Two-Dimensional Clusters of Colloidal Spheres: Ground States, Excited States, and Structural Rearrangements.

We study experimentally what is arguably the simplest yet nontrivial colloidal system: two-dimensional clusters of six spherical particles bound by depletion interactions. These clusters have multiple, degenerate ground states whose equilibrium distribution is determined by entropic factors, principally the symmetry. We observe the equilibrium rearrangements between ground states as well as all of the low-lying excited states. In contrast to the ground states, the excited states have soft modes and low symmetry, and their occupation probabilities depend on the size of the configuration space reached through internal degrees of freedom, as well as a single "sticky parameter" encapsulating the depth and curvature of the potential. Using a geometrical model that accounts for the entropy of the soft modes and the diffusion rates along them, we accurately reproduce the measured rearrangement rates. The success of this model, which requires no fitting parameters or measurements of the potential, shows that the free-energy landscape of colloidal systems and the dynamics it governs can be understood geometrically.

Tetrahedral colloidal clusters from random parking of bidisperse spheres.

Using experiments and simulations, we investigate the clusters that form when colloidal spheres stick irreversibly to--or "park" on--smaller spheres. We use either oppositely charged particles or particles labeled with complementary DNA sequences, and we vary the ratio α of large to small sphere radii. Once bound, the large spheres cannot rearrange, and thus the clusters do not form dense or symmetric packings. Nevertheless, this stochastic aggregation process yields a remarkably narrow distribution of clusters with nearly 90% tetrahedra at α = 2.45. The high yield of tetrahedra, which reaches 100% in simulations at α = 2.41, arises not simply because of packing constraints, but also because of the existence of a long-time lower bound that we call the "minimum parking" number. We derive this lower bound from solutions to the classic mathematical problem of spherical covering, and we show that there is a critical size ratio α(c) = (1 + sqrt[2]) ≈ 2.41, close to the observed point of maximum yield, where the lower bound equals the upper bound set by packing constraints. The emergence of a critical value in a random aggregation process offers a robust method to assemble uniform clusters for a variety of applications, including metamaterials.