Spiral diffusion of rotating self-propellers with stochastic perturbation

Translationally diffusive behavior arising from the combination of orientational diffusion and powered motion at microscopic scales is a known phenomenon, but the peculiarities of the evolution of expected position conditioned on initial position and orientation have been neglected. A theory is given of the spiral motion of the mean trajectory depending upon propulsion speed, angular velocity, orientational diffusion, and rate of random chirality reversal. We demonstrate the experimental accessibility of this effect using both tadpole-like and Janus sphere dimer rotating motors. Sensitivity of the mean trajectory to the kinematic parameters suggest that it may be a useful way to determine those parameters.

Figure 1

Traces of mean trajectories (solid blue) of synthetic ensembles constructed from video of a single motor trajectory, along with fits to the theory (dashed red). Two trajectories for each of the motor types, (a) Janus sphere dimers and (b) tadpoles, were selected for this investigation. The fit parameters $(\omega [\text{rad}/\text{s}],D_{\text{o}}[{\text{rad}}^2/\text{s}])$ to trajectories for dimers are left: (0.86, 0.176) and right: (1.07, 0.037), and for tadpole-like swimmers are left: (4.26, 0.446) and right: (6.15, 0.321).

Figure 2

Traces of the mean position $\langle \mathit{r}\left(t\right)\rangle$ of clockwise rotary self-propellers conditioned on initial position at the origin and initial velocity directed along ${\widehat{e}}_y$, for dimensionless orientational diffusivity $D_{\text{o}}/\omega =0.01,0.05$, and 0.1 and dimensionless flipping rate $f/\omega =0,0.2,0.5,0.7,1$, and 1.2. According to Eqs. (3), (4), and (6), $\langle \mathit{r}\left(t\right)\rangle$ spirals into its asymptotic value when $f<\omega$, but the approach is nonoscillatory for $f>\omega$. Watch the videos in the Supplemental Material [22] for a comparison of theory and simulation.

Figure 3

(a) The visibility ratio $\mathrm{vis}\left(t\right)={\left|\langle \mathit{r}\left(t\right)\rangle \right|}^2/\mathrm{var}(\mathit{r}\left(t\right))$ measures the ease of discerning the mean behavior of $\mathit{r}\left(t\right)$ against the background of its statistical spread. It drops to near zero more quickly the larger $f$ and $D_{\text{o}}$. Inset: Mean-squared displacement reaches the diffusive regime faster as $(f+D_{\text{o}})/\omega$ increases. (b) Left: ${|\langle \mathrm{\Delta }\mathit{r}\rangle }_{\infty }|/R=2(D_{\text{eff}}/R^2)\sqrt{{\omega }^{-2}+{(D_{\text{o}}+2f)}^{-2}}$ as a function of $D_{\text{o}}/\omega$ and $f/\omega$. Right: $|{\varphi }_{\infty }|={tan}^{-1}[\omega /(D_{\text{o}}+2f)]$, the magnitude of the angle between ${\langle \mathrm{\Delta }\mathit{r}\rangle }_{\infty }$ and ${\widehat{v}}_0$, as a function of $D_{\text{o}}/\omega$ and $f/\omega$.