Brownian Dynamics of Confined Rigid Bodies

TL;DR

This paper presents robust numerical methods for simulating the Brownian motion of arbitrarily shaped rigid bodies in viscous fluids, accounting for hydrodynamics and constraints, with applications to colloidal particles near boundaries.

Contribution

It introduces quaternion-based Langevin equations and two efficient integration schemes for simulating rigid body diffusion with hydrodynamic effects.

Findings

Accurate simulation of colloidal particle diffusion near boundaries.

Identification of optimal tracking points for linear MSD measurements.

Efficient long-time diffusion coefficient estimation using Monte Carlo methods.

Abstract

We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the other based on a random finite difference approach, both of which ensure the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary, and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long timescales.