Eigenmodes of a hydrodynamically coupled micron-size multiple-particle ring

We use a continuous acquisition, high-speed camera with integrated centroid tracking to simultaneously measure the positions of a ring of micron-sized particles held in holographic optical tweezers. Hydrodynamic coupling between the particles gives a set of eigenmodes, each one independently relaxing with a characteristic decay rate (eigenvalue) that can be measured using our positional data. Despite the finite particle size, we find an excellent agreement between the measured eigenvalues and those numerically predicted by Oseen theory applied to the two-dimensional (2D) ring geometry. Particle motions are also analyzed in terms of the alternative eigenmode set obtained by wrapping onto the ring the eigenmodes of a 1D periodic chain. We identify the modes for which the periodic chain is a good approximation to the ring and those for which it is not.

Figure 1

(Color online) Ring of eight particles held by optical tweezers. The squares show regions of interest within which the particles’ positions were monitored using a high speed camera.

Figure 2

The 16 eigenmodes predicted using the Oseen approximation for the ring, a periodic 1D chain, and the chain coordinate mapped onto a ring. The ring eigenmodes are labeled a–j with $(*)$ denoting degeneracy. These modes are cross referenced to those of the periodic chain, many of which occur in a different position within the eigenmode spectrum.

Figure 3

(Color online) The 16 measured autocorrelation functions for the predicted eigenmodes for the eight particle ring and the eight particle periodic chain. The slowest decaying eigenmodes correspond to common motion of the particles, and hence are strongly perturbed by the mechanical noise in the laboratory. The lower panels show the, near-zero, cross correlation functions between the eigenmodes.

Figure 4

Experimental (white squares) and predicted (black circles) decay rates for the eigenmodes of a ring and a 1D periodic chain of eight trapped particles subject to hydrodynamic coupling.