Mean-field theory of random close packings of axisymmetric particles

Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis, in particular, for shapes like dimers, spherocylinders and ellipsoids of revolution. Here, we present a mean-field formalism to estimate the packing density of axisymmetric non-spherical particles. We derive an analytic continuation from the sphere that provides a phase diagram predicting that, for the same coordination number, the density of monodisperse random packings follows the sequence of increasing packing fractions: spheres $<$ oblate ellipsoids $<$ prolate ellipsoids $<$ dimers $<$ spherocylinders. We find the maximal packing densities of $73.1\%$ for spherocylinders and $70.7\%$ for dimers, in good agreement with the largest densities found in simulations. Moreover, we find a packing density of $73.6\%$ for lens-shaped particles, representing the densest random packing of the axisymmetric objects studied so far.