Introduction to the Theories and Modelling of Active Colloids
Date Issued
December 20, 2024
DOI
10.1039/9781837674589-00315
Abstract
This chapter will introduce the mathematics of modelling of active (autophoretic) colloids. It is intended to be something of a beginners’ guide, rather than an extensive literature review, and hopefully has useful information for theorist and experimentalist alike. The focus will be on modelling active colloids at the particle level, rather than at the suspension level via coarse-grained continuum methods. We first take a particle-centered view, whereby we consider the forces acting on a single sphere, to get Langevin dynamics – an ordinary differential equation (ODE) for the Janus particle’s motion. We discuss how the dynamics of a collection of such particles can be simulated by solving these ODEs together, and about how to add in physics – such as pair-wise fluid interactions between particles – to make the modelling more realistic. We then switch viewpoints to focus on what is going on outside the particle in the fluid, looking at the partial differential equations that govern the interactions of the solute fuel, particle, and the propulsive flows. We discuss some numerical techniques for studying autophoretic systems within this framework, with a focus on the Boundary Element Method. We present a method of simplifying this framework for slender autophoretic filaments and loops with arbitrary 3D shape and chemical patterning. In doing so, we see that the particle viewpoint and the fluid viewpoint “meet in the middle”, as we describe the interacting particles as a set of fundamental solutions with increasingly fast decay – point sources, dipoles, forces, torques, and stresses. We finish with a discussion of some potential future directions.