Hydrodynamically-bound states of a pair of microrollers: a dynamical system insight

Recent work has identified persistent cluster states which were shown to be assembled and held together by hydrodynamic interactions alone [Driscoll \textit{et al.} (2017) Nature Physics, 13(4), 375]. These states were seen in systems of colloidal microrollers; microrollers are colloidal particles which rotate about an axis parallel to the floor and generate strong, slowly decaying, advective flows. To understand these bound states, we study a simple, yet rich, model system of two microrollers. Here we show that pairs of microrollers can exhibit hydrodynamic bound states whose nature depends on a dimensionless number, denoted $B$, that compares the relative strength of gravitational forces and external torques. Using a dynamical system framework, we characterize these various states in phase space and analyze the bifurcations of the system as $B$ varies. In particular, we show that there is a critical value, $B^*$, above which active flows can beat gravity and lead to stable motile orbiting, or "leapfrog", trajectories, reminiscent of the self-assembled motile structures, called "critters", observed by Driscoll \textit{et al}. We identify the conditions for the emergence of these trajectories and study their basin of attraction. This work shows that a wide variety of stable bound states can be obtained with only two particles. Our results aid in understanding the mechanisms that lead to spontaneous self-assembly in hydrodynamic systems, such as microroller suspensions, as well as how to optimize these systems for particle transport.