Phoretic swimming with bulk absorption

TL;DR

This paper investigates how chemically active colloids propel themselves when solute consumption occurs both at their surface and within the bulk, revealing how key parameters influence their velocity and boundary-layer behavior.

Contribution

It introduces a comprehensive model linking surface and bulk solute consumption via Damk"ohler numbers, and analyzes the asymptotic behavior and boundary-layer structure of phoretic propulsion.

Findings

Velocity scales with Damk"ohler numbers in different regimes.

Boundary-layer analysis matches eigenfunction solutions.

Transition regions resemble wave diffraction problems.

Abstract

We consider phoretic self-propulsion of a chemically active colloid where solute is consumed at both the colloid boundary and within the bulk solution. Assuming first-order kinetics, the dimensionless transport problem is governed by the surface Damk\"ohler number ${\mathcal{S}}$ and the bulk Damk\"ohler number ${\mathcal B}$. The dimensionless colloid velocity $U$, normalized by a self-phoretic scale, is a nonlinear function of these two parameters. We identify two scenarios where these numbers are linked. When the controlling physical parameter is colloid size, ${\mathcal{S}}$ is proportional to ${\mathcal B}^{1/2}$; when the controlling parameter is solute diffusivity, ${\mathcal{S}}$ is proportional to ${\mathcal B}$. In the limit of small Damk\"ohler numbers, $U$ adopts the same asymptotic limit in both scenarios, proportional to ${\mathcal{S}}$. In the limit of large Damk\"ohler numbers, the deviations of solute concentration from the equilibrium value are restricted to a narrow layer about the active portion of the colloid boundary. The asymptotic predictions of the associated boundary-layer problem are corroborated by an eigenfunction solution of the exact problem. The boundary-layer structure breaks down near the transition between the active and inactive portions of the boundary. The transport problem in that local region partially resembles the classical Sommerfeld problem of wave diffraction from an edge.