下载:
-
拓扑荷是有序介质中形成的拓扑缺陷, 在超流体、玻色-爱因斯坦凝聚物[1,2] (Bose-Einstein condensate)以及卡拉比-丘流形[3] (Calabi-Yau manifold)中都有出现. 它们影响病毒颗粒的分布[4-6]、影响液晶的光电性质[7-10]、影响液晶共聚物纳米囊泡的自组织成型[11-13]. 利用拓扑荷的特殊性质, 装配[14]和分离微小颗粒[15]、预先设定人真皮纤维细胞的生长纹理及方向[16]. 在向列相液晶(NLqc)中拓扑荷间有着类电荷的相互作用[17]. 在科学研究中制造并控制拓扑荷, 以制作拓扑复合材料[18,19]. 拓扑荷是物理学中应用最广泛的概念之一. 液晶中的缺陷也可以用拓扑荷来描述. 液晶及其聚合物等软物质体系中, 液晶分子在空间中占据位置, 而且其分子取向也有丰富多变的排布方式. 其他体系中能够出现的拓扑现象, 在液晶体系中都有存在, 而且由于液晶的光学特性, 拓扑现象在液晶体系中十分便于观测. 因此液晶等软材料是研究此类拓扑现象的优良研究对象.
本工作模拟拓扑荷在不同尺寸的圆盘型向列相液晶薄膜中的空间分布[20,21], 模拟格点大小128 × 128 × 4, 格点尺寸与模拟圆盘的半径相关. 并且模拟了液晶圆盘的偏光光学显微镜(polarizing optical microscope, POM)视图. 基于自由能最小化的数值模拟结果表明, 向列相液晶薄膜中的二维拓扑荷, 有近似固定的平衡位置, 平衡位置随圆盘大小浮动, 并由自由能平面分布图来解释这一现象.
-
液晶中的缺陷可以用缺陷的拓扑荷来描述, Landau-de Gennes模型[22]可以解释在物理上观察到的整数拓扑荷和半整数拓扑荷. 在本研究采用Landau-de Gennes模型, 通过相场方法模拟向列相液晶中的拓扑荷.
Landau-de Gennes模型通过对称的无迹张量Qij, 作为序参量[22]:
$ {Q}_{ij}=S\left({n}_{i}{n}_{j}-\frac{1}{3}{\delta }_{ij}\right)\text{,} $ $ {\rm{T}}{\rm{r}} { Q}=0\text{,} $ 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Figure 1. (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.体积自由能密度表达式[22]为
$\begin{split} {f}_{{\rm{b}}{\rm{u}}{\rm{l}}{\rm{k}}}=\;&\frac{1}{2}a{Q}_{ij}{Q}_{ij}-\frac{1}{3}b{Q}_{ij}{Q}_{jk}{Q}_{kl}\\ &+{\frac{1}{4}c}_{1}{\left({Q}_{ij}{Q}_{ij}\right)}^{2}+{\frac{1}{4}c}_{2}{Q}_{ij}{Q}_{jk}{Q}_{kl}{Q}_{li}\text{,} \end{split}$ 其中, a=(a*(T-T*)), T是温度, T*为相变点, b, c1和c2与具体材料有关[23]. 弹性能表达式为
$\begin{split} {f}_{{\rm{g}}{\rm{r}}{\rm{a}}{\rm{d}}}=\;&\frac{1}{2}{L}_{1}{\nabla }_{i}{Q}_{jk}{\nabla }_{i}{Q}_{jk}+\frac{1}{2}{L}_{2}{\nabla }_{i}{Q}_{ik}{\nabla }_{j}{Q}_{jk}\\ &+{L}_{3}{\epsilon }_{ijk}{Q}_{il}{\nabla }_{k}{Q}_{jl}\text{,} \end{split}$ 其中Landau-de Gennes模型的弹性能系数[23]
