Positionally ordered liquid crystals on curved manifolds
Typically liquid crystals exhibit a rich bulk phase diagram with various mesophases involving both positional and orientational order. Their equilibrium phase behaviour has been extensively studied in flat space by statistical theories and computer simulation. Liquid crystals have also been confined in various ways on curved manifolds. While the topology of defects has been widely studied for nematic layers on curved surfaces, much less is known regarding the morphology for positionally ordered phases (such as smectic, plastic crystalline and full crystalline phases) constrained to manifolds.
In this project, we shall use a microscopic (i.e. particle-resolved) density functional approach to tackle positionally ordered liquid crystalline phases on curved manifolds. We shall systematically derive the corresponding equations for the full density field, which depends both on position and orientation, and the corresponding phase field crystal approximation, to determine the morphology of the liquid crystal on manifolds of different topology. New numerical schemes are required to solve these equations. Different approaches based on parametric finite elements, discrete exterior calculus and diffuse interface approximations will be developed.
We shall apply our theory and algorithms to different kinds of manifolds like cylinders, spheres, tori and hyperbolic planes for various apolar and polar particles and compare it to particle-resolved Monte-Carlo simulations.
Project duration: 07/2015 - 06/2018
Funded by: DFG
together with Hartmut Löwen (Heinrich-Heine-Universität Düsseldorf)