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 and 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, Academic Press, 2003.

[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), 944-976. doi: 10.1137/050630829.

[3]

P. Bauman and Phillips, Analysis and stability of bent-core liquid crystal fibers, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1707-1728. doi: 10.3934/dcdsb.2012.17.1707.

[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), 243-257. doi: 10.3934/dcdss.2015.8.243.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.

[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), 1202-1207. doi: 10.1103/PhysRevA.14.1202.

[7]

A. Eremin and A. Jákli, Polar bent-shape liquid crystals - from molecular bend to layer splay and chirality, Soft Matter, 9 (2013), 615-637. doi: 10.1039/C2SM26780B.

[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), 1087-1093. doi: 10.1002/cphc.200400623.

[9]

S. Joo and D. Phillips, Chiral nematic toward smectic liquid crystals, Comm. Math. Phys., 269 (2007), 369-399. doi: 10.1007/s00220-006-0132-z.

[10]

S. T. Lagerwall, Ferroelectric and antiferroelectric liquid crystals, Encyclopedia of Materials: Science and Technology, (2001), 3044-3063. doi: 10.1016/B0-08-043152-6/00545-3.

[11]

I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases, Phys. Rev. E, 57 (1998), 574-581.

[12]

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

[13]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, $2^{nd}$ edition, Clarendon Press, Oxford, 1993.

[14]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for non convex discrete systems, Math. Models Methods Appl. Sci., 21 (2011), 777-817. doi: 10.1142/S0218202511005210.

[15]

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

[16]

N. Vaupotič and M. Čopič, Polarization modulation instability in liquid crystals with spontaneous chiral symmetry breaking, Phys. Rev. E, 72 (2005), 031701.

[17]

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

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, $2^{nd}$ edition, Academic Press, 2003.

[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), 944-976. doi: 10.1137/050630829.

[3]

P. Bauman and Phillips, Analysis and stability of bent-core liquid crystal fibers, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1707-1728. doi: 10.3934/dcdsb.2012.17.1707.

[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), 243-257. doi: 10.3934/dcdss.2015.8.243.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.

[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), 1202-1207. doi: 10.1103/PhysRevA.14.1202.

[7]

A. Eremin and A. Jákli, Polar bent-shape liquid crystals - from molecular bend to layer splay and chirality, Soft Matter, 9 (2013), 615-637. doi: 10.1039/C2SM26780B.

[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), 1087-1093. doi: 10.1002/cphc.200400623.

[9]

S. Joo and D. Phillips, Chiral nematic toward smectic liquid crystals, Comm. Math. Phys., 269 (2007), 369-399. doi: 10.1007/s00220-006-0132-z.

[10]

S. T. Lagerwall, Ferroelectric and antiferroelectric liquid crystals, Encyclopedia of Materials: Science and Technology, (2001), 3044-3063. doi: 10.1016/B0-08-043152-6/00545-3.

[11]

I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases, Phys. Rev. E, 57 (1998), 574-581.

[12]

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

[13]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, $2^{nd}$ edition, Clarendon Press, Oxford, 1993.

[14]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for non convex discrete systems, Math. Models Methods Appl. Sci., 21 (2011), 777-817. doi: 10.1142/S0218202511005210.

[15]

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

[16]

N. Vaupotič and M. Čopič, Polarization modulation instability in liquid crystals with spontaneous chiral symmetry breaking, Phys. Rev. E, 72 (2005), 031701.

[17]

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

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