Sphere packing

Abstract: The search for the most efficient way of filling a container with balls is one of the oldest of mathematical puzzles. Aside from its intrinsic interest, the problem has practical relevance in systems as varied as granular processing, fruit packing, colloid behaviour and in living cells. Experiments have shown that the loosest way to pack spheres (random loose packing) gives a density of about 55% and that the most compact (random close packing, or RCP) gives a maximum density of about 64%. These values appear robust, but there is as yet no physical interpretation for them. Now Chaoming Song et al. show analytically that, indeed, spheres cannot pack in three dimensions above the 63.4% limit found by experiment. The limit arises from a statistical picture of the jammed states in which the RCP can be defined as the ground state of the ensemble of jammed matter. These results ultimately lead to a phase diagram for jammed matter that provides a unifying view of the sphere-packing problem. This paper presents a statistical description of jammed states in which random close packing can be interpreted as the ground state of the ensemble of jammed matter. The approach demonstrates that random packings of hard spheres in three dimensions cannot exceed a limit of ∼63.4 per cent. A phase diagram provides a common view of the hard sphere packing problem and illuminates various data, including the random loose packing state. 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.

Abstract: Models of randomly packed hard spheres exhibit some features of the properties of simple liquids, eg the packing density and the radial distribution The value of the maximum packing density of spheres can be determined from models if care is taken to ensure random packing at the boundary surfaces and if correction is made for volume errors at the boundaries Experiments for both the random `loose' and the random close-packed densities are reported with fraction one-eighth in plexiglass, nylon and steel balls in air, and also with steel balls immersed in oil A series of measurements for the random close-packed density has been made with up to 80 000 steel balls and with the aid of a mechanical vibrator A computer analysis of the results permits a one-step, two-parameter extrapolation to infinite volume The figure so obtained for the random close-packed density is 06366±00005, which represents an improvement in precision over previous results by an order of magnitude