Local origin of global contact numbers in frictional ellipsoid packings

Evident structural anisotropies arising from near-zero particle asphericity in granular spherocylinder packings

With magnetic resonance imaging experiments, we study packings of granular spherocylinders with merely 2% asphericity. Evident structural anisotropies across all length scales are identified. Most interestingly, the global nematic order decreases with increasing packing fraction, while the local contact anisotropy shows an opposing trend. We attribute this counterintuitive phenomenon to a competition between gravity-driven ordering aided by frictional contacts and a geometric frustration effect at the marginally jammed state. It is also surprising to notice that such slight particle asphericity can trigger non-negligible correlations between contact-level and mesoscale structures, manifested in drastically different nonaffine structural rearrangements upon compaction from that of granular spheres. These observations can help improve statistical mechanical models for the orientational order transformation of nonspherical granular particle packings, which involves complex interplays between particle shape, frictional contacts, and external force field.

Spheroid models to elaborate the broken symmetry and equivalent volume of molecules in crystalline phase

Dense packing of particles has provided powerful models to elaborate the important structural features of matter in various systems such as liquid, glassy, and crystalline phases. The simplest sphere packing models can represent and capture salient properties of the building blocks for covalent, metallic, and ionic crystals; it, however, becomes insufficient to reflect the broken symmetry of the commonly anisotropic molecules in molecular crystals. Here, we develop spheroid models with a minimal degree of anisotropy, which serve as a simple geometrical representation for a rich spectrum of molecules-including both isotropic and anisotropic, convex and concave ones-in crystalline phases. Our models are determined via an inverse packing approach: Given a molecular crystal, an optimal spheroid model is constructed using a contact diagram, which depicts the packing relationship between neighboring molecules within the crystal. The spheroid models are capable of accurately capturing the broken symmetry and characterizing the equivalent volume of molecules in the crystalline phases. Moreover, our model retrieves such molecular information from low-quality x-ray diffraction data with poorly resolved structures, and by using soft spheroids, it can also describe the packing behavior in cocrystals.

Mixing properties of bi-disperse ellipsoid assemblies: mean-field behaviour in a granular matter experiment

The structure and spatial statistical properties of amorphous ellipsoid assemblies have profound scientific and industrial significance in many systems, from cell assays to granular materials. This paper uses a fundamental theoretical relationship for mixture distributions to explain the observations of an extensive X-ray computed tomography study of granular ellipsoidal packings. We study a size-bi-disperse mixture of two types of ellipsoids of revolutions that have the same aspect ratio of α ≈ 0.57 and differ in size, by about 10% in linear dimension, and compare these to mono-disperse systems of ellipsoids with the same aspect ratio. Jammed configurations with a range of packing densities are achieved by employing different tapping protocols. We numerically interrogate the final packing configurations by analyses of the local packing fraction distributions calculated from the Voronoi diagrams. Our main finding is that the bi-disperse ellipsoidal packings studied here can be interpreted as a mixture of two uncorrelated mono-disperse packings, insensitive to the compaction protocol. Our results are consolidated by showing that the local packing fraction shows no correlation beyond their first shell of neighbours in the binary mixtures. We propose a model of uncorrelated binary mixture distribution that describes the observed experimental data with high accuracy. This analysis framework will enable future studies to test whether the observed mean-field behaviour is specific to the particular granular system or the specific parameter values studied here or if it is observed more broadly in other bi-disperse non-spherical particle systems.

Approximate Packing of Binary Mixtures of Cylindrical Particles

Particle packing plays an essential role in industry and chemical engineering. In this work, the discrete element method is used to generate the cylindrical particles and densify the binary cylindrical particle mixtures under the poured packing conditions. The influences of the aspect ratio and volume fraction of particles on the packing structure are measured by planar packing fraction. The Voronoi tessellation is used to quantify the porous structure of packing. The cumulative distribution functions of local packing fractions and the probability distributions of the reduced free volume of Voronoi cells are calculated to describe the local packing characteristics of binary mixtures with different volume fractions. As a result, it is observed that particles with larger aspect ratios in the binary mixture tend to orient randomly, and the particles with smaller aspect ratios have a preferentially horizontal orientation. Results also show that the less dense packings are obtained for mixtures with particles of higher aspect ratios and mixtures with a larger fraction of elongated cylindrical particles.

