Hydrodynamic synchronization of autonomously oscillating optically trapped particles

Ellipsoidal micron-sized colloidal particles can oscillate spontaneously when trapped in a focused laser beam. If two oscillating particles are held in proximity their oscillations synchronize through hydrodynamic interactions. The degree of synchronization depends on the distance between the oscillators and on their orientation. Due to the anisotropic nature of hydrodynamic coupling the synchronization is strongest when particles are arranged along the direction of oscillations. Similar behavior is observed for many oscillating particles arranged in a row. Experimental observations are well reproduced with a model that uses a phenomenological description of the optical force and hydrodynamic interactions. Our results show that oscillating ellipsoidal particles can serve as a model system for studying hydrodynamic synchronization between biological cilia.

Figure 1

(a) SEM image of a particle with major radius $8\mu \mathrm{m}$, minor radius $2.5\mu \mathrm{m}$, and thickness $2.2\mu \mathrm{m}$. (b) $\phi$ denotes the angle between the particle's long axis and the $z$ axis; $x$ is the distance between its center of reaction and the center of the laser beam.

Figure 2

(a) Time sequence of an oscillatory motion of an ellipsoidal particle trapped by a laser beam. (b) Dynamics of the oscillating particle in the $y$ direction. (c) A particle's oscillation frequency as a function of laser power. The solid line represents a linear fit to the data. (d) Oscillatory period of an ellipsoidal particle comparing the experimental and theoretical data for $\alpha =0.8$ and $\beta =1.5$. (e) Stable limit cycle for the same values of $\alpha$ and $\beta$ as in (d).

Figure 3

Bifurcation diagram for a single particle described by Eqs. (3). Depending on the parameters $\alpha$ and $\beta$, the model can exhibit one or three fixed points and a limit cycle that describes autonomous oscillations. In addition, the diagram shows (dashed line) the regions in which two particles show in-phase and antiphase synchronization for parameter values corresponding to longitudinal arrangement at large distances $\mathrm{[}{c}_{RR}/c_{TT}=0.5,c_{TR}/\left(\epsilon c_{TT}\right)=0.5,c_{RT}\epsilon /c_{TT}=1.5]$.

Figure 4

(a) Position oscillations of two particles in longitudinal arrangement [inset in (b)] at a distance of $8\mu \mathrm{m}$, laser power 56 mW. (b)–(d) Synchronization strength at distance $15\mu \mathrm{m}$ for three arrangements: (b) longitudinal, (c) diagonal, and (d) lateral. The insets show the position of the particles. Laser power: 56 mW. (e)–(f) Correlation coefficient $r_x$ as a function of the interparticle distance $d$ for different arrangements showing (e) in-phase synchronization (laser power 28 mW) and (f) antiphase synchronization (laser power 42 mW). (g),(h) Theoretically calculated correlation coefficients with parameters $\alpha =0.8,\beta =1.5,\epsilon =14\mu \text{m}$ (g), and $\epsilon =6\mu \text{m}$ (h).

Figure 5

(a) Particles arranged in a row. (b) Phase profile: laser power 40 mW per particle and distance between adjacent particles 8–$9\mu \mathrm{m}$. (c) Simulation of the same system using parameters $\alpha =0.8,\beta =1.5$, and $\epsilon =12\mu \mathrm{m}$. (d) Correlation matrix of experimentally measured particle positions. (e) Equivalent correlation matrix from simulation.