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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

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 and 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), 031701. doi: 10.1103/PhysRevE.75.031701.

[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), 161-186. doi: 10.1007/s00205-002-0223-8.

[3]

P. Bauman and D. Phillips, Stability of bent core liquid crystal fibers, Molecular Crystals and Liquid Crystals Journal, 510 (2009), 1135-1145.

[4]

H. R. Brand, P. E. Cladis and H. Pleiner, Fluid biaxial banana smectics: Symmetry at work, Liquid Crystal Today, 9 (1999), 1-10.

[5]

H. R. Brand, P. E. Claids and H. Pleiner, Macroscopic properties of smectic $C_G$ liquid crystals, Eur. Phys. J. B, 6 (1998), 347-353.

[6]

M. Cagnon and G. Durand, Positional anchoring of smectic liquid crystals, Phys. Rev. Lett., 70 (1993), p. 2742. doi: 10.1103/PhysRevLett.70.2742.

[7]

C. M. Chen and F. C. Mackintosh, Theory of modulated phases in lipid bilayers and liquid crystal films, Phys. Rev. E, 53 (1996), 4933-4943. doi: 10.1103/PhysRevE.53.4933.

[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), p. 1202. doi: 10.1103/PhysRevA.14.1202.

[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), 1204-1211. doi: 10.1126/science.1084956.

[10]

P. G. de Gennes, An anology between superconductivity and smectic-A, Solid State Commun., 10 (1972), 753-756.

[11]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993.

[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), 1-9.

[13]

G. B. Folland, Real Analysis, John Wiley & Sons, Inc., 1984.

[14]

V. Girault and P. Raviart, Finite Element Methods for Navier Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[15]

A. E. Jacobs, G. Goldner and D. Mukamel, Modulated structures in tilt chiral smectic films, Phys. Rev. A, 45 (1992), 5783-5788. doi: 10.1103/PhysRevA.45.5783.

[16]

A. Jákli, C. Bailey and J. Harden, Thermotropic Liquid Crystals, Springer, 2007.

[17]

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.

[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), p. 2940. doi: 10.1103/PhysRevE.50.2940.

[19]

S. T. Lagerwall, Ferroelectric and Antiferroelectric Liquid Crystals, Wiley-VCH, 1999.

[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), p. 4392. doi: 10.1103/PhysRevA.41.4392.

[21]

I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases, Phys. Rev. E, 57 (1994), 574-581. doi: 10.1103/PhysRevE.57.574.

[22]

I. Muševič, R. Blinc and B. Žekš, The Physics of Ferroelectric and Antiferroelectric Liquid Crystals, World-Scientific, Singapore, New Jersey, London, Hong Kong, 2000.

[23]

M. A. Osipov and S. A. Pikin, Dipolar and quadrapolar ordering in ferroelectric liquid crystals, J. Phys. II France, 5 (1995), 1223-1240.

[24]

J. Park and M. C. Calderer, Analysis of nonlocal electrostatic effects in chiral smectic c liquid crystals, SIAM J. Appl. Math., 66 (2006), 2107-2126. doi: 10.1137/050641120.

[25]

S. Renn, Multicritical behavior of abrikosov vortex lattices near the cholesteric - smectic A - smectic C* point, Phys. Rev. A, 45 (1992), p. 953. doi: 10.1103/PhysRevA.45.953.

[26]

A. Shalaginov, L. Hazelwood and T. Sluckin, Dynamics of chevron formation, Phys. Rev. E, 58 (1998), p. 7455.

[27]

M. Slavinec, S. Kralj, S. Zumer and T. Sluckin, Surface depinning of smectic-A edge dislocations, Phys. Rev. E, 63 (2001), p. 031705. doi: 10.1103/PhysRevE.63.031705.

[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), p. 061705.

[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), p. 247802.

[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), p. 2317.

[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), p. 3783.

show all references

References:
[1]

C. Bailey, E. C. Gartland and A. Jákli, Structure and stability of bent core liquid crystal fibers, Phys. Rev. E, 75 (2007), 031701. doi: 10.1103/PhysRevE.75.031701.

