A generalized approach to solving the mixed boundary value problem governing self-diffusiophoresis

Active matter is an emerging area of interest but the broken symmetry necessary for motion increases the complexity of mathematical modeling. However, these models can be critical for the interpretation of experimental data and efficient optimization of these systems, particularly towards the practical realization of commercial applications. Here, we present a linearized solution (small Péclet and thin-interaction layer) to a generalized mixed boundary-value problem that can be used to describe a variety of active colloidal systems consisting of partially coated (Janus-type) chemically active spherical microparticles, where the size of the active patch is left arbitrary (although axisymmetric). The solution is derived using the Fourier–Legendre-Collocation approach and the accuracy of this methodology is validated against known exact solutions to some limiting cases. We demonstrate the application of the general formulation by solving the mobility of a catalytic “patchy” particle which has two surfaces of differing chemical reactivity which can be characterized by either a fixed-flux or fixed-rate reaction (characterized by an arbitrary Damköhler number (Da)). The analytic solution is shown to agree with limiting cases in the literature presented for both small and infinite Da. Comparison of the different models to experiments published in the literature, indicate that the characterization of the surface reaction by a Dirichlet boundary condition, as in the case of infinite Da is uniquely able to capture the dependence of the mobility on the size of the active patch. This finding highlights the role of mathematical modeling in experimental characterization and optimization of active colloid systems.