The packing of hard-core particles in contact with their neighbors offers the statically determinate problem which allows analytical investigation of the stress tensor distribution. We construct the stress probability functional and derive the complete set of equations for the tensor components in two-dimensions.
Disordered jammed packings under confinement have received considerably less attention than their \textit{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 {\bf 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 s…
Disordered hyperuniform packings are unusual amorphous states of two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large hyperuniform samples due to practical limitations of conventional methods. To overcome these limitations, we introduce a general theoretical method…
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 $ϕ_{\rm MRJ} \simeq 0.73$. This value is only a few percent lower than $ϕ_{\rm DKP} = 0.76210\dots$, the packing fraction of the corresponding densest-known crystalline (degenerate) packings. 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 …
A new tetrahedral structure model was developed and the geometrical structure of jammed disordered packings of monodisperse spheres with different friction coefficients was systematically characterized. An intrinsic structure feature is revealed for all jammed disordered packings, which is determined only by the packing fraction, independent of interparticle friction. Moreover, the structural configurations associated with crystal phases are more prevalent than the geometric frustrated ones. In addition, irregular simplexes are found to play a key role in an increase in density, rather than re…
Dynamical signatures are known to precede jamming in hard-particle systems, but static structural signatures have proven more elusive. The observation that compressing hard-particle packings towards jamming causes growing hyperuniformity has paved the way for the analysis of jamming as an "inverted critical point" in which the direct correlation function $c(r)$ diverges. We establish quantitative relationships between various singularities in $c(r)$ and the total correlation function $h(r)$ that provide a concrete means of identifying features that must be expressed in $c(r)$ if one hopes to r…
Jammed disordered packings of non-spherical particles show significant variation in the packing density as a function of particle shape for a given packing protocol. Rotationally symmetric elongated shapes such as ellipsoids, spherocylinders, and dimers, e.g., pack significantly denser than spheres over a narrow range of aspect ratios, exhibiting a characteristic peak at aspect ratios of $α_{\rm max}\approx 1.4-1.5$. However, the structural features that underlie this non-monotonic behaviour in the packing density are unknown. Here, we study disordered packings of frictionless dimers in three …
We investigate the mechanical response of packings of purely repulsive, frictionless disks to quasistatic deformations. The deformations include simple shear strain at constant packing fraction and at constant pressure, "polydispersity" strain (in which we change the particle size distribution) at constant packing fraction and at constant pressure, and isotropic compression. For each deformation, we show that there are two classes of changes in the interparticle contact networks: jump changes and point changes. Jump changes occur when a contact network becomes mechanically unstable, particles …
Recent diffraction experiments on metallic glasses have unveiled an unexpected non-cubic scaling law between density and average interatomic distance, which lead to the speculations on the presence of fractal glass order. Using X-ray tomography we identify here a similar non-cubic scaling law in disordered granular packing of spherical particles. We find that the scaling law is directly related to the contact neighbors within first nearest neighbor shell, and therefore is closely connected to the phenomenon of jamming. The seemingly universal scaling exponent around 2.5 arises due to the isost…
In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of correlation functions that can be rigorously related to effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, ela…
In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles' images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy min…
In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here, we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical appr…
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 orde…
Disordered hyperuniform dispersions are exotic amorphous two-phase materials characterized by an anomalous suppression of long-wavelength volume-fraction fluctuations, endowing them with novel physical properties. While such unusual materials have received considerable attention, a stumbling block has been an inability to create large samples that are truly hyperuniform due to current computational and experimental limitations. To overcome such barriers, we introduce a new, simple construction procedure that guarantees perfect hyperuniformity for very large sample sizes. It involves tessellati…
We study packings of bidispersed spherical particles on a spherical surface. The presence of curvature necessitates defects even for monodispersed particles; bidispersity either leads to a more disordered packing for nearly equal radii, or a higher fill fraction when the smaller particles are accomodated in the interstices of the larger spheres. Variation in the packing fraction is explained by a percolation transition, as chains of defects or scars previously discovered in the monodispersed case grow and eventually disconnect the neighbor graph.
Maximally random jammed (MRJ) particle packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The prediction of the MRJ packing density phi, among other packing properties of frictionless particles, still poses many theoretical challenges, even for congruent spheres or disks. Using the geometric-structure approach, we derive for the first time a highly accurate formula for MRJ densities for a very wide class of twodimensional frictionless packings, namely, binary convex superdisks, with shapes that continuously interp…
Packing spheres efficiently in large dimension $d$ is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be effectively constructed by this method, up to a packing fraction close to $7\, d\, 2^{-d}$. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space o…
Spatially ordered systems confined to surfaces such as spheres exhibit interesting topological structures because of curvature induced frustration in orientational as well as translational order. The study of these structures is important for investigating the interplay between geometry, topology, and elasticity, and for their potential applications in materials science. In this work we numerically simulate a spherical monolayer of soft repulsive spherocylinders (SRS) and study the packing of rods and their ordering transition as a function of the packing fraction. In the model that we study, …
We analyze the fluctuations in particle positions and inter-particle forces in disordered jammed crystals in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by…
Paradigmatic model systems, which are used to study the mechanical response of matter, are random networks of point-atoms, random sphere packings, or simple crystal lattices, all of these models assume central-force interactions between particles/atoms. Each of these models differs in the spatial arrangement and the correlations among particles. In turn, this is reflected in the widely different behaviours of the shear (G) and compression (K) elastic moduli. The relation between the macroscopic elasticity as encoded in G, K and their ratio, and the microscopic lattice structure/order, is not u…