${L}_{1}=4.20\times {10}^{-12}~\left({\rm{N}}\right)$ ,$ {L}_{2}=5.51\times {10}^{-12}\left({\rm{N}}\right) $ ,${L}_{3}= 1.02\times {10}^{-12}~\left({\rm{N}}\right)$ 它们的值和展曲弹性系数${k}_{11}= 6.70 {10}^{-12} ~ ({\rm{N}})$ , 扭曲弹性系数${k}_{22}\!=\!3.60\times {10}^{-12}~\left({\rm{N}}\right)$ , 弯曲弹性系数${k}_{33}=9.00\times {10}^{-12}~\left({\rm{N}}\right)$ [24]以及 S 相关[25,26], (5CB(LC 1264)[27]的展曲、扭曲和弯曲弹性能系数$ {\epsilon }_{ijk} $ 是列维-奇维塔符号:$ {L}_{1}=\left({k}_{33}+{2k}_{22}-{k}_{11}\right)/\left(9{S}^{2}\right)\text{,} $ $ {L}_{2}=4\left({k}_{11}-{k}_{22}\right)/\left(9{S}^{2}\right)\text{,} $ $ {L}_{3}=\left({k}_{33}-{k}_{11}\right)/\left(9{S}^{3}\right)\text{.} $ 表面锚定能密度[28]为
$ {f}_{{\rm{s}}}={W}_{1}\left({\tilde Q}_{ij}-{\tilde Q}_{ij}^{\parallel }\right)+{W}_{2}{({\tilde Q}_{ij}^{2}-{S}^{2})}^{2}\text{,} $ 其中
$ {W}_{1}>0 $ (W1= 9.00 × 10–8(N))对应锚定强度, 有利于指向矢沿着边界的切线方向,$ {W}_{2}>0 $ 保证了表面标量参数的最小值.${\tilde Q}_{ij}={Q}_{ij}+\dfrac{1}{3}S{\delta }_{ij}$ ,${\tilde Q}_{ij}^{\parallel }= {P}_{ij}{Q}_{kl}{P}_{lj}$ , 其中${P}_{ij}{=\delta }_{ij}-{t{v}}_{i}{{v}}_{j}$ ,${\overrightarrow{{{v} }}}$ 是边界切线方向的单位矢量.系统总自由能为
$ F=F_{0}+\int_{\Omega}\left(f_{\text {bulk }}+f_{\text {grad }}\right) {\rm{d}}^{3} x+\int_{\partial \Omega} f_{s} {\rm{~d}} s\text{.} $ 在相场模拟中, 相场参数的演化是由含时Ginzburg-Landau方程控制:
$ \frac{\partial {Q}_{ij}\left(r,t\right)}{\partial t}=- \varGamma \frac{\delta F}{\delta {Q}_{ij}\left(r,t\right)},\left(i,j=x,y,z\right)\text{,} $ 其中Γ 是向列相液晶的黏度系数. 数值方法求解方程(10), 得到向列相液晶的指向矢随时间的空间分布, 从而得到向列相液晶中的缺陷的演化. 模型所用参数如表1所列.
Constants/N 5CB $ {k}_{11} $ 6.70 × 10–12 $ {k}_{22} $ 3.60 × 10–12 $ {{\rm{k}}}_{33} $ 9.00 × 10–12 $ {L}_{1} $ 4.20 × 10–12 $ {L}_{2} $ 5.51 × 10–12 $ {L}_{3} $ 1.02 × 10–12 表 1 5CB(LC 1264)的弹性常数[27]
Table 1. Elastic constants of 5CB (LC 1264).
-
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
Figure 2. (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
Figure 3. (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.
基于以上的结果, 固定一个+1/2的拓扑荷在距离圆心0.55 R0处, 另一个+1/2的拓扑荷遍历整个圆面, 通过数值方法计算出体系的总自由能随其位置变化的热值图, 如图3 (e) 所示. 在图中左侧蓝色月牙状区域体系自由能最小, 因此平衡时拓扑荷优先占据此区域. 这一结果与 Duclos 等[29]拍摄的486个圆盘结果一致. 图3 (e) 中自由能较低的区域, 正是第二个拓扑荷出现概率最大的区域. 两个拓扑荷的夹角在140°—180°之间.
