Competition between Frank elasticity and tilt coupling determines how chiral membranes respond to curvature

An external field tends to align the rod-like constituents of a liquid crystal. A sufficiently strong field can untwist a chiral liquid crystal, leading to a continuous transition from a cholesteric state to a nematic state. Motivated by colloidal membranes in solution with non-adsorbing polymer depletants, we study an analogous phenomenon in a chiral liquid crystal on a curved surface. To model the effects of the depletion interaction, there is a 'tilt coupling' in the free energy that favors the alignment of the rods with the surface normal. Thus, curvature together with the tilt coupling acts like a spatially varying external field tending to untwist the cholesteric. Using Morpho, an open-source software package for modeling soft materials, we systematically study the interplay of curvature, the tilt coupling, and liquid crystal elasticity on a cylinder. We show that when the tilt coupling is sufficiently large, the curvature of the cylinder leads to a discontinuous transition between the twisted state and the state with all directors parallel to the local surface normal. We also study the metastable twisted conformations above the transition corresponding to two sharp twist domain walls (π-walls) that wrap around the cylinder in helical paths. Our numerical work is supported by analytic calculations that approximately capture the transition as well as the trend of increasing pitch with increasing penalty for tilt away from the surface normal. Finally, we explore the role of the Gaussian curvature of the surface by determining the conformation of a chiral liquid crystal on a family of unduloids, which smoothly vary from the cylinder to a string of spheres.