American Institute of Mathematical Sciences

Advanced
June  2007, 7(4): 907-929. doi: 10.3934/dcdsb.2007.7.907

Nonparallel solutions of extended nematic polymers under an external field

Hong Zhou 1, , Hongyun Wang 2, and  Qi Wang 3,

1. 

Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216
2. 

Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA 95064, United States
3. 

Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510

Received  December 2005 Revised  February 2007 Published  March 2007

We continue the study on equilibria of the Smoluchowski equation for dilute solutions of rigid extended (dipolar) nematics and dispersions under an imposed electric or magnetic field [25]. We first provide an alternative proof for the theorem that all equilibria are dipolar with the polarity vector parallel to the external field direction if the strength of the permanent dipole ($\mu$) is larger than or equal to the product of the external field ($E$) and the anisotropy parameter ($\alpha_0$) (i.e. $\mu \ge |\alpha_0| E$). Then, we show that when $\mu < |\alpha_0| E$, there is a critical value $\alpha^$*$\geq 1$ for the intermolecular dipole-dipole interaction strength ($\alpha$) such that all equilibria are either isotropic or parallel to the external field if $\alpha \le \alpha^$*; but nonparallel dipolar equilibria emerge when $\alpha > \alpha^$*. The nonparallel equilibria are analyzed and the asymptotic behavior of $\alpha^$* is studied. Finally, the asymptotic results are validated by direct numerical simulations.
Keywords: equilibrium solutions., dipolar nematics, liquid crystals, magnetic dispersions, polymers, kinetic theory.
Mathematics Subject Classification: Primary: 35A99, 35C15; Secondary: 76W05, 82D6.
Citation: Hong Zhou, Hongyun Wang, Qi Wang. Nonparallel solutions of extended nematic polymers under an external field. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 907-929. doi: 10.3934/dcdsb.2007.7.907
[1]

Carlos J. García-Cervera, Sookyung Joo. Reorientation of smectic a liquid crystals by magnetic fields. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1983-2000. doi: 10.3934/dcdsb.2015.20.1983
[2]

Pierre Degond, Amic Frouvelle, Jian-Guo Liu. From kinetic to fluid models of liquid crystals by the moment method. Kinetic and Related Models, 2022, 15 (3) : 417-465. doi: 10.3934/krm.2021047
[3]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[4]

Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic and Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235
[5]

Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591
[6]

Xiaoyu Zheng, Peter Palffy-Muhoray. One order parameter tensor mean field theory for biaxial liquid crystals. Discrete and Continuous Dynamical Systems - B, 2011, 15 (2) : 475-490. doi: 10.3934/dcdsb.2011.15.475
[7]

Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445
[8]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757
[9]

Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243
[10]

Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303
[11]

Pierre Degond, Hailiang Liu. Kinetic models for polymers with inertial effects. Networks and Heterogeneous Media, 2009, 4 (4) : 625-647. doi: 10.3934/nhm.2009.4.625
[12]

Wenya Ma, Yihang Hao, Xiangao Liu. Shape optimization in compressible liquid crystals. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1623-1639. doi: 10.3934/cpaa.2015.14.1623
[13]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[14]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[15]

Jinhae Park, Feng Chen, Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1419-1440. doi: 10.3934/dcds.2010.26.1419
[16]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[17]

Zhenlu Cui, M. Carme Calderer, Qi Wang. Mesoscale structures in flows of weakly sheared cholesteric liquid crystal polymers. Discrete and Continuous Dynamical Systems - B, 2006, 6 (2) : 291-310. doi: 10.3934/dcdsb.2006.6.291
[18]

Zhenlu Cui, Qi Wang. Permeation flows in cholesteric liquid crystal polymers under oscillatory shear. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 45-60. doi: 10.3934/dcdsb.2011.15.45
[19]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[20]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy