Defect patterns of two-dimensional nematic liquid crystals in confinement

Geometry-structure models for liquid crystal interfaces, drops and membranes: wrinkling, shape selection and dissipative shape evolution

We review our recent contributions to anisotropic soft matter models for liquid crystal interfaces, drops and membranes, emphasizing validations with experimental and biological data, and with related theory and simulation literature. The presentation aims to illustrate and characterize the rich output and future opportunities of using a methodology based on the liquid crystal-membrane shape equation applied to static and dynamic pattern formation phenomena. The geometry of static and kinetic shapes is usually described with dimensional curvatures that co-mingle shape and curvedness. In this review, we systematically show how the application of a novel decoupled shape-curvedness framework to practical and ubiquitous soft matter phenomena, such as the shape of drops and tactoids and bending of evolving membranes, leads to deeper quantitative insights than when using traditional dimensional mean and Gaussian curvatures. The review focuses only on (1) statics of wrinkling and shape selection in liquid crystal interfaces and membranes; (2) kinetics and dissipative dynamics of shape evolution in membranes; and (3) computational methods for shape selection and shape evolution; due to various limitations other important topics are excluded. Finally, the outlook follows a similar structure. The main results include: (1) single and multiple wavelength corrugations in liquid crystal interfaces appear naturally in the presence of surface splay and bend orientation distortions with scaling laws governed by ratios of anchoring-to-isotropic tension energy; adding membrane elasticity to liquid crystal anchoring generates multiple scales wrinkling as in tulips; drops of liquid crystals encapsulates in membranes can adopt, according to the ratios of anchoring/tension/bending, families of shapes as multilobal, tactoidal, and serrated as observed in biological cells. (2) Mapping the liquid crystal director to a membrane unit normal. The dissipative shape evolution model with irreversible thermodynamics for flows dominated by bending rates, yields new insights. The model explains the kinetic stability of cylinders, while spheres and saddles are attractors. The model also adds to the evolving understanding of outer hair cells in the inner ear. (3) Computational soft matter geometry includes solving shape equations, trajectories on energy and orientation landscapes, and shape-curvedness evolutions on entropy production landscape with efficient numerical methods and adaptive approaches.

Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals

Observing and characterizing the complex ordering phenomena of liquid crystals subjected to external constraints constitutes an ongoing challenge for chemists and physicists alike. To elucidate the delicate balance appearing when the intrinsic positional order of smectic liquid crystals comes into play, we perform Monte-Carlo simulations of rod-like particles in a range of cavities with a cylindrical symmetry. Based on recent insights into the topology of smectic orientational grain boundaries in two dimensions, we analyze the emerging three-dimensional defect structures from the perspective of tetratic symmetry. Using an appropriate three-dimensional tetratic order parameter constructed from the Steinhardt order parameters, we show that those grain boundaries can be interpreted as a pair of tetratic disclination lines that are located on the edges of the nematic domain boundary. Thereby, we shed light on the fine structure of grain boundaries in three-dimensional confined smectics.