Screw Symmetry, Chiral Hydrodynamics and Odd Instability in Active Cholesterics

TL;DR

This work develops a geometric Ericksen-Leslie formulation for active cholesterics, leveraging their continuous screw symmetry to derive a passive chiral curl force and to reveal how rotation-translation coupling permits chiral activity to influence linear pseudolayer dynamics. It identifies a new active instability that arises when chiral activity opposes the passive screw-symmetry term, altering the Helfrich-Hurault threshold and selecting a distinct wavevector, with handedness-dependent behavior. The analysis extends to a three-dimensional isotropic generalization, predicting linear-in-$q$ modes and isotropic rotation-translation couplings, and discusses potential experimental realizations in active nematics within cholesteric and blue-phase contexts. Overall, the paper provides a unified, geometry-based framework for chiral hydrodynamics in screw-symmetric materials, with broad implications for active matter and soft robotics in chiral media.

Abstract

Active cholesterics are chiral in both their structure, which has continuous screw symmetry, and their active stresses, which include contributions from torque dipoles. Both expressions of chirality give rise to curl forces in the hydrodynamics, which we derive from the active Ericksen-Leslie equations using a geometric approach. This clarifies the hydrodynamics of continuous screw symmetry and provides an example of generalised odd elastic forces that originate from an equilibrium free energy. We discuss also the nonlinear structure of the active hydrodynamics in terms of the Eulerian displacement field of the cholesteric pseudolayers. For the active instability, screw symmetry generates a contribution of chiral activity to the linearised pseudolayer hydrodynamics that is absent in materials with chiral activity but achiral structure. When the two forms are sufficiently antagonistic, this term produces a new active instability with threshold and characteristic wavevector distinct from those of the active Helfrich-Hurault instability in chiral active smectics. Finally, we comment on the isotropic chiral hydrodynamics of materials with three-dimensional screw symmetry.

Figures (3)

• Figure 1: (a) Local structure of a (pseudo)layered material with adapted orthonormal frame. The (pseudo)layers correspond to a (local) discrete translational symmetry along the pitch axis or layer normal ${\bf N}$. (b) In a cholesteric each pseudolayer corresponds to a full $2\pi$ rotation of the director field, shown by the blue cylinders. The grey discs provide a guide to the eye. The structure has continuous screw symmetry (here right-handed) along the pitch axis. (c) In smectic A the director points along the layer normal and the layers correspond to a genuine density modulation. There is continuous rotational symmetry about the layer normal but only discrete translational symmetry along it.
• Figure 2: Linearised growth rate $g_{\bf q}$ (made dimensionless) and active instabilities in cholesterics. The solid blue line shows the full function \ref{['eq:gq_full']} and the dashed red line shows the approximation \ref{['eq:gq_approx']}. The minimum in the full function at small $q_{\perp}/q_0$ corresponds to the scale of confinement along the pitch axis, $q_{\perp} \simeq |q_z|$. (a) Passive cholesteric, $\zeta = \zeta_c = 0$. (b) Helfrich-Hurault-type instability for $\zeta > \zeta_{\textrm{th}}$; the most unstable wavevector $q_{\perp}^{\star}$ is indicated. (c) Chiral instability for $|\zeta_c| > |\zeta_c^{\textrm{th}}|$; the most unstable wavevector $q_{\perp}^{\star}$ is indicated.
• Figure 3: Odd active instability in cholesterics. Pseudolayer undulations, here a square lattice, are accompanied by in-layer vortical flows generated by the chiral activity, shown for $\zeta_c > 0$, that rotate the local director. This rotation is equivalent to a translation (layer displacement) along the pitch axis that opposes the initial displacement for $q_0 > 0$ (right-handed) but amplifies it for $q_0 < 0$ (left-handed).