Huygens' clocks at the microscale

Abstract

Weakly coupled oscillators adjust their dynamics to work in unison: they synchronize. This ubiquitous phenomenon is observed for oscillating pendulum, electronic devices, as well as clapping crowds or flashing fireflies. In effect, synchronization constitutes an efficient mean to translate microscopic into large scale dynamics. While broadly studied theoretically, experimental investigations of synchronization of systems at the microscale are limited. Here we devise and study a model system of noisy and "measurably imperfect" colloidal oscillators: autonomous clocks made of an active swimmer revolving around a passive sphere. The distribution of natural frequency of the clock is achieved using passive spheres of various sizes, thus without altering the (phoretic) coupling between clocks. We observe that pairs of oscillators lock phases before slipping and returning to sync, and we characterize the synchronicity of the pair. We rationalize our findings with a stochastic model, formalizing synchronization as a classical Kramers escape problem in an adequate potential. This provides an analytical expression for the rate of synchronization of a pair set by the ratio between differences of natural frequency and environmental noise, and agrees qualitatively with the experiment. Our results set a blueprint for synchronization with micrometric autonomous systems.

Figures (4)

• Figure 1: Colloidal clocks. (a) Scheme of the colloidal clock. A colloidal microswimmer (inset) revolves around a passive sphere of radius $R$. The white bar is 5 µ m. (b) Normalized cumulated phase angle $\theta / 2 \pi$ for a single colloidal clock with passive colloid radius $R=5µ m$, as a function of time. The slope provides the natural frequency of the clock, $\omega$, with high accuracy. (b-inset) The variance $\mathrm{Var}[\omega_{\Delta t}]\propto D_r/\Delta t$ quantifies the rotational noise $D_r=0.02~\mathrm{s^{-1}}$ for the clock. (c) Distribution of natural frequencies for this study obtained by changing the size $R=7, 8, 10, 12, 20$ µ m of the central sphere. (d) Constant tangential speed $v_0 = \omega r_0$ for colloidal clocks of radii $r_0 \approx \sqrt{2 R d_s}$. (e) Constant translational diffusion $D_0$ for colloidal clocks of radii $r_0 \approx \sqrt{2 R d_s}$.
• Figure 2: Pairs of colloidal clocks. (a) Timelapse formation of a pair of co-rotating colloidal clocks. Clocks are assembled initially, far apart (not presented). They are approached using optical traps and released ($t = 0\,$s). They then attract phoretically until contact ($t_\mathrm{c}=70\,$s). The equators remain in contact and the distance of the centers of the colloidal clocks is constant $d=2R$. (b) Cumulated angle of rotating clocks before they come in contact. The slope is the natural frequency, that appear close but not identical. (c) Dynamics of the relative phase $\delta$. Delta increases linearly before contact $t_\mathrm{c}$: the clocks are not synchronized. After $t_\mathrm{c}$, $\delta$ shows plateaus of phase-locking near odd values of $\pi$ indicative of synchronicity.
• Figure 3: Synchronization of colloidal clocks. (a) Temporal dynamics of the phases for co-rotor and counter-rotor clock pairs. For both experimental and simulation results, plateaus are manifested for the clock pair with smaller colloid radius, suggesting strong synchronization. Conversely, the pair with larger radius manifest phase slip, pointing at little to no synchronization. (b) Experimental and (c) simulated probability density function (p.d.f.) of the relative phases, $\delta$ for co-rotors and $\phi$ for counter-rotors, for the clock pairs from panel (a). Insets contain the joint distributions.
• Figure 4: Synchronization for co-rotors with different colloidal radii. (a) Synchronization ratio $\Gamma(\Delta \omega)$ between two colloidal clocks vanishing for $\Delta \omega\gtrsim0.3~\mathrm{s^{-1}}$. (b) Collapse of the experimental data onto the theoretical prediction $\Gamma=\exp(-\lambda)$ without fitting parameter (dashed line). (c) Simulation data of probability of being in a synchronized state for a broad range of attractive coupling strengths $k$, swimming speed $\omega r_0$, and radii $R$, collapsed as predicted by theoretical modeling. (inset) Simulation data prior collapse.