Collective behaviors of self-propelled particles with tunable alignment angles

TL;DR

Addresses how nonzero alignment angles in collision rules modify collective motion of self-propelled particles. A cone-shaped particle model with apex angle $\alpha$ introduces frustration, revealing four regimes: homogeneous nematic order, anti-parallel and parallel polar bands, and metastable chaotic nematic bands. A Boltzmann continuum framework with Fourier-mode truncation to $k_{\max}=16$ provides a stability diagram in $(\alpha,\rho)$ and explains metastability via density-driven longitudinal instabilities, showing qualitative agreement with the microscopic model. These results underscore the importance of many-body interactions beyond binary collisions and point toward experimental tests with cone-shaped colloids to probe frustration-induced emergent patterns.

Abstract

We present a novel aligning active matter model by extending the nematic alignment rule in self-propelled rods to tunable alignment angles, as represented by collision of cone-shaped particles. Non-vanishing alignment angles introduce frustration in the many-body interactions, and we investigate its effect on the collective behavior of the system. Through numerical simulations of an agent-based microscopic model, we found that the system exhibits distinct phenomenology compared to the original self-propelled rods. In particular, anti-parallel bands are observed in an intermediate parameter range. The linear stability analysis of the continuum description derived from the Boltzmann approach demonstrates qualitative consistency with the microscopic model, while frustration due to many-body interactions in the latter destabilizes homogeneous nematic order over a wide range of the alignment angle.

Figures (5)

• Figure 1: Illustration of the alignment rule for cone-shaped particles characterized by the apex angle $\alpha$. (a) A collision at a small angle (with the heading angle difference smaller than $(\alpha + \pi)/2$) causes the angle difference $\alpha$. (b) A collision at a large angle results in the anti-parallel alignment.
• Figure 2: Typical snapshots of collective states at different values of $\alpha=0.05$. (a) $\alpha=0.05$: homogeneous nematic ordered state. (b) $\alpha=0.35$: anti-parallel polar bands. (c) $\alpha=0.66$: parallel polar bands. (d) $\alpha=0.74$: chaotic nematic bands. (e) $\alpha=0.75$: homogeneous disordered state. The particle orientation is represented by color. (f)-(j) Schematic illustrations of the different states corresponding to (a)-(e). Note that for the anti-parallel polar bands, the red and blue bands move in opposite directions and penetrate each other.
• Figure 3: (a) Variation of the global nematic order parameter $Q$ and its root mean square (rms) fluctuation $\Delta Q$ with $\alpha$. The average density is fixed at $\bar{\rho} = N/L^2 = 0.5$. Region I: homogeneous nematic state. Region II: polar bands (either anti-parallel or parallel). Region III: metastable chaotic nematic bands. Region IV: homogeneous disordered state where $Q$ is close to zero. In region III, instead of plotting the order parameter, we present the average lifetime of the chaotic nematic band state in (b). (b) Average lifetime $\tau$ of the chaotic nematic bands. The maximum timestep of each simulation is $5 \times 10^5$. The red dot represents the average taken over 10 samples, while the blue dots show all data from these 10 samples. (c) Numerical simulation results at other densities.
• Figure 4: Snapshots of the time evolution in region III are shown, illustrating the transition from chaotic nematic bands to anti-parallel polar bands. In this simulation $\alpha = 0.73$. The timestep is indicated above each snapshot, and the particle orientation is represented by color.
• Figure 5: Stability diagram in the $\alpha$-$\rho$ parameter space with $\sigma=0.1$ fixed. The linear stability analysis of the homogeneous solution reveals four distinct regions: (1) stable nematic solutions, (2) unstable nematic solutions with the most unstable wavevector parallel to the nematic order (parallel instability), (3) unstable nematic solutions with the most unstable wavevector perpendicular to the nematic order (perpendicular instability), and (4) stable disordered solutions.