Positionally ordered liquid crystals on curved manifolds

本计画的中心目标是研究曲面上位置有序液晶相（如近晶相和结晶相）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个结果，但我们将继续使用微观（即粒子分辨）密度泛函理论（DFT）来解决第二个资助期内弯曲流形上的位置有序液晶相。相应的方程的全密度场，这取决于位置和取向，以及相应的相位场晶体（PFC）近似，将通过各种方法推导。在第一个资助期内，我们已经获得了圆柱体上棒的一些密度泛函结果和球体上棒的模拟数据，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，如各种非极性和极性粒子的环面和双曲曲面，并对球体上的棒进行DFT。此外，需要在DFT和PFC之间建立桥梁。由此产生的PFC方程耦合标量，矢量和张量值的表面偏微分方程，需要新的数值方案。对于矢量和张量值方程，我们将遵循第一个资助期开发的方法，该方法基于三维公式和法向分量的惩罚。这种近似允许使用表面有限元或扩散界面方法在一个组件明智的方式，并适用于一般的流形。与位置排序耦合的初步结果已经存在，但详细的调查将是第二个资助期的中心点。最后，我们将考虑时间相关的流形（如周期性波动的圆柱和呼吸球），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。这将需要在数值方法中进行额外的建模工作和调整。

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.