Packing fraction of trimodal spheres with small size ratio: an analytical expression

In previous papers analytical expressions were derived and validated for the packing fraction of bimodal hard spheres with small size ratio, applicable to ordered (crystalline) [H. J. H. Brouwers, Phys. Rev. E 76, 041304 (2007);H. J. H. Brouwers, Phys. Rev. E 78, 011303 (2008)] and disordered (random) packings [H. J. H. Brouwers, Phys. Rev. E 87, 032202 (2013)]. In the present paper the underlying statistical approach, based on counting the occurrences of uneven pairs, i.e., the fraction of contacts between unequal spheres, is applied to trimodal discretely sized spheres. The packing of such ternary packings can be described by the same type of closed-form equation as the bimodal case. This equation contains the mean volume of the spheres and of the elementary cluster formed by these spheres; for crystalline arrangements this corresponds to the unit cell volume. The obtained compact analytical expression is compared with empirical packing data concerning random close packing of spheres, taken from the literature, comprising ternary binomial and geometric packings; good agreement is obtained. The presented approach is generalized to ordered and disordered packings of multimodal mixes.