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) no. 6, pp. 1009-1026.

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.

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