Manipulating chiral microswimmers in a channel

We numerically simulate the diffusion of overdampd pointlike Janus particles along narrow two-dimensional periodically corrugated channels with reflecting walls. The self-propulsion velocity of the particle is assumed to rotate subject to an intrinsic bias modeled by a torque. Breaking the mirror symmetry of the channel with respect to its axis suffices to generate a directed particle flow with orientation and magnitude which depend on the channel geometry and the particle swimming properties. This means that chiral microswimmers drift autonomously along a narrow channel under more general asymmetry conditions than previously reported, a property of potential impact on their fabrication and technological applications.

Figure 1

Chiral Janus particle in a narrow channel: (a) particle with velocity ${\mathbf{v}}_0$ and torque $\Omega$, Eq. (1), and examples of periodically corrugated channels with different symmetry properties: (b) upside-down asymmetric and right-left symmetric; (c) right-left asymmetric and upside-down symmetric, and (d) centrosymmetric. Channel walls: (b) and (d) as in Eqs. (2) with $\epsilon$ and $x_0$ given in the legends; (c) the triangular wall profile of Ref. [23]. In (b) the open boundary trajectories and a closed circular trajectory of radius $R_{\Omega }$ are drawn for explanatory purposes (see Fig. 2 for actual simulation data).

Figure 2

Logarithmic contour plots of the stationary particle density, $P(x,y)$, in a compartment of Eq. (2) with $x_L=y_L=1$, $x_0=0$, $\Delta =0.12$, and $\epsilon =0.25$, for different values of $\Omega$. Other simulation parameters are ${\tau }_{\theta }=10$, $v_0=1$, and $D_0=0.01$. For $\Omega =2$, central panel, the boundary flows are the strongest, as $P(x,y)$ gets uniformly depleted at the center of the compartment. This condition corresponds to a maximum in the rectification power of Fig. 4.

Figure 3

Rectification of a levogyre JP in the channel of Eq. (2): (a) $\overline{v}/v_0$ vs $\epsilon$ for $\Delta =0.12$, $D_0=0.01$, and different $\Omega$; (b) $\eta =|\overline{v}|/v_0$ vs $\Delta$ for $\Omega =1$ and different $\epsilon$, with $\epsilon <1$, and $D_0$ (see legends). Note that here $\overline{v}=-\left|\overline{v}\right|$ and for a right-left symmetric channel, $\overline{v}(-\Omega )=-\overline{v}\left(\Omega \right)$. Other simulation parameters are $D_{\theta }=0.3$, $v_0=1$, $x_0=0$, and $x_L=y_L=1$.

Figure 4

Optimization of the rectification of a levogyre JP with $v_0=1$ in the channel of Eq. (2) with $\epsilon =0.25$, $x_0=0$, and $x_L=y_L=1$: $\eta$ vs $\Omega {\tau }_X$ for $\Delta =0.12$, ${\tau }_{\theta }=10$, and different $D_0$ (see legend). Here ${\tau }_X$ is the compartment crossing time, ${\tau }_X=x_L/v_0$, and the vertical arrow points to ${\Omega }_M{\tau }_X$.

Figure 5

(a) Rectification of a Janus particle in a $x\to -x$ mirror symmetric channel in the presence of torque: $\overline{v}/v_0$ vs $\epsilon$ for $D_{\theta }=0.3$, $\gamma =0.2$, $F_0=0.5$, $kT=0.05$, and different values of $B$. The channel boundaries are described by Eq. (2) with $x_L=y_L=1$, $\Delta =0.1$, and $x_0=0$. Inset: ¯$\overline{v}/v_0$ vs $y_L$ for $B=1$, $\epsilon =0$ (half a sinusoidal channel) and different $\Delta$. The remaining parameters are as in the main panel; (b) $\overline{v}$ vs $B$ for $D_{\theta }=0.03$ and different values of $\gamma$, $F_0$, and $kT$ (see legend). Units are as in Eq. (4) and the channel boundaries as in Eq. (2) with $\epsilon =0$ (half a sinusoidal channel), $x_L=y_L=1$, and $\Delta =0.12$; (c) rectification of an inertial Janus particle in an asymmetric channel at zero torque, $B=0$: $\overline{v}/v_0$ vs $x_0$ for different $\gamma$. $w_{\pm }\left(x\right)$ are given by Eq. (2) for $x_L=y_L=1$, $\epsilon =0.25$, and $\Delta =0.12$; other simulation parameters are $F_0=0.5$, $kT=0.05$, and $D_{\theta }=0.3$.

Figure 6

Rectification of a levogyre JP in an annulus of outer radius $R_1=10$ and width $\Delta R=R_1-R_2$ (see sketch) as a function of $\Omega$: net angular velocity, ${\overline{\omega }}_{\theta }$ vs $R_1/R_{\Omega }$, for ${\tau }_{\theta }=20$, $D_0=0.01$, $v_0=1$, and different $\Delta R$. Here, $R_{\Omega }=v_0/\Omega$ and ${\omega }_1=v_0/R_1=0.1$, i.e., $R_1/R_{\Omega }=\Omega /{\omega }_1$.

Figure 7

Sketch of a chiral sieve consisting of cascaded rings: As a mixture of levogyre (L) and dextrogyre JPs (D) is pumped from left to right, the levogyre JPs are selectively removed at the links.