Fréedericksz transition on a phenomenological model for a nematic inhomogeneous superfluid in presence of an electric field

Abstract

In this article we derive a Ginzburg–Landau energy functional for a nematic inhomogeneous superfluid in presence of an electric field. The molecules occupy an infinite cylinder Ω with cross section D. We suppose vacuum in R3∖Ω, with the possibility that an external electric field can be applied parallel to D. The Helmholtz free energy is obtained by taking the London limit of a Ginzburg–Landau nematic superconducting model in absence of magnetic fields, and by including an appropriate contribution of the electric potential energy. We show that the critical parameter inside Ω, which defines the Fréedericksz transition on the molecular alignment, is not only influenced by the effects of the electric field in the sample, but also by the additional contribution of the superfluid current. We take a particular solution for the Ginzburg–Landau equations, where the superfluid phase does not have circulation. Then, we demonstrate that the corresponding Fréedericksz threshold can be calculated, on an arbitrary domain, by using the notion of the first positive eigenvalue of the Laplacian. This eigenvalue depends on the chosen geometry and the boundary conditions on the nematic phase in the sample. Next, we apply our results in an infinite slab and in an infinite cylinder with circular cross section, where the nematic superfluid system is subjected to Dirichlet or Neumann boundary conditions in each case. We deduce a modified Fréedericksz threshold, for each configuration mentioned before, in a uniform electric field. In these instances we notice the remarkable fact that, for specific values and regimes of the intrinsic parameters, the critical fields are different than the ones obtained in the pure nematic case. Finally, we also study a Fréedericksz type threshold in a long hollow cylinder with uniform charge density, where molecules are reoriented by the electric field produced only by the internal charges of the sample. This setting suggests that, if molecules are oriented radially at the boundary of the region, a Fréedericksz type threshold appears in order to maintain the radial molecular distribution, which varies with the typical radii of the domain.