Hydrodynamic coupling in polygonal arrays of colloids: experimental and analytical results

Colloidal particles are trapped harmonically on the vertices of planar regular polygons, using optical tweezers. The particles interact with each other via hydrodynamic coupling, which can be described adequately by Oseen's tensor. Because of the interaction, the dynamics of any individual sphere is complex. Thermal motion results in a spectrum of relaxation times. The configuration of a system of N particles can be decomposed into 2N normal modes. In this work it is shown how to calculate these modes and their relaxation time scale analytically. The mathematical structure of the matrix of interaction leads to general properties for the symmetry of the normal modes and their dynamics, differing between the cases of even and odd N. The theory is compared to experiments performed on a range of rings with 3 ≤ N ≤ 10, varying also the trap stiffness and the distance between particles.