Emergent vortices in populations of colloidal rollers

Abstract

Coherent vortical motion has been reported in a wide variety of populations including living organisms (bacteria, fishes, human crowds) and synthetic active matter (shaken grains, mixtures of biopolymers), yet a unified description of the formation and structure of this pattern remains lacking. Here we report the self-organization of motile colloids into a macroscopic steadily rotating vortex. Combining physical experiments and numerical simulations, we elucidate this collective behaviour. We demonstrate that the emergent-vortex structure lives on the verge of a phase separation, and single out the very constituents responsible for this state of polar active matter. Building on this observation, we establish a continuum theory and lay out a strong foundation for the description of vortical collective motion in a broad class of motile populations constrained by geometrical boundaries.

Figure 1. Experimental setup.

(a) Sketch of the setup. Five5-micrometre PMMA colloids roll in a microchannel made of two ITO-coated glass slides assembled with double-sided scotch tape. An electrokinetic flow confines the rollers at the centre of the device in a circular chamber of radius R_c. (b) Superimposed fluorescence pictures of a dilute ensemble of rollers (E₀/E_Q=1.1, φ₀=6 × 10⁻³). The colloids propel only inside a circular disc of radius R_c=1 mm and follow persistent random walks.

Figure 2. Dynamics of an isolated colloidal roller.

(a) Local packing fraction φ(r), averaged over the azimuthal angle φ, plotted as a function of the radial distance. The dashed line indicates the radius of the circular chamber. (b) Probability distribution function of the roller velocities measured from the individual tracking of the trajectories. (c) Autocorrelation of the roller velocity 〈v_i(t)·v_i(t+T)〉 plotted as a function of v₀T for packing fractions ranging from φ₀=6 × 10⁻³ to φ₀=10⁻². Full line: best exponential fit. (d) Superimposed trajectories of colloidal rollers bouncing off the edge of the confining circle. Time interval: 5.3 ms (E₀/E_Q=1.1, φ₀=6 × 10⁻³). Same parameters for the four panels: R_c=1.4 mm, E₀/E_Q=1.1, φ₀=6 × 10⁻³.

Figure 3. Collective-dynamics experiments.

(a) Snapshot of a vortex of rollers. The dark dots show the position of one half of the ensemble of rollers. The blue vectors represent their instantaneous speed (R_c=1.35 mm, φ₀=5 × 10⁻²). (b) Average polarization plotted versus the average packing fraction for different confinement radii. Open symbols: experiments. Full line: best fit from the theory. Filled circles: numerical simulations (b=3a, R_c=1 mm). (c) Time-averaged polarization field (R_c=1.35 mm, φ₀=5 × 10⁻²). (d) Time average of the local packing fraction (R_c=1.35 mm, φ₀=5 × 10⁻²). (e) Time-averaged packing fraction at the centre of the disc, normalized by and plotted versus the average packing fraction. Error bars: one standard deviation. (f) Fraction of the disc where versus the average packing fraction. Open symbols: experiments. Full line: theoretical prediction with no free fitting parameter. Filled circles: numerical simulations (b=3a, R_c=1 mm). (g) Radial density profiles plotted as a function of the distance to the disc centre r. All the experiments correspond to φ₀=0.032±0.002, error bars: 1σ. (h) Open symbols: same data as in g. The radial density profiles are rescaled by and plotted versus the rescaled distance to the centre r/R_c. All the profiles are seen to collapse on a single master curve. Filled symbols: Numerical simulations. Solid line: theoretical prediction. All the data correspond to E₀/E_Q=1.1.

Figure 4. Collective-dynamics simulations.

(a) The numerical phase diagram of the confined population is composed of three regions: isotropic gas (low φ₀, small b), swarm coexisting with a gaseous phase (intermediate φ₀ and b) and vortex state (high φ₀ and b). R_c=0.5 mm. (b) Snapshot of a vortex state. Numerical simulation for φ₀=0.1 and b=5a. (c) Snapshot of a swarm. Numerical simulation for φ₀=4.5 × 10⁻² and b=2a. (d) Variation of the density correlation length as a function of R_c. Above R_c=1 mm, ξ plateaus and a vortex is reached (φ₀=3 × 10⁻², b=3a). (e) Four numerical snapshots of rollers interacting via: alignment interactions only (A), alignment interactions and repulsive torques (A+B, where the magnitude of B is five times the experimental value), alignment and excluded volume interactions (A+b, where the repulsion distance is b=5a), alignment and the C-term in equation 3 (A+C). Polarized vortices emerge solely when repulsive couplings exist (A+B and A+b).