1.Department of Physics, University of Science and Technology Beijing, Beijing 100083, China. 2.Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China.
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11174030, 51271157, 11504020)
Received Date:30 September 2020
Accepted Date:16 October 2020
Available Online:08 February 2021
Published Online:20 February 2021
Abstract:Algebraic topology, algebraic geometry, and category theory are new branches of mathematics that have developed in the last hundred years and have had profound collisions with modern physics in recent decades. A large number of topological phenomena are found in systems such as viruses, bacteria, fingerprints, fish school, typhoons, and the galaxies. Topological phenomena play a significant role in the spatial distribution of viral particles, the formation of nanovesicles of polymer, and Bose-Einstein condensates. In this paper, based on Landau-de Gennes theory, models have been constructed to simulate the topological charge distribution and other topological phenomena in liquid crystals. The research indicates that as the radius of the liquid crystal panel grows, the ratio of the optimal distance between the topological charge to the radius gradually increases and tends to stabilize. The size of the disc affects the equilibrium position of the topological load. The relative equilibrium position of topological load is between 0.542 and 0.558, in which the ratio of the distance between the two +1/2 topological loads in the 0–5 mm disc increases from 0.542 to 0.558, and then in the 5–12 mm section the ratio is almost stable at 0.558. As the size of the disc increases, the influence of the boundary anchoring energy decreases, and the equilibrium position, i.e. the distance between the two topological charges and the diameter of the disc, approaches a constant value. This equilibrium position is the result of the repulsive force of the disc boundary on the +1/2 topological load and the repulsive force between the two topological loads. The angle between two topological charges in a liquid crystal disc is between 140° and 180°. The trajectory of the topological charge is the process of finding the lowest free energy point, and the end of the trajectory is in the region of minimum free energy. The result is instructive significance in the design of classification containers by using topological charge condensate effect. And it is helpful to further understand the topological phenomena in soft materials including topological colloids, liquid crystals, and liquid crystal copolymers. Keywords:nematic liquid crystal/ phase-field/ topological charge/ Landau-de Gennes model
其中, S是标量序参量, 其范围是$ -1/2<S<1 $. 室温下, S在0.40—0.60之间, 本文取 S =0.50. 向列相液晶指向矢 ${ { n}}=\left({\rm{sin}}\;\theta\; {\rm{cos}}\;\varphi , {\rm{sin}}\;\theta\; {\rm{sin}}\;\varphi , {\rm{cos}}\;\theta \right)$, 其中, θ 是指向矢与 z 轴的夹角, φ 是指向矢在 x-y 平面内的投影与 x 轴之间的夹角, 如图1 (a)所示. 图 1 (a) 液晶指向矢与空间坐标轴之间夹角的示意图; (b) 液晶圆盘直径D0和两个大小为 $ + {1}/{2} $ 拓扑荷之间距离d的示意图, 红色标记表示 $ + {1}/{2} $ 拓扑荷 Figure1. (a) Schematic of the angle between director of liquid crystal and the spatial axis; (b) schematic of the NLqc disc diameter D0 and d the distance between two topological charges, $ + {1}/{2} $ topological charges represented by red markers.
表15CB(LC 1264)的弹性常数[27] Table1.Elastic constants of 5CB (LC 1264).
3.结果与讨论对于不同半径的液晶圆盘, 使用128 × 128 × 4的网格系统, 计算不同指向矢分布的体系总自由能大小. 如图1 (b)所示, 液晶盘直径为D0 (取0.4—12 mm), 其中格点尺寸对应为(0.0031, 0.0055, 0.0078, 0.0102, 0.0141, 0.0234 0.0391, 0.0547, 0.0703, 0.0938) mm, 在每个圆盘中对称分布有两个值为 1/2 的拓扑荷, 两个拓扑荷的距离为d, 计算总自由能, 得到不同尺寸圆盘状液晶薄膜总自由能随拓扑荷之间的距离变化的彩色曲线如图2 (a)所示, 其中曲线上标注的自由能最小值点即为两个拓扑荷的最优距离. 其中, 在两个拓扑荷间距 d/D0 在0.542—0.559时, 为二者最优距离, 此时自由能最小. 图 2 (a) 直径分别为0.4?12 mm圆盘中液晶薄膜自由能随中心两个拓扑荷的间距变化曲线; (b) 两个拓扑荷的最优位置随液晶圆盘直径变化的趋势图 Figure2. (a) The free energy of liquid crystal film in a disk with diameters ranging from 0.4 mm to 12 mm as a function of the distance between the two topological charges; (b) the trend of the optimal position of the two topological charges as a function of the diameter of the liquid crystal disk.
图2 (b) 中曲线上的点是由图2 (a)每条曲线上的自由能最小值点得到的. 自由能最小值点, 随圆盘直径变化, 其变化规律如图2 (b)所示, 随直径由0.4 mm到12 mm逐渐增大, 在0—5 mm段内, 两个+1/2缺陷平衡位置的距离与圆盘直径的比值逐渐增大, 由0.542增大到趋近于0.558, 5—12 mm段, 这一比值保持在0.559附近. 这种现象是因为, 边界锚定能作用区域较小, 随着圆盘直径增大, 两个拓扑荷距离边界越来越远, 边界对其排斥作用越来越小. 图3 (a), 和图3(b)分别为液晶圆盘直径为0.4和12 mm模拟超长时间的POM偏光显微镜下的显影. 可以看出, 直径为0.4 mm时两个+1/2的拓扑荷比直径为12 mm时更靠近圆心. 图3 (c), 和图3(d) 是模拟过程中的POM图像, 模拟过程中两个拓扑荷的夹角在非常长的时间内都在140°—180°之间不断变化. 图 3 (a)?(d)偏光镜图片 (a), (b) 圆盘直径为0.4和 12 mm时得到的平衡位置POM图; (c), (d)计算模拟的接近最终平衡位置的偏光显微镜图片. (e) 自由能随角度和位置变化的分布图 Figure3. (a)?(d) are polarizing optical microscope images: (a), (b) are POM images of the optimal positions for disk diameters of 0.4 and 12 mm, respectively; (c), (d) POM images of a near-final optimal position obtained from computational simulation. (e) Free energy as a function of position.