Ericksen's bar and modeling of the smectic A-nematic phase transition

We consider free energy functionals to model equilibrium smectic A liquid c rystal configurations in the neighborhood of the nematic phase transition. We begin with the functional proposed by de Gennes based on the Ginzburg La ndau model for superconductivity and consider its covariant formulations. E xploring qualitative analogies with the nonlinear elastic bar of Ericksen, we motivate a revision of the liquid crystal energy so as to include a nonc onvex constraint. We study boundary-value problems corresponding to Neumann and Dirichlet bou ndary conditions for smectic A liquid crystals conned between two parallel plates. We show that the nonconvex term of the free energy density causes t he presence in the solutions of nematic defects known to occur near the pha se transition from smectic A to nematic. The latter are reminiscent of the dislocations occurring in higher dimensional configurations. We also determ ine parameter values that give rise to nucleation of nematic defects for bo undary conditions consistent with externally imposed winding of the smectic phase field. The resulting energy also allows us to sort out liquid-like a nd solid-like behaviors, respectively.