Diffusivity of E. coli-like microswimmers in confined geometries: The role of the tumbling rate

We analyzed the effect of confinement on the effective diffusion of a run-and-tumble E. coli-like flagellated microswimmer. We used a simulation protocol where the run phases are obtained via a fully resolved swimming problem, i.e., Stokes equations for the fluid coupled with rigid-body dynamics for the microorganism, while tumbles and collisions with the walls are modeled as random reorientation of the microswimmer. For weak confinement, the swimmer is trapped in circular orbits close to the solid walls. In this case, optimal diffusivity is observed when the tumbling frequency is comparable with the angular velocity of the stable orbits. For strong confinement, stable circular orbits disappear and the diffusion coefficient monotonically decreases with the tumbling rate. Our findings are generic and can be potentially applied to other natural or artificial chiral microswimmers that follow circular trajectories close to an interface or in confined geometries.

Figure 1

Flagellated bacterium model and channel geometry. The flagellated microswimmer is modeled as a system of two rigid bodies: the head (a prolate ellipsoid) and the tail (an helical tubular structure). The head and tail axes lie along the same direction, and the tail can rotate with respect to the head. The system has 7 degrees of freedom, namely the position ${\mathbf{x}}_{\mathbf{j}}$ of the junction between the head and the tail, the orientation of the head with respect to the fixed reference frame, and the rotation angle $\varphi$ of the tail with respect to the head. In the plane channel geometry, the only relevant degrees of freedom are the distance between the junction and the wall $z_j$ and the angle $\mathrm{\Theta }$ between ${\widehat{\mathbf{e}}}_{\mathbf{1}}$ and $\widehat{\mathbf{x}}$.

Figure 2

Deterministic microswimmer trajectory for a boundary accumulating case. Panels (a) and (b) show $xz$ and $xy$ projections of the 3D trajectory for two initial conditions. Red (dashed) line, the swimmer stabilizes on a closed circular orbit; black (solid) line, the swimmer collides with the wall. The reduced $(\mathrm{\Theta },z_j)$ phase space is represented in panel (c). The color refers to the different manifolds. Trajectories originating from the gray regions (I and II) eventually collide with the wall. Trajectories originating from green (III) and yellow (IV) regions are attracted to one of the two stable orbits (red points close to the walls). Channel height, $h=15$. Microswimmer geometry: tail length $L=5.31$, tail amplitude $A=0.286$, tail wavelength $\lambda =1.8$, tail radius $a_t=0.05$, cell equivalent radius $a=1.0$, with $a_2/a_1=0.33$.

Figure 3

(a) Effective diffusion coefficient $D$ as a function of the tumbling rate $\alpha$ for boundary accumulating microswimmers (same geometry as in Fig. 2) for tumbling angle $\psi =\pi /3$. The tumbling rate $\alpha$ is normalized with the frequency ${\omega }_s$ corresponding to the stable circular motion observed for the same microswimmer in the absence of tumbling. (b),(c) Examples of corresponding trajectories for low tumbling rate (blue), optimal tumbling rate (red), and high tumbling rate (green).

Figure 4

(a) Optimal tumbling rate ${\alpha }^{*}$ as a function of the tumbling angle $\psi$ for a boundary accumulating case (same geometry as in Fig. 2). The red dashed line corresponds to the relation ${\alpha }^{*}/{\omega }_s=1/\psi$. (b) Sketch of the geometrical argument discussed in the text. In a 2D model where the swimmer moves along circles during the run phase, the maximum possible displacement after two consecutive run phases is obtained when the displacement vectors are aligned (red lines correspond to the chord of the circular trajectories).

Figure 5

Deterministic microswimmer trajectory in a boundary escaping case. Panels (a) and (b) show, respectively, $xz$ and $xy$ projections of the 3D trajectory for two initial conditions. Red (dashed) line, the swimmer stabilizes on a trajectory parallel to the channel midplane; black (solid) line, the swimmer collides with the wall. The reduced $(\mathrm{\Theta },z_j)$ phase space is represented in panel (c). The color refers to the different manifolds. Trajectories originating from the gray regions (I and II) eventually collide with the wall. Trajectories originating from blue (III) region asymptotically move on a straight line parallel to the channel midplane, red dot in the $\mathrm{\Theta },z_j$ phase space. Channel height, $h=3.75$. Same swimmer geometry as in Fig. 2.

Figure 6

Effective diffusion coefficient $D$ as a function of the tumbling rate $\alpha$ for boundary escaping condition for different tumbling angle $\psi$. For the sake of clarity, $D$ is normalized with the value obtained for $\alpha =4.4\times {10}^{-3}$. The solid line $D\sim {\alpha }^{-1}$ is drawn to guide the eye. The prefactor $c$ is shown in the inset, where the dotted line corresponds to the 2D random flight model prediction, Eq. (5).

Figure 7

Channel height $h=7.5$. Same swimmer geometry as in Fig. 2. Panel (a) shows the $xz$ projection of the 3D trajectory that oscillates around the midplane. The reduced $(\mathrm{\Theta },z_j)$ phase space is shown in panel (b). Trajectories originating from the gray regions (I and II) eventually collide with the wall. Trajectories originating from pink regions (${\mathrm{III}}_a$ and ${\mathrm{III}}_b$) asymptotically approach the limit cycle that corresponds to oscillating trajectories such as the one shown in panel (a). Panel (c) shows the effective diffusion coefficient $D$ as a function of the tumbling rate $\alpha$ for different tumbling angle $\psi$. $D$ is normalized with the value obtained for $\alpha =4.4\times {10}^{-3}$. The solid line $D\sim {\alpha }^{-1}$ is drawn to guide the eye. The prefactor $c$ is shown in the inset, the dotted line corresponds to the 2D random flight model prediction, Eq. (5).