for superellipses by Fourier transform and application to superellipse packing

This paper presents a method to establish explicit force models for identical superellipses. Currently for two superellipses in contact, the overlap parameters must be solved by using numerical iteration but cannot be explicitly calculated. Our method is firstly to numerically solve these solutions in the comprehensive range of the orientation angles with the angles varying discretely. Then using two-dimensional (2D) discrete Fourier transform (DFT), the discrete solutions as a function of two angles are transformed to the frequency domain. The transformed terms are sorted according to their magnitudes, and those below a certain value are filtered in the inverse discrete Fourier transform (IDFT). This finally results in simplified Fourier series that can approximate overlap parameters explicitly. The method was realized for a wide range of superellipse shapes and the errors between the approximate Fourier series and the original solutions were analyzed and shown to be small. In addition, the established explicit force model was applied to the simulation of the packing of superellipses. The results show that the explicit force model established by this method can be effectively used in DEM simulations. And the packing of superellipses is complicatedly dependent on the squareness, aspect ratio and friction. These findings provide a novel way to establish the explicit force models for non-spherical particles and also improve our understanding on the packing of non-spherical particles.

Two-dimensional densely ordered packings of non-convex bending and assembled rods

The investigation of the problem of particle packing has provided basic insights into the structure, symmetry, and physical properties of condensed matter. Dense packings of non-spherical particles have many applications, both in research and industry. We report the two-dimensional dense packing patterns of bending and assembled rods, which are non-convexly deformed from simple objects and modeled as entangled particles. Monte Carlo simulations and further analytical constructions are carried out to explore possible densely packed structures. Two typical densely packed structures of C-bending rods are found, and their packing densities are identified as being functions of the aspect ratio and central angle. Six shapes of assembled rods, representing the combined deformations of rods, are employed in simulations with the packing structures classified into three types. The dense packing density of each packing pattern is derived as a function of different shape parameters. In contrast with the case of disordered packings, both the shape and order are verified to affect the packing density.

An Equivalent Spherical Particle System to Describe Characteristics of Flow in a Dense Packing of Non-spherical Particles

Effective characteristics, such as the effective particle diameter and hydraulic radius, are usually used to determine the pressure loss associated with single-phase flow through an isotropic, dense non-spherical granular packing. These quantities can be considered as those of an equivalent spherical particle system. This paper reviews different methods to select effective particle diameters taking into account the effect of particle shape. A new method is proposed to determine an equivalent spherical particle system with both the hydraulic radius and the pressure loss equivalent to those in the actual non-spherical particle packing. This equivalent system can be used to determine the characteristics of flow in the actual system. The analyses provide theoretical justification for adoption of proper effective particle diameters and porosity as well as the dependency of Ergun–Kozeny constant on the shape factor of particles. It is also shown theoretically that the critical Reynolds number for Darcian flow depends on the porosity and the shape factor of particles.

Random-packing properties of spheropolyhedra

It is of both theoretical and engineering significance to understand the random packings of non-spherical particles. However, apart from the well-discussed aspect ratio, studies on particle shapes continuously evolved from sphere to ideal polyhedra are still lacking, which represents the roundness effect. In this work, we investigate two packing states, namely random close packing (RCP) and saturated random packing (SRP), of spheropolyhedra (SPP), including three shape families, namely spherotetrahedron (SPT), spherocube (SPC), and spherooctahedron (SPO). We observe a common density peak phenomenon of these two states for all the families with respect to sphericity. Specifically, the RCP densities can reach ~0.746, ~0.750, and ~0.731 for the SPT, SPC, and SPO respectively, comparable with the crystalline density ~0.74 for spheres. Density peaks of the SRP locate at the sphericity ~0.96 for all the families. Additionally, the local structural analysis reveals the complex dependencies of order parameters on the roundness, including positional order q6 and facet alignment Δ. The SRP states are more random in particle position than orientation. The dimer clusters formed by particles sharing common facets are also explored. We find that the facet number of a single particle is positively correlated with the q6 yet negatively correlated with the Δ and the cluster ratio for all the polyhedron-like shapes at the RCP. Furthermore, the mechanism of excluded volume can explain the density peak of both the RCP and SRP for all the families and even partly reproduce the general trend of the RCP density for SPT.

