The structure of the densest crystal packings is determined for a variety of concave shapes in 2D constructed by the overlap of two or three disks. The maximum contact number per particle pair is defined and proposed as a useful means of categorizing particle shape. We demonstrate that the densest packed crystal exhibits a maximum in the number of contacts per particle but does not necessarily include particle pairs with the maximum contact number. In contrast, amorphous structures, generated by energy minimization of high temperature liquids, typically do include maximum contact pairs. The amorphous structures exhibit a large number of contacts per particle corresponding to over-constrained structures. Possible consequences of this over-constraint are discussed.
Shearing granular materials induces non-affine displacements. Such non-affine displacements have been studied extensively, and are known to correlate with plasticity and other mechanical features of amorphous packings. A well known example is shear transformation zones as captured by the local deviation from affine deformation, $D^2_{min}$, and their relevance to failure and stress fluctuations. We analyze sheared frictional athermal disk packings and show that there exists at least one additional mesoscopic transport mechanism that superimposes itself on top of local diffusive motion. We evid…
Microstructural dynamics in amorphous particle packings is commonly probed by quantifying particle displacements. While rigidity in particle packings emerges when displacement of particles are hindered, it is not obvious how the typically disordered displacement metrics connect to mechanical response. Particle rotations, in contrast, are much less sensitive to confinement effects, while still sensitive to the mechanics of the packing. So far, little attention has been paid to connect microscopic rotational motion to mechanics of athermal amorphous packings. We demonstrate through experimental …
Dense packings of hard particles have important applications in many fields, including condensed matter physics, discrete geometry and cell biology. In this paper, we employ a stochastic search implementation of the Torquato-Jiao Adaptive-Shrinking-Cell optimization scheme [Nature 460, 876 (2009)] to find maximally dense particle packings in d-dimensional Euclidean space $\mathbb{R}^d$. While the original implementation was designed to study spheres and convex polyhedra in $d\ge3$, our implementation focuses on $d=2$ and extends the algorithm to include both concave polygons and certain convex…
The close packing density of log-normal and bimodal distributed, surface-adsorbed particles or discs in 2D is studied by numerical simulation. For small spread in particle size, the system orders in a polycrystalline structure of hexagonal domains. The domain size and the packing density both decrease as the spread in particle size is increased up to 10.5+/-0.5%. From this point onwards the system becomes amorphous, and the close packing density increases again with spread in particle size. We argue that the polycrystalline and amorphous regions are separated by a Kosterlitz-Thouless-type phas…
We examine the regime between crystalline and amorphous packings of anisotropic objects on surfaces of different genus by continuously varying their size distribution or shape from monodispersed spheres to bidispersed mixtures or monodispersed ellipsoidal particles; we also consider an anisotropic variant of the Thomson problem with a mixture of charges. With increasing anisotropy, we first observe the disruption of translational order with an intermediate orientationally ordered hexatic phase as proposed by Nelson, Rubinstein and Spaepen, and then a transition to amorphous state. By analyzing…
In particulate systems with short-range interactions, such as granular matter or simple fluids, local structure plays a pivotal role in determining the macroscopic physical properties. Here, we analyse local structure metrics derived from the Voronoi diagram of configurations of oblate ellipsoids, for various aspect ratios $α$ and global volume fractions $ϕ_g$. We focus on jammed static configurations of frictional ellipsoids, obtained by tomographic imaging and by discrete element method simulations. In particular, we consider the local packing fraction $ϕ_l$, defined as the particle's volume…
The aim of this paper is to review and discuss qualitatively some results on the properties of amorphous packings of hard spheres that were recently obtained by means of the replica method. The theory gives predictions for the equation of state of the glass, the complexity of the metastable states, the scaling of the pressure close to jamming, the coordination of the packing and the pair correlation function in any space dimension d. The predictions compare very well with numerical simulations in d=3 and somehow also in d=2. The asymptotic predictions for d->oo are within the rigorous bounds. …
