Fluctuating hydrodynamics of an autophoretic particle near a permeable interface

TL;DR

This paper develops a comprehensive theoretical framework for the stochastic motion of autophoretic particles near permeable interfaces, providing exact solutions and Green's functions for complex boundary conditions in fluctuating environments.

Contribution

It introduces a boundary-domain integral equation approach for chemohydrodynamics of autophoretic particles near permeable interfaces, including explicit solutions and stochastic dynamics equations.

Findings

Exact solutions for unbounded domain chemohydrodynamics.

Explicit Green's functions for permeable interfaces.

Analysis of particle behavior near interfaces with varying permeability.

Abstract

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.