Effect of substrate-product competition on the swimming velocity of catalytic Janus spheres

We explore spherical Janus particles in which a chemical reaction occurs on one face, depleting a substrate in the suspending fluid and synthesizing a product, while no reaction occurs on the other face. The steady state substrate concentration field is governed by Laplace's equation with mixed boundary conditions. We use the collocation method to obtain numerical solutions to the equation in spherical coordinates and we show that the product concentration field can be expressed in terms of the substrate concentration field. The asymmetry of the reaction gives rise to a slip velocity that causes the particle to move spontaneously in the fluid through a process known as self-diffusiophoresis. Using the Lorentz reciprocal theorem, we obtain the swimming velocity of the particle. The magnitude and direction of propulsion depends on competition between the substrate and product mobilities and diffusivities. We extend the results to Janus particles with arbitrary surface coverage, maximizing the swimming velocity as a function of the size of the reaction site.