Reduced Landau–de Gennes functional and surface smectic state of liquid crystals

In [X.B. Pan, Landau–de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1–2) (2003) 343–382], based on the de Gennes analogy between liquid crystals and superconductivity [P.G. de Gennes, An analogy between superconductors and smectics A, Solid State Commun. 10 (1972) 753–756], the second author introduced the critical wave number Qc3 (which is an analog of the upper critical field Hc3 for superconductors) and predicted the existence of a surface smectic state, which was supposed to be an analogy of the surface superconducting state. In a surface smectic state, the bulk liquid crystal is in the nematic state, and a thin layer of smectic appears in a helical strip on the surface of the sample. In this paper we study an approximate form of the Landau–de Gennes model of liquid crystals, and examine the behavior of minimizers, in particular the boundary layer behavior. Our work shows the importance of the joint chirality constant qτ, which is the product of wave number q and chirality τ and also appears in the work of [P. Bauman, M. Calderer, C. Liu, D. Phillips, The phase transition between chiral nematic and smectic A ∗ liquid crystals, Arch. Rational Mech. Anal. 165 (2002) 161–186] and [X.B. Pan, Landau–de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1–2) (2003) 343–382]. The joint chirality constant of a liquid crystal is useful to predict whether the liquid crystal is of type I or type II, and it is also useful to examine whether the liquid crystal is in a surface smectic state. The results in this paper suggest that a liquid crystal with large Ginzburg–Landau parameter κ and large joint chirality constant qτ exhibits type II behavior, and it will be in the surface smectic state if qτ ∼ bκ2 for some β0