Self-reverting vortices in chiral active matter

Chirality—non-superimposability of an object with its mirror image—leads many microorganisms and synthetic microswimmers to follow circular trajectories due to self-rotation. While chiral active particles with explicit alignment interactions exhibit rich collective phenomena (e.g., rotating micro-flocks, traveling waves, chimera states), systems without alignment typically show limited collective effects; circular motion mainly reduces persistence, suppresses clustering, and can yield hyperuniform states and demixing. Recent reports identified non-permanent vortex-like velocity patterns at high chirality that did not produce net vorticity. The research question addressed here is whether, in the absence of explicit alignment, attractions combined with chirality can induce robust, coherent vortex states with non-zero global vorticity and, if so, what dynamical regimes and mechanisms govern them. The study aims to reveal and characterize new classes of collective motion in chiral active matter driven by attractions, including persistent vortices and time-periodic, self-reverting vortices, and to provide theoretical understanding and scaling relations relevant for designing active materials and micromotors.

Prior work has shown that chirality in active systems with alignment can produce complex patterns including rotating micro-flocks, chiral self-recognition, traveling waves, and chimera states. Without alignment, chirality generally reduces effective persistence, suppressing motility-induced clustering and promoting hyperuniform phases and demixing. Other observations include oscillatory caging in chiral active glasses and transient vortex patterns in velocity fields at significant chirality, which, however, did not yield global vorticity (as also seen in chiral rollers). These studies highlight the gap: robust, sustained vorticity at the collective level without explicit alignment remained elusive. The present work builds on theoretical mappings developed for attractive active particles (velocity alignment emerging effectively) and extends them to include chirality as an effective Lorentz-like term, predicting selection of vortex/antivortex states and global vorticity.

Model: N interacting chiral active Brownian particles (2D) with underdamped dynamics for positions x_i and velocities v_i: m v̇_i = −γ v_i + F_i + γ v_0 n_i + √(2Tγ) η_i; orientations obey θ̇_i = √(2D_r) ξ_i + ω, where n_i = (cos θ_i, sin θ_i). Chirality ω sets a single-particle circular radius v_0/ω. Interactions are via a truncated and shifted attractive Lennard-Jones potential U(r) = 4ε[(σ/r)^12 − (σ/r)^6] for r ≤ 3σ, shifted such that U(r)=U_LJ(r)−U_LJ(3σ), and zero otherwise. No explicit aligning torques are present; F_i = −∇i U_tot with U_tot = Σ{ij} U(r_ij). Time scales: inertial time τ_ι = m/γ, persistence time τ = 1/D_r, and chiral time 1/ω. The study targets the large persistence regime τ/τ_ι ≫ 1 (effectively overdamped) and strong attraction relative to thermal and active kinetic energies to maintain stable clusters. Theory: Using an exact change of variables, Ito calculus for activity in Cartesian form, and a lattice approximation appropriate for solid-like, hexagonally ordered clusters (first-neighbor interactions on a triangular lattice), the dynamics map to an effective velocity equation valid at large persistence: v̇_i = (1/τ) Σ_j J_{ij} (v_j − v_i) + ω v_i × z. The first term is an emergent effective velocity-alignment interaction due to activity–interaction coupling, minimized by (i) full alignment, (ii) vortex, or (iii) antivortex configurations. The second term is a chirality-induced effective Lorentz force that selects vortex or antivortex depending on the sign of ω, thereby breaking the degeneracy and predicting spontaneous global vorticity. A torque-based argument considers the total torque M ≈ γ v_0 Σ_i (r_i × n_i) dominated by the outer layer at radius R; competition between the chiral rotation period (~π/ω) of n_i and the cluster rotation determines whether torque maintains its sign (permanent vortex) or periodically changes sign (self-reverting). Simulations: Particle-based simulations in a periodic box of size L at fixed area fraction (main text: φ ≈ 0.3) forming a single attraction-stabilized cluster. The large-persistence regime τ/τ_ι ≫ 1 yields effectively overdamped dynamics. Parameters (typical): Pe = τ v_0/σ = 50; reduced chirality ωτ varied; strong attraction (ε) and low thermal noise T to prevent evaporation; cluster sizes N = 113, 226, 452, 904, 1809 (L ≈ σ√N). In Methods, dimensionless integration via Euler with step dt' = 10^-6, and example parameters include τ_ι/τ = 10^-6, reduced energy and temperature sufficiently small, and φ = 0.35 for some runs. Vorticity characterization: compute spatially averaged vorticity Ω(t) from the coarse-grained velocity field; classify states based on time-average vs fluctuations and presence of periodic sign reversals. Order parameters include ⟨Ω⟩ and ⟨|Ω|⟩ and the oscillation period of Ω(t).

