Figure 1

(a) Two colloidal “rotors” are made by driving particles along predefined closed circular trajectories with optical trapping potentials. The driving force $F$, pointing towards the trap’s minimum (marked by red crosses), may be maintained constant or modulated as a predefined function of the phase angle $\varphi$. The time dependence of $\varphi$ is not predefined and evolves under the net action of the driving force and any other forces acting on the particle. (b) The orbital velocity can be either held constant or modulated so that it is anisotropic: the functional form $F(\varphi )=F_0[1-A_2\sin (2\varphi )]$ has been chosen in this work, studying the effect of the modulation parameter $A_2$ following 24. The velocity of a single orbiting particle is shown here for $A_2=$, 0.3 and 0.85 (increasing modulation). Markers are measured velocities (proportional to force at low Re) and lines are the expected shape, proportional to $[1-A_2\sin (2\varphi )]$.Reuse & Permissions

Figure 2

Relaxation time $\tau$ depending on parameters $k_r$ and $A_2$. Experimental results (markers), simulations (dotted line), and theoretical value (solid line) from Eq. (4), well defined only for weak coupling. In both simulations and the analytical formula, a correction has been applied to $k_r$, as described in the main text. In (a) the experimental $k_r$ is varied for $A_2=0$ (plusses), 0.4 (circle), and 0.7 (square). In (b) $A_2$ is varied for $k_r=1.7$ (plusses), 4.0 (circle), and 9.9 (square) $\mathrm{pN}/\mu \mathrm{m}$. Decreasing $k_r$ and increasing $A_2$ produces stronger synchronization. The inset shows the transient to phase locking between ${\varphi }_1$ and ${\varphi }_2$, for different initial conditions, in the HOT setup (shown here with $A_2=0$, $k_r\sim 10\mathrm{pN}/\mu \mathrm{m}$, and $F_0\sim 8\mathrm{pN}$). It typically takes a few orbits for the phase difference to decay into the stable in-phase synchronized state. When both $k_r$ is small and $A_2$ is close to 1, the strength of synchronization $\Gamma$ is not small compared to 1: theory becomes inaccurate and the argument of the log in Eq. (4) can even become negative, as in some sections for $k_r=1.7\mathrm{pN}/\mu \mathrm{m}$ in (b).Reuse & Permissions

Figure 3

Relaxation time depending on the Lenz (${\Gamma }_L$) and Golestanian (${\Gamma }_G$) dimensionless control parameters. Simulations (circles) are in excellent agreement with the theoretical model in Eqs. (3, 4) (mesh surface). The thick line separates the regions in which synchronization is dominated by the flexibility of the rotor (Lenz synchronization, LS) or by the angular modulation of the driving force (Golestanian synchronization, GS). For ${\Gamma }_L+{\Gamma }_G$ close to 0, synchronization can be lost because of thermal fluctuations; simulations were performed in a range in which rotors are synchronized, for which the relaxation time can be easily extracted.Reuse & Permissions

Figure 4

Thermal noise leads to fluctuations of $\Delta \varphi$ around 0 (the in-phase state). (a) Simulations for variable values of $A_2$, $k_r$, $a$, $r$, $F_0$, $d$, and $\eta$ in a regime of small fluctuations ($T=10\mathrm{K}$). The variance of $\Delta \varphi$ exhibits a linear dependence on $t_0{k}_{B}T/(\Gamma \gamma r^2)$, with a coefficient that depends only on $A_2$ (as shown in the inset). (b) For $a=1.74\mu \mathrm{m}$, $d=35\mu \mathrm{m}$, $r=3.2\mu \mathrm{m}$, $F_0=5.5\mathrm{pN}$, $k_r=12.97\mathrm{pN}/\mu \mathrm{m}$, and $A_2=0.0$, the simulation temperature is varied from 0 to $1.5\times {10}^5\mathrm{K}$. The line shows the variance from Eq. (5) multiplied by the coefficient obtained from (a). The inset is an enlarged portion of the main graph (plusses), with experiment results (square, experimental $k_r=5.9\mathrm{pN}/\mu \mathrm{m}$) obtained by varying the rotor separation between 20 and $44\mu \mathrm{m}$. Varying $d$ is equivalent to varying $T$ when the fluctuations are well described by Eq. (5). Synchronization was lost between $d=44\mu \mathrm{m}$ (last point on the graph) and $d=48\mu \mathrm{m}$ (not represented: phase slips were observed and $⟨\Delta {\varphi }^2⟩=1.7{\mathrm{rad}}^2$).Reuse & Permissions