Abstract

At thermal equilibrium, chiral molecules form a range of liquid-crystalline phases, such as the cholesteric which presents a helical structure of the molecular orientation. Chirality, though essential to the construction of the cholesteric, is totally absent in its long-wavelength hydrodynamics, which is identical to that of the achiral smectic  A liquid crystal. This cloaking of chirality, however, relies on the existence of an energy function for the dynamics.  We show here that the macroscopic mechanics of active layered phases carries striking chiral signatures. We start by constructing, in two and three dimensions, the chiral and active variants of model H. Thanks to the mix of solidand liquid-like directions, we predict that chiral active stresses create a force density tangent to contours of constant mean curvature of the layers. This non-dissipative force in a fluid direction – odder than odd elasticity – leads, in the presence of an undulational instability created by non-chiral active stresses normal to the layers, to spontaneous vortical flows arranged in a two-dimensional array with vorticity aligned along the pitch axis and alternating in sign in the plane. This vortex-lattice state can be switched on or off by means of an externally imposed uniaxial stress, due to an exact equivalence of the active buckling instability and the passive Helfrich-Hurault instability of achiral layered phases.We further show that two-dimensional odd elasticity, an effect that is attracting much current attention, is naturally realised in three-dimensional material for polar and chiral columnar systems. The resulting oscillatory mode, thanks to the Stokesian hydrodynamic interaction, has a nonzero frequency on macroscopic scales, set by the ratio of the coefficient of chiral and polar active stress and the viscosity. In effect, the two components of the in-plane displacement field mimic a position-momentum pair. We also show that a bulk active columnar phase is spontaneously unstable to an extensile activity along the column direction via a buckling instability and predict singular stiffening or softening – depending on whether the active achiral stress is contractile or extensile – of the buckling of fluid columns in all active columnar materials, irrespective of whether they are chiral or polar. The instability is mediated by a twist-bend mode resulting in helical columns — the same as those that arise from a Helfrich-Hurault instability of passive columnarmaterial. If the active units composing the columnar state are, in addition, chiral, the buckled and twisted state beyond the spontaneous Helfrich-Hurault instability in an apolar system hosts large-scale shear flows due to a new form of odd elasticity.