December 4, 2009
The most efficient way to pack equally sized spheres isotropically in 3D is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution, based upon a mapping onto a one-dimensional problem. To test this theory we performed extensive simulations for mixtures of elastic spheres with hydrodynamic friction. The simulations show a general (but weak) dependence of the final (essentially hard sphere) packing density on fluid viscosity and on particle size, but this can be eliminated by choosing a specific relation between mass and particle size, making the random close packed volume fraction well-defined. Our theory agrees well with the simulations for bidisperse, tridisperse and log-normal distributions, and correctly reproduces the exact limits for large size ratios.
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We apply a recent one-dimensional algorithm for predicting random close packing fractions of polydisperse hard spheres [Farr and Groot, J. Chem. Phys. 133, 244104 (2009)] to the case of lognormal distributions of sphere sizes and mixtures of such populations. We show that the results compare well to two much slower algorithms for directly simulating spheres in three dimensions, and show that the algorithm is fast enough to tackle inverse problems in particle packing: designin...
June 6, 2010
We develop a model to describe the properties of random assemblies of polydisperse hard spheres. We show that the key features to describe the system are (i) the dependence between the free volume of a sphere and the various coordination numbers between the species, and (ii) the dependence of the coordination numbers with the concentration of species; quantities that are calculated analytically. The model predicts the density of random close packing and random loose packing o...
January 23, 2023
We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in $d=3$ dimensions, and thus obtain an estimate of the random close packing (RCP) volume fraction, $\phi_{\textrm{RCP}}$, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributi...
March 19, 2013
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...
March 5, 2009
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...
February 14, 2014
A recent proposal in which the equation of state of a polydisperse hard-sphere mixture is mapped onto that of the one-component fluid is extrapolated beyond the freezing point to estimate the jamming packing fraction $\phi_\text{J}$ of the polydisperse system as a simple function of $M_1M_3/M_2^2$, where $M_k$ is the $k$th moment of the size distribution. An analysis of experimental and simulation data of $\phi_\text{J}$ for a large number of different mixtures shows a remark...
May 4, 2022
We present a theoretical prediction on random close packing factor \phi_RCP^b of binary granular packings based on the hard-sphere fluid theory. An unexplored regime is unravelled, where the packing fraction \phi_RCP^b is smaller than that of the mono-sized one \phi_RCP^m, i.e., the so-called loose jamming state. This is against our common perception that binary packings should always reach a denser packing than mono-sized packings at the jamming state. Numerical evidence fur...
May 3, 2021
By generalizing a geometric argument for frictionless spheres, a model is proposed for the jamming density $\phi_J$ of mechanically stable packings of bidisperse, frictional spheres. The monodisperse, $\mu_s$-dependent jamming density $\phi_J^{\mathrm{mono}}(\mu_s)$ is the only input required in the model, where $\mu_s$ is the coefficient of friction. The predictions of the model are validated by robust estimates of $\phi_J$ obtained from computer simulations of up to $10^7$ ...
November 29, 1998
We consider the effect of intermolecular interactions on the optimal size-distribution of $N$ hard spheres that occupy a fixed total volume. When we minimize the free-energy of this system, within the Percus-Yevick approximation, we find that no solution exists beyond a quite low threshold ($\eta \thickapprox 0.260$). Monte Carlo simulations reveal that beyond this density, the size-distribution becomes bi-modal. Such distributions cannot be reproduced within the Percus-Yevic...
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We study jammed configurations of hard spheres as a function of compression speed using an event-driven molecular dynamics algorithm. We find that during the compression, the pressure follows closely the metastable liquid branch until the system gets arrested into a glass state as the relaxation time exceeds the compression speed. Further compression yields a jammed configuration that can be regarded as the infinite pressure configuration of that glass state. Consequently, we...