September  2015, 20(7): 2001-2026. doi: 10.3934/dcdsb.2015.20.2001

Analysis of a model for bent-core liquid crystals columnar phases

1. 

Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, United States, United States

Received  April 2014 Revised  March 2015 Published  July 2015

We consider a model originally introduced to study layer-undulated structures in bent-core molecule liquid crystals. We first prove existence of minimizers, then analyze a simplified version used to study how in columnar phases the width of the column affects the type of switching, which occurs under an applied electric field. We show via $\Gamma$-convergence that as the width of the column tends to infinity, rotation around the tilt cone is favored, provided the coefficient of the coupling term, between the polar parameter, the nematic parameter, and the layer normal is large.
Citation: Tiziana Giorgi, Feras Yousef. Analysis of a model for bent-core liquid crystals columnar phases. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2001-2026. doi: 10.3934/dcdsb.2015.20.2001
References:
[1]

R. Adams and J. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). Google Scholar

[2]

N. Ansini, A. Braides and V. Valente, Multiscale analysis by $\Gamma$-convergence of a one-dimensional nonlocal functional related to a shell-membrane transition,, SIAM J. Math. Anal., 38 (2006), 944. doi: 10.1137/050630829. Google Scholar

[3]

P. Bauman and Phillips, Analysis and stability of bent-core liquid crystal fibers,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1707. doi: 10.3934/dcdsb.2012.17.1707. Google Scholar

[4]

P. Bauman, D. Phillips and J. Park., Existence of solutions to boundary value problems for smectic liquid crystals,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 243. doi: 10.3934/dcdss.2015.8.243. Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar

[6]

J.-H. Chen and T. C. Lubensky, Landau-Ginzburg mean-field theory for the nematic to smectic-C and nematic to smectic-A phase transitions,, Phys. Rev. A, 14 (1976), 1202. doi: 10.1103/PhysRevA.14.1202. Google Scholar

[7]

A. Eremin and A. Jákli, Polar bent-shape liquid crystals - from molecular bend to layer splay and chirality,, Soft Matter, 9 (2013), 615. doi: 10.1039/C2SM26780B. Google Scholar

[8]

E. Gorecka, N. Vaupotič, D. Pociecha, M. Čepič and J. Mieczkowski, Switching mechanism in polar columnar mesophases made of bent-core molecules,, ChemPhysChem, 6 (2005), 1087. doi: 10.1002/cphc.200400623. Google Scholar

[9]

S. Joo and D. Phillips, Chiral nematic toward smectic liquid crystals,, Comm. Math. Phys., 269 (2007), 369. doi: 10.1007/s00220-006-0132-z. Google Scholar

[10]

S. T. Lagerwall, Ferroelectric and antiferroelectric liquid crystals,, Encyclopedia of Materials: Science and Technology, (2001), 3044. doi: 10.1016/B0-08-043152-6/00545-3. Google Scholar

[11]

I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases,, Phys. Rev. E, 57 (1998), 574. Google Scholar

[12]

I. Muševič, R. Blinc and B. Žekš, The Physics of Ferroelectric and Antiferroelectric Liquid Crystals,, World Scientific Publishing Company, (2000). Google Scholar

[13]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, $2^{nd}$ edition, (1993). Google Scholar

[14]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for non convex discrete systems,, Math. Models Methods Appl. Sci., 21 (2011), 777. doi: 10.1142/S0218202511005210. Google Scholar

[15]

I. W. Stewart, The Static and Dynamic Continuum theory of Liquid Crystals,, Taylor & Francis, (2004). Google Scholar

[16]

N. Vaupotič and M. Čopič, Polarization modulation instability in liquid crystals with spontaneous chiral symmetry breaking,, Phys. Rev. E, 72 (2005). Google Scholar

[17]

E. G. Virga., Variational Theories for Liquid Crystals,, Chapman & Hall, (1994). doi: 10.1007/978-1-4899-2867-2. Google Scholar

