Bypassing slip velocity: rotational and translational velocities of autophoretic colloids in terms of surface flux

TL;DR

This paper presents a method to calculate the propulsion velocities of autophoretic colloids directly from surface flux data, bypassing the need to compute slip velocities, thus simplifying analysis of such systems.

Contribution

It introduces explicit integral formulas linking surface flux to colloid velocities, extending analysis to spheroidal and slender-body geometries.

Findings

Derived formulas for translational velocities

Derived formulas for rotational velocities

Applicable to spheroidal and slender-body colloids

Abstract

A standard approach to propulsion velocities of autophoretic colloids with thin interaction layers uses a reciprocity relation applied to the slip velocity. But the surface flux (chemical, electrical, thermal, etc.), which is the source of the field driving the slip is often more accessible. We show how, under conditions of low Reynolds number and a field obeying the Laplace equation in the outer region, the slip velocity can be bypassed in velocity calculations. In a sense, the actual slip velocity and a normal field proportional to the flux density are equivalent for this type of calculation. Using known results for surface traction induced by rotating or translating an inert particle in a quiescent fluid, we derive simple and explicit integral formulas for translational and rotational velocities of arbitrary spheroidal and slender-body autophoretic colloids.