Self-propelled colloidal particle near a planar wall: A Brownian dynamics study

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 to 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 we 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. The Peclet number scales the ratio of deterministic to Brownian forces, where Pe = pi mu a(2) (nu) over tilde (c)/k(B)T and a denotes the colloid radius, mu the continuous phase viscosity, (nu) over tildec the characteristic diffusiophoretic velocity, and k(B)T the thermal energy. The skimming and stationary states are found to persist for Pe above 10(3). At Pe below 200, the trajectory of a locomotor approaching the wall is unpredictable. We present representative individual trajectories along with probability distributions for statistical ensembles of particles, quantifying the effects of thermal fluctuations and illustrating the transition from unpredictable to passively guided motion.