Preparation and Tracking of Oblate Core-Shell Polymethyl-Methacrylate Ellipsoids

Although single-particle level studies on prolate ellipsoidal colloids are relatively abundant, similar studies on oblate ellipsoids are rare because suitable model systems are scarcely available. Here, we present the preparation of monodisperse hard core-shell oblate ellipsoids that can be imaged and tracked in 3D with confocal laser scanning microscopy. Using a thermomechanical squeezing method, we transform spherical core-shell polymethyl-methacrylate (PMMA) particles into oblate ellipsoids. We show how the shape polydispersity as well as the aspect ratio of the obtained oblate ellipsoids can be controlled. In addition, we discuss how the core-shell geometry limits the range of aspect ratios because of the different viscoelastic properties of the cross-linked PMMA core and linear PMMA shell. We further demonstrate imaging of the core-shell oblate dispersions on a single-particle level in real space and time and the tracking of position and orientation using our recently developed tracking algorithm for anisotropic core-shell colloids. Our results thus provide the tools for the future investigation of the behavior of oblate ellipsoids, especially in dense suspensions.

X-ray nanotomography of dry colloidal packings

Random packings are crucial in understanding arrangement and geometry of particles. Random packings of dry small particles may be subject to adhesion or friction, as expected theoretically and numerically. We explore experimentally random packings of dry colloids with X-ray nanotomography that directly provides three-dimensional structural and geometric information of dry colloidal packings. We find that dry colloidal packings, as characterized by contact number and packing density, are quite consistent with adhesive loose packings that significantly deviate from random loose packings for hard spheres. This study may offer direct evidence for adhesive loose packings comprising dry small particles, as proven by X-ray nanotomography.

Structural universality in disordered packings with size and shape polydispersity

We numerically investigate disordered jammed packings with both size and shape polydispersity, using frictionless superellipsoidal particles. We implement the set Voronoi tessellation technique to evaluate the local specific volume, i.e., the ratio of cell volume over particle volume, for each individual particle. We focus on the average structural properties for different types of particles binned by their sizes and shapes. We generalize the basic observation that the larger particles are locally packed more densely than the smaller ones in a polydisperse-sized packing into systems with coupled particle shape dispersity. For this purpose, we define the normalized free volume vf to measure the local compactness of a particle and study its dependency on the normalized particle size A. The definition of vf relies on the calibrated monodisperse specific volume for a certain particle shape. For packings with shape dispersity, we apply the previously introduced concept of equivalent diameter for a non-spherical particle to define A properly. We consider three systems: (A) linear superposition states of mixed-shape packings, (B) merely polydisperse-sized packings, and (C) packings with coupled size and shape polydispersity. For (A), the packing is simply considered as a mixture of different subsystems corresponding to monodisperse packings for different shape components, leading to A = 1, and vf = 1 by definition. We propose a concise model to estimate the shape-dependent factor αc, which defines the equivalent diameter for a certain particle. For (B), vf collapses as a function of A, independent of specific particle shape and size polydispersity. Such structural universality is further validated by a mean-field approximation. For (C), we find that the master curve vf(A) is preserved when particles possess similar αc in a packing. Otherwise, the dispersity of αc among different particles causes the deviation from vf(A). These findings show that a polydisperse packing can be estimated as the combination of various building blocks, i.e., bin components, with a universal relation vf(A).

Universality of internal structure characteristics in granular media under shear

We examine the signatures of internal structure emerged from quasistatic shear responses of granular materials based on three-dimensional discrete element simulations. Granular assemblies consisting of spheres or nonspherical particles of different polydispersity are sheared from different initial densities and under different loading conditions (drained or undrained) steadily to reach the critical state (a state featured by constant stress and constant volume). The radial distribution function used to measure the packing structure is found to remain almost unchanged during the shearing process, regardless of the initial states or loading conditions of an assembly. Its specific form, however, varies with polydispersities in both grain size and grain shape. Set Voronoi tessellation is employed to examine the characteristics of local volume and anisotropy, and deformation. The local inverse solid fraction and anisotropy index are found following inverse Weibull and log-normal distributions, respectively, which are unique at the critical states. With further normalization, an invariant distribution for local volume and anisotropy is observed whose function can be determined by the polydispersities in both particle size and grain shape but bears no relevance to initial densities or loading conditions (or paths). An invariant Gaussian distribution is found for the local deformation for spherical packings, but no invariant distribution can be found for nonspherical packings where the distribution of normalized local volumetric strain is dependent on initial states.