演化过程中两个拓扑荷的位置不断变化, 在统计路径时, 通过旋转液晶圆盘, 把其中一个拓扑荷固定在圆心指向右侧的半径上, 它可以在此半径上左右平移. 模拟十种不同的相对位置, 得到十条曲线, 如图4 所示. 图中黑色圆点表示轨迹的起点, 蓝色圆点表示轨迹的终点;每个彩色线代表一组拓扑荷的相对运动轨迹, 红色到蓝色的变化表示时间. 图中拓扑荷的运动轨迹最后都走向图3 (e) 所示的蓝色区域, 此区域是自由能较低的状态.
图 4 十个不同相对位置的拓扑荷演化过程的运动迹图
Figure 4. Motion traces of the topological charges evolution process for 10 different relative positions.
-
圆盘的尺寸对拓扑荷的平衡位置有影响. 拓扑荷的相对平衡位置在0.542—0.558之间, 其中0—5 mm液晶圆盘中两个+1/2拓扑荷的间距与圆盘直径的比值由0.542增大到0.558, 之后在5—12 mm段这一比值基本稳定在0.558. 随着圆盘尺寸的增大边界锚定能的影响越小, 平衡位置即两拓扑荷的间距与圆盘直径的比值趋近于恒定值. 这一平衡位置是圆盘边界对+1/2拓扑荷的斥力和这两个拓扑荷之间排斥力平衡的结果. 液晶圆盘中两个拓扑荷的夹角在140°—180°之间. 拓扑荷的运动轨迹, 是其寻找自由能最低点的过程, 轨迹的终点在自由能最小值区域.
-
拓扑现象对于病毒颗粒的空间分布、高分子聚合物纳米囊泡的成型以及玻色-爱因斯坦凝聚物等方面都发挥着重要作用. 本文利用Landau-de Gennes理论, 构建模型来模拟液晶中拓扑荷分布及其他现象. 通过对数值模型序参量场的演化, 以及模拟液晶薄膜中所生成的拓扑荷之间的相互作用来分析液晶(Lqc)薄膜的尺寸对拓扑荷的影响. 研究结果表明,随着液晶盘半径增大, 拓扑荷间最优距离与半径之比渐增并趋于稳定. 此研究结论对利用拓扑荷凝聚颗粒物效应设计分离容器有指导意义, 有助于进一步理解拓扑胶体和液晶以及液晶共聚物等软物质中的拓扑现象.
-
关键词:
- 向列相液晶 /
- 相场 /
- 拓扑荷 /
- Landau-de Gennes模型
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
[1] Crawford G P, Zumer S 1996 Liquid Crystals in Complex Geometries: Formed by Polymer and Porous Networks (CRC Press: Boca Raton) p521
[2] Takeuchi H, Tsubota M 2006 J. Phys. Soc. Jpn. 75 063601
Google Scholar
[3] Joyce D D 2000 Compact Manifolds with Special Holonomy (Oxford: Oxford University Press) p395.
