Variations around disordered close packing

The most suitable paradigms and tools for investigating the structure of granular packings are reviewed and discussed in the light of some recent empirical results. I examine the ‘typical’ local configurations, their relative occurrences, their correlations, their organization and the resulting overall hierarchical structure. The analysis of very large samples of monosized spheres packed in a disorderly fashion, with packing fractions ρ ranging from 0.58 to 0.64, demonstrates the existence of clear relations between the geometrical structure and the packing density ρ. It is observed that the local Delaunay and Voronoı̈ volumes have distributions which decay exponentially at large volumes, and have characteristic coefficients proportional to ρ. The average number of contacts per sphere grows linearly with ρ. The topological density increases with ρ and it is larger than the corresponding one for lattice sphere packing. The heights and the shapes of the peaks in the radial distribution function also depend on ρ. Moreover, the study of several other quantities, such as the dihedral angle distribution, the fractions of four-, five-and six-membered rings and the shape of the Voronoı̈ cells, shows that the local organization has characteristic patterns which depend on the packing density. All the empirical evidence excludes the possibility of the presence of any crystalline order, even at the minimal correlation length (one bead diameter). Moreover, icosahedral order and other close packed configurations have no statistical significance, excluding therefore the possibility that geometrical frustration could play any significant role in the formation of such amorphous structures. On the other hand, the analysis of the local geometrical ‘caging’ discloses that, at densities above 0.601, the system can no longer explore the phase space by means of local moves only and it becomes trapped either in the basin of attractions of inherent states, with limiting densities in the range 0.61–0.64,or in the crystalline branch. (Some figures in this article are in colour only in the electronic version)