Positionally ordered liquid crystals on curved manifolds

• Applicants: Professor Dr. Hartmut Löwen; Professor Dr. Axel Voigt
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics ; Mathematics
• Term: from 2015 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 280671903

The central goal of this project concerns the morphology and defect structure of positionally ordered liquid crystalline phases (such as smectic and crystalline ones) on curved manifolds. While first results were obtained in the first funding period for rods on cylinders and spheres, we shall continue to use microscopic (i.e. particle-resolved) density functional theory (DFT) to tackle positionally ordered liquid crystalline phases on curved manifolds in the second funding period. The corresponding equations for the full density field, which depends both on position and orientation, and the corresponding phase field crystal (PFC) approximation, will be derived by various approaches. Some density functional results for rods on cylinders and simulation data for rods on a sphere were already obtained in the first funding period, but we still need to apply our theory and algorithms to further manifolds like tori and hyperbolic surfaces for various apolar and polar particles and perform the DFT for rods on the sphere. Furthermore, the bridge between DFT and PFC needs to be established. The resulting PFC equations are coupled scalar-, vector- and tensor-valued surface partial differential equations for which new numerical schemes are required. For the vector- and tensor-valued equations we will follow the developed methods in the first funding period, which are based on a three-dimensional formulation and a penalization of the normal components. This approximation allows to use surface finite elements or diffuse interface methods in a component-wise fashion and is applicable for general manifolds. Preliminary results for the coupling with positional ordering already exist, however a detailed investigation will be a central point for the second funding period. Finally we shall consider time-dependent manifolds (such as periodically undulated cylinders and breathing spheres) and use dynamical DFT and PFC modeling to study the dynamic response of liquid crystals upon change of curvature. This will require additional modeling efforts and adaptations in the numerical approaches.

• DFG Programme: Research Grants