Jammed packings of 3D superellipsoids with tunable packing fraction, coordination number, and ordering

We carry out numerical studies of static packings of frictionless superellipsoidal particles in three spatial dimensions. We consider more than 200 different particle shapes by varying the three shape parameters that define superellipsoids. We characterize the structural and mechanical properties of both disordered and ordered packings using two packing-generation protocols. We perform athermal quasi-static compression simulations starting from either random, dilute configurations (Protocol 1) or thermalized, dense configurations (Protocol 2), which allows us to tune the orientational order of the packings. In general, we find that superellipsoid packings are hypostatic, with coordination number zJ < ziso, where ziso = 2df and df = 5 or 6 depending on whether the particles are axi-symmetric or not. Over the full range of orientational order, we find that the number of quartic modes of the dynamical matrix for the packings always matches the number of missing contacts relative to the isostatic value. This result suggests that there are no mechanically redundant contacts for ordered, yet hypostatic packings of superellipsoidal particles. Additionally, we find that the packing fraction at jamming onset for disordered packings of superellipsoidal depends on at least two particle shape parameters, e.g. the asphericity A and reduced aspect ratio β of the particles.

Geometric effects in random assemblies of ellipses

Assemblies of anisotropic particles commonly appear in studies of active many-body systems. However, in two dimensions, the geometric ramifications of the finite density of such objects are not entirely understood. To fully characterize these effects, we perform an in-depth study of random assemblies generated by a slow compression of frictionless elliptical particles. The obtained configurations are then analysed using the Set Voronoi tessellation, which takes the particle shape into account. Not only do we analyse most scalar and vectorial morphological measures, which are commonly discussed in the literature or which have recently been addressed in experiments, but we also systematically explore the correlations between them. While in a limited range of parameters similarities with findings in 3D assemblies could be identified, important differences are found when a broad range of aspect ratios and packing fractions are considered. The data discussed in this study should thus provide a unique reference set such that geometric effects and differences from random assemblies could be clearly identified in more complex systems, including ones with soft and active particles that are typically found in biological systems.

Vibrational properties of two-dimensional dimer packings near the jamming transition

Jammed particulate systems composed of various shapes of particles undergo the jamming transition as they are compressed or decompressed. To date, sphere packings have been extensively studied in many previous works, where isostaticity at the transition and scaling laws with the pressure of various quantities, including the contact number and the vibrational density of states, have been established. Additionally, much attention has been paid to nonspherical packings, and particularly recent work has made progress in understanding ellipsoidal packings. In this work, we study the dimer packings in two dimensions, which have been much less understood than systems of spheres and ellipsoids. We first study the contact number of dimers near the jamming transition. It turns out that packings of dimers have "rotational rattlers," each of which still has a free rotational motion. After correcting this effect, we show that dimers become isostatic at the jamming, and the excess contact number obeys the same critical law and finite-size scaling law as those of spheres. We next study the vibrational properties of dimers near the transition. We find that the vibrational density of states of dimers exhibits two characteristic plateaus that are separated by a peak. The high-frequency plateau is dominated by the translational degree of freedom, while the low-frequency plateau is dominated by the rotational degree of freedom. We establish the critical scaling laws of the characteristic frequencies of the plateaus and the peak near the transition. In addition, we present detailed characterizations of the real space displacement fields of vibrational modes in the translational and rotational plateaus.

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.

Extending the range and physical accuracy of coarse-grained models: Order parameter dependent interactions

Order parameters (i.e., collective variables) are often used to describe the behavior of systems as they capture different features of the free energy surface. Yet, most coarse-grained (CG) models only employ two- or three-body non-bonded interactions between the CG particles. In situations where these interactions are insufficient for the CG model to reproduce the structural distributions of the underlying fine-grained (FG) model, additional interactions must be included. In this paper, we introduce an approach to expand the basis sets available in the multiscale coarse-graining (MS-CG) methodology by including order parameters. Then, we investigate the ability of an additive local order parameter (e.g., density) and an additive global order parameter (i.e., distance from a hard wall) to improve the description of CG models in interfacial systems. Specifically, we study methanol liquid-vapor coexistence, acetonitrile liquid-vapor coexistence, and acetonitrile liquid confined by hard-wall plates, all using single site CG models. We find that the use of order parameters dramatically improves the reproduction of structural properties of interfacial CG systems relative to the FG reference as compared with pairwise CG interactions alone.

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.

Charging changes contact composition in binary sphere packings

Equal volume mixtures of small and large polytetrafluorethylene spheres are shaken in an atmosphere of controlled humidity which allows one to also control their tribocharging. We find that the contact numbers are charge dependent: As the charge density of the beads increases, the number of same-type contacts decreases and the number of opposite-type contacts increases. This change is not caused by a global segregation of the sample. Hence, tribocharging can be a way to tune the local composition of a granular material.

