Miniaturized, self-propelled locomotors use chemo-mechanical transductionmechanisms to convert fuel in the environment to autonomous motion. Recentexperimental and theoretical studies demonstrate that these autonomous enginescan passively follow the contours of solid boundaries they encounter. Boundaryguidance, however, is not necessarily stable: Mechanical disturbances can causethe 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 andthese stochastic forces can randomize passively-steered trajectories. Here we examine theoretically the stability of boundary guided motion ofmicromotors along infinite planar walls to mechanical disturbances and toBrownian forces. Our aim is to understand under what conditions this passivelyguided motion is stable. We choose a locomotor design in which sphericalcolloids are partially coated with a catalytic cap that reacts with solute toproduce a product. The product is repelled from the particle surface, causingthe particle to move with the inert face at the front (autonomous motion viaself-diffusiophoresis). When propelled towards a planar wall, deterministichydrodynamic studies demonstrate that these locomotors can exhibit, for largeenough cap sizes, steady trajectories in which the particle either skimsunidirectionally along the surface at a constant distance from the wall, orbecomes stationary. We first investigate the linear hydrodynamic stability ofthese states by expanding the equations of motion about the states, and findthat linear perturbations decay exponentially in time. We then study theeffects of thermal fluctuations by formulating a Langevin equation for theparticle motion which includes the Brownian stochastic force...
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