Gapped gapless packing structures

The assumption of a gapless packing structure has previously been used to obtain the density and partial coordination numbers of a random mixture of hard spheres in the maximally dense regime. Here we extend the notion of a gapless packing structure to allow the determination of the characteristics of a packing away from maximal density by adding an appropriate number of void spherical elements. A gapless packing is then considered in which the void and solid spherical elements are assumed to be indistinguishable except for the purposes of calculating packing fraction and coordination number. We utilize the notion of specific volume to generate a one-parameter family of void distributions to obtain a set of coupled integral equations, which are solved numerically. Monodisperse and bi-disperse packings are investigated for packing fractions ranging from rho=0.26 to 0.78. Results are shown to be comparable to experiments and the effect of varying packing fraction on coordination numbers is shown to be invariant with respect to number distribution. A linear relationship between coordination number and packing fraction is elucidated for moderate to low packing fractions. Maximum and minimum random packing fractions are also discussed.