Random packing fraction of binary hyperspheres with small or large size difference: A geometric approach

The random packing fraction of binary particles in D-dimensional Euclidean space R^{D} is studied using a geometric approach. First, the binary packing fraction of assemblies with small size difference are studied, using a recently developed model that has its foundations in the excluded volume model by Onsager for cylinders and spherocylinders (D=3). According to this model the packing increase by bidispersity is proportional to (1-f)(u^{D}-1)^{2}, with f as monosized packing fraction, u as size ratio and D as space dimension. The model predictions are compared with computational results for disks in two dimensions (D=2) and hyperspheres in the large-dimension limit (D→∞), yielding very good agreement. Subsequently, the packing of hyperspheres with large size difference is modeled, employing the classic theory of Furnas. This theory, developed for three dimensions, starts from an infinite size ratio of larger and smaller particles (u→∞). Here, the pertaining equations are applied to hyperspheres, and successfully compared with computational results for hyperspheres in the large-dimension limit. Furthermore, an asymptotic approximation of the binary packing fraction for large size ratio is derived, which shows that the first order variation of the Furnas packing fraction (u^{-1}=0) is proportional to (2-f)u^{-1}. Finally, a normalized D-dimensional binary packing graph is presented, governing a simplified phase diagram that borders the binary random packing fraction of amorphous assemblies. To summarize, basic space-filling and geometric ("athermal") theories on "simple" hard spheres appear to be a valuable tool for the study of hyperspheres' random packing and amorphization.