One sentence summary: Experiments reveal that small clusters of hard spheres with short-range attractions favor equilibrium structures determined by geometrical rules.The study of clusters has provided the most tangible link between local geometry and bulk condensed matter. But experiments have not yet systematically explored the thermodynamics of even the smallest clusters. Here we present experimental measurements of the structures and free energies of colloidal clusters in which the particles act as hard spheres with short-range attractions. We find that highly symmetric clusters are strongly suppressed by rotational entropy, while the most stable clusters have anharmonic vibrational modes or extra bonds. Many of these are subsets of closepacked lattices. As the number of particles increases from 6 to 10 we observe the emergence of a complex free energy landscape with a small number of ground states and many local minima.An isolated system of 10 interacting atoms or molecules will in general adopt a structure that differs in symmetry and average energy from that of a bulk liquid, solid, or even a system containing 100 particles. Yet the study of such small clusters has shed light on a wide variety of phenomena in condensed matter physics and physical chemistry. Since Frank first predicted (1) that icosahedral short-range order * This is the authors version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published

The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. 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 À 6 contacts. Interestingly, for n ¼ 10 and possibly at n ¼ 11 and n ¼ 12, the ground states of this system are subsets of hexagonal close-packed crystals. The ground states are not icosahedra at n ¼ 12 and n ¼ 13. We relate our results to the structure and thermodynamics of suspensions of colloidal particles with short-ranged attractions.

Abstract. Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R 3 satisfying minimal rigidity constraints (≥ 3 contacts per sphere and ≥ 3n − 6 total contacts). We derive such packings for n ≤ 10, and provide a preliminary set of maximum contact packings for 10 < n ≤ 20. The resultant set of packings has some striking features, among them: (i) all minimally-rigid packings for n ≤ 9 have exactly 3n − 6 contacts, (ii) non-rigid packings satisfying minimal rigidity constraints arise for n ≥ 9, (iii) the number of ground states (i.e. packings with the maximum number of contacts) oscillates with respect to n, (iv) for 10 ≤ n ≤ 20 there are only a small number of packings with the maximum number of contacts, and for 10 ≤ n < 13 these are all commensurate with the HCP lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdös repeated distance problem and Euclidean distance matrix completion problems.