title: Can phoretic particles swim in two dimensions?
abstract: Artificial phoretic particles swim using self-generated gradients in chemical species (self-diffusiophoresis) or charges and currents (self-electrophoresis). These particles can be used to study the physics of collective motion in active matter and might have promising applications in bioengineering. In the case of self-diffusiophoresis, the classical physical model relies on a steady solution of the diffusion equation, from which chemical gradients, phoretic flows, and ultimately the swimming velocity may be derived. Motivated by disk-shaped particles in thin films and under confinement, we examine the extension to two dimensions. Because the two-dimensional diffusion equation lacks a steady state with the correct boundary conditions, Laplace transforms must be used to study the long-time behavior of the problem and determine the swimming velocity. For fixed chemical fluxes on the particle surface, we find that the swimming velocity ultimately always decays logarithmically in time. In the case of finite Péclet numbers, we solve the full advection-diffusion equation numerically and show that this decay can be avoided by the particle moving to regions of unconsumed reactant. Finite advection thus regularizes the two-dimensional phoretic problem.
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
authors.forename: David
authors.initials: D
authors.lastname: Sondak
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
authors.forename: Cory
authors.initials: C
authors.lastname: Hawley
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
authors.forename: Siyu
authors.initials: S
authors.lastname: Heng
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
authors.forename: Rebecca
authors.initials: R
authors.lastname: Vinsonhaler
authors.affiliation: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom.
authors.forename: Eric
authors.initials: E
authors.lastname: Lauga
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA.
authors.forename: Jean-Luc
authors.initials: JL
authors.lastname: Thiffeault