Collective Behavior of Chiral Active Matter: Pattern Formation and Enhanced Flocking

We generalize the Vicsek model to describe the collective behavior of polar circle swimmers with local alignment interactions. While the phase transition leading to collective motion in 2D (flocking) occurs at the same interaction to noise ratio as for linear swimmers, as we show, circular motion enhances the polarization in the ordered phase (enhanced flocking) and induces secondary instabilities leading to structure formation. Slow rotations promote macroscopic droplets with late time sizes proportional to the system size (indicating phase separation) whereas fast rotations generate patterns consisting of phase synchronized microflocks with a controllable characteristic size proportional to the average single-particle swimming radius. Our results defy the viewpoint that monofrequent rotations form a vapid extension of the Vicsek model and establish a generic route to pattern formation in chiral active matter with possible applications for understanding and designing rotating microflocks.

Figure 1

Trajectory of an isolated linear [(a), $\mathrm{\Omega }=0$] and circle swimmer [(b), $\mathrm{\Omega }=3$]. Trajectories of circle swimmers in the macrodroplet [(c), $g{\rho }_0=2.8$, $\mathrm{\Omega }=0.2$] and the microflock phase [(d), $g{\rho }_0=2.8$, $\mathrm{\Omega }=3$]. (e) and (f) are cartoons illustrating the mechanism underlying macrodroplet and microflock formation: for slow rotations, circle swimmers phase-lock and follow circular orbits allowing for aligned configurations and the formation of large rotating clusters (e), whereas fast rotations frustrate the alignment interactions (f).

Figure 2

Simulation snapshots for $N=32000$ particles with colors encoding particle orientations. [(a), $\mathrm{\Omega }=0$]: Traveling bands; [(b), $\mathrm{\Omega }=0.2<1$]: rotating macrodroplet (phase separation) (c)–(h): Microflock pattern at $g{\rho }_0=2.8$, $\mathrm{\Omega }=3$, and ${\mathrm{Pe}}_r=0.2$ (c), ${\mathrm{Pe}}_r=1.0$ (d), and ${\mathrm{Pe}}_r=2$ (e), and at ${\mathrm{Pe}}_r=0.2$, $\mathrm{\Omega }=3$, and $g{\rho }_0=2.4$ (f), 3.6 (g), and 6 (h). (i), (j): Microflock length scale $l$ for $g{\rho }_0=2.8$; for $\mathrm{\Omega }=3$ as a function of ${\mathrm{Pe}}_r$ (i), and for ${\mathrm{Pe}}_r=0.2$ as a function of $\mathrm{\Omega }$ (j) for the system sizes shown in the key.

Figure 3

Global polarization over $g{\rho }_0$ and $\mathrm{\Omega }$ showing invariance of the flocking transition against rotations (left) and enhanced flocking (right) as predicted in the text.

Figure 4

Nonequilibrium phase diagram. The solid red line (obtained by linear stability analysis) and red symbols (simulations; bars represent numerical uncertainty) separate the macrodroplet phase (left, blue domain) induced by a long wavelength instability (LWI) from the microflock phase (pink domain) following an oscillatory, short wavelength instability (SWI) [33]. Overlaying colors ($\mathrm{LWI}+\mathrm{SWI}$) indicate phase coexistence. The light gray domain (bottom) represents stability of the disordered uniform phase. Black symbols show the location of the flocking transition from simulations. Filled symbols show parameters of Fig. 2: (a), (b) blue squares; (c)–(e) brown dot, (f)–(g) gray triangles.