Directed autonomous motion of active Janus particles induced by wall-particle alignment interactions

TL;DR

This work demonstrates a non-periodic ratchet mechanism that rectifies active particle motion in a narrow channel through wall-induced alignment interactions. By introducing asymmetry between the top and bottom wall couplings or a transverse gravitational drag, chiral Janus particles acquire a directed drift with rectification powers exceeding 60%, and the direction can be flipped by changing chirality or wall coupling. For achiral particles, an unbiased Couette flow provides the requisite torque to generate directed motion, with analytic-like expressions showing how the drift scales as $\overline{v} \approx - \frac{v_0 u_0}{y_L\omega_a} + ...$ depending on $\kappa$. The findings are robust across a broad parameter range and offer a practical route to transport and sort microswimmers without periodic substrates, with potential applications in controlling artificial and biological active matter.

Abstract

We propose a highly efficient mechanism to rectify the motion of active particles by exploiting particle-wall alignment interactions. Through numerical simulations of active particles' dynamics in a narrow channel, we demonstrate that a slight difference in alignment strength between the top and bottom walls or a small gravitational drag suffices to break upside-down symmetry, leading to rectifying the motion of chiral active particles with over 60% efficiency. In contrast, for achiral swimmers to achieve rectified motion using this protocol, an unbiased fluid flow is necessary that can induce orbiting motion in the particle's dynamics. Thus, an achiral particle subject to Couette flow exhibits spontaneous directed motion due to an upside-down asymmetry in particle-wall alignment interaction. The rectification effects caused by alignment we report are robust against variations in self-propulsion properties, particle's chirality, and the most stable orientation of self-propulsion velocities relative to the walls. Our findings offer insights into controlled active matter transport and could be useful to sort artificial as well as natural microswimmers (such as bacteria and sperm cells) based on their chirality and self-propulsion velocities.

Figures (6)