Analyzing X-ray tomographies of granular packings

Starting from three-dimensional volume data of a granular packing, as, e.g., obtained by X-ray Computed Tomography, we discuss methods to first detect the individual particles in the sample and then analyze their properties. This analysis includes the pair correlation function, the volume and shape of the Voronoi cells, and the number and type of contacts formed between individual particles. We mainly focus on packings of monodisperse spheres, but we will also comment on other monoschematic particles such as ellipsoids and tetrahedra. This paper is accompanied by a package of free software containing all programs (including source code) and an example three-dimensional dataset which allows the reader to reproduce and modify all examples given.

Dense crystalline packings of ellipsoids

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

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.

Upper bound on the Edwards entropy in frictional monodisperse hard-sphere packings

We extend the Widom particle insertion method [B. Widom, J. Chem. Phys., 1963, 39, 2808-2812] to determine an upper bound sub on the Edwards entropy in frictional hard-sphere packings. sub corresponds to the logarithm of the number of mechanically stable configurations for a given volume fraction and boundary conditions. To accomplish this, we extend the method for estimating the particle insertion probability through the pore-size distribution in frictionless packings [V. Baranau, et al., Soft Matter, 2013, 9, 3361-3372] to the case of frictional particles. We use computer-generated and experimentally obtained three-dimensional sphere packings with volume fractions φ in the range 0.551-0.65. We find that sub has a maximum in the vicinity of the Random Loose Packing Limit φRLP = 0.55 and decreases then monotonically with increasing φ to reach a minimum at φ = 0.65. Further on, sub does not distinguish between real mechanical stability and packings in close proximity to mechanical stable configurations. The probability to find a given number of contacts for a particle inserted in a large enough pore does not depend on φ, but it decreases strongly with the contact number.

Structural characterization of the packings of granular regular polygons

By using a recently developed method for discrete modeling of nonspherical particles, we simulate the random packings of granular regular polygons with three to 11 edges under gravity. The effects of shape and friction on the packing structures are investigated by various structural parameters, including packing fraction, the radial distribution function, coordination number, Voronoi tessellation, and bond-orientational order. We find that packing fraction is generally higher for geometrically nonfrustrated regular polygons, and can be increased by the increase of edge number and decrease of friction. The changes of packing fraction are linked with those of the microstructures, such as the variations of the translational and orientational orders and local configurations. In particular, the free areas of Voronoi tessellations (which are related to local packing fractions) can be described by log-normal distributions for all polygons. The quantitative analyses establish a clearer picture for the packings of regular polygons.

Modeling the arrangement of particles in natural swelling-clay porous media using three-dimensional packing of elliptic disks

Swelling clay minerals play a key role in the control of water and pollutant migration in natural media such as soils. Moreover, swelling clay particles' orientational properties in porous media have significant implications for the directional dependence of fluid transfer. Herein we investigate the ability to mimic the organization of particles in natural swelling-clay porous media using a three-dimensional sequential particle deposition procedure [D. Coelho, J.-F. Thovert, and P. M. Adler, Phys. Rev. E 55, 1959 (1997)]. The algorithm considered is first used to simulate disk packings. Porosities of disk packings fall onto a single master curve when plotted against the orientational scalar order parameter value. This relation is used to validate the algorithm used in comparison with existing ones. The ellipticity degree of the particles is shown to have a negligible effect on the packing porosity for ratios ℓ(a)/ℓ(b) less than 1.5, whereas a significant increase in porosity is obtained for higher values. The effect of the distribution of the geometrical parameters (size, aspect ratio, and ellipticity degree) of particles on the final packing properties is also investigated. Finally, the algorithm is used to simulate particle packings for three size fractions of natural swelling-clay mineral powders. Calculated data regarding the distribution of the geometrical parameters and orientation of particles in porous media are successfully compared with experimental data obtained for the same samples. The results indicate that the obtained virtual porous media can be considered representative of natural samples and can be used to extract properties difficult to obtain experimentally, such as the anisotropic features of pore and solid phases in a system.

Structural and mechanical features of the order-disorder transition in experimental hard-sphere packings

Here we present an experimental and numerical investigation on the grain-scale geometrical and mechanical properties of partially crystallized structures made of macroscopic frictional grains. Crystallization is inevitable in arrangements of monosized hard spheres with packing densities exceeding Bernal's limiting density ϕ(Bernal)≈0.64. We study packings of monosized hard spheres whose density spans over a wide range (0.59<ϕ<0.72). These experiments harness x-ray computed tomography, three-dimensional image analysis, and numerical simulations to access precisely the geometry and the 3D structure of internal forces within the sphere packings. We show that clear geometrical transitions coincide with modifications of the mechanical backbone of the packing both at the grain and global scale. Notably, two transitions are identified at ϕ(Bernal)≈0.64 and ϕ(c)≈0.68. These results provide insights on how geometrical and mechanical features at the grain scale conspire to yield partially crystallized structures that are mechanically stable.