Creation of a topological defect array in liquid crystals has been a notable focus in recent years, because the defect array can be utilized as precision optics, templates of self-assembled microstructures, and elastomer actuators. So far, the defect arrays are created intuitively by trial and error. Systematic rules to arrange defects into stable long-ranged arrays are in demand. A model of two-dimensional square and hexagonal defect array was developed based on previous experimental results. The model is generalized for defect crystals and quasicrystals in this research. A crystal is the periodic repetition of a unit cell. A stable defect crystal must have minimum free energy, and the arrangement of the defects must obey the topological conservation laws. By solving the Euler–Lagrange equation of the director field of a unit cell and by integrating the topological rules into the boundary conditions, the director field of a defect crystal can be easily obtained. A large variety of defect crystals and quasicrystals are derived. The lattices are rectangular, triangular, square, pentagonal, and hexagonal. The defects can be either radial or azimuthal (vortex-like). The nematic and vector orders are both considered. The collection of defect crystals is presented here as a catalog for the designers.

中文翻译：

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向列液晶中的类晶体拓扑缺陷阵列

近年来，在液晶中创建拓扑缺陷阵列一直是一个值得注意的焦点，因为缺陷阵列可用作精密光学器件、自组装微结构的模板和弹性体致动器。到目前为止，缺陷阵列是通过反复试验直观地创建的。需要将缺陷排列成稳定的远程阵列的系统规则。基于先前的实验结果开发了二维正方形和六边形缺陷阵列模型。该模型在本研究中适用于缺陷晶体和准晶体。晶体是晶胞的周期性重复。一个稳定的缺陷晶体必须具有最小的自由能，并且缺陷的排列必须服从拓扑守恒定律。通过求解晶胞指向矢场的欧拉-拉格朗日方程，并将拓扑规则整合到边界条件中，可以很容易地获得缺陷晶体的指向矢场。衍生出多种缺陷晶体和准晶体。格子有矩形、三角形、正方形、五边形和六边形。缺陷可以是径向的或方位角的（涡旋状）。向列和矢量顺序都被考虑。此处为设计人员提供了缺陷晶体的集合作为目录。缺陷可以是径向的或方位角的（涡旋状）。向列和矢量顺序都被考虑。此处为设计人员提供了缺陷晶体的集合作为目录。缺陷可以是径向的或方位角的（涡旋状）。向列和矢量顺序都被考虑。此处为设计人员提供了缺陷晶体的集合作为目录。