Programmable topotaxis of magnetic rollers in time-varying fields.

We describe how spatially uniform, time-periodic magnetic fields can be designed to power and direct the migration of ferromagnetic spheres up (or down) local gradients in the topography of a solid substrate. Our results are based on a dynamical model that considers the time-varying magnetic torques on the particle and its motion through the fluid at low Reynolds number. We use both analytical theory and numerical simulation to design magnetic fields that maximize the migration velocity up (or down) an inclined plane. We show how "topotaxis" of spherical particles relies on differences in the hydrodynamic resistance to rotation about axes parallel and perpendicular to the plane. Importantly, the designed fields can drive multiple independent particles to move simultaneously in different directions as determined by gradients in their respective environments. Experiments on ferromagnetic spheres provide evidence for topotactic motions up inclined substrates. The ability to program the autonomous navigation of driven particles within anisotropic environments is relevant to the design of colloidal robots.