• Figure 1: (color online) A schematic illustration depicts various stable self-propulsion velocity orientations, shown by red arrows. The associated interaction potential $V(\theta)$ [refer to Eqs. (\ref{['top_V']})] is represented by solid and dotted lines for the top and bottom walls, respectively. (a) Configuration I: The particle is most stable at angles $\theta = {\pi}/{2}$ (near the top wall) and $\theta = {3\pi}/{2}$ (near the bottom wall). For this configuration, the interaction potential with the top wall is given by Eq. (\ref{['top_V']}), where ${q} = 1$ and $\phi = 0$. For the bottom wall the alignment interaction potential, $V_b(\theta) = -\kappa V_t(\theta)$. (b) Configuration II: In this case, the stable orientation of $\mathbf{v_0}$ occurs at $\theta = {\pi}/{4}$ and $\theta = {3\pi}/{4}$ near the top wall. Near the bottom wall, the stable orientations correspond to $\theta = {5\pi}/{4}$ and $\theta = {7\pi}/{4}$. For this configuration, ${q} = 4$ and $\phi = 0$. Note that for the top wall, $V(\theta)$ is given by Eq. (\ref{['top_V']}) for $0 \leq \theta \leq \pi$; otherwise, $V(\theta) = 0$. For the bottom wall, $V_b(\theta) = \kappa\omega(y) \cos(4\theta)$ for $\pi \leq \theta \leq 2\pi$; otherwise, $V(\theta) = 0$. (c) Configuration III: Here, the particle-wall interaction energy reaches its minimum when $\mathbf{v_0}$ is parallel to the wall. The interaction potential is described by Eq. (\ref{['top_V']}), with ${q} = 2$ and $\phi = {\pi}/{2}$. Further, for this configuration, the interaction potential for both the top and bottom wall alignments is the same for $\kappa =1$, [that is $V_b(\theta)=\kappa V_t(\theta)$ ].
• Figure 2: (Color online) (a) The schematic depicts a typical orientation of a chiral active particle of the Janus kind, illustrating the tilting angle ($\overline{\theta}$), the direction of the self-propulsion velocity ${\mathbf{v_0}}$, and its normal ($v_0 \cos\overline{\theta}$) and tangential components ($v_0 \sin\overline{\theta}$). The blue dotted line represents the locus of the particle's center of mass. (b) Average velocity $\overline{v}$ as a function of alignment-induced coupling strength ($\omega_a$) for different alignment ratios of alignment-interaction strength, $\omega_t/\omega_b = \kappa$ (see legends). (c) Similar to the panel, $\overline{v} \; vs. \; \omega_a$ for different intrinsic torques $\Omega_I$ (see legends). The plots with hollow and solid symbols, respectively, correspond to dextogyre (-ve $\Omega_I$) and levogyre (+ve $\Omega_I$) active particles. In the panels (b) and (c), the dotted lines and dashed lines indicate analytical estimation based on the Eq. (\ref{['drift-approx-1a']}-\ref{['drift-approx-1b']}) and Eq. (\ref{['drift-2nd-peak']}), respectively. Simulation parameters (unless reported otherwise in the legends): $D_{\rm \theta} = 0.01 \; s^{-1}$, $D_0 = 0.01 \; \mu m^2/s$, $v_0 = 1\; \mu m/s$, $\Omega_I = 1 \; s^{-1}$, $\kappa = 0.125, \; u_s = 0,\; \Omega_s = 0, \; \; g = 0,\; \lambda = 0.05\; \mu m, \; y_L = 1\; \mu m$.
• Figure 3: (Color online) $\overline{v}$ as a function of alignment induced coupling strength ($\omega_a$) for different péclet number $Pe$ (see legends). To get different $Pe$ values, we have varied $v_0$ for $D_{\rm \theta} = 0.01 \, s^{-1}$ and $D_0 = 0.01 \, \mu m^2/s$. The dotted lines and dashed lines indicate analytical estimation based on the Eq. (\ref{['drift-approx-1a']}-\ref{['drift-approx-1b']}) and Eq. (\ref{['drift-2nd-peak']}), respectively. Simulation parameters (unless reported otherwise in the legends): $\Omega_I = 1 \; s^{-1}$, $\kappa = 0.125, \; u_s = 0,\; \Omega_s = 0, \; \; g = 0,\; \lambda = 0.05\; \mu m, \; y_L = 1\; \mu m$.
• Figure 4: (Color online) Rectification of motion of chiral active particles when upside-down symetry is broken due to particle's apparent weight. (a) $\overline{v}$ as a function of alignment induced coupling strength ($\omega_a$) for various apparent weights $g$ (see legends). (b) $\overline{v} \; vs. \; \omega_a$ for different intrinsic torque values ($\Omega_I$), as shown in the legends. (c) $\overline{v} \; vs. \; \omega_a$ for different $Pe$. To achieve different $Pe$ values, we have varied $v_0$ for $D_{\rm \theta} = 0.01 \, s^{-1}$ and $D_0 = 0.01 \, \mu m^2/s$. In all three panels, the dotted lines represent the estimated drift velocity based on Equations (\ref{['g4']} - \ref{['g5']}). Simulation parameters (unless reported otherwise in the legends): $D_{\rm \theta} = 0.01 \, s^{-1}$, $D_0 = 0.01 \, \mu m^2/s$, $v_0=1\;\mu m/s$, $\Omega_I = 1 \; s^{-1}$, $\kappa = 0, \; u_s = 0,\; \Omega_s = 0, \; \; g = 0.1 \; \mu m/s,\; \lambda = 0.05\; \mu m, \; y_L = 1\; \mu m$
• Figure 5: (Color online) (a) Schematic diagrams illustrate the possible stable orientations of self-propulsion velocity near the walls for configurations II and III. Small arrows indicate the direction of $\mathbf{v_0}$ when $\Omega_I = 0$, while larger arrows represent the self-propulsion direction under positive chiral torque. The four states, ($\tau_{tL},v_{tL}$), ($\tau_{tR},v_{tR}$), ($\tau_{bL},v_{bL}$) and ($\tau_{bR},v_{bR}$) are characterized by their waiting times $\tau_{ij}$ ($i=t,b;\;j=L, R$) and the components of self-propulsion velocities $v_{ij}$ along the channel axis (as explained in the text). In panel (b), $\overline{v}$ is plotted against $\omega_a/\Omega_I$ for configuration II with varying values of Pe ($D_\theta$ values are varied for $v_0=1\; \mu m/s$ and $D_0=0.01 \;\mu m^2/s$). Dashed lines are predictions based on Eq. (\ref{['con-2_3']}). Panels (c) and (d) show rectification efficiency $\eta$ as a function of $\omega_a/\Omega_I$ for configuration III, considering different values of $\Omega_I$ and $Pe$ (refer to the legends for details). To obtain different $Pe$, here we varied $v_0$ for $D_{\rm \theta} = 0.01 s^{-1}$, and $D_0 = 0.01 \mu m^2/s$. Dashed lines are predictions based on Eq. (\ref{['con-3_2']}). Simulation parameters (unless reported otherwise in the legends): $\Omega_I = 1.0 s^{-1}$, $v_0 = 1.0 \mu m/s$, $\kappa = 0.125, \; u_s = 0, \; \Omega_s = 0, \; \; g = 0,\; \lambda = 0.05\;\mu m, \; y_L = 1\;\mu m$.