Optimal packings of superballs.

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|2p+|x2|2p+|x3|2por=0.5) with both cubic-like and octahedral-like shapes as well as concave particles (0or=0.5) are most likely the optimal ones. The maximal packing density as a function of p is nonanalytic at the sphere point (p=1) and increases dramatically as p moves away from unity. Two more nontrivial nonanalytic behaviors occur at pc*=1.150 9... and po*=ln 3/ln 4=0.792 4... for "cubic" and "octahedral" superballs, respectively, where different Bravais lattice packings possess the same densities. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional "superdisk" counterparts [Y. Jiao, Phys. Rev. Lett. 100, 245504 (2008)] and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.