[4] Dammone O J, Zacharoudiou I, Dullens R P, Yeomans J M, Lettinga M P, Aarts D G 2012 Phys. Rev. Lett. 109 108303
Google Scholar
[5] Duclos G, Adkins R, Banerjee D, Peterson M S E, Varghese M, Kolvin I, Baskaran A, Pelcovits R A, Powers T R, Baskaran A, Toschi F, Hagan M F, Streichan S J, Vitelli V, Beller D A, Dogic Z 2020 Science 367 1120
Google Scholar
[6] Manyuhina O V, Lawlor K B, Marchetti M C, Bowick M J 2015 Soft Matter 11 6099
Google Scholar
[7] Smalyukh I I, Lansac Y, Clark N A, Trivedi R P 2010 Nat. Mater. 9 139
Google Scholar
[8] Senyuk B, Liu Q, He S, Kamien R D, Kusner R B, Lubensky T C, Smalyukh Ⅱ 2013 Nature 493 200
Google Scholar
[9] 刘永军, 孙伟民, 刘晓颀, 姚丽双, 鲁兴海, 宣丽 2012 物理学报 61 114211
Google Scholar
Liu Y J, Sun W M, Liu X Q, Yao L S, Lu X H, Xuan L 2012 Acta Phys. Sin. 61 114211
Google Scholar
[10] 吕月兰, 尹向宝, 杨月, 刘永军, 苑立波 2017 物理学报 66 154205
Google Scholar
Lü Y L, Yin X B, Yang Y, Liu Y J, Yuan L B 2017 Acta Phys. Sin. 66 154205
Google Scholar
[11] Ackerman P J, van de Lagemaat J, Smalyukh Ⅱ 2015 Nat. Commun. 6 6012
Google Scholar
[12] Xing X, Shin H, Bowick M J, Yao Z, Jia L, Li M H 2012 Proc. Natl. Acad. Sci. U. S. A 109 5202
Google Scholar
[13] Opathalage A, Norton M M, Juniper M P N, Langeslay B, Aghvami S A, Fraden S, Dogic Z 2019 Proc. Natl. Acad. Sci. U. S. A. 116 4788
Google Scholar
[14] Dolganov P V, Cluzeau P, Dolganov V K 2019 Liq. Cryst. Rev. 7 1
Google Scholar
[15] Peng C, Lavrentovich O D 2019 Micromachines-Basel 10 02187
[16] Turiv T, Krieger J, Babakhanova G, Yu H, Shiyanovskii S V, Wei Q H, Kim M H, Lavrentovich O D 2020 Sci. Adv. 6 eaaz6485
Google Scholar
[17] Giomi L, Kos Z, Ravnik M, Sengupta A 2017 Proc. Natl. Acad. Sci. U. S. A. 11 4
[18] Liu Q, Senyuk B, Tasinkevych M, Smalyukh Ⅱ 2013 Proc. Natl. Acad. Sci. U. S. A. 110 9231
Google Scholar
[19] Solodkov N V, Shim J U, Jones J C 2019 Nat. Commun. 10 198
Google Scholar
[20] Liang D, Ma X, Liu Z, Jafri H M, Cao G, Huang H, Shi S, Chen L Q 2020 J. Appl. Phys. 128 124701
Google Scholar
[21] Tang X, Selinger J V 2017 Soft Matter 13 5481
Google Scholar
[22] De Gennes P G, Prost J 1993 The Physics of Liquid Crystals (Vol. 83) (New York: Oxford University Press) p597.
[23] Pelka R, Saito K 2006 Phys. Rev. E 74 041705
Google Scholar
[24] Frank F C 1958 Discuss. Faraday Soc. 25 19
Google Scholar
[25] Berreman D W, Meiboom S 1984 Phys. Rev. A 30 1955
Google Scholar
[26] Longa L, Monselesan D, Trebin H R 1987 Liq. Cryst. 2 769
Google Scholar
[27] Inukai T, Miyazawa K 1917 Outline of Development of Nematic Liquid Crystal Compounds for LCD (The Museum: King's Printer for British Columbia) p559.
[28] Fournier J B, Galatola P 2005 Europhys. Lett. 72 403
Google Scholar
[29] Duclos G, Erlenkämper C, Joanny J F, Silberzan P 2017 Nat. Phys. 13 58
Google Scholar
-
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Fig. 1. (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.
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
Fig. 2. (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 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
Fig. 3. (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.
图 4 十个不同相对位置的拓扑荷演化过程的运动迹图
Fig. 4. Motion traces of the topological charges evolution process for 10 different relative positions.
-
[1] Crawford G P, Zumer S 1996 Liquid Crystals in Complex Geometries: Formed by Polymer and Porous Networks (CRC Press: Boca Raton) p521
[2] Takeuchi H, Tsubota M 2006 J. Phys. Soc. Jpn. 75 063601
Google Scholar
[3] Joyce D D 2000 Compact Manifolds with Special Holonomy (Oxford: Oxford University Press) p395.