Weak Shape Anisotropy Leads to a Nonmonotonic Contribution to Crowding, Impacting Protein Dynamics under Physiologically Relevant Conditions.

The effect of a nonspherical particle shape on the dynamics in crowded solutions presents a significant challenge for a comprehensive understanding of interaction and structural relaxation in biological and soft matter. We report that small deviations from a spherical shape induce a nonmonotonic contribution to the crowding effect on the short-time cage diffusion compared with spherical systems, using molecular dynamics simulations with mesoscale hydrodynamics of a multiparticle collision dynamics fluid in semidilute systems with volume fractions smaller than 0.35. We show that the nonmonotonic effect due to anisotropy is caused by the combination of a reduced relative mobility over the entire concentration range and a looser and less homogeneous cage packing of nonspherical particles. Our finding stresses that nonsphericity induces new complexity, which cannot be accounted for in effective sphere models, and is of great interest in applications such as formulations as well as for the fundamental understanding of soft matter in general and crowding effects in living cells in particular.

Transport properties of concrete-like granular materials interacted by their microstructures and particle components

Granular materials as typical soft matter, their transport properties play significant roles in durability and service life in relevant practical engineering structures. Physico-mechanical properties of materials are generally dependent of their microstructures including interfacial and porous characteristics. The formation of such microstructures is directly related to particle components in granular materials. Understanding the interactive mechanism of particle components, microstructures, and transport properties is a problem of great interest in materials research community. The resulting rigorous component-structure-property relations are also valuable for material design and microstructure optimization. This review article describes state-of-the-art progresses on modeling particle components, interfacial and porous configurations and incorporating these internal structural characteristics into modeling transport properties of granular materials. We mainly focus on three issues involving the simulation for geometrical components, the quantitative characterization for interfacial and porous microstructures, and the modeling strategies for diffusive behaviors of granular materials. In the first aspect, in-depth reviews are presented to realize complex morphologies of geometrical particles, to detect the overlap between adjacent nonspherical particles, and to simulate the random packings of nonspherical particles. In the second aspect, we emphasize the development progresses on the interfacial thickness and porosity distribution, the interfacial volume fraction, and the continuum percolation of soft particles representing compliant interfaces and discrete pores. In the final aspect, a literature review is also provided on modeling of transport properties on the forefront of the effective diffusion and anomalous diffusion in multiphase granular materials. Finally, some conclusions and perspectives for future studies are provided.

Hypostatic jammed packings of frictionless nonspherical particles.

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.

Numerical Analysis of Packings of Nonspherical Particles

Numerical Analysis of Packings of Nonspherical Particles

Pressure drop study on packings of differently shaped particles in milli-structured channels

Pressure drop measurements and calculations were carried out for differently shaped packings of particles (315–1000µm) in milli-structured reactors with slit widths of 1.5–3mm. It could be shown that the wall effect in these packings with D/d ratios between 6 and about 14 must be taken into account while laminar flow dominates for the employed Reynolds numbers in the range 0.5<Reε<12. Although developed for cylindrical packings, earlier proposed pressure drop correlations including wall effect terms can be also applied to slit-type channels. For prediction of pressure drops over packings of non-spherical particles it is recommended to evaluate the results with the volume-equivalent sphere diameter which is corrected by the sphericity instead of the Sauter diameter. An exact prediction of the pressure drop, however, is only possible if the sphericity is used as a fitting parameter. Furthermore µ-tomographic scans of real packings were used to describe the geometry for pore scale resolving simulations of the flow and pressure drop calculations using the lattice-Boltzmann method. While an excellent agreement between measurements and simulations was achieved for the non-spherical catalyst particles, the simulations slightly overestimated the pressure drop for the glass beads. These deviations are most probably due to artefacts occurring during the binarization procedure of the scans.

Voronoi analysis of the packings of non-spherical particles

We present a structural analysis of the packings of identical non-spherical particles based on Voronoi cells. The packings are generated by discrete element method (DEM) simulations. The particles include axisymmetric ellipsoidal particles from oblates to prolates and cylindrical particles from disks to rods. The Voronoi cells are constructed by virtue of space discretization and surface reconstruction, which is shown to be universal for different shapes. The effects of particle aspect ratio and sliding friction coefficient on the properties of Voronoi cells, including the reduced volume, reduced surface area and sphericity, are quantified. The reduced volume and surface area are found to observe log-normal distributions, while their mean values and standard deviations have different dependencies on particle shape and friction. By analyzing the correlations and using inherent relationships between different Voronoi cell properties, we establish a group of universal equations to predict these distributions according to particle sphericity and overall packing fraction. Such findings can not only improve our understanding on the packings of non-spherical particles but also provide a basis for evaluating the transport properties and advancing the statistical mechanics theory for granular matter composed of non-spherical particles.

