# American Institute of Mathematical Sciences

September  2015, 20(7): 1983-2000. doi: 10.3934/dcdsb.2015.20.1983

## Reorientation of smectic a liquid crystals by magnetic fields

 1 Mathematics Department, University of California, Santa Barbara, CA 93106, United States 2 Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, United States

Received  December 2013 Revised  March 2015 Published  July 2015

We consider the de Gennes' smectic A free energy with a complex order parameter in order to study the influence of magnetic fields on the smectic layers in the strong field limit as well as near the critical field. In previous work by the authors [6], the critical field and a description of the layer undulations at the instability were obtained using $\Gamma$-convergence and bifurcation theory. It was proved that the critical field is lowered by a factor of $\sqrt{\pi}$ compared to the classical Helfrich Hurault theory by using natural boundary conditions for the complex order parameter, but still with strong anchoring condition for the director. In this paper, we present numerical simulations for undulations at the critical field as well as the layer and director configurations well above the critical field. We show that the estimate of the critical field and layer configuration at the critical field agree with the analysis in [6]. Furthermore, the changes in smectic order density as well as layer and director will be illustrated numerically as the field increases well above the critical field. This provides the smectic layers' melting along the bounding plates where the layers are fixed. In the natural case, at a high field, we prove that the directors align with the applied field and the layers are homeotropically aligned in the domain, keeping the smectic order density at a constant in $L^2$.
Citation: 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
##### References:
 [1] V. G. Čigrinov, Electrooptic Effects in Liquid Crystal Materials,, Springer, (1996).   Google Scholar [2] P. G. de Gennes, An analogy between superconductors and smectics A,, Solid State Communications, 10 (1972), 753.   Google Scholar [3] P. G. de Gennes, The Physics of Liquid Crystals,, International Series of Monographs on Physics, (1974).   Google Scholar [4] W. E and X. P. Wang, Numerical methods for the Landau-Lifshitz equation,, SIAM J. Numer. Anal., 38 (2000), 1647.  doi: 10.1137/S0036142999352199.  Google Scholar [5] M. Frigo and S. G. Johnson, The design and implementation of FFTW3,, Proceedings of the IEEE, 93 (2005), 216.  doi: 10.1109/JPROC.2004.840301.  Google Scholar [6] C. J. García-Cervera and S. Joo, Analytic description of layer undulations in smectic $A$ liquid crystals,, Arch. Ration. Mech. Anal., 203 (2012), 1.  doi: 10.1007/s00205-011-0442-y.  Google Scholar [7] C. J. García-Cervera and S. Joo, Analysis and simulations of the Chen-Lubensky energy for smectic liquid crystals: Onset of undulations,, Commun. Math. Sci., 12 (2014), 1155.  doi: 10.4310/CMS.2014.v12.n6.a7.  Google Scholar [8] T. Giorgi, C. J. García-Cervera and S. Joo, Sawtooth profile in smectic $A$ liquid crystals,, submitted., ().   Google Scholar [9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monographs in Mathematics, (1984).  doi: 10.1007/978-1-4684-9486-0.  Google Scholar [10] W. Helfrich, Electrohydrodynamic and dielectric instabilities of cholesteric liquid crystals,, The Journal of Chemical Physics, 55 (1971), 839.  doi: 10.1063/1.1676151.  Google Scholar [11] J. P. Hurault, Static distortions of a cholesteric planar structure induced by magnetic or ac electric fields,, The Journal of Chemical Physics, 59 (1973), 2068.  doi: 10.1063/1.1680293.  Google Scholar [12] T. Ishikawa and O. D. Lavrentovich, Undulations in a confined lamellar system with surface anchoring,, Phys. Rev. E, 63 (2001).  doi: 10.1103/PhysRevE.63.030501.  Google Scholar [13] T. Ishikawa and O. D. Lavrentovich, Defects and undulation in layered liquid crystals,, in Defects in Liquid Crystals: Computer Simulations, (2001), 271.  doi: 10.1007/978-94-010-0512-8_11.  Google Scholar [14] T. Ishikawa and O. Lavrentovich, Dislocation profile in cholesteric finger texture,, Physical Review E, 60 (1999).  doi: 10.1103/PhysRevE.60.R5037.  Google Scholar [15] O. Lavrentovich, M. Kleman and V. M. Pergamenshchik, Nucleation of focal conic domains in smectic a liquid crystals,, Journal de Physique II, 4 (1994), 377.  doi: 10.1051/jp2:1994135.  Google Scholar [16] F. Lin and X. B. Pan, Magnetic field-induced instabilities in liquid crystals,, SIAM J. Math. Anal., 38 (): 1588.  doi: 10.1137/050638643.  Google Scholar [17] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital., 14 (1977), 285.   Google Scholar [18] L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rational Mech. Anal., 98 (1987), 123.  doi: 10.1007/BF00251230.  Google Scholar [19] B. I. Senyuk, I. I. Smalyukh and O. D. Lavrentovich, Undulations of lamellar liquid crystals in cells with finite surface anchoring near and well above the threshold,, Phys. Rev. E, 74 (2006).  doi: 10.1103/PhysRevE.74.011712.  Google Scholar [20] M. Struwe, Heat-flow methods for harmonic maps of surfaces and applications to free boundary problems,, in Partial Differential Equations (Rio de Janeiro, (1986), 293.  doi: 10.1007/BFb0100801.  Google Scholar

