Positionally ordered liquid crystals on curved manifolds

本项目的中心目标是关注曲面流形上位置有序液晶相（如近晶和结晶）的形态和缺陷结构。虽然在第一个资助期内获得了圆柱体和球体上的棒的第一批结果，但我们将在第二个资助期内继续使用微观（即粒子分解）密度泛函理论（DFT）来处理弯曲流形上的位置有序液晶相。本文将通过各种方法推导出与位置和方向有关的全密度场方程以及相应的相场晶体（PFC）近似。在第一个资助期内，我们已经获得了一些柱面棒的密度泛函结果和球面棒的模拟数据，但我们仍然需要将我们的理论和算法应用于进一步的流形，如各种极性和极性粒子的环面和双曲曲面，并对球面上的棒进行DFT。此外，需要建立DFT和PFC之间的桥梁。所得的PFC方程是耦合的标量值、向量值和张量值的曲面偏微分方程，需要新的数值格式。对于矢量值和张量值方程，我们将遵循在第一个资助期内开发的方法，这些方法基于三维公式和法向分量的惩罚。这种近似允许以组件方式使用表面有限元或扩散界面方法，并且适用于一般流形。与位置排序耦合的初步结果已经存在，但是详细的调查将是第二个供资期的中心点。最后，我们将考虑时变流形（如周期波动圆柱体和呼吸球），并使用动态DFT和PFC建模来研究液晶在曲率变化时的动态响应。这将需要额外的建模工作和数值方法的调整。

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.