Quantitative structural analysis of simulated granular packings of non-spherical particles

A set of computationally generated granular packings of frictionless grains is statistically analyzed using tools from stochastic geometry. We consider both the graph of the solid phase (formed using the particle mid-points) and the pore-phase. Structural characteristics rooted in the analysis of random point processes are seen to yield valuable insights into the underlying structure of granular systems. The graph of the solid phase is analyzed using traditional measures such as edge length and coordination number, as well as more instructive measures of the overall transport properties such as geometric tortuosity, where significant differences are observed in the windedness of paths through the different particle graphs considered. In contrast, the distributions of pore-phase characteristics have a similar shape for all considered granular packings. Interestingly, it is found that prolate and oblate ellipsoid packings show a striking similarity between their solid-phase graphs as well as between their pore-phase graphs.

Modified collective rearrangement sphere-assembly algorithm for random packings of nonspherical particles: Towards engineering applications

A modified collective rearrangement algorithm, based on the sphere-assembly representation of the particle shape, is presented for simulating random packings of nonspherical particles and arbitrary shapes. Compared to other collective rearrangement methods, the modification introduced by this algorithm consists in i) avoiding the particle-to-container shrinking procedure by allowing particles to sequentially settle down and ii) in a stability constraint applied to all the particles in the packing. The coupling of these criteria ensures that all the particles are stable and contacting each other, allowing for an unambiguous detection of contacts, which is important in the evaluation of the effective properties desired in many engineering applications, such as percolation thresholds and effective conductivity. The effect of the internal parameters of the algorithm is investigated, showing that random close packings can be obtained. The algorithm is applied to simulate packings of rigid ellipsoids and cylinders with different aspect ratios, which are compared with simulation results provided by other packing algorithms, showing the consistency of our method. Simulations of inter-penetrating particles, mixtures of particles with different shapes and packings of agglomerates are shown, which confirm the applicability of the method to a broad range of packing problems of practical interest and, in particular, for fuel cell applications.

Precise algorithm to generate random sequential addition of hard hyperspheres at saturation

The study of the packing of hard hyperspheres in d-dimensional Euclidean space R^{d} has been a topic of great interest in statistical mechanics and condensed matter theory. While the densest known packings are ordered in sufficiently low dimensions, it has been suggested that in sufficiently large dimensions, the densest packings might be disordered. The random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time saturation limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for nearly saturated configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. U. Uche, and F. H. Stillinger, Phys. Rev. E 74, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space in a large simulation box using finite computational time with heretofore unattained precision and across the widest range of dimensions (2≤d≤8). We have also calculated the packing and covering densities, pair correlation function g(2)(r), and structure factor S(k) of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed "decorrelation" principle, and the degree of "hyperuniformity" (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles.

Mean-field theory of random close packings of axisymmetric particles

Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis, in particular, for shapes like dimers, spherocylinders and ellipsoids of revolution. Here we present a mean-field formalism to estimate the packing density of axisymmetric non-spherical particles. We derive an analytic continuation from the sphere that provides a phase diagram predicting that, for the same coordination number, the density of monodisperse random packings follows the sequence of increasing packing fractions: spheres <oblate ellipsoids <prolate ellipsoids <dimers <spherocylinders. We find the maximal packing densities of 73.1% for spherocylinders and 70.7% for dimers, in good agreement with the largest densities found in simulations. Moreover, we find a packing density of 73.6% for lens-shaped particles, representing the densest random packing of the axisymmetric objects studied so far.

Organizing principles for dense packings of nonspherical hard particles: Not all shapes are created equal

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids and certain smoothly shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. All of our organizing principles are applied to and tested against the most comprehensive set of both convex and concave particle shapes examined to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres, and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, "lens-shaped" particles, square pyramids, and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles.