Existence of solutions to boundary value problems for smectic liquid crystals

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April 2015, 8(2): 243-257. doi: 10.3934/dcdss.2015.8.243

Existence of solutions to boundary value problems for smectic liquid crystals

Patricia Bauman ^1, , Daniel Phillips ^1, and Jinhae Park ^2,

1.	Department of Mathematics, Purdue University, West Lafayette, IN 47906, United States
2.	Department of Mathematics, Chungnam National University, 99 Daehak-ro, Gung-Dong Yuseong-gu, Daejeon 305-764

Received April 2013 Revised October 2013 Published July 2014

Abstract
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We prove lower semicontinuity and lower bounds for a Chen-Lubensky energy describing nematic/smectic liquid crystals with physically realistic boundary conditions. The Chen-Lubensky energy captures stable phases of the liquid crystal material, ranging from purely nematic or smectic states to coexisting nematic/smectic states. By including appropriate additional terms, the model includes the effects of applied electric or magnetic fields, and/or electrical self-interactions in the case of polarized liquid crystals. As a consequence of our results, we establish existence of minimizers with weak or strong anchoring of the director field (describing molecular orientation) at the boundary, and Dirichlet or Neumann boundary conditions on the smectic order parameter for the liquid crystal material.

Keywords: ferroelectrics, Calculus of variations, boundary value problem., smectic liquid crystal, elliptic equations and systems.

Mathematics Subject Classification: Primary: 49J40, 35J50; Secondary: 35J57, 35J6.

Citation: 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

References:

[1]	C. Bailey, E. C. Gartland and A. Jákli, Structure and stability of bent core liquid crystal fibers,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.031701. Google Scholar
[2]	P. Bauman, M. C. Calderer, C. Liu and D. Phillips, The phase transition between nematic and smectic A* liquid crystals,, Arch. Rational Mech. Anal., 165* (2002), 161. doi: 10.1007/s00205-002-0223-8. Google Scholar*
[3]	P. Bauman and D. Phillips, Stability of bent core liquid crystal fibers,, Molecular Crystals and Liquid Crystals Journal, 510* (2009), 1135. Google Scholar*
[4]	H. R. Brand, P. E. Cladis and H. Pleiner, Fluid biaxial banana smectics: Symmetry at work,, Liquid Crystal Today, 9 (1999), 1. Google Scholar
[5]	H. R. Brand, P. E. Claids and H. Pleiner, Macroscopic properties of smectic $C_G$ liquid crystals,, Eur. Phys. J. B, 6 (1998), 347. Google Scholar
[6]	M. Cagnon and G. Durand, Positional anchoring of smectic liquid crystals,, Phys. Rev. Lett., 70 (1993). doi: 10.1103/PhysRevLett.70.2742. Google Scholar
[7]	C. M. Chen and F. C. Mackintosh, Theory of modulated phases in lipid bilayers and liquid crystal films,, Phys. Rev. E, 53 (1996), 4933. doi: 10.1103/PhysRevE.53.4933. Google Scholar
[8]	J. Chen and T. Lubensky, Landau-Ginzburg mean-field theory for the nematic to smectic-C and nematic to smectic-A phase transitions,, Phys. Rev. A, 14 (1976). doi: 10.1103/PhysRevA.14.1202. Google Scholar
[9]	D. A. Coleman, J. Fernsler, N. Chattham, M. Nakata, Y. Takanishi, E. Köblova, D. R. Link, R. F. Shao, W. G. Jang, J. E. Maclennan, O. Mondainn-Monval, C. Boyer, W. Weissflog, G. Petzel, L. C. Chien, J. Zasadzinski, J. Watanabe, D. M. Walba, H. Takezoe and N. A. Clark, Polarization-Modulated liquid crystal phases,, Science, 301* (2003), 1204. doi: 10.1126/science.1084956. Google Scholar*
[10]	P. G. de Gennes, An anology between superconductivity and smectic-A,, Solid State Commun., 10 (1972), 753. Google Scholar
[11]	P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, Clarendon Press, (1993). Google Scholar
[12]	R. De Vita and I. W. Stewart, Influence of alignnment of smectic A liquid crystals with surface pretilt,, J. Phys.: Condensed Matter, 20 (2008), 1. Google Scholar
[13]	G. B. Folland, Real Analysis,, John Wiley $&$ Sons, (1984). Google Scholar
[14]	V. Girault and P. Raviart, Finite Element Methods for Navier Stokes Equations,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61623-5. Google Scholar
[15]	A. E. Jacobs, G. Goldner and D. Mukamel, Modulated structures in tilt chiral smectic films,, Phys. Rev. A, 45 (1992), 5783. doi: 10.1103/PhysRevA.45.5783. Google Scholar
[16]	A. Jákli, C. Bailey and J. Harden, Thermotropic Liquid Crystals,, Springer, (2007). Google Scholar
[17]	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*
[18]	S. Kralj and T. Sluckin, Landau-de Gennes theory of the chevron structure in a smectic-A liquid crystal,, Phys. Rev. E, 50 (1994). doi: 10.1103/PhysRevE.50.2940. Google Scholar
[19]	S. T. Lagerwall, Ferroelectric and Antiferroelectric Liquid Crystals,, Wiley-VCH, (1999). Google Scholar
[20]	T. Lubensky and S. Renn, Twist-grain-boundary phases near the nematic-smectic-A-smectic-C point in liquid crystals,, Phys. Rev. E, 41 (1990). doi: 10.1103/PhysRevA.41.4392. Google Scholar
[21]	I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases,, Phys. Rev. E, 57 (1994), 574. doi: 10.1103/PhysRevE.57.574. Google Scholar
[22]	I. Muševič, R. Blinc and B. Žekš, The Physics of Ferroelectric and Antiferroelectric Liquid Crystals,, World-Scientific, (2000). Google Scholar
[23]	M. A. Osipov and S. A. Pikin, Dipolar and quadrapolar ordering in ferroelectric liquid crystals,, J. Phys. II France, 5 (1995), 1223. Google Scholar
[24]	J. Park and M. C. Calderer, Analysis of nonlocal electrostatic effects in chiral smectic c liquid crystals,, SIAM J. Appl. Math., 66 (2006), 2107. doi: 10.1137/050641120. Google Scholar
[25]	S. Renn, Multicritical behavior of abrikosov vortex lattices near the cholesteric - smectic A - smectic C* point,, Phys. Rev. A, 45 (1992). doi: 10.1103/PhysRevA.45.953. Google Scholar
[26]	A. Shalaginov, L. Hazelwood and T. Sluckin, Dynamics of chevron formation,, Phys. Rev. E, 58 (1998). Google Scholar
[27]	M. Slavinec, S. Kralj, S. Zumer and T. Sluckin, Surface depinning of smectic-A edge dislocations,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.031705. Google Scholar
[28]	N. Vaupotič and M. Čopič, Effect of spontaneous polarization and polar surface anchoring on the director and layer structure in surface stabilized liquid crystal cells,, Phys. Rev. E, 68 (2003). Google Scholar
[29]	N. Vaupotič, M. Čopič, E. Gorecka and D. Pociecha, Modulated structures in bent-core liquid crystals: Two faces of one phase,, Phys. Rev. Lett., 98 (2007). Google Scholar
[30]	N. Vaupotič, V. Grubelnik and M. Čopič, Influence of an external field on structure in surface-stabilized smectic-C chevron cells,, Phys. Rev. E, 62 (2000). Google Scholar
[31]	N. Vaupotič, S. Kralj, M. Čopič and T. Sluckin, Landau-de Gennes theory of the chevron structure in a smectic liquid crystal,, Phys. Rev. E, 54 (1996). Google Scholar