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). Google Scholar

[2]

N. Ansini, A. Braides and V. Valente, Multiscale analysis by $\Gamma$-convergence of a one-dimensional nonlocal functional related to a shell-membrane transition,, SIAM J. Math. Anal., 38 (2006), 944. doi: 10.1137/050630829. Google Scholar

[3]

P. Bauman and Phillips, Analysis and stability of bent-core liquid crystal fibers,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1707. doi: 10.3934/dcdsb.2012.17.1707. Google Scholar

[4]

P. Bauman, D. Phillips and J. Park., Existence of solutions to boundary value problems for smectic liquid crystals,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 243. doi: 10.3934/dcdss.2015.8.243. Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar

[6]

J.-H. Chen and T. C. Lubensky, Landau-Ginzburg mean-field theory for the nematic to smectic-C and nematic to smectic-A phase transitions,, Phys. Rev. A, 14 (1976), 1202. doi: 10.1103/PhysRevA.14.1202. Google Scholar

[7]

A. Eremin and A. Jákli, Polar bent-shape liquid crystals - from molecular bend to layer splay and chirality,, Soft Matter, 9 (2013), 615. doi: 10.1039/C2SM26780B. Google Scholar

[8]

E. Gorecka, N. Vaupotič, D. Pociecha, M. Čepič and J. Mieczkowski, Switching mechanism in polar columnar mesophases made of bent-core molecules,, ChemPhysChem, 6 (2005), 1087. doi: 10.1002/cphc.200400623. Google Scholar

[9]

S. Joo and D. Phillips, Chiral nematic toward smectic liquid crystals,, Comm. Math. Phys., 269 (2007), 369. doi: 10.1007/s00220-006-0132-z. Google Scholar

[10]

S. T. Lagerwall, Ferroelectric and antiferroelectric liquid crystals,, Encyclopedia of Materials: Science and Technology, (2001), 3044. doi: 10.1016/B0-08-043152-6/00545-3. Google Scholar

[11]

I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases,, Phys. Rev. E, 57 (1998), 574. Google Scholar

[12]

I. Muševič, R. Blinc and B. Žekš, The Physics of Ferroelectric and Antiferroelectric Liquid Crystals,, World Scientific Publishing Company, (2000). Google Scholar

[13]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, $2^{nd}$ edition, (1993). Google Scholar

[14]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for non convex discrete systems,, Math. Models Methods Appl. Sci., 21 (2011), 777. doi: 10.1142/S0218202511005210. Google Scholar

[15]

I. W. Stewart, The Static and Dynamic Continuum theory of Liquid Crystals,, Taylor & Francis, (2004). Google Scholar

[16]

N. Vaupotič and M. Čopič, Polarization modulation instability in liquid crystals with spontaneous chiral symmetry breaking,, Phys. Rev. E, 72 (2005). Google Scholar

[17]

E. G. Virga., Variational Theories for Liquid Crystals,, Chapman & Hall, (1994). doi: 10.1007/978-1-4899-2867-2. Google Scholar

[1]

Patricia Bauman, Daniel Phillips. Analysis and stability of bent-core liquid crystal fibers. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1707-1728. doi: 10.3934/dcdsb.2012.17.1707

[2]

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

[3]

Kyungkeun Kang, Jinhae Park. Partial regularity of minimum energy configurations in ferroelectric liquid crystals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1499-1511. doi: 10.3934/dcds.2013.33.1499

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017

[10]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Xian-Gao Liu, Jianzhong Min, Kui Wang, Xiaotao Zhang. Serrin's regularity results for the incompressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5579-5594. doi: 10.3934/dcds.2016045

[17]

Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1

[18]

Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357

[19]

Shijin Ding, Junyu Lin, Changyou Wang, Huanyao Wen. Compressible hydrodynamic flow of liquid crystals in 1-D. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 539-563. doi: 10.3934/dcds.2012.32.539

[20]

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

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]