Structural defects within amorphous packings of symmetric particles can be characterized using a machine learning approach that incorporates structure functions of radial distances and angular arrangement. This yields a scalar field, \emph{softness}, that correlates with the probability that a particle is about to rearrange. However, when particle shapes are elongated, as in the case of dimers and ellipses, we find the standard structure functions produce imprecise softness measurements. Moreover, ellipses exhibit deformation profiles in stark contrast to circular particles. In order to accoun…
A free-energy functional that contains both the symmetry conserved and symmetry broken parts of the direct pair correlation function has been used to investigate the freezing of a system of hard spheres into crystalline and amorphous structures. The freezing parameters for fluid-crystal transition have been found to be in very good agreement with the results found from simulations. We considered amorphous structures found from the molecular dynamics simulations at packing fractions $η$ lower than the glass close packing fraction $η_{J}$ and investigated their stability compared to that of a ho…
The channel size distribution in hard sphere systems, based on the local neighbor correlation of four particle positions, is investigated for all volume fractions up to jamming. For each particle, all three particle combinations of neighbors define channels, which are relevant for the concept of caging. The analysis of the channel size distribution is shown to be very useful in distinguishing between gaseous, liquid, partially and fully crystallized, and glassy (random) jammed states. A common microstructural feature of four coplanar particles is observed in crystalline and glassy jammed state…
Mechanically stable sphere packings are generated in three-dimensional space using the discrete element method, which span a wide range in structural order, ranging from fully amorphous to quasi-ordered structures, as characterized by the bond orientational order parameter. As the packing pressure, $p$, varies from the marginally rigid limit at the jamming transition ($p \approx 0$) to that of more robust systems ($p \gg 0$), the coordination number, $z$, follows a familiar scaling relation with pressure, namely, $Δz = z - z_c \sim p^{1/2}$, where $z_c = 2d = 6$ ($d=3$ is the spatial dimension…
It has been recently pointed out by Farr and Groot (arXiv:0912.0852) and by Kyrylyuk and Philipse (Prog. Colloid Polym. Sci., 2010, in press) that our theoretical result for the jamming density of a binary mixture of hard spheres (arXiv:0903.5099) apparently violates an upper bound that is obtained by considering the limit where the diameter ratio r = DA/DB goes to infinity. We believe that this apparent contradiction is the consequence of a misunderstanding, which we try to clarify here.
We construct the densest known two-dimensional packings of superdisks in the plane whose shapes are defined by |x^(2p) + y^(2p)| <= 1, which contains both convex-shaped particles (p > 0.5, with the circular-disk case p = 1) and concave-shaped particles (0 < p < 0.5). The packings of the convex cases with p 1 generated by a recently developed event-driven molecular dynamics (MD) simulation algorithm [Donev, Torquato and Stillinger, J. Comput. Phys. 202 (2005) 737] suggest exact constructions of the densest known packings. We find that the packing density (covering fraction of the particles) inc…
We relate the structure factor $S(\mathbf{k} \to \mathbf{0})$ in a system of jammed hard spheres of number density $ρ$ to its complexity per particle $Σ(ρ)$ by the formula $S(\mathbf{k} \to \mathbf{0})=-1/ [ρ^2Σ''(ρ)+2ρΣ'(ρ)]$. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find $Σ(ρ)$ and $S(\mathbf{k})$ analytically. Hyperuniformity, which is the vanishing of $S(\mathbf{k} \to \mathbf{0})$, will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperun…
We numerically study the local stress distribution within athermal, isotropically stressed, mechanically stable, packings of bidisperse frictionless disks above the jamming transition in two dimensions. Considering the Fourier transform of the local stress, we find evidence for algebraically increasing fluctuations in both isotropic and anisotropic components of the stress tensor at small wavenumbers, contrary to recent theoretical predictions. Such increasing fluctuations imply a lack of self-averaging of the stress on large length scales. The crossover to these increasing fluctuations define…
We present a multiscale simulation algorithm for amorphous materials, which we illustrate and validate in a canonical case of dense granular flow. Our algorithm is based on the recently proposed Spot Model, where particles in a dense random packing undergo chain-like collective displacements in response to diffusing "spots" of influence, carrying a slight excess of interstitial free volume. We reconstruct the microscopic dynamics of particles from the "coarse grained" dynamics of spots by introducing a localized particle relaxation step after each spot-induced block displacement, simply to enf…
Steady Granular Flow in a Rotating Drum: Universal description of stress, velocity and packing fraction profiles covering grain shape effects from convex to very concave
The three dimensional structure of large packings of monosized spheres with volume fractions ranging between 0.58 and 0.64 has been studied with X-ray Computed Tomography. We search for signatures of organization, we classify local arrangements and we explore the effects of local geometrical constrains on the global packing. This study is the largest and the most accurate empirical analysis of disordered packings at the grain-scale to date with over 140,000 sphere coordinates mapped. We discuss topological and geometrical ways to characterize and classify these systems, and discuss implication…
The diffusion and flow of amorphous materials, such as glasses and granular materials, has resisted a simple microscopic description, analogous to defect theories for crystals. Early models were based on either gas-like inelastic collisions or crystal-like vacancy diffusion, but here we propose a cooperative mechanism for dense random-packing dynamics, based on diffusing ``spots'' of interstitial free volume. Simulations with the Spot Model can efficiently generate realistic flowing packings, and yet the model is simple enough for mathematical analysis. Starting from a non-local stochastic dif…
Studies of random close packing of spheres have advanced our knowledge about the structure of systems such as liquids, glasses, emulsions, granular media, and amorphous solids. When these systems are confined their structural properties change. To understand these changes we study random close packing in finite-sized confined systems, in both two and three dimensions. Each packing consists of a 50-50 binary mixture with particle size ratio 1.4. The presence of confining walls significantly lowers the overall maximum area fraction (or volume fraction in three dimensions). A simple model is pres…
Memory encoding by cyclic shear is a reliable process to store information in jammed solids, yet its underlying mechanism and its connection to the amorphous structure are not fully understood. When a jammed sphere packing is repeatedly sheared with cycles of the same strain amplitude, it optimizes its mechanical response to the cyclic driving and stores a memory of it. We study memory by cyclic shear training as a function of the underlying stability of the amorphous structure in marginally stable and highly stable packings, the latter produced by minimizing the potential energy using both po…
Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a versatile family of convex particles (p >= 0.5) with both cubic- and octahedral-like shapes as well as concave particles (0 < p < 0.5) with octahedral-lik…
Amorphous packings prepared in the vicinity of the jamming transition play a central role in theoretical studies of the vibrational spectrum of glasses. Two mean-field theories predict that the vibrational density of states $g(ω)$ obeys a characteristic power law, $g(ω)\simω^2$, called the non-Debye scaling in the low-frequency region. Numerical studies have however reported that this scaling breaks down at low frequencies, due to finite dimensional effects. In this study, we prepare amorphous packings of up to $128000$ particles in spatial dimensions from $d=3$ to $d=9$ to characterise the ra…
We show numerically that the response of simple amorphous solids (elastic networks and particle packings) to a local force dipole is characterized by a lengthscale $\ell_c$ that diverges as unjamming is approached as $\ell_c \sim (z - 2d)^{-1/2}$, where $z \ge 2d$ is the mean coordination, and $d$ is the spatial dimension, at odds with previous numerical claims. We also show how the magnitude of the lengthscale $\ell_c$ is amplified by the presence of internal stresses in the disordered solid. Our data suggests a divergence of $\ell_c\sim (p_c-p)^{-1/4}$ with proximity to a critical internal s…
We present the results of simulation studies of a model binary metal-metalloid alloy in which we characterize and explain the local coordination structure, the intermediate structure associated with the packing of these coordination polyhedra and the thermal stability of the various structural elements of this model amorphous solid.
Amorphous systems of soft particles above jamming have more contacts than are needed to achieve mechanical equilibrium. The force network of a granular system with a fixed contact network is thus underdetermined, and can be characterized as a random instantiation within the space of the force network ensemble. In this work, we show that by examining the boundaries of this space of allowed force networks, defect contacts which are not necessary for stability of the system can be uniquely identified. We further show that in the near jamming limit, this identification is nearly always correct and…
At the jamming transition, amorphous packings are known to display anomalous vibrational modes with a density of states (DOS) that remains constant at low frequency. The scaling of the DOS at higher densities remains, however, unclear. One might expect to find simple Debye scaling, but recent results from effective medium theory and the exact solution of mean-field models both predict an anomalous, non-Debye scaling. Being mean-field solutions, however, these solutions are only strictly applicable to the limit of infinite spatial dimension, and it is unclear what value they have for finite-dim…
The densest amorphous packing of rigid particles is known as random close packing. It has long been appreciated that higher densities are achieved by using collections of particles with a variety of sizes. The variety of sizes is often quantified by the polydispersity of the particle size distribution: the standard deviation of the radius divided by the mean radius. Several prior studies quantified the increase of the packing density as a function of polydispersity. Of course, a particle size distribution is also characterized by its skewness, kurtosis, and higher moments, but the influence of…
We numerically investigate the mechanical properties of static packings of ellipsoidal particles in 2D and 3D over a range of aspect ratio and compression $Δϕ$. While amorphous packings of spherical particles at jamming onset ($Δϕ=0$) are isostatic and possess the minimum contact number $z_{\rm iso}$ required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming ($z < z_{\rm iso}$) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interpar…
A wide range of materials can exist in microscopically disordered solid forms, referred to as amorphous solids or glasses. Such materials -- oxide glasses and metallic glasses, to polymer glasses, and soft solids such as colloidal glasses, emulsions and granular packings -- are useful as structural materials in a variety of contexts. Their deformation and flow behaviour is relevant for many others. Apart from fundamental questions associated with the formation of these solids, comprehending their mechanical behaviour is thus of interest, and of significance for their use as materials. In parti…
Dense packing of particles has provided important models to study the structure of matter in various systems such as liquid, glassy and crystalline phase, etc. The simplest sphere packing models are able to 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 complex molecular crystals. Here we develop spheroid models with the minimal degree of anisotropy, which serve as a simple geometrical representation for a rich spectrum of molecules…
The process of colloidal drying gives way to particle self-assembly in numerous elds including photonics or biotechnology. Yet, the mechanisms and conditions driving the nal particle arrangement in dry colloidal layers remain elusive. Here, we examine how the drying rate selects the nanostructure of thick dried layers in four dierent suspensions of silica nanospheres. Depending on particle size and dispersity, either an amorphous arrangement, a crystalline arrangement, or a rate-dependent amorphous-to-crystalline transition occurs at the drying surface. Amorphous arrangements are observed in t…
Characterizing structural inhomogeneity is an essential step in understanding the mechanical response of amorphous materials. We introduce a threshold-free measure based on the field of vectors pointing from the center of each particle to the centroid of the Voronoi cell in which the particle resides. These vectors tend to point in toward regions of high free volume and away from regions of low free volume, reminiscent of sinks and sources in a vector field. We compute the local divergence of these vectors, for which positive values correspond to overpacked regions and negative values identify…
Several mean-field theories predict that Hessian matrices of amorphous solids can be written by using the random matrix in the limit of the large spatial dimensions $d\to\infty$. Motivated by these results, we here propose a way to map a Hessian of the amorphous solid to a random matrix. This is possible by determining the coefficients of a random matrix so that the trace of the random matrix coincides with the Hessian of the original system. We compare our result with that of previous numerical simulations of harmonic spheres in several spatial dimensions $d=3$, $5$, and $9$. For small pressu…
Hard spheres are ubiquitous in condensed matter: they have been used as models for liquids, crystals, colloidal systems, granular systems, and powders. Packings of hard spheres are of even wider interest, as they are related to important problems in information theory, such as digitalization of signals, error correcting codes, and optimization problems. In three dimensions the densest packing of identical hard spheres has been proven to be the FCC lattice, and it is conjectured that the closest packing is ordered (a regular lattice, e.g, a crystal) in low enough dimension. Still, amorphous pac…
Obtaining general relations between macroscopic properties of random assemblies, such as density, and the microscopic properties of their constituent particles, such as shape, is a foundational challenge in the study of amorphous materials. By leveraging existing understanding of the random packing of spherical particles, we estimate the random packing density for all sufficiently spherical shapes. Our method uses the ensemble of random packing configurations of spheres as a reference point for a perturbative calculation, which we carry to linear order in the deformation. A fully analytic calc…
Soft, amorphous solids such as tissues, foams, and emulsions are composed of deformable particles. However, the effect of single-particle deformability on the collective behavior of soft solids is still poorly understood. We perform numerical simulations of two-dimensional jammed packings of explicitly deformable particles to study the mechanical response of model soft solids. We find that jammed packings of deformable particles with excess shape degrees of freedom possess low-frequency quartic vibrational modes that stabilize the packings even though they possess fewer interparticle contacts …
