Shape minimisation problems in liquid crystals

We consider a class of liquid crystal free-boundary problems for which both the equilibrium shape and internal configuration of a system must simultaneously be determined, for example in films with air–liquid or fluid–liquid crystal interfaces and elastomers. We develop a finite element algorithm to solve such problems with dynamic mesh control, achieved by supplementing the free energy with an auxiliary functional that promotes mesh quality and is minimised in the null space of the energy. We apply this algorithm to a flexible capacitor, as well as to determine the shape of liquid crystal tactoids as a function of the surface tension and elastic constants. These are compared with theoretical predictions and experimental observations of tactoids from the literature.

我们研究一类液晶自由边界问题，此类问题需同时确定体系的平衡形貌与内部构型，例如带有气-液界面、液-液晶界面的薄膜以及弹性体中的相关体系。我们开发了一种结合动态网格控制的有限元算法以求解此类问题：通过在自由能中补充一项辅助泛函来优化网格质量，且该泛函在原自由能的零空间内实现最小化。我们将该算法应用于柔性电容器场景，同时用于求解液晶 tactoid（liquid crystal tactoid）的形貌随表面张力与弹性常数的变化规律，并将所得结果与已有文献中关于液晶 tactoid 的理论预测与实验观测结果进行对比。