Ericksen\u27s Bar and Modeling of the Smectic A-Nematic Phase Transition

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