A Smoluchowski Model of Colloidal Crystallization Dynamics

1. Introduction Understanding concentrated colloidal dynamics in the presence of different pairwise interactions and external fields provides a basis to predict the temporal evolution of colloidal microstructures in diverse phenomena including suspension rheology and colloidal crystallization. However, a microscopic theory of concentrated colloidal dynamics does not yet exist that rigorously includes both statistical mechanical (configuration dependent free energy changes) and fluid mechanical (configuration dependent multi-body hydrodynamic interactions) contributions. The goal of the present work is to generate a coarse-grained dynamic, as well as equilibrium, model for the phase behavior of a model colloidal crystallizing system. The basis of our model is the Smoluchowski equation (SE), a partial differential equation whose solution is the probability density of a stochastic system in a set of observable (order parameter) variables. Several order parameters are studied to describe the dynamic crystallization process. One-dimensional and two-dimensional descriptions of the model system are studied. The values of the free energies (i.e. the free energy landscape, FEL) and diffusivities (i.e. the diffusivity landscapes) of the SE, which provide a complete dynamic description of the system, were determined by analyzing, according to previously described statistical procedures, [1,2] ensembles of short-time trajectories from Brownian Dynamics (BD) and Stokesian Dynamics (SD) simulations on the system under the conditions of interest. The SE models are then compared with Monte Carlo-Umbrella Sampling (MC-US) and BD and SD simulations (FEL from equilibrium analyses and mean first passage times, MFPT, from dynamic analyses).