Densest local sphere-packing diversity: General concepts and application to two dimensions

The densest local packings of N identical nonoverlapping spheres within a radius R-min(N) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of configuration space. The knowledge of R-min(N) in d-dimensional Euclidean space R-d allows for the construction both of a realizability condition for pair-correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings in R-d. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding R-min(N) for selected values of N up to N = 348 and use this knowledge to construct such a realizability condition and an upper bound. We additionally analyze the properties and characteristics of the maximally dense packings, finding significant variability in their symmetries and contact networks, and that the vast majority differ substantially from the triangular lattice even for large N. Our work has implications for packaging problems, nucleation theory, and surface physics.