Hydrodynamically synchronized states in active colloidal arrays

Colloidal particles moving in a low Reynolds number fluid interact via the induced velocity field, described to a good approximation by the Oseen tensor. We consider the collective dynamic states for a class of actively forced colloids, driven by harmonic potentials via a non-linear switching rule that couples forces with configurations, establishing oscillations between prescribed positions. Experiments, simulations and theoretical arguments show that these states are determined by the equilibrium eigenmode structure of the Oseen interaction matrix and thus by the system's geometry. The stable dynamical state is predominantly formed by the eigenmode with the longest relaxation time. This has the surprising consequence that while 2 particles, or polygonal arrays of 4 or more colloids, synchronize with the nearest neighbors that are in the anti-phase, 3 equally spaced colloids synchronize in-phase. Odd-numbered arrangements with 5 or more particles sustain traveling waves. The emerging complex dynamical state can therefore be predicted from the simple mean spatial configuration of the active colloids, or equivalently from an analysis of their fluctuations near equilibrium.