A phase diagram for jammed matter

The problem of finding the most efficient way to pack spheres has a long history, dating back to the crystalline arrays conjectured(1) by Kepler and the random geometries explored(2) by Bernal. Apart from its mathematical interest, the problem has practical relevance(3) 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 similar to 55 per cent(4-6). On the other hand, the most compact way to pack spheres ( random close packing) results in a maximum density of similar to 64 per cent(2,4,6). Although these values seem to be robust, there is as yet no physical interpretation for them. Here we present a statistical description of jammed states(7) 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 similar to 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.