Large Scale Zigzag Pattern Emerging from Circulating Active Shakers

We report the emergence of large zigzag bands in a population of reversibly actuated magnetic rotors that behave as active shakers, namely squirmers that shake the fluid around them without moving. The shakers collectively organize into dynamic structures displaying self-similar growth and generate topological defects in the form of cusps that connect vortices of rolling particles with alternating chirality. By combining experimental analysis with particle-based simulation, we show that the special flow field created by the shakers is the only ingredient needed to reproduce the observed spatiotemporal pattern. We unveil a self-organization scenario in a collection of driven particles in a viscoelastic medium emerging from the reduced particle degrees of freedom, as here the frozen orientational motion of the shakers.

Figure 1

(a) Trajectories of a dilute suspension of shakers after $N=68$ cycles of the rotating field, Movie S1 in Ref. [47]. Top inset shows scanning electron microscope image of one hematite particle with the permanent magnetic moment $\mathit{m}$. Small scheme on the right-hand side shows lateral view of one magnetic roller. (b) Mean displacement $⟨x⟩$ versus time $t$ of shakers. Dashed line denotes the field direction of rotation. (c) Sequence of images illustrating the band formation from an initially disordered suspension ($t=0$) and under a magnetic modulation with $f=80\mathrm{Hz}$ and $\delta t=1/8\mathrm{s}$; see also Movie S3 in Ref. [47]. The image at $t=240\mathrm{s}$ shows two bands at a distance $D$ with wavelength $\lambda$, bond angle $\theta$, and thickness $e$. (d) Average horizontal velocity $⟨v_y⟩$ of the shakers in bands. Red arrows denote the direction of the circulating particles, Movie S4 in Ref. [47].

Figure 2

(a) Normalized flow field created by a shaker $v_r/v_M$ in the $(x,y)$ plane. The color codes for the radial velocity $v_r$ are normalized by the maximal radial velocity $v_M$. It shows regions of attractions ($v_r<0$) and repulsion ($v_r>0$) separated by ${\theta }_l$. Left-hand (right-hand) panel refers to experimentally measured (simulated) flow field ($v_M=2.2\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$ for the experiments, $v_M=22\mathrm{\mu }\mathrm{m}{\mathrm{s}}^{-1}$ for the simulation; see also the SM [47]). (b) Normalized first normal stress difference in the $(x,y)$ plane with ${\sigma }_0=({\eta }_s+{\eta }_p)2\pi f$; see text. (c) Schematics of two microrotors in three configurations: attraction (repulsion) arises when $\theta >{\theta }_l$ ($\theta <{\theta }_l$) with ${\theta }_l=31°$. (d)–(f) Evolution with time of (d) the average radial velocity $⟨v_r⟩$, (e) the relative angle $⟨\theta ⟩$, and (f) the radial distance $⟨r⟩$ between two approaching microrotors. In all images blue [black] (yellow [gray]) arrows indicate attraction (repulsion) between the pair, Movie S5 in Ref. [47]. (g) Top view of the flow velocity generated from the interaction of two shakers. (h) Assembled structure from the flow produced by the shakers. (i) Sequence of snapshots showing a one-particle thin band with superimposed two particle’s trajectories, Movie S6 in Ref. [47].

Figure 3

(a) Sequence of snapshots showing the annihilation of a cusp (total duration $\mathrm{\Delta }t=30\mathrm{s}$), Movie S7 in Ref. [47]. Coarsening dynamics: time evolution of the mean angle $⟨\theta ⟩$ (b), wavelength $⟨\lambda ⟩$ (c), thickness $⟨e⟩$ (d), and distance $⟨D⟩$ (e). After a short transitory, $\lambda$, $e$, and $D$ all scale linearly with time (red line) while $\theta$ set to a stationary value of ${\theta }_l=31°$. In (b) $⟨\theta ⟩$ corresponds to the full average over the experimental data, weighted by the size of the zigzag bands, while $⟨{\theta }_M⟩$ is its maximal value over the bands.

Figure 4

(a) Average radial velocity $⟨v_r⟩$ between two particles as a function of their relative distance $r$ and for different angle $\theta$ (color bar). Scattered data are experiments, while continuous lines are nonlinear regression using Eq. (9) in Ref. [47]. (b),(c) Average radial distance (b) and relative angle (c) as function of time for pairs of interacting particles. In both graphs, experiments (simulation) are represented by the red [gray] (black) curve. (d) Image showing the zigzag state obtained after a time $t=50\mathrm{s}$, with $N=601$ particles, Movie S8 in Ref. [47]. The inset shows the initial random configuration of the particles. (e) Corresponding color-coded plot showing the average horizontal velocity $⟨v_y⟩$ with arrows denoting the direction of the circulating particles.