Memory-Induced Transition from a Persistent Random Walk to Circular Motion for Achiral Microswimmers

We experimentally study the motion of light-activated colloidal microswimmers in a viscoelastic fluid. We find that, in such a non-Newtonian environment, the active colloids undergo an unexpected transition from enhanced angular diffusion to persistent rotational motion above a critical propulsion speed, despite their spherical shape and stiffness. We observe that, in contrast to chiral asymmetric microswimmers, the resulting circular orbits can spontaneously reverse their sense of rotation and exhibit an angular velocity and a radius of curvature that nonlinearly depend on the propulsion speed. By means of a minimal non-Markovian Langevin model for active Brownian motion, we show that these nonequilibrium effects emerge from the delayed response of the fluid with respect to the self-propulsion of the particle without counterpart in Newtonian fluids.

Figure 1

(a) Image of a spherical colloid, which is half-coated with carbon (dark area). The arrow pointing from the capped to the uncapped hemisphere represents its orientation $\mathbf{n}$, which defines the angle $\theta$ with respect to the $x$ axis. Some exemplary trajectories $\mathbf{r}=(x,y)$ (solid lines) and orientations $\mathbf{n}=(\cos \theta ,\sin \theta )$ (arrows) at different propulsion speeds are plotted in: (a) $v=0.075\mu \mathrm{m}{\mathrm{s}}^{-1}$, (b) $v=0.100\mu \mathrm{m}{\mathrm{s}}^{-1}$, (c) $v=0.250\mu \mathrm{m}{\mathrm{s}}^{-1}$, (d) $v=0.350\mu \mathrm{m}{\mathrm{s}}^{-1}$, and (e) $v=0.500\mu \mathrm{m}{\mathrm{s}}^{-1}$. Insets: Time evolution of the corresponding angle $\theta$. The inset of Fig. 1 also shows the time evolution of $\theta$ for a passive colloid ($v=0$, upper curve). (f) Log-log representation of the mean-squared angular displacements for different propulsion speeds, increasing from bottom to top. Inset: Expanded view of the mean-squared angular displacement for a passive colloid, showing the change from subdiffusive to diffusive behavior with increasing $t$. The dotted and dashed lines are guides to the eye to indicate diffusive and ballistic behavior, respectively.

Figure 2

(a) Dependence of the angular speed $|\omega |$ of the circular orbits on the propulsion speed $v$ for an active colloid moving in a PAAm solution at 0.05 wt % (circle) and 0.06 wt % (square). The vertical dotted line depicts the critical velocity $v_c$ in the former case, below which only enhanced rotational diffusion occurs. (b) Dependence of the corresponding radius of curvature $R$ on $v$, same symbols as in 2. The solid lines in 2 and 2 represent the dependence on $v$ given by Eq. (3), while the error bars correspond to the standard deviation over several cycles with the same sense of rotation.

Figure 3

(a) Mean-squared angular displacements computed from numerical solutions of Eqs. (1) and (2) at different propulsion speeds $v<v_c=0.240\mu \mathrm{m}{\mathrm{s}}^{-1}$, increasing from bottom to top. (b) Corresponding effective rotational diffusion coefficient as a function $v$. The vertical dashed line represents $v_c$. Inset: Time evolution of $\theta$ for a numerical passive trajectory ($v=0$, bottom) and an active one ($v=0.200\mu \mathrm{m}{\mathrm{s}}^{-1}$, top). Examples of numerical trajectories at different propulsion speeds above $v_c$: (c) $v=0.250\mu \mathrm{m}{\mathrm{s}}^{-1}$, and (d) $v=0.600\mu \mathrm{m}{\mathrm{s}}^{-1}$. The thick arrows represent the orientation $\mathbf{n}$, whereas the thin arrows show the corresponding propulsion force ${\mathbf{F}}_v$. (e) Time evolution of the phase difference between $\mathbf{n}$ and ${\mathbf{F}}_v$ for a particle moving at $v=0.250\mu \mathrm{m}{\mathrm{s}}^{-1}$ (inner curve) and at $v=0.600\mu \mathrm{m}{\mathrm{s}}^{-1}$ (outer curve). The straight lines represent the steady-state values ${\varphi }_{\pm }$.