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 $|x_1|^{2p}+|x_2|^{2p}+|x_3|^{2p}\leq1$, where $p>0$ is a deformation parameter indicating to what extent the shape deviates from a sphere. As $p$ increases from the sphere point ($p=1$), the su…
The origin of boson peak -- an excess of density of states over Debye's model in glassy solids -- is still under intense debate, among which some theories and experiments suggest that boson peak is related to van-Hove singularity. Here we show that boson peak and van-Hove singularity are well separated identities, by measuring the vibrational density of states of a two-dimensional granular system, where packings are tuned gradually from a crystalline, to polycrystals, and to an amorphous material. We observe a coexistence of well separated boson peak and van-Hove singularities in polycrystals,…
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones syste…
Granular packings display a wealth of mechanical features which are of widespread significance. One of these features is creep: the slow deformation under applied stress. Creep is common for many other amorphous materials such as many metals and polymers. The slow motion of creep is challenging to understand, probe and control. We probe the creep properties of packings of soft spheres with a sinking ball viscometer. We find that in our granular packings, creep persists up to large strains and has a power law form, with diffusive dynamics. The creep amplitude is exponentially dependent on both …
In amorphous solids subject to shear or thermal excitation, so-called structural indicators have been developed that predict locations of future plasticity or particle rearrangements. An open question is whether similar tools can be used in dense active materials, but a challenge is that under most circumstances, active systems do not possess well-defined solid reference configurations. We develop a computational model for a dense active crowd attracted to a point of interest, which does permit a mechanically stable reference state in the limit of infinitely persistent motion. Previous work on…
We report a combined experimental and simulation study of deformation-induced diffusion in compacted two-dimensional amorphous granular pillars, in which thermal fluctuations play negligible role. The pillars, consisting of bidisperse cylindrical acetal plastic particles standing upright on a substrate, are deformed uniaxially and quasistatically by a rigid bar moving at a constant speed. The plastic flow and particle rearrangements in the pillars are characterized by computing the best-fit affine transformation strain and non-affine displacement associated with each particle between two stage…
The random sequential adsorption of various particle shapes is studied in order to determine the influence of particle anisotropy on the saturated random packing. For all tested particles there is an optimal level of anisotropy which maximizes the saturated packing fraction. It is found that a concave shape derived from a dimer of disks gives a packing fraction of 0.5833, which is comparable to the maximum packing fraction of ellipsoids and spherocylinders and higher than any other studied shape. Discussion why this shape is so beneficial for random sequential adsorption is given.
Numerous soft materials jam into an amorphous solid at high packing fraction. This non-equilibrium phase transition is best understood in the context of a model system in which particles repel elastically when they overlap. Recently, however, it was shown that introducing any finite amount of attraction between particles changes the universality class of the transition. The properties of this new ``sticky jamming'' class remain almost entirely unexplored. We use molecular dynamics simulations and scaling analysis to determine the shear modulus, bulk modulus, and coordination of marginal solids…
Amorphous packings of frictionless, spherical particles are isostatic at jamming onset, with the number of constraints (contacts) equal to the number of degrees of freedom. Their structural and mechanical properties are controlled by the interparticle contact network. In contrast, amorphous packings of frictional particles are typically hyperstatic at jamming onset. We perform extensive numerical simulations in two dimensions of the geometrical asperity (GA) model for static friction, to further investigate the role of isostaticity. In the GA model, interparticle forces are obtained by summing…
The yield stress is a defining feature of amorphous materials which is difficult to analyze theoretically, because it stems from the strongly non-linear response of an arrested solid to an applied deformation. Mode-coupling theory predicts the flow curves of materials undergoing a glass transition, and thus offers predictions for the yield stress of amorphous solids. We use this approach to analyse several classes of disordered solids, using simple models of hard sphere glasses, soft glasses, and metallic glasses for which the mode-coupling predictions can be directly compared to the outcome o…
We study the size-topology relations in random packings of dry adhesive polydisperse microspheres with Gaussian and lognormal size distributions through a geometric tessellation. We find that the dependence of the neighbour number on the centric particle size is always quasilinear, independent of the size distribution, the size span or interparticle adhesion. The average local packing fraction as a function of normalized particle size for different size variances is well regressed on the same profile, which grows to larger values as the relative strength of adhesion decreases. As for the local…
We propose a simple route to evaluate the static structure, in terms of average coordination, of completely disordered solids with spherical constituents, from ca. 55% volume fraction up to random close packing, in the absence of structural heterogeneities. Based on the current understanding, according to which the structure-determining interaction in amorphous solids is the hard-core repulsion while weaker, longer-range interactions are mere perturbations, the model yields the average coordination in the solid as a result of a hyperquenching process where the instantaneous structure of the pr…
