Singular limits in polymer-stabilized liquid crystals

Abstract

We investigate equilibrium configurations for a polymer-stabilized liquid-crystal material subject to an applied magnetic field. The configurations are determined by energy minimization, where the energies of the system include those of bulk, surface and external field. The Euler–Lagrange equation is a nonlinear partial differential equation with nonlinear boundary conditions defined on a perforated domain modelling the cross-section of the liquid-crystal–polymer-fibre composite. We analyse the critical values for the external magnetic field representing Fredericks transitions and describe the equilibrium configurations under any magnitude of the external field. We also discuss the limit of the critical values and configurations as the number of polymer fibres approaches infinity. In the case where, away from the boundary of the composite, the fibres are part of a periodic array, we prove that non-constant configurations develop order-one oscillations on the scale of the array's period. Furthermore, we determine the small-scale structure of the configurations as the period tends to zero.