Self-Propelled Colloidal Particle Near a Planar Wall: A Brownian Dynamics Study

TL;DR

This study analyzes the stability of boundary-guided motion of self-propelled colloidal particles near planar walls, considering hydrodynamic stability and Brownian fluctuations, to understand conditions for stable autonomous locomotion.

Contribution

It provides a theoretical framework combining hydrodynamic stability analysis and stochastic modeling to assess the passivity and stability of micromotor trajectories near surfaces.

Findings

Steady boundary-guided trajectories are linearly stable against small perturbations.

Brownian forces can cause significant deviations from guided paths at small scales.

Hydrodynamic stability persists despite thermal fluctuations under certain conditions.

Abstract

Miniaturized, self-propelled locomotors use chemo-mechanical transduction mechanisms to convert fuel in the environment to autonomous motion. Recent experimental and theoretical studies demonstrate that these autonomous engines can passively follow the contours of solid boundaries they encounter. Boundary guidance, however, is not necessarily stable: Mechanical disturbances can cause the motor to hydrodynamically depart from the passively guided pathway. Furthermore, given the scaled-down size of micromotors (typically 100 nm -10 $\mu$m), Brownian thermal fluctuation forces are necessarily important and these stochastic forces can randomize passively-steered trajectories. Here we examine theoretically the stability of boundary guided motion of micromotors along infinite planar walls to mechanical disturbances and to Brownian forces. Our aim is to understand under what conditions this passively guided motion is stable. We choose a locomotor design in which spherical colloids are partially coated with a catalytic cap that reacts with solute to produce a product. The product is repelled from the particle surface, causing the particle to move with the inert face at the front (autonomous motion via self-diffusiophoresis). When propelled towards a planar wall, deterministic hydrodynamic studies demonstrate that these locomotors can exhibit, for large enough cap sizes, steady trajectories in which the particle either skims unidirectionally along the surface at a constant distance from the wall, or becomes stationary. We first investigate the linear hydrodynamic stability of these states by expanding the equations of motion about the states, and find that linear perturbations decay exponentially in time. We then study the effects of thermal fluctuations by formulating a Langevin equation for the particle motion which includes the Brownian stochastic force...