Hydrodynamic synchronization of colloidal oscillators

Abstract

Two colloidal spheres are maintained in oscillation by switching the position of an optical trap when a sphere reaches a limit position, leading to oscillations that are bounded in amplitude but free in phase and period. The interaction between the oscillators is only through the hydrodynamic flow induced by their motion. We prove that in the absence of stochastic noise the antiphase dynamical state is stable, and we show how the period depends on coupling strength. Both features are observed experimentally. As the natural frequencies of the oscillators are made progressively different, the coordination is quickly lost. These results help one to understand the origin of hydrodynamic synchronization and how the dynamics can be tuned. Cilia and flagella are biological systems coupled hydrodynamically, exhibiting dramatic collective motions. We propose that weakly correlated phase fluctuations, with one of the oscillators typically processing the other, are characteristic of hydrodynamically coupled systems in the presence of thermal noise.

Fig. 1.

The physical system. Driven oscillations of fixed amplitude, but free inphase, are obtained using optical traps. The trap alternates between two minima with a “geometric switch” triggered by the position of the colloidal particle. (A) and (B) illustrate the experimental parameters, showing the trap separation in a pair λ = 1 μm, the distance from the minimum at which the traps are switched ξ = 0.248 μm. The distance between trap pairs is in the range 4 μm ≤ d ≤ 40 μm. (C) Two particles lock in antiphase. Particle positions overlayed on an image sequence of a pair of particles undergoing driven oscillations controlled by the geometric switch. Antiphase motion can be seen. Images are shown every 0.1 s, and data points are shown every 0.01 s. Scale bar, 5 μm.

Fig. 2.

Antiphase synchronization. The beating frequency in the synchronized state depends on the hydrodynamic coupling strength. (A) Particle displacement in an experiment with d = 6 μm. (B) The power spectrum of position of a particle during the active feedback experiment, for two different values of d. The peak can be fitted with a Lorentzian function to obtain ω_sync. (C) As the distance d between beads is reduced, the drag increases and ω_sync decreases. Experiments (circles and error bars) compare well with the theoretical estimate based on the deterministic relaxation time of hydrodynamically coupled beads (solid lines). The inset shows the dependence of ω_sync on the trap stiffness, with κ₁ = κ₂.

Fig. 3.

Loss of synchonization. As the coupling is reduced, either by increasing d or by detuning the characteristic beating times of the two oscillators, synchronization is lost due to thermal noise. (A) Heat map showing the histogram of the synchronization order parameter Q as a function of distance, and thus at varying coupling strength. A strong antiphase correlation (Q ≃ 0.8) can be seen in both experiments (Top) and numerical simulation (Bottom). Q < 1 is principally determined by a time delay Δt which results from correlated fluctuations. As d increases, synchronization is lost via the process shown in (B). (B) The switch time difference is plotted as a function of the period index (each panel is obtained a distance d, as labeled). At small d this quantity fluctuates around the locked state, while phase-slips and drift emerge as the coupling weakens. (Bottom) Noise level in flat regions is plotted and shows a linear increase with d, up to a level that is comparable to the half-period of the motion (solid line). This is the process by which synchronization is lost in this system. (C) (Top) and (Middle) Histogram of Q at varying ratio of the stiffness for the two traps, and thus of the intrinsic frequency of the two oscillators. Maximal antiphase correlation is observed for oscillators of equal intrinsic frequency. (Bottom) Loss of synchrony (defined by a threshold at Q = 0.6) at different coupling strengths. The numerical data (black circles) provides a measure of the Arnold tongues for this system and are compatible with the experimental observations at d = 10 μm. Error bar is the uncertainty in the trap stiffness ratio.

Fig. 4.

The fingerprint of hydrodynamic synchronization. In the presence of thermal noise, a characteristic delay time emerges between subsequent switches. The bead oscillations are delayed by a finite time Δt. (A) Heat-map for the distribution of |Δt| at varying distance, measured in the experiment (Top) and by numerical simulation (Middle), shows the peak value of |Δt| → 0 and also a broadening of the distribution with increasing d. (B) Heat map of the cross-correlation of switch positions as a function of the delay time, showing strong correlation at finite time intervals.