Self and directed assembly of small clusters of colloidal particles is an area of major scientific and technological interest. Of particular interest is the assembly of clusters of colloidal particles into highly ordered crystalline structures. These clusters are termed thermodynamically small as they contain a small number of particles, from O(10) to O(100) colloidal particles. The phase behavior of these clusters is qualitatively different from that of the corresponding bulk phases. Specifically, such thermodynamically small assemblies exhibit coexistence of phases or configurations over a broad range of physical conditions. In this presentation, we report results of a systematic investigation of the phase behavior of thermodynamically small assemblies of colloidal particles, which interact via an experimentally validated pair potential consisting of two terms: an electrostatic repulsion and Asakura-Oosawa (AO) depletion attraction.
In order to study the phase behavior of small clusters of colloidal particles, we conducted Monte Carlo (MC) simulations and employed windowed Monte Carlo-umbrella sampling (MC-US) to generate free-energy landscapes (FELs). These FELs were generated with respect to a well justified coarse-variable space, which has been shown to be able to capture both order-to-disorder and polymorphic solid-to-solid structural transitions. The dimensionality of coarse-variable space was determined by analysis based on the macine learning technique of diffusion mapping on dynamic data sets generated by Brownian-dynamics simulations of the colloidal clusters of interest. We constructed FELs over a range of inter-particle interaction strength, inter-particle interaction range, and cluster size and obtained a comprehensive picture over this parameter space regarding the possible stable configurations of such colloidal assemblies at equilibrium, as well as the phase changes observed between them. In particular, we find an order-to-disorder transition that occurs between fluid-like and crystalline phases, as well as a polymorphic transition that occurs between relaxed face-centered cubic and relaxed hexagonal close-packed structures. The analysis of these FELs yields phase-diagram information, which can describe the complex phase behavior arising from the system smallness inherent to these colloidal clusters.