We model the packing structure of a marginally jammed bulk ensemble of polydisperse spheres. To this end we expand on the granocentric model [Clusel et al., Nature (London) 460, 611 (2009)], explicitly taking into account rattlers. This leads to a relationship between the characteristic parameters of the packing, such as the mean number of neighbors and the fraction of rattlers, and the radial distribution function g(r). We find excellent agreement between the model predictions for g(r) and packing simulations, as well as experiments on jammed emulsion droplets. The observed quantitative agreement opens the path towards a full structural characterization of jammed particle systems for imaging and scattering experiments.
We have studied the contact network properties of two and three dimensional polydisperse, frictionless sphere packings at the random closed packing density through simulations. We observe universal correlations between particle size and contact number that are independent of the polydispersity of the packing. This allows us to formulate a mean field version of the granocentric model to predict the contact number distribution P (z). We find the predictions to be in good agreement with a wide range of discrete and continuous size distributions. The values of the two parameters that appear in the model are also independent of the polydispersity of the packing. Finally we look at the nearest neighbour spatial correlations to investigate the validity of the granocentric approach. We find that both particle size and contact number are anti-correlated which contrasts with the assumptions of the granocentric model. Despite this shortcoming, the correlations are sufficiently weak which explains the good approximation of P (z) obtained from the model.
We investigate next-nearest-neighbor correlations of the contact number in simulations of polydisperse, frictionless packings in two dimensions. We find that disks with few contacting neighbors are predominantly in contact with disks that have many neighbors and vice versa at all packing fractions. This counterintuitive result can be explained by drawing a direct analogy to the Aboav-Weaire law in cellular structures. We find an empirical one parameter relation similar to the Aboav-Weaire law that satisfies an exact sum rule constraint. Surprisingly, there are no correlations in the radii between neighboring particles, despite correlations between contact number and radius. We address this question through simulations of a twodimensional model system with polydisperse, frictionless soft disks. At the random close-packing density φ c , the disks just touch and have an average contact number z close to 4 as required for mechanical stability [1,12,13]. Due to disorder the individual contact numbers z are distributed according to some distribution P (z) that depends on the polydispersity of the disk size distribution [2]. Here, we investigate whether spatial correlations in the contact network exist.In our simulations we find that disks with many contacts favor neighbors with fewer contacts and vice versa. These correlations persist for all packing densities. This result is a direct analog to the well-known Aboav-Weaire law in the field of cellular structures which states that cells with fewer neighbors are surrounded by cells with many neighbors [14][15][16][17][18]. We show that our results are in excellent agreement with a modified Aboav-Weaire law. Since geometrical constraints in the packing dictate that smaller particles have fewer contacts on average [2], one may expect similar correlations for the size distribution in the packing, namely, that larger particles are surrounded by smaller ones. Surprisingly, we find that the size of the central particle is uncorrelated to the average size of the contacting neighboring particles.We simulate the disordered packings by using Durian's soft disk model [9], as implemented by Langlois et al. [19]. The disks have a harmonic repulsion proportional to their overlap and experience viscous tangential drag. We use 1500 polydisperse disks whose radii are drawn from a Gaussian distribution with a mean r and a variance σ 2 = (0.304 r ) 2 . The polydispersity allows us to access a wide range of contact numbers, which is important to measure next-nearest-neighbor correlations. The disks are randomly placed in a periodic box at low packing fraction and then allowed to relax while their radii are slowly increased. The simulation terminates when the total elastic energy due to overlaps reaches a steady state at a predefined packing fraction. For each packing density, up to 10 different packings are created to increase the statistics of our correlation measures. Upon reaching mechanical equilibrium, disks with fewer than three contacts (rattlers) are removed for the analysis of the ...
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