The phase transition between chiral nematic and smectic A* liquid crystals

In this paper we study the Landau-de Gennes free energy used to describe the transition between chiral nematic and smectic A liquid crystal phases. We consider the phenomenology of the transition and discuss the behavior of the material constants. Within the present mathematical framework, the physically observed growth behavior of the twist and bend Frank constants, K₂ and K₃ respectively, plays a major role in determining the transition regime. We show existence of minimizers in a large class of admissible fields. Then, under the hypothesis that K₂ and K₃ are large, we establish estimates for the transition regime separating the two phases. The work emphasizes the interplay between two competing effects: the layer formation of the smectic A phase and the twist tendency of the chiral nematic phase. Our discussion also illustrates the analogies as well as the discrepancies in modeling and behavior between smectic A* liquid crystals and superconducting materials described by the Ginzburg-Landau theory.