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##### References:
 [1] V. G. Čigrinov, Electrooptic Effects in Liquid Crystal Materials,, Springer, (1996).   Google Scholar [2] P. G. de Gennes, An analogy between superconductors and smectics A,, Solid State Communications, 10 (1972), 753.   Google Scholar [3] P. G. de Gennes, The Physics of Liquid Crystals,, International Series of Monographs on Physics, (1974).   Google Scholar [4] W. E and X. P. Wang, Numerical methods for the Landau-Lifshitz equation,, SIAM J. Numer. Anal., 38 (2000), 1647.  doi: 10.1137/S0036142999352199.  Google Scholar [5] M. Frigo and S. G. Johnson, The design and implementation of FFTW3,, Proceedings of the IEEE, 93 (2005), 216.  doi: 10.1109/JPROC.2004.840301.  Google Scholar [6] C. J. García-Cervera and S. Joo, Analytic description of layer undulations in smectic $A$ liquid crystals,, Arch. Ration. Mech. Anal., 203 (2012), 1.  doi: 10.1007/s00205-011-0442-y.  Google Scholar [7] C. J. García-Cervera and S. Joo, Analysis and simulations of the Chen-Lubensky energy for smectic liquid crystals: Onset of undulations,, Commun. Math. Sci., 12 (2014), 1155.  doi: 10.4310/CMS.2014.v12.n6.a7.  Google Scholar [8] T. Giorgi, C. J. García-Cervera and S. Joo, Sawtooth profile in smectic $A$ liquid crystals,, submitted., ().   Google Scholar [9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monographs in Mathematics, (1984).  doi: 10.1007/978-1-4684-9486-0.  Google Scholar [10] W. Helfrich, Electrohydrodynamic and dielectric instabilities of cholesteric liquid crystals,, The Journal of Chemical Physics, 55 (1971), 839.  doi: 10.1063/1.1676151.  Google Scholar [11] J. P. Hurault, Static distortions of a cholesteric planar structure induced by magnetic or ac electric fields,, The Journal of Chemical Physics, 59 (1973), 2068.  doi: 10.1063/1.1680293.  Google Scholar [12] T. Ishikawa and O. D. Lavrentovich, Undulations in a confined lamellar system with surface anchoring,, Phys. Rev. E, 63 (2001).  doi: 10.1103/PhysRevE.63.030501.  Google Scholar [13] T. Ishikawa and O. D. Lavrentovich, Defects and undulation in layered liquid crystals,, in Defects in Liquid Crystals: Computer Simulations, (2001), 271.  doi: 10.1007/978-94-010-0512-8_11.  Google Scholar [14] T. Ishikawa and O. Lavrentovich, Dislocation profile in cholesteric finger texture,, Physical Review E, 60 (1999).  doi: 10.1103/PhysRevE.60.R5037.  Google Scholar [15] O. Lavrentovich, M. Kleman and V. M. Pergamenshchik, Nucleation of focal conic domains in smectic a liquid crystals,, Journal de Physique II, 4 (1994), 377.  doi: 10.1051/jp2:1994135.  Google Scholar [16] F. Lin and X. B. Pan, Magnetic field-induced instabilities in liquid crystals,, SIAM J. Math. Anal., 38 (): 1588.  doi: 10.1137/050638643.  Google Scholar [17] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza,, Boll. Un. Mat. Ital., 14 (1977), 285.   Google Scholar [18] L. Modica, The gradient theory of phase transitions and the minimal interface criterion,, Arch. Rational Mech. Anal., 98 (1987), 123.  doi: 10.1007/BF00251230.  Google Scholar [19] B. I. Senyuk, I. I. Smalyukh and O. D. Lavrentovich, Undulations of lamellar liquid crystals in cells with finite surface anchoring near and well above the threshold,, Phys. Rev. E, 74 (2006).  doi: 10.1103/PhysRevE.74.011712.  Google Scholar [20] M. Struwe, Heat-flow methods for harmonic maps of surfaces and applications to free boundary problems,, in Partial Differential Equations (Rio de Janeiro, (1986), 293.  doi: 10.1007/BFb0100801.  Google Scholar
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