Concerning Maximal Packing Arrangements of Binary Disk Mixtures

Obioma Uche, Princeton University

Photo of Obioma Uche

The determination of the maximal packing arrangements of two-dimensional, binary hard disks of radii RS and RL (with RSRL ) for sufficiently small RS amounts to finding the optimal arrangement of the small disks within a tricusp (the nonconvex cavity between three close-packed large disks). We present a non-conventional Monte Carlo algorithm for the generation of geometric packings of equi-sized hard disks within such a tricusp. The first nineteen members of an infinite sequence of maximal density structures thus produced are reported. In addition, the non-conventional Monte Carlo algorithm is applied to the geometric packing of disks within a flat-sided equilateral triangle and compared to published results for the packing problem. We perform an analysis of geometric properties, e.g. covering fraction of structures confined to both containers. During our investigation, we observe a non-monotonic increase in the covering fraction for increasing number of disks packed within both flat-sided triangle and tricusp. It is important to note that for disk packings in a flat-sided equilateral triangle, this non-monotonic increase of the covering fraction has never been explicitly reported in previously published works. For the flat-sided equilateral triangle, local maxima occur at the triangular integers NS = 1, 3, 6, 10, 15, ... where NS is the number of disks in each packing. However, local maxima for packings within the tricusp exist at NS = 1, 3, 6, 10, 18, ... . Finally, we analyze the asymptotic approach to the upper bound on the packing fraction of the infinite sequence of maximal structures of disks confined to the tricusp.

Abstract Author(s): Obioma U. Uche<br />Frank H. Stillinger<br />Salvatore Torquato