液晶模型中一些高阶张量的计算
Calculation of Some Higher-Order Tensors in Liquid Crystal Models
DOI: 10.12677/PM.2023.1311324, PDF,   
作者: 周陆纤：贵州大学数学与统计学院，贵州 贵阳
关键词: 向列相液晶；平移扩散；高阶张量；Nematic Liquid Crystal； Translational Diffusion； High-Order Tensors
摘要: 本文针对具有 C2v对称性的液晶分子形成的向列相，基于液晶的 Onsager 分子理论，通过对自由能求变分，建立了多张量模型。多张量模型中含有较多的高阶张量，需对其进行封闭近似，得到由对称迹零张量表示的多张量模型，可用于描述不同液晶相之间的相变。
Abstract: In this paper, for the nematic phase formed by liquid crystal molecules with C2v symmetry, a multi-tensor model is established based on Onsager molecular theory of liquid crystals, and by variational of free energy. The multi-tensor model contains more high-order tensors, which need to be closed approximation, and a multi-tensor model represented by symmetric traceless tensors is obtained, which can be used to describe the phase transition between different liquid crystal phases.
文章引用：周陆纤. 液晶模型中一些高阶张量的计算[J]. 理论数学, 2023, 13(11): 3126-3138. https://doi.org/10.12677/PM.2023.1311324

参考文献

[1] Li, S.,Wang, W. and Zhang, P. (2015) Local Posedness and Small Deborah Limit of a Molecule- Based Q-Tensor System. Discrete and Continuous Dynamical Systems B, 20, 2611-2655.
https://doi.org/10.3934/dcdsb.2015.20.2611
[2] Yang, X.F., Forest, M.G., Liu, C. and Shen, J. (2011) Shear Cell Rupture of Nematic Liquid Crystal Droplets in Viscous Fluids. Journal of Non-Newtonian Fluid Mechanics, 166, 487-499.
https://doi.org/10.1016/j.jnnfm.2011.02.004
[3] Zhao, J., Yang, X.F., Shen, J. and Wang, Q. (2016) A Decoupled Energy Stable Scheme for a Hydrodynamic Phase-Field Model of Mixtures of Nematic Liquid Crystals and Viscous Fluids. Journal of Computational Physics, 305, 539-556.
https://doi.org/10.1016/j.jcp.2015.09.044
[4] Han, J.Q., Luo, Y., Wang, W., Zhang, P.W. and Zhang, Z.F. (2015) From Microscopic Theory to Macroscopic Theory: Systematic Study on Modeling for Liquid Crystals. Archive for Rational Mechanics and Analysis, 215, 741-809.
https://doi.org/10.1007/s00205-014-0792-3
[5] Fatkullin, I. and Slastikov, V. (2005) Critical Points of the Onsager Functional on a Sphere. Nonlinearity, 18, 2565-2580.
https://doi.org/10.1088/0951-7715/18/6/008
[6] Liu, H.L., Zhang, H. and Zhang, P.W. (2005) Axial Symmetry and Classification of Stationary Solutions of Doi-Onsager Equation on the Sphere with Maier-Saupe Potential. Communications in Mathematical Sciences, 3, 201-218.
https://doi.org/10.4310/CMS.2005.v3.n2.a7
[7] Govers, E. and Vertogen, G. (1985) Fluid Dynamics of Biaxial Nematics. Physica A: Statistical Mechanics and Its Applications, 133, 337-344.
https://doi.org/10.1016/0378-4371(85)90073-1
[8] Liu, M. (1981) Hydrodynamic Theory of Biaxial Nematics. Physical Review A, 24, 2720-2726.
https://doi.org/10.1103/PhysRevA.24.2720
[9] Xu, J. and Zhang, P.W. (2018) Onsager-Theory-Based Dynamical Model for Nematic Phases of Bent-Core Molecules and Star Molecules. Journal of Non-Newtonian Fluid Mechanics, 251, 43-55.
https://doi.org/10.1016/j.jnnfm.2017.11.005
[10] Xu, J., Ye, F. and Zhang, P.W. (2018) A Tensor Model for Nematic Phases of Bent-Core Molecules Based on Molecular Theory. Multiscale Modeling and Simulation, 16, 1581-1602.
https://doi.org/10.1137/16M1099789
[11] Li, S.R. and Xu, J. (2021) Frame Hydrodynamics of Biaxial Nematics from Molecular-Theory- Based Tensor Models. SIAM Journal on Applied Mathematics, to appear.
[12] Xu, J. (2022) Quasi-Entropy by Log-Determinant Covariance Matrix and Application to Liquid Crystals. Physica D: Nonlinear Phenomena, 435, Article 133308.
https://doi.org/10.1016/j.physd.2022.133308
[13] Xu, J. (2020) Classifying Local Anisotropy Formed by Rigid Molecules: Symmetries and Ten- sors. SIAM Journal on Applied Mathematics, 80, 2518-2546.
https://doi.org/10.1137/20M134071X
[14] Xu, J. (2022) Symmetry-Consistent Expansion of Interaction Kernels between Rigid Molecules. CSIAM Transactions on Applied Mathematics, 3, 383-427.
https://doi.org/10.4208/csiam-am.SO-2021-0034
[15] Mayer, J.E. and Mayer, M.G. (1940) Statistical Mechanics. John Wiley and Sons, Hoboken.

为你推荐