A Smoluchowski model of crystallization dynamics of small colloidal clusters

Abstract

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.

Colloidal crystal grain boundary formation and motion

“…Statistical methods reported in the literature23, and further developed by us for application to colloidal assembly32, were used to analyze large numbers of BD simulated trajectories to construct W ( x ) and D ( x ) (see more details in Supplementary Methods and our previous work2932). In brief, the displacement and mean squared displacement of reaction coordinate vs. time trajectories can be used to measure drift and diffusion at each value of x , which ultimately yield W ( x ) and D ( x ).To assess the quantitative accuracy of candidate low-dimensional dynamic models, we compared first passage time distributions for ensembles of trajectories between different starting and ending states from particle-scale BD simulations and low-dimensional Langevin dynamic (LDLD) simulations.…”

“…BD simulations in the canonical ensemble were performed for 210 colloidal particles at constant voltage using numerical methods described in previous papers3132404142. A 0.1 ms time step was used for at least 2 × 10 7 steps, and reaction coordinates were stored every 1250 steps for subsequent analysis.…”

Colloidal cluster crystallization dynamics

“…(1), we determined W(X) and D(X) from 3N-dimensional BD simulation data using Bayesian inference (BI). 17,30,31 BI analysis follows a method that was reported previously for a system with dynamical behavior captured adequately by a single order parameter, which was clearly physically justified. 30,32,33 Although Smoluchowski-equation coefficients can also be obtained via linear fits to short time slopes of the average and variance of order parameter displacements, 22 this approach is not viable when short time linear regions do not exist in displacement data.…”

“…However, we found that a one-dimensional Smoluchowski model did not quantitatively capture crystallization dynamics (fluid-solid phase changes) in a cluster. 17 In this work, we report a multi-dimensional Smoluchowski model of colloidal crystallization dynamics using the same model system from our previous work. 12,17 We use diffusion mapping to identify the correct number of order parameters required in the model description, and then determine the corresponding Smoluchowski-equation coefficients using Bayesian inference.…”

“…17 In this work, we report a multi-dimensional Smoluchowski model of colloidal crystallization dynamics using the same model system from our previous work. 12,17 We use diffusion mapping to identify the correct number of order parameters required in the model description, and then determine the corresponding Smoluchowski-equation coefficients using Bayesian inference. We then validate the resulting dynamic model against Brownian dynamic simulations.…”