Dynamics of a Brownian circle swimmer

Self-propelled particles move along circles rather than along a straight line when their driving force does not coincide with their propagation direction. Examples include confined bacteria and spermatozoa, catalytically driven nanorods, active, anisotropic colloidal particles and vibrated granulates. Using a non-Hamiltonian rate theory and computer simulations, we study the motion of a Brownian “circle swimmer” in a confining channel. A sliding mode close to the wall leads to a huge acceleration as compared to the bulk motion, which can further be enhanced by an optimal effective torque-to-force ratio.

Figure 1

(Color online) Trajectories of the mean position $\overline{\mathbf{r}}$ of the self-propelling rod for fixed $\beta FL=10$, $\beta M=0.2,1,5,25$ (${\mathbf{r}}_0=0$, ${\varphi }_0=0$). Left inset: the mean-square displacement $\overline{\Delta {\mathbf{r}}^2}$ for the same force and torques, but also for $\beta FL=0$, $\beta M=0$ (lowermost curve). Right inset: a typical trajectory of the rod for $\beta FL=25$, $\beta M=10$, for times $0<t<{\tau }_B$. Lower right inset: Sketch of the self-propelled circle swimmer.

Figure 2

(Color online) Mean-square displacement $\overline{\Delta {\mathbf{r}}^2}$ in confinement ($\beta FL=60$, $L_x=8L$) without gravity and with zero torque (black), without gravity and with finite torque (red), and with torque and gravity (blue). The left inset displays the rod sliding along the walls. The right inset shows the confining potential without (left) and with (right) gravity.

Figure 3

(Color online) Long-time diffusion coefficient $D_L$ as a function of the torque $M$ for the swimmer in the bulk and in confinement for $\beta FL=60$ and $L_x=6,8,10L$. (a) Computer simulation, (b) rate theory.

Figure 4

(Color online) Top panel: the paths governing the flipping rate: (a) turning in the direction of the torque, (b) turning against the direction of the torque, (c) three-stage event—first, turning in the direction of the torque and then turning against it. Bottom panel: The rates of the three different paths as a function of $M∕LF$ for $\beta FL=60$, $L_x=8L$.