Analysis of a Ginzburg–Landau type energy model for smectic C* liquid crystals with defects
Colbert-Kelly, Sean ; Phillips, Daniel
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 1009-1026 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Dans ce travail on étudie les propriétés dʼun film de cristaux liquides smectiques C* contenant des défauts qui produisent des motifs en spirale dans la texture du film. Les phénomènes sont décrits par un modèle de type Ginzburg–Landau dans un domaine borné du plan, et cet article fournit une analyse détaillée des configurations dʼénergie minimale du champ de directions du film. On montre lʼexistence dʼune configuration limite pour les défauts (tourbillons) qui minimise une énergie renormalisée. On démontre que si le degré du champ sur le bord du domaine est positif, alors les tourbillons sont dans lʼintérieur du domaine et sont chacun de degré +1. On prouve que quand le paramètre ε de Ginzburg–Landau tend vers zéro, pour une suite de minimiseurs, la limite de lʼénergie moins la somme des énergies autour des tourbillons est égale à lʼénergie renormalisée de lʼétat limite.

This work investigates properties of a smectic C* liquid crystal film containing defects that cause distinctive spiral patterns in the filmʼs texture. The phenomena are described by a Ginzburg–Landau type model and the investigation provides a detailed analysis of minimal energy configurations for the filmʼs director field. The study demonstrates the existence of a limiting location for the defects (vortices) so as to minimize a renormalized energy. It is shown that if the degree of the boundary data is positive then the vortices each have degree +1 and that they are located away from the boundary. It is proved that the limit of the energies for a sequence of minimizers minus the sum of the energies around their vortices, as the G–L parameter ε tends to zero, is equal to the renormalized energy for the limiting state.

@article{AIHPC_2013__30_6_1009_0,
     author = {Colbert-Kelly, Sean and Phillips, Daniel},
     title = {Analysis of a Ginzburg--Landau type energy model for smectic C* liquid crystals with defects},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {1009-1026},
     doi = {10.1016/j.anihpc.2012.12.010},
     mrnumber = {3132414},
     zbl = {1288.35443},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_6_1009_0}
}

Colbert-Kelly, Sean; Phillips, Daniel. Analysis of a Ginzburg–Landau type energy model for smectic C* liquid crystals with defects. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 1009-1026. doi : 10.1016/j.anihpc.2012.12.010. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_6_1009_0/

[1] F. Bethuel, H. Brezis, F. Hélein, Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and Their Applications vol. 13, Birkhäuser, Boston (1994) | MR 1269538 | Zbl 0802.35142

[2] F. Lin, Solutions of Ginzburg–Landau equations and critical points of the renormalized energy, Analyse Non Linéaire 12 no. 5 (1995), 599-622 | Numdam | MR 1353261 | Zbl 0845.35052

[3] M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg–Landau model in 2 dimensions, Differential and Integral Equations 7 no. 6 (1994), 1613-1624 | MR 1269674 | Zbl 0809.35031

[4] R. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, SIAM Journal on Mathematical Analysis 30 no. 4 (1999), 721-746 | MR 1684723 | Zbl 0928.35045

[5] E. Sandier, Lower bounds for the energy of unit vector fields and applications, Journal of Functional Analysis 152 no. 2 (1998), 379-403 | MR 1607928 | Zbl 0908.58004

[6] J.-B. Lee, D. Konovalov, R.B. Meyer, Textural transformations in islands on free standing smectic-C* liquid crystal films, Physical Review E 73 (2006), 051705

[7] B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press (2009) | MR 2480025

[8] S.T. Lagerwall, Ferroelectric and Antiferroelectric Liquid Crystals, Wiley-VCH, Weinheim (1999)

[9] R.B. Meyer, L. Liébert, L. Strzelecki, P. Keller, Ferroelectric liquid crystals, Le Journal de Physique Lettres 36 no. 3 (1975), L69-L71

[10] J.-B. Lee, R.A. Pelcovits, R.B. Meyer, Role of electrostatics in the texture of islands in free-standing ferroelectric liquid crystal films, Physical Review E 75 (2007), 051701

[11] I. Kraus, R.B. Meyer, Polar smectic films, Physical Review Letters 82 no. 19 (1999), 3815-3818 | MR 1670055

[12] R.B. Meyer, D. Konovalov, I. Kraus, J.-B. Lee, Equilibrium size and textures of islands in free-standing smectic C* films, Molecular Crystals and Liquid Crystals 364 no. 1 (2001), 123-131

[13] N. Silvestre, P. Patricio, M. Telo De Gama, A. Pattanaporkratans, C. Park, J. Maclennan, N. Clark, Modeling dipolar and quadrupolar defect structures generated by chiral islands in freely suspended liquid crystal films, Physical Review E 80 no. 4 (2009), 041708

[14] F. Lin, Vortex dynamics for the nonlinear wave equation, Communications on Pure and Applied Mathematics 52 (1999), 0737-0761 | MR 1676761 | Zbl 0929.35076

[15] P. Bauman, J. Park, D. Phillips, Analysis of nematic liquid crystals with disclination lines, Archive for Rational Mechanics and Analysis 205 no. 3 (2012), 795-826 | MR 2960033 | Zbl 1281.76020

[16] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, Princeton University Press, Princeton (1983) | MR 717034 | Zbl 0516.49003

[17] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg (2001) | MR 1814364 | Zbl 0691.35001

[18] F. Lin, Static and moving vortices in Ginzburg–Landau theories, Progress in Nonlinear Differential Equations and Their Applications 29 (1997), 71-111 | MR 1437152 | Zbl 0867.35039

[19] F. Lin, Some dynamical properties of Ginzburg–Landau vortices, Communications on Pure and Applied Mathematics 49 (1996), 323-359 | MR 1376654 | Zbl 0853.35058

[20] F. Lin, T.-C. Lin, Vortices in p-wave superconductivity, SIAM Journal on Mathematical Analysis 34 no. 5 (2003), 1105-1127 | MR 2001661 | Zbl 1126.82343

[21] L.C. Evans, Weak convergence methods for nonlinear partial differential equations, A.M. Society (ed.), Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics vol. 74 (1990) | MR 1034481

[22] R. Hardt, D. Kinderlehrer, F.H. Lin, The variety of configurations of static liquid crystals, Variational Methods, Paris, 1988, Progress in Nonlinear Differential Equations and Their Applications vol. 4, Birkhäuser Boston, Boston, MA (1990), 115-131

[23] S. Colbert-Kelly, Theoretical and computational analysis of a Ginzburg–Landau type energy model for smectic C* liquid crystals, PhD thesis, Purdue University, 2012.