• Discovery of two coherent collective states in attractive chiral active particles without explicit alignment: (i) permanent vorticity states with time-independent nonzero global vorticity aligned with single-particle chirality; (ii) self-reverting vortices where global vorticity periodically switches sign, yielding alternating vortex/antivortex configurations.
• Mechanism: Effective velocity alignment (from attractions and activity) is degenerate between aligned, vortex, and antivortex configurations. Chirality acts as an effective Lorentz force (∝ ω) that lifts this degeneracy and selects a handedness. A simple anchored-particle argument explains that when the chiral radius v_0/ω exceeds the anchoring length scale (∼σ or cluster radius scale), particles complete rotations, producing permanent vorticity; when v_0/ω is smaller, the propulsion reverses direction before half a turn, driving back-and-forth angular motion and self-reversion.
• State diagram: In the plane of reduced chirality ωτ and cluster size L, three regimes are observed: negligible vorticity, permanent vorticity, and self-reverting vorticity. The cross-over between permanent and self-reverting states occurs when the cluster size L ≃ v/ω (chiral radius comparable to cluster size), consistent with a scaling law that fits simulations. Larger clusters favor self-reverting over permanent vorticity and suppress the no-vorticity regime due to reduced vorticity fluctuations with N.
• Vorticity dynamics: For small ωτ, Ω(t) fluctuates around zero (no preferred vorticity). With increasing ωτ, Ω(t) fluctuates weakly around a positive mean (permanent vorticity). At higher ωτ, Ω(t) exhibits periodic oscillations with alternating sign (self-reverting), with oscillation period scaling as ~1/ω.
• Order parameters: ⟨Ω⟩ vs ωτ is non-monotonic—growing as permanent vorticity emerges, peaking before the self-reverting regime, then dropping toward zero due to symmetric oscillations. The amplitude ⟨|Ω|⟩ increases with ωτ and saturates in the self-reverting regime, indicating oscillation amplitude that is relatively insensitive to cluster size.
• Regime dependence: In the small-persistence limit (τ/τ_ι ≪ 1), the active force reduces to effective white noise and none of the three coherent states appear (no collective motion), consistent with theory.

The findings answer the central question by demonstrating that chirality together with attractive interactions—without any explicit alignment rules—suffices to generate coherent vortex patterns at the collective scale. The emergent effective alignment due to attractions/activity provides the scaffold for coherent velocity structures, while chirality acts as an effective magnetic field selecting a rotational sense and inducing global vorticity. The competition between the chiral rotation period and the time scale for collective rotation underlies the transition from permanent to self-reverting vortices when the chiral radius becomes smaller than the cluster size. The state diagram and scaling L ∼ v/ω rationalize how cluster size and chirality control the dynamical regime. These results connect chiral active matter to odd-viscous and odd-elastic behaviors in active materials by providing a microscopic mechanism for sustained chiral oscillations and torque generation. Practically, they suggest routes to design spontaneously rotating gears and tunable self-reverting patterns in experiments with chiral active colloids or granular spinners.

Attractive chiral active particles without explicit alignment self-organize into coherent vortices with either permanent or self-reverting global vorticity. A theoretical mapping shows that attractions plus activity generate effective velocity alignment, while chirality contributes a Lorentz-like term that selects vortex handedness. Simulations validate these predictions and reveal a state diagram controlled by reduced chirality and cluster size, with a transition governed by the condition that the chiral radius is comparable to the cluster size. The work links microscopic chirality to macroscopic torque and vorticity generation, offering design principles for micromotors and chiral active materials. Future directions include: incorporating explicit alignment mechanisms at high density to explore interplay with the attraction–chirality mechanism; testing robustness under different interaction potentials and noise levels; investigating hydrodynamic interactions; and experimental realizations in colloidal and granular chiral systems to engineer spontaneous gear-like rotation and self-reverting patterns.

• Regime constraints: Coherent vortex states are observed in the large-persistence regime (τ/τ_ι ≫ 1) with strong attractions and low thermal noise; in the small-persistence regime the active force becomes effectively white noise and no collective motion is found.
• Cluster stability: Results rely on attraction strong enough to maintain a single stable cluster; insufficient attraction leads to evaporation and precludes collective rotation.
• Approximations: The theoretical mapping assumes solid-like hexagonal order, a lattice approximation with first-neighbor truncation, linearization of interparticle forces, and differentiable short-range potentials (e.g., truncated LJ). These assumptions may limit quantitative accuracy outside crystalline or near-crystalline clusters.
• Smooth crossovers: The boundaries between regimes in the state diagram are crossovers defined by threshold criteria rather than sharp phase transitions; precise locations may depend on thresholds and finite-size effects.
• Parameter specificity: Simulations focus on specific ranges (e.g., Pe ≈ 50, selected ωτ, φ around 0.3–0.35, and chosen N). Generalization to other densities, interaction models, or inclusion of hydrodynamics is not addressed here.
• No explicit alignment or hydrodynamics: The model neglects explicit alignment torques and hydrodynamic interactions; the impact of these mechanisms on the identified states remains to be explored.