Hydrodynamic Synchronization of Chiral Microswimmers

We study synchronization in bulk suspensions of spherical microswimmers with chiral trajectories using large scale numerics. The model is generic. It corresponds to the lowest order solution of a general model for self-propulsion at low Reynolds numbers, consisting of a nonaxisymmetric rotating source dipole. We show that both purely circular and helical swimmers can spontaneously synchronize their rotation. The synchronized state corresponds to velocity alignment with high orientational order in both the polar and azimuthal directions. Finally, we consider a racemic mixture of helical swimmers where intraspecies synchronization is observed while the system remains as a spatially uniform fluid. Our results demonstrate hydrodynamic synchronization as a natural collective phenomenon for microswimmers with chiral trajectories.

Figure 1

Model for rotational squirmers. (a) The surface slip flows corresponding to the different modes: $B_1$, $C_1$, and ${\tilde{B}}_{11}$ in the particle frame. The magnitude of the normalized surface velocity (slip flow) for each mode is represented by a color-code and the streamlines are colored black. (b) The particle trajectories in the lab frame, corresponding to circular (left) and helical (right) swimming. The unit vectors $\mathit{m}$ and $\mathit{s}$ correspond to the particle polar and azimuthal axes, respectively, and $\psi$ is the inclination angle with respect to $\mathit{m}$. (c) Swimmer flow field obtained from the simulations, corresponding to a source dipole $\mathbf{B}$ (neutral squirmer). The magnitude of the fluid velocity is coloured using a logarithmic scale and overlaid by black streamlines.

Figure 2

Synchronization diagram for circular swimmers ($\psi =90°$) as a function of the volume fraction $\varphi$ and particle trajectory radius $r_t$. Green circles indicate global synchronization, and the red triangles mark isotropic states. The synchronization region is colored according to a waiting time $t_{\mathrm{sync}}/T_0$ corresponding to the total time elapsed from the start of the simulation until synchronization is reached. The white curve corresponds to $\varphi ={\varphi }_c^{\prime }[(4/3\pi a^3)/2\pi r_t^{2}a]$ with ${\varphi }_c^{\prime }=70\%$. (see text and Supplemental Material [85] for details).

Figure 3

(a) Circular swimmers: Example of a typical time evolution of the azimuthal $P_s$ (red), velocity $P_v$ (blue), and polar $P_m$ (black) order parameters. (b) Radial distribution function $g(r)$ of the system at the beginning (gray) and at the end (black) of the simulation. The $g(r_{\perp })$ ($g(r_{||})$) are calculated perpendicular (parallel) to the polar director ${\mathit{P}}_M$. Probability distribution of the (c) angular velocities $\omega$ and (d) phase lag angle $\alpha$ between all particle pairs, at the start and end of the simulation. (e) Snapshots of 25 selected particles at the beginning (left) and end (right) of the simulation. The particles are colored according to the $y$ component of their $\mathit{s}$ vector. The trajectories are shown over one period and colored according to the $x$ component of the swimmer’s $\mathit{m}$ vector. (The data correspond to $\varphi \approx 0.15$ and $r_t\approx 3.33a$).

Figure 4

State diagram for helical swimmers as a function of $\varphi$ and $\lambda =r_t/p$. The green circles correspond to global synchronization, and red triangles to isotropic states. The blue diamonds mark polar order for classic linear neutral squirmers and yellow diamonds correspond to finite polar order in the absence of synchronization for chiral swimmers. (data correspond to $p\approx 21a$).

Figure 5

Helical swimmers: Time evolution of the order parameters $P_s$ (red), $P_v$ (blue), and $P_m$ (black) for (a) $\varphi \approx 0.083$, $\lambda \approx 0.19$, and (b) $\varphi \approx 0.125$, $\lambda \approx 0.1$. (c) Snapshots of the system in the synchronized state. The uwrapped trajectories of all the $N=286$ helical swimmers colored according to ${\mathit{m}}_x$ (left). 8 selected microswimmers with their trajectories colored as a function of time (right). The particles are colored according to ${\mathit{s}}_y$. [The snapshots in (c) correspond to $\varphi \approx 0.15$, $\lambda \approx 0.16$].

Figure 6

Racemic mixture: Distributions of the (a) spinning frequency $\omega$ and (b) the phase-angle difference $\alpha$ calculated for all swimmer pairs (black), for clockwise (blue) and counterclockwise (red) rotating populations, as well as for the cross population (orange). (c) Total (black), homochiral (violet), and heterochiral (orange) radial distribution functions. (d) Snapshot of the unwrapped trajectories at the steady state after $20T_0$. (The data correspond to $\varphi \approx 0.1$ and $\lambda \approx 0.16$).