Particle-size distribution and packing fraction of geometric random packings

This paper addresses the geometric random packing and void fraction of polydisperse particles. It is demonstrated that the bimodal packing can be transformed into a continuous particle-size distribution of the power law type. It follows that a maximum packing fraction of particles is obtained when the exponent (distribution modulus) of the power law function is zero, which is to say, the cumulative finer fraction is a logarithmic function of the particle size. For maximum geometric packings composed of sieve fractions or of discretely sized particles, the distribution modulus is positive (typically 0<alpha<0.37). Furthermore, an original and exact expression is derived that predicts the packing fraction of the polydisperse power law packing, and which is governed by the distribution exponent, size width, mode of packing, and particle shape only. For a number of particle shapes and their packing modes (close, loose), these parameters are given. The analytical expression of the packing fraction is thoroughly compared with experiments reported in the literature, and good agreement is found.