The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured1 by Kepler and the random geometries explored2 by Bernal. Apart from its mathematical interest, the problem has practical relevance3 in a wide range of fields, from granular processing to fruit packing. There are currently numerous experiments showing that the loosest way to pack spheres (random loose packing) gives a density of ∼55 per cent4,5,6. On the other hand, the most compact way to pack spheres (random close packing) results in a maximum density of ∼64 per cent2,4,6. Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states7 in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. Our approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a density limit of ∼63.4 per cent. We construct a phase diagram that provides a unified view of the hard-sphere packing problem and illuminates various data, including the random-loose-packed state.
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This work is supported by the National Science Foundation, CMMT Division and the US Department of Energy, Office of Basic Energy Sciences, Geosciences Division. We are grateful to J. Brujić, A. Yupanqui and M. Makse for stimulating discussions.
The file contains Supplementary Notes which describe the details of the calculations leading to the phase diagram of jammed matter. The main theoretical result is the deffinition of the volume function based on the Voronoi volume of a particle explained in Section I. Section II describes the isostatic condition that deffines the ensemble of jammed matter through the W function. Section III explains the difference between the geometrical, z, and mechanical, Z, coordination number, which is important to deffine the canonical partition function. Section IV deffines the partition function for the ensemble of jammed states under the isostatic condition, Qiso. Section V explains how to solve the partition function Qiso which leads to the equations of state (4) and (5), and the phase diagram of Fig. 1. Finally, Section VI explains the numerical studies to test the theoretical predictions. The file also includes Supplementary Figures 1-14 with Legends. (PDF 930 kb)
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Song, C., Wang, P. & Makse, H. A phase diagram for jammed matter. Nature 453, 629–632 (2008). https://doi.org/10.1038/nature06981
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