Sedimentation and polar order of active bottom-heavy particles

Self-propelled particles in an external gravitational field have been shown to display both an increased sedimentation length and polar order even without particle interactions. Here, we investigate self-propelled particles which additionally are bottom-heavy, that is they feel a torque aligning them to swim against the gravitational field. For bottom-heavy particles the gravitational field has the two opposite effects of i) sedimentation and ii) upward alignment of the particles' swimming direction. We perform a multipole expansion of the one-particle distribution of non-interacting particles with respect to orientation and derive expressions for sedimentation length and mean particle orientation which we check against Brownian Dynamics simulations. For large strength of gravity or small particle speeds and aligning torque, we observe sedimentation with increased sedimentation length compared with passive colloids but also active colloids without bottom-heaviness. Increasing, for example, swimming speed the sedimentation profile is inverted and the particles swim towards the top wall of the enclosing box. We find maximal orientational order at intermediate swimming speeds for both cases of particles with bottom-heaviness and those without. Ordering unsurprisingly is increased for the bottom-heavy particles, but this difference disappears at higher levels of activity and for very high activities ordering goes to zero in both cases.