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References:

[1]	C. Bailey, E. C. Gartland and A. Jákli, Structure and stability of bent core liquid crystal fibers,, Phys. Rev. E, 75 (2007). doi: 10.1103/PhysRevE.75.031701. Google Scholar
[2]	P. Bauman, M. C. Calderer, C. Liu and D. Phillips, The phase transition between nematic and smectic A* liquid crystals,, Arch. Rational Mech. Anal., 165* (2002), 161. doi: 10.1007/s00205-002-0223-8. Google Scholar*
[3]	P. Bauman and D. Phillips, Stability of bent core liquid crystal fibers,, Molecular Crystals and Liquid Crystals Journal, 510* (2009), 1135. Google Scholar*
[4]	H. R. Brand, P. E. Cladis and H. Pleiner, Fluid biaxial banana smectics: Symmetry at work,, Liquid Crystal Today, 9 (1999), 1. Google Scholar
[5]	H. R. Brand, P. E. Claids and H. Pleiner, Macroscopic properties of smectic $C_G$ liquid crystals,, Eur. Phys. J. B, 6 (1998), 347. Google Scholar
[6]	M. Cagnon and G. Durand, Positional anchoring of smectic liquid crystals,, Phys. Rev. Lett., 70 (1993). doi: 10.1103/PhysRevLett.70.2742. Google Scholar
[7]	C. M. Chen and F. C. Mackintosh, Theory of modulated phases in lipid bilayers and liquid crystal films,, Phys. Rev. E, 53 (1996), 4933. doi: 10.1103/PhysRevE.53.4933. Google Scholar
[8]	J. Chen and T. Lubensky, Landau-Ginzburg mean-field theory for the nematic to smectic-C and nematic to smectic-A phase transitions,, Phys. Rev. A, 14 (1976). doi: 10.1103/PhysRevA.14.1202. Google Scholar
[9]	D. A. Coleman, J. Fernsler, N. Chattham, M. Nakata, Y. Takanishi, E. Köblova, D. R. Link, R. F. Shao, W. G. Jang, J. E. Maclennan, O. Mondainn-Monval, C. Boyer, W. Weissflog, G. Petzel, L. C. Chien, J. Zasadzinski, J. Watanabe, D. M. Walba, H. Takezoe and N. A. Clark, Polarization-Modulated liquid crystal phases,, Science, 301* (2003), 1204. doi: 10.1126/science.1084956. Google Scholar*
[10]	P. G. de Gennes, An anology between superconductivity and smectic-A,, Solid State Commun., 10 (1972), 753. Google Scholar
[11]	P. G. de Gennes and J. Prost, The Physics of Liquid Crystals,, Clarendon Press, (1993). Google Scholar
[12]	R. De Vita and I. W. Stewart, Influence of alignnment of smectic A liquid crystals with surface pretilt,, J. Phys.: Condensed Matter, 20 (2008), 1. Google Scholar
[13]	G. B. Folland, Real Analysis,, John Wiley $&$ Sons, (1984). Google Scholar
[14]	V. Girault and P. Raviart, Finite Element Methods for Navier Stokes Equations,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61623-5. Google Scholar
[15]	A. E. Jacobs, G. Goldner and D. Mukamel, Modulated structures in tilt chiral smectic films,, Phys. Rev. A, 45 (1992), 5783. doi: 10.1103/PhysRevA.45.5783. Google Scholar
[16]	A. Jákli, C. Bailey and J. Harden, Thermotropic Liquid Crystals,, Springer, (2007). Google Scholar
[17]	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*
[18]	S. Kralj and T. Sluckin, Landau-de Gennes theory of the chevron structure in a smectic-A liquid crystal,, Phys. Rev. E, 50 (1994). doi: 10.1103/PhysRevE.50.2940. Google Scholar
[19]	S. T. Lagerwall, Ferroelectric and Antiferroelectric Liquid Crystals,, Wiley-VCH, (1999). Google Scholar
[20]	T. Lubensky and S. Renn, Twist-grain-boundary phases near the nematic-smectic-A-smectic-C point in liquid crystals,, Phys. Rev. E, 41 (1990). doi: 10.1103/PhysRevA.41.4392. Google Scholar
[21]	I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases,, Phys. Rev. E, 57 (1994), 574. doi: 10.1103/PhysRevE.57.574. Google Scholar
[22]	I. Muševič, R. Blinc and B. Žekš, The Physics of Ferroelectric and Antiferroelectric Liquid Crystals,, World-Scientific, (2000). Google Scholar
[23]	M. A. Osipov and S. A. Pikin, Dipolar and quadrapolar ordering in ferroelectric liquid crystals,, J. Phys. II France, 5 (1995), 1223. Google Scholar
[24]	J. Park and M. C. Calderer, Analysis of nonlocal electrostatic effects in chiral smectic c liquid crystals,, SIAM J. Appl. Math., 66 (2006), 2107. doi: 10.1137/050641120. Google Scholar
[25]	S. Renn, Multicritical behavior of abrikosov vortex lattices near the cholesteric - smectic A - smectic C* point,, Phys. Rev. A, 45 (1992). doi: 10.1103/PhysRevA.45.953. Google Scholar
[26]	A. Shalaginov, L. Hazelwood and T. Sluckin, Dynamics of chevron formation,, Phys. Rev. E, 58 (1998). Google Scholar
[27]	M. Slavinec, S. Kralj, S. Zumer and T. Sluckin, Surface depinning of smectic-A edge dislocations,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.031705. Google Scholar
[28]	N. Vaupotič and M. Čopič, Effect of spontaneous polarization and polar surface anchoring on the director and layer structure in surface stabilized liquid crystal cells,, Phys. Rev. E, 68 (2003). Google Scholar
[29]	N. Vaupotič, M. Čopič, E. Gorecka and D. Pociecha, Modulated structures in bent-core liquid crystals: Two faces of one phase,, Phys. Rev. Lett., 98 (2007). Google Scholar
[30]	N. Vaupotič, V. Grubelnik and M. Čopič, Influence of an external field on structure in surface-stabilized smectic-C chevron cells,, Phys. Rev. E, 62 (2000). Google Scholar
[31]	N. Vaupotič, S. Kralj, M. Čopič and T. Sluckin, Landau-de Gennes theory of the chevron structure in a smectic liquid crystal,, Phys. Rev. E, 54 (1996). Google Scholar

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