The crystallization dynamics of a colloidal cluster is modeled using a low-dimensional Smoluchowski equation. Diffusion mapping shows that two order parameters are required to describe the dynamics. Using order parameters as metrics for condensation and crystallinity, free energy, and diffusivity landscapes are extracted from brownian dynamics simulations using bayesian inference. Free energy landscapes are validated against Monte Carlo simulations, and mean first-passage times are validated against dynamic simulations. The resulting model enables a low-dimensional description of colloidal crystallization dynamics.

We investigate the dynamics of colloidal crystallization in a 32-particle system at a fixed value of interparticle depletion attraction that produces coexisting fluid and solid phases. Free energy landscapes (FELs) and diffusivity landscapes (DLs) are obtained as coefficients of 1D Smoluchowski equations using as order parameters either the radius of gyration or the average crystallinity. FELs and DLs are estimated by fitting the Smoluchowski equations to Brownian dynamics (BD) simulations using either linear fits to locally initiated trajectories or global fits to unbiased trajectories using Bayesian inference. The resulting FELs are compared to Monte Carlo Umbrella Sampling results. The accuracy of the FELs and DLs for modeling colloidal crystallization dynamics is evaluated by comparing mean first-passage times from BD simulations with analytical predictions using the FEL and DL models. While the 1D models accurately capture dynamics near the free energy minimum fluid and crystal configurations, predictions near the transition region are not quantitatively accurate. A preliminary investigation of ensemble averaged 2D order parameter trajectories suggests that 2D models are required to capture crystallization dynamics in the transition region.

A computational study is presented in which real-time manipulation of the interaction potential between particles in a colloidal system is used to control their assembly into a close-packed crystalline object. The basic model used throughout the study is a highfidelity representation of a real experimental system in which 32 colloidal silica particles are suspended in aqueous solution with polymer hydrogel providing a temperaturetunable attractive force between the particles. Diffusion mapping is used to determine a set of coarse variables that provide an appropriate low-dimensional representation of this system at four discrete values of the attraction strength. In this case the diffusion mapping process identified two dimensions; one correlates well with the radius of gyration of the entire set of particles and the other correlates well with the average distance between distinct clusters of particles. Two different stochastic models are then built in the two-dimensional (2D) space of these variables, using data from a large number of short Brownian dynamics simulations of the full 32-particle system. The first 2D model is based on a Smoluchowski framework and is used to characterize the overall equilibrium and diffusive properties of the system. The second 2D model is based on a transition rate matrix and is used for process control. A control policy based on an infinite-horizon Markov decision process is developed using the four different attraction strengths as the input variables. The resulting policy is non-trivial; rather than simply selecting the strongest level of attraction, some mix of weak and strong attractions generally provides the optimal approach to the target close-packed state. This study, while focused on the particular mechanism of tunable depletion attraction, suggests a general strategy that could be adapted to different mechanisms of actuating colloidal assembly.

Total internal reflection microscopy (TIRM) and video microscopy (VM) are methods for nonintrusively measuring weak colloidal interactions important to many existing and emerging applications. Existing analyses of TIRM measured single particle trajectories can be used to extract particle-surface potentials and average particle diffusion coefficients. Here we develop a Fokker-Planck (FP) formalism to simultaneously extract both particle-surface interaction potentials and position dependent diffusion coefficients. The FP analysis offers several advantages including capabilities to measure separation dependent hydrodynamic interactions and nonequilibrium states that are not possible with existing analyses. The FP analysis is implemented to analyze Brownian dynamic simulations of single particle TIRM and VM experiments in several configurations. Relative effects of spatial and temporal sampling on the correct interpretation of both conservative and dissipative forces are explored and show a broad range of applicability for accessible experimental systems. Our results demonstrate the ability to extract both static and dynamic information from microscopy measurements of isolated particles near surfaces, which provides a foundation for further investigation of particle ensembles and nonequilibrium systems.

We report the findings of a computational study designed to determine the onset of a stable crystalline phase in assemblies of small numbers (13–32) of colloidal particles that interact via a depletion-based short-ranged attractive potential. Using Monte Carlo umbrella sampling with coarse graining in two order parameters, we generate free-energy landscapes that can indicate coexistence between fluid-like and crystalline phases. The emergence of a stable crystalline phase is observed as the number of particles in the assembly increases beyond a critical value. We find that the critical cluster size for crystallization onset decreases with increasing strength of the interparticle attraction.