Kinetics and Statistics of Topological Defects in the Phase Ordering of Liquid Crystals

数值研究了在混合边界条件下薄膜中的平衡态及其形成过程。顶面边界是指矢垂直于薄膜的垂面边界，底面边界是指矢平行于薄膜的均质边界。Frank能中的表面积分所代表的能量表面项对形成斑图有重要作用，但在体问题中却被忽略了。我们提出的模型是一个改进的细胞动力学系统模型，它不仅包含了表面项，而且还能代表体弹性常数的各向异性。其主要思想是不对Frank能量进行离散，而对TDGL方程进行离散，表面能和体能之间的竞争产生周期性的调制图案。模拟结果表明，在表面常数K_ =0.35的起始值以上，周期条纹是稳定<24>的，当K_=较大时，周期与（K_<24>-K_<13>/2）^<-1>成正比<24>，晶格各向异性影响条纹的方向。为了避免这种非物理效应，必须考虑次近邻相互作用。单向场使被发现，使串状纹影图案之间的boojum和antiboojum类似的实验模式在膜的厚度约为10微米。

Equilibrium patterns and those formation process in thin nematic films under hybrid boundary conditions are numerically investigated. The top boundary is homeotropic one where the director is perpendicular to the film and the bittom boundary is homogeneous one where the director is parallel to the film. The surface terms of energy represented by surface integrals in the Frank energy have an important role for forming patterns, thourh those have been neglected in bulk problems. Our present model is an improved cell-dynamical-system model, which does not only includes the surface terms but alsobe able to represent the anisotropy of bulk elastic constants. A main idea is that we do not discretize the Frank energy but does the TDGL equation.The competition between the surface energy and the bulk energy makes periodically modulated patterns. The present simulation shows that periodic pattern is stable above the onset value of the surface constant K_<24>=0.35 and the period is proportional to (K_<24>-K_<13>/2)^<-1> for large K_<24>.The anisotropy due to the lattice affects the direction of the striped pattern. To avoid this unphysical effect, the second nearest interaction must be taken into account.The effect of anisotropic on-site fields at the homogeneous boundary is examined. The unidirectional field makes is found to make string-like Schlieren patterns between a boojum and an antiboojum similar to experimental patterns in films of thickness about 10mum.