Csub2v_sub对称性下液晶多张量模型的取向弹性推导.pdf

Pure Mathematics n,2023,13(9),2499-2515Published Online September 2023 in Hans.https:/www.hanspub.org/journal/pmhttps:/doi.org/10.12677/pm.2023.139256C2v5e?.?5?nnn!BO?B B?vF2023c7?28FF2023c8?30FuF2023c9?8F?k C2v5?f/?uNU-:gdU?L?C2v?5LU?3N5?5?Xfkk(?nc?5?5Derivation of Orientational Elasticity ofLiquid Crystal Multitensor Model with C2vSymmetryLuqian Zhou,Xinxin FengSchool of Mathematics and Statistics,Guizhou University,Guiyang GuizhouReceived:Jul.28th,2023;accepted:Aug.30th,2023;published:Sep.8th,2023AbstractIn this paper,the orientational elasticity of C2vnematic phase is derived based on the:n,!.C2v5e?.?5?J.n,2023,13(9):2499-2515.DOI:10.12677/pm.2023.139256n!expression of the stability points of volume energy and free energy for the nematicphase formed by liquid crystal molecules with C2vsymmetry.This expression canreflect the symmetry of local anisotropy of liquid crystal phase to a certain extent,and the coefficients are related to molecular parameters,which has clear physicalsignificance.KeywordsNematic Liquid Crystal,Symmetry,Orientational ElasticityCopyright c?2023 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.0u?m?,?k5?A:.3,:+f5?5 1.?5!?-5,3?LX-?.5?3G?e?A?,3dG?e,?5?u.?u)?/C.f/?,?5U?/.k C2v5?f(?E,?5,U?/?,?/?k C2v5?5.?5Ly?5,n?.?C/?,?UC?O,d?x?,=n=n(x).?UL Oseen-Frank UL,u n(x)?,nOL-,-,?-?)?U 2.Oseen-Frank U?5 Ki?k-.3,Frederiks?Gf?5 35,d?Kaur?q?d/f/?A 68,Sathyanarayana?.A?A,Ed?kmCz?/.z1 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/10.1137/20M134071X2 Oseen,C.W.(1933)The Theory of Liquid Crystals.Transactions of the Faraday Society,29,883-899.https:/doi.org/10.1039/tf93329008833 Durand,G.,Lger,L.,Rondelez,F.,et al.(1969)Quasielastic Rayleigh Scattering in NematicLiquid Crystals.Physical Review Letters,22,1361-1363.https:/doi.org/10.1103/PhysRevLett.22.13614 Prost,J.and Gasparoux H.(1971)Determination of Twist Viscosity Coefficient in the NematicMesophases.Physics Letters A,36,245-246.https:/doi.org/10.1016/0375-9601(71)90443-95 Frederiks,V.and Zolina,V.(1933)Forces Causing the Orientation of an Anisotropic Liquid.Transactions of the Faraday Society,29,919-930.https:/doi.org/10.1039/TF9332900919DOI:10.12677/pm.2023.1392562513nn!6 Kaur,S.,Addis,J.,Greco,C.,et al.(2012)Understanding the Distinctive Elastic Constantsin an Oxadiazole Bent-Core Nematic Liquid Crystal.Physical Review E,86,Article 041703.https:/doi.org/10.1103/PhysRevE.86.0417037 Majumdar,M.,Salamon,P.,J akli,A.,et al.(2011)Elastic Constants and OrientationalViscosities of a Bent-Core Nematic Liquid Crystal.Physical Review E,83,Article 031701.https:/doi.org/10.1103/PhysRevE.83.0317018 Sathyanarayana,P.,Mathew,M.,Li,Q.,et al.(2010)Splay Bend Elasticity of a Bent-CoreNematic Liquid Crystal.Physical Review E,81,Article 010702.https:/doi.org/10.1103/PhysRevE.81.0107029 Sathyanarayana,P.,Radhika,S.,Sadashiva,B.K.,et al.(2012)Structure-Property Correlationof a Hockey Stick-Shaped Compound Exhibiting N-SmA-SmCaPhase Transitions.Soft Matter,8,2322-2327.https:/doi.org/10.1039/c2sm06767f10 Sathyanarayana,P.,Varia,M.C.,Prajapati,A.K.,et al.(2010)Splaybend Elasticity of aNematic Liquid Crystal with T-Shaped Molecules.Physical Review E,82,Article 050701.https:/doi.org/10.1103/PhysRevE.82.05070111 Acharya,B.R.,Primak,A.and Kumar,S.(2004)Biaxial Nematic Phase in Bent-Core Ther-motropic Mesogens.Physical Review Letters,92,Article 145506.https:/doi.org/10.1103/PhysRevLett.92.14550612 Madsen,L.A.,Dingemans,T.J.,Nakata,M.,et al.(2004)Thermotropic Biaxial NematicLiquid Crystals.Physical Review Letters,92,Article 145505.https:/doi.org/10.1103/PhysRevLett.92.14550513 Govers,E.and Vertogen,G.(1984)Elastic Continuum Theory of Biaxial Nematics.PhysicalReview A,30,1998-2000.https:/doi.org/10.1103/PhysRevA.30.199814 Liu,M.(1981)Hydrodynamic Theory of Biaxial Nematics.Physical Review A,24,2720-2726.https:/doi.org/10.1103/PhysRevA.24.272015 Stallinga,S.and Vertogen,G.(1994)Theory of Orientational Elasticity.Physical Review E,49,1483-1494.https:/doi.org/10.1103/PhysRevE.49.148316 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.17 Xu,J.and Zhang,P.W.(2018)Calculating Elastic Constants of Bent-Core Molecules fromOnsager-Theory-Based Tensor Model.Liquid Crystals,45,22-31.https:/doi.org/10.1080/02678292.2017.129028518 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-2021-0034DOI:10.12677/pm.2023.1392562514nn!19 Xu,J.and Zhang,P.W.(2018)Onsager-Theory-Based Dynamical Model for Nematic Phasesof Bent-Core Molecules and Star Molecules.Journal of Non-Newtonian Fluid Mechanics,251,43-55.https:/doi.org/10.1016/j.jnnfm.2017.11.00520 Xu,J.and Zhang,P.W.(2017)The Transmission of Symmetry in Liquid Crystals.Physics,15,185-195.https:/doi.org/10.4310/CMS.2017.v15.n1.a821 Xu,J.(2020)Quasi-Entropy by Log-Determinant Covariance Matrix and Application to LiquidCrystals.arXiv:2007.15786DOI:10.12677/pm.2023.1392562515n