Autophoretic motion in three dimensions.
Janus particles with the ability to move phoretically in self-generated chemical concentration gra- dients are model systems for active matter. Their motion typically consists of straight paths with rotational diffusion being the dominant reorientation mechanism. In this paper, we show theoreti- cally that by a suitable surface coverage of both activity and mobility, translational and rotational motion can be induced arbitrarily in three dimensions. The resulting trajectories are in general helical, and their pitch and radius can be controlled by adjusting the angle between the transla- tional and angular velocity. Building on the classical mathematical framework for axisymmetric self-phoretic motion under fixed-flux chemical boundary condition, we first show how to calculate the most general three-dimensional motion for an arbitrary surface coverage of a spherical particle. After illustrating our results on surface distributions, we next introduce a simple intuitive patch model to serve as a guide for designing arbitrary phoretic spheres.