Two Perspectives on the Phase Behavior of Small Clusters of Colloidal Particles

The directed assembly of a small ensemble of colloidal particles into a structured object is a topic of technological interest, for example as a route to construct a finite piece of meta-material. Such assembly processes are also of fundamental scientific interest because they involve systems with a number of particles N far below the bulk limit, e.g. N = 10–100, which are considered “thermodynamically small” in the terminology introduced by T.L. Hill. The relationships between thermodynamic variables in a small system can be qualitatively different from those in a bulk system, and phase behavior is a prime example. In this work we study the phase behavior of thermodynamically small colloidal clusters interacting via the Asakura-Oosawa pair potential, in which the concentration of a depletant species controls the degree of attraction, or effective osmotic pressure, experienced by the main colloidal species of interest. Experimental data and Brownian Dynamics simulations show that, at a fixed temperature, small clusters may exhibit coexistence between fluid-like and solid-like phases over a range of osmotic pressure values, rather than at a single value. We use parallel tempering Monte Carlo to map out the range of fluid-solid coexistence for clusters of several different N in the range 10-50, employing histograms of the potential energy as the most fundamental way to identify distinct phases, and compare the phase diagrams with their bulk counterparts. We also quantitatively compare our results with previously published work using free energy landscapes constructed in a two-dimensional order parameter space identified by a technique called diffusion maps.