Minimal mechanism for flocking in phoretically interacting active particles

TL;DR

This work identifies a minimal, deterministic mechanism for global polar order in phoretically interacting active particles, arising purely from chemo-repulsive torques and short-range excluded-volume repulsion to produce Chemorepulsive Liquid Flocks (CLF); adding long-range translational repulsion yields Chemorepulsive Crystalline Flocks (CCF). The authors develop a pair-collision sliding criterion, quantify density- and translation-dependent phase boundaries via dimensionless groups Λ_r and Λ_t, and supplement particle simulations with a continuum hydrodynamic stability analysis that reproduces the observed transition lines. They show distinct structural signatures in CLF versus CCF through pair correlations and hexatic order, and demonstrate robustness to long-range attractive translational interactions. The findings reveal a purely dynamical route to global polar order without explicit alignment, with potential implications for migrating cells and phoretic colloids in experimental settings.

Abstract

Coherent collective motion is a widely observed phenomenon in active matter systems. Here, we report a flocking transition mechanism in a system of chemically interacting active colloidal particles sustained purely by chemo-repulsive torques at low to medium densities. The basic requirements to maintain the global polar order are excluded volume repulsions and long-ranged repulsive torques. This mechanism requires that the time scale individual colloids move a unit length to be dominant with respect to the time they deterministically respond to chemical gradients, or equivalently, pair colloids sliding together a minimal unit length before deterministically rotating away from each other. Switching on the translational repulsive forces renders the flock a crystalline structure. Furthermore, liquid flocks are observed for a range of chemo-attractive inter-particle forces. Various properties of these two distinct flocking phases are contrasted and discussed. We complement these results with stability analysis of a hydrodynamic model, which admits the transition corresponding to destabilization of the flocking state observed in particle-based simulations.

Figures (8)

• Figure 1: Schematic description of the chemical interactions between the particles based on Eq.\ref{['eq:mainLE']}. The black dot on the orange circles indicate the orientation of the particles. Black curved arrows show rotations from chemical interactions, while solid red arrows shows translations from chemical interactions. The particles turn away from each other if $\chi_r>0$, while they turn towards each other if $\chi_r<0$. Similarly, particles translate away from each other if $\chi_t>0$, while they translate towards each other if $\chi_t<0$.
• Figure 2: CLF for $\Lambda_t=0$. (a) Phase diagram in the $(\phi, \Lambda_r)$ plane, where $\phi$ is area fraction of particles. Here, $\Lambda_r^{-}$ is annotated as a dotted red line, while the density dependent $\Lambda_r^{+}$ in dashed black line. Symbols here and throughout the paper denoting a flock (yellow triangle), bands (green circles), and disordered (blue squares). Marked (I)-(III) scenario and transition lines (black and red) are described in panels (f-h). Snapshots of the respective phases are shown in (b)-(d) respectively (color coding for orientation of each particle is in middle panel). (e) The evolution of the polarization is displayed for the corresponding states (similarly colored). Panels (f-h) are series of scenarios depicted corresponding to parameters choices labelled in panel (a). Scenarios (I) and (III) have chemical response that are either too slow or too quick, such that a stable flock cannot be formed. Panel (g)- scenario (II) - shows the regime in which the chemical response is in an optimal range such the pair collision mechanism that would produce a flock. See \ref{['sec:threeCases']} for details of the three distinct cases and mechanism of flocking.
• Figure 3: (a) Phase diagram in the $(\Lambda_r, \Lambda_t)$ plane for the case of $\chi_t > 0$. Dashed line refers to lower bound on $\chi_r$, solid line for lower bound on $\chi_r$ from Eq. (\ref{['eq:ineq_chit_chir']}). (b) shows a snapshot of the CCF phase (same color scheme to indicate the orientation of the particles as in Fig (\ref{['fig_CLF']})).
• Figure 4: (a) and (b) compare $g(r, \varphi)$ for the case of the CLF and the CCF. In (c), a comparison of the flocking phase, for CLF (red) and CCF (blue), is shown for the pair correlation $g(r)$.
• Figure 5: Phase diagrams in terms of (left) global hexatic order ($\psi_6$) and (right) density variance. The top panel (a) show these for the CLF, whereas the bottom panel (b) for the CCF. Dashed curves in each plot indicate order-disorder (polarization) transition line as in Figures \ref{['fig_CLF']}(a) and \ref{['fig_CCF']}(a).