Random packing fraction of bimodal spheres : an analytical expression

In previous papers analytical equations were derived and validated for the packing fraction of crystalline structures consisting of bimodal randomly placed hard spheres [Phys. Rev. E 76, 041304 (2007); Phys. Rev. E 78, 011303 (2008)]. In this article it will be demonstrated that the bimodal random packing fraction of spheres with small size ratio can be described by the same type of closed-form equation. This equation contains the volume of the spheres and of the elementary cluster formed by these spheres. The obtained compact analytical expression appears to be in good agreement with a large collection of empirical and computer-generated packing data, taken from literature. By following a statistical approach of the number of uneven pairs in a binary packing, and the associated packing reduction (compared to the monosized limit), the number fraction of hypostatic spheres is estimated to be 0.548.