[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), 161-186. doi: 10.1007/s00205-002-0223-8.

[3]

P. Bauman and D. Phillips, Stability of bent core liquid crystal fibers, Molecular Crystals and Liquid Crystals Journal, 510 (2009), 1135-1145.

[4]

H. R. Brand, P. E. Cladis and H. Pleiner, Fluid biaxial banana smectics: Symmetry at work, Liquid Crystal Today, 9 (1999), 1-10.

[5]

H. R. Brand, P. E. Claids and H. Pleiner, Macroscopic properties of smectic $C_G$ liquid crystals, Eur. Phys. J. B, 6 (1998), 347-353.

[6]

M. Cagnon and G. Durand, Positional anchoring of smectic liquid crystals, Phys. Rev. Lett., 70 (1993), p. 2742. doi: 10.1103/PhysRevLett.70.2742.

[7]

C. M. Chen and F. C. Mackintosh, Theory of modulated phases in lipid bilayers and liquid crystal films, Phys. Rev. E, 53 (1996), 4933-4943. doi: 10.1103/PhysRevE.53.4933.

[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), p. 1202. doi: 10.1103/PhysRevA.14.1202.

[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), 1204-1211. doi: 10.1126/science.1084956.

[10]

P. G. de Gennes, An anology between superconductivity and smectic-A, Solid State Commun., 10 (1972), 753-756.

[11]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993.

[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), 1-9.

[13]

G. B. Folland, Real Analysis, John Wiley & Sons, Inc., 1984.

[14]

V. Girault and P. Raviart, Finite Element Methods for Navier Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[15]

A. E. Jacobs, G. Goldner and D. Mukamel, Modulated structures in tilt chiral smectic films, Phys. Rev. A, 45 (1992), 5783-5788. doi: 10.1103/PhysRevA.45.5783.

[16]

A. Jákli, C. Bailey and J. Harden, Thermotropic Liquid Crystals, Springer, 2007.

[17]

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.

[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), p. 2940. doi: 10.1103/PhysRevE.50.2940.

[19]

S. T. Lagerwall, Ferroelectric and Antiferroelectric Liquid Crystals, Wiley-VCH, 1999.

[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), p. 4392. doi: 10.1103/PhysRevA.41.4392.

[21]

I. Luk'yanchuk, Phase transition between the cholesteric and twist grain boundary C phases, Phys. Rev. E, 57 (1994), 574-581. doi: 10.1103/PhysRevE.57.574.

[22]

I. Muševič, R. Blinc and B. Žekš, The Physics of Ferroelectric and Antiferroelectric Liquid Crystals, World-Scientific, Singapore, New Jersey, London, Hong Kong, 2000.

[23]

M. A. Osipov and S. A. Pikin, Dipolar and quadrapolar ordering in ferroelectric liquid crystals, J. Phys. II France, 5 (1995), 1223-1240.

[24]

J. Park and M. C. Calderer, Analysis of nonlocal electrostatic effects in chiral smectic c liquid crystals, SIAM J. Appl. Math., 66 (2006), 2107-2126. doi: 10.1137/050641120.

[25]

S. Renn, Multicritical behavior of abrikosov vortex lattices near the cholesteric - smectic A - smectic C* point, Phys. Rev. A, 45 (1992), p. 953. doi: 10.1103/PhysRevA.45.953.

[26]

A. Shalaginov, L. Hazelwood and T. Sluckin, Dynamics of chevron formation, Phys. Rev. E, 58 (1998), p. 7455.

[27]

M. Slavinec, S. Kralj, S. Zumer and T. Sluckin, Surface depinning of smectic-A edge dislocations, Phys. Rev. E, 63 (2001), p. 031705. doi: 10.1103/PhysRevE.63.031705.

[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), p. 061705.

[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), p. 247802.

[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), p. 2317.

[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), p. 3783.

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