Clusters of colloidal crystals exhibit a broad range of emergent, size dependent properties. Tuning such properties requires a strong fundamental understanding of the thermodynamics and kinetics of colloidal clusters. A first step in developing this understanding is to accurately describe the equilibrium structure and morphology of these colloidal assemblies. In this presentation, we report the results of a new computational approach, which we term “generalized Wulff construction,” that can accurately describe the equilibrium, i.e., of minimum free energy, shape of a crystalline assembly of colloidal particles. The colloidal system that we consider is modeled according to an inter-particle interaction potential consisting of two terms, an electrostatic repulsion and an Asakura-Oosawa (AO) depletion attraction. This inter-particle potential has been validated experimentally and used for accurate analyses of the thermodynamics and kinetics of these colloidal clusters, including disorder-to-order transitions as well as polymorphic transitions of crystalline clusters.
Our implementation of the generalized Wulff construction is based on lattice site exchange-Monte Carlo (LSE-MC) simulations performed within a parallel tempering framework. These LSE-MC simulations are carried out for the face-centered cubic as well as the hexagonally close-packed crystal structures. This approach differs from the conventional Wulff construction, which considers surface free energy effects on the equilibrium structure through contributions to the surface free energy by surface facets only, in that it automatically includes contributions from all other surface features (such as edges and vertices). The result of our generalized approach is a minimum-free-energy configuration for given crystal size (volume or number of atoms). We carry out these calculations over a range of crystal sizes to systematically analyze size effects on the morphological stability of colloidal clusters. Based on the findings of this analysis, we can determine the existence of “magic” cluster sizes, for which improved colloidal cluster stability is exhibited, i.e., magic clusters have lower internal energy compared to other clusters of similar size. Such analyses also provide useful information for the development of coarse-grained theories of colloidal crystal nucleation and growth.