Hydrodynamic interactions in two dimensions

We measure hydrodynamic interactions between colloidal particles confined in a thin sheet of fluid. The reduced dimensionality, compared to a bulk fluid, increases dramatically the range of couplings. Using optical tweezers we force a two body system along the eigenmodes of the mobility tensor and find that eigenmobilities change logarithmically with particle separation. At a hundred radii distance, the mobilities for rigid and relative motions differ by a factor of 2, whereas in bulk fluids, they would be practically indistinguishable. A two dimensional counterpart of Oseen hydrodynamic tensor quantitatively reproduces the observed behavior, once the relevant boundary conditions are recognized. These results highlight the importance of dimensionality for transport and interactions in colloidal systems and proteins in biological membranes.

Figure 1

(Color online) Trapping geometry. Two beads of radius $a$ are optically trapped at a distance $r$ in a liquid film of thickness $h$.

Figure 2

(Color online) Relaxation of eigencoordinates. For each of the selected distances, we report the time evolution of the four eigencoordinates relaxing to equilibrium after the applied perturbation. Solid lines are exponential fits. Dashed line is the average, single particle dynamics.

Figure 3

Eigenmobilities. The four eigenmobilities of a two particle system arranged at different particles separations. To correct for variations in trapping power, for each distance mobilities have been normalized to their average.