Lattice-based random jammed configurations for hard particles

A nontrivial subset of the jammed packings for rigid disks and spheres are those that can be obtained by sequential removal of particles from periodic crystalline arrays. This paper considers the enumeration problems presented by such packings that are based on the close-packed triangular disk lattice, and the face-centered and body-centered cubic sphere lattices. Three distinct categories of packings have been distinguished, depending on their behavior with respect to nonoverlap geometric constraints and/or externally imposed virtual displacements: locally jammed, collectively jammed, and strictly jammed. Each of these possesses an upper limiting vacancy concentration beyond which no packings of the types considered can exist. For each of the three lattices, specific vacancy clusters have been identified whose presence would destroy local jamming, and some of the corresponding patterns that would destroy collective jamming in the triangular disk lattice have also been found. Within the allowable range of vacancy concentration for each case, the number of distinct jammed packings is expected to rise exponentially with system size. By using the concept of local attrition factors, approximate enumerations have been constructed for the three lattice classes of locally jammed packings. In the interests of later extension of this work, we stress that at least some aspects of these enumeration problems might benefit from the formal transcription to a lattice-gas/Ising-model representation with vacancy interactions chosen to enforce the packing category of interest.