Minimal Energy Clusters of Hard Spheres with Short Range Attractions

We calculate the ground states of hard-sphere clusters, in which n identical hard spherical particles bind by isotropic short-ranged attraction. Combining graph theoretic enumeration with basic geometry, we analytically solve for clusters of n = 10 particles satisfying minimal rigidity constraints. For n = 9 the ground state degeneracy increases exponentially with n, but for n > 9 the degeneracy decreases due to the formation of structures with >3n - 6 contacts. Interestingly, for n = 10 and possibly at n = 11 and n = 12, the ground states of this system are subsets of hexagonal close-packed crystals. The ground states are not icosahedra at n = 12 and n = 13. We relate our results to the structure and thermodynamics of suspensions of colloidal particles with short-ranged attractions.