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 Q(C3) (which is an analog of the upper critical field H-C3 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 q tau, which is the product of wave number q and chirality tau 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 11, 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 K and large joint chirality constant q tau exhibits type 11 behavior, and it will be in the surface smectic state if q tau similar to bk(2) for some beta(0) < b < 1, where beta(0) is the lowest eigenvalue of the Schrodinger operator with a unit magnetic field in the half space, and 0 < beta(0) < 1. We also show that a liquid crystal with small q tau exhibits type I behavior. (C) 2008 Elsevier Inc. All rights reserved.