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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.