[4] Dammone O J, Zacharoudiou I, Dullens R P, Yeomans J M, Lettinga M P, Aarts D G 2012 Phys. Rev. Lett. 109 108303
Google Scholar
[5] Duclos G, Adkins R, Banerjee D, Peterson M S E, Varghese M, Kolvin I, Baskaran A, Pelcovits R A, Powers T R, Baskaran A, Toschi F, Hagan M F, Streichan S J, Vitelli V, Beller D A, Dogic Z 2020 Science 367 1120
Google Scholar
[6] Manyuhina O V, Lawlor K B, Marchetti M C, Bowick M J 2015 Soft Matter 11 6099
Google Scholar
[7] Smalyukh I I, Lansac Y, Clark N A, Trivedi R P 2010 Nat. Mater. 9 139
Google Scholar
[8] Senyuk B, Liu Q, He S, Kamien R D, Kusner R B, Lubensky T C, Smalyukh Ⅱ 2013 Nature 493 200
Google Scholar
[9] 刘永军, 孙伟民, 刘晓颀, 姚丽双, 鲁兴海, 宣丽 2012 物理学报 61 114211
Google Scholar
Liu Y J, Sun W M, Liu X Q, Yao L S, Lu X H, Xuan L 2012 Acta Phys. Sin. 61 114211
Google Scholar
[10] 吕月兰, 尹向宝, 杨月, 刘永军, 苑立波 2017 物理学报 66 154205
Google Scholar
Lü Y L, Yin X B, Yang Y, Liu Y J, Yuan L B 2017 Acta Phys. Sin. 66 154205
Google Scholar
[11] Ackerman P J, van de Lagemaat J, Smalyukh Ⅱ 2015 Nat. Commun. 6 6012
Google Scholar
[12] Xing X, Shin H, Bowick M J, Yao Z, Jia L, Li M H 2012 Proc. Natl. Acad. Sci. U. S. A 109 5202
Google Scholar
[13] Opathalage A, Norton M M, Juniper M P N, Langeslay B, Aghvami S A, Fraden S, Dogic Z 2019 Proc. Natl. Acad. Sci. U. S. A. 116 4788
Google Scholar
[14] Dolganov P V, Cluzeau P, Dolganov V K 2019 Liq. Cryst. Rev. 7 1
Google Scholar
[15] Peng C, Lavrentovich O D 2019 Micromachines-Basel 10 02187
[16] Turiv T, Krieger J, Babakhanova G, Yu H, Shiyanovskii S V, Wei Q H, Kim M H, Lavrentovich O D 2020 Sci. Adv. 6 eaaz6485
Google Scholar
[17] Giomi L, Kos Z, Ravnik M, Sengupta A 2017 Proc. Natl. Acad. Sci. U. S. A. 11 4
[18] Liu Q, Senyuk B, Tasinkevych M, Smalyukh Ⅱ 2013 Proc. Natl. Acad. Sci. U. S. A. 110 9231
Google Scholar
[19] Solodkov N V, Shim J U, Jones J C 2019 Nat. Commun. 10 198
Google Scholar
[20] Liang D, Ma X, Liu Z, Jafri H M, Cao G, Huang H, Shi S, Chen L Q 2020 J. Appl. Phys. 128 124701
Google Scholar
[21] Tang X, Selinger J V 2017 Soft Matter 13 5481
Google Scholar
[22] De Gennes P G, Prost J 1993 The Physics of Liquid Crystals (Vol. 83) (New York: Oxford University Press) p597.
[23] Pelka R, Saito K 2006 Phys. Rev. E 74 041705
Google Scholar
[24] Frank F C 1958 Discuss. Faraday Soc. 25 19
Google Scholar
[25] Berreman D W, Meiboom S 1984 Phys. Rev. A 30 1955
Google Scholar
[26] Longa L, Monselesan D, Trebin H R 1987 Liq. Cryst. 2 769
Google Scholar
[27] Inukai T, Miyazawa K 1917 Outline of Development of Nematic Liquid Crystal Compounds for LCD (The Museum: King's Printer for British Columbia) p559.
[28] Fournier J B, Galatola P 2005 Europhys. Lett. 72 403
Google Scholar
[29] Duclos G, Erlenkämper C, Joanny J F, Silberzan P 2017 Nat. Phys. 13 58
Google Scholar
计量
- 文章访问数: 3002
- PDF下载量: 49
- 被引次数: 0
