摘要
Vector-based continuous models for nematic liquid crystals such as the Oseen-Frank model and the Ericksen model are relatively simpler compared with tensor-based models such as the Landau-de Gennes model.However,these vector models do not respect head-to-tail symmetry.As a result,they cannot predict configurations corresponding to non-orientable line fields,particularly the half-integer defects.This paper confirms a significant discrepancy between the transition dynamics predicted by the Oseen-Frank vector model and Landau-de Gennes tensor model for liquid crystals confined in a two-dimensional square well.The so-called inner product weighted Laplacian operator is introduced as an anisotropic diffusion operator to evolve the Euler-Lagrange equations corresponding to the modified Oseen-Frank model.Numerical results show that both the predicted equilibrium configurations and the transition dynamics from one equilibrium states to another satisfies head-to-tail symmetry and can accommodate half-integer defects.The connections of anisotropic diffusion operator to the graph Laplacian and the discrete Lebwohl-Lasher model are also discussed.The numerical trick proposed in this paper can be considered a simple remedy to restore head-to-tail symmetry in vector models of liquid crystals,making them more applicable in situations such as systems containing half-integer defects where the traditional numerical approach would fail.
与基于张量的Landau-de Gennes模型相比,基于向量的向列相液晶连续模型,例如OseenFrank和Ericksen模型等相对更简单。然而这些向量模型并不遵循头尾对称性原则。因此,其不能预测不可定向线场的构型,特别是对于半整数缺陷。本文证实了由Oseen-Frank矢量模型和Landau-de Gennes张量模型预测的二维方形阱液晶的过渡动力学之间的显著差异;引入所谓的内积加权拉普拉斯算子作为各向异性扩散算子,并演化对应于修正的Oseen-Frank模型的EulerLagrange方程。数值结果表明:预测的平衡态和从一种平衡态到另一种平衡态的过渡动力学都满足头尾对称,并能适应半整数缺陷;讨论了各向异性扩散算符与图Laplacian和离散LebwohlLasher模型的联系。本文提出的数值方法可以被认为是恢复液晶矢量模型头尾对称的一种简单方法,使其更适用于含有半整数缺陷的系统,而传统的数值方法将失效。
出处
《首都师范大学学报（自然科学版）》
2024年第6期9-20,共12页
Journal of Capital Normal University：Natural Science Edition
基金
国家自然科学基金项目(12271375)
北京市自然科学基金项目(Z230009)
广东省基础与应用基础研究基金项目(2023A1515010157)。
