Orientable and non-orientable line field models for uniaxial nematic liquid crystals

Uniaxial nematic liquid crystals are often modeled using the Oseen-Frank theory, in which the mean orientation of the rod-like molecules is described through a unit vector field n. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which n should be equivalent to-n; that is, instead of n taking values in the unit sphere S2, it should take values in the sphere with opposite points identified, i.e., in the real projective plane RP2. The Landau-de Gennes theory respects this symmetry by working with the tensor Q=s (n⊗n-1/3Id). In the case of a non-zero constant scalar order parameter s the Landau-de Gennes theory is equivalent to that of Oseen-Frank when the director field is orientable. We report on a general study of when the director fields can be oriented, described in terms of the topology of the domain filled by the liquid crystal, the boundary data and the rate of blow-up of possible singularities. We also analyze the circumstances in which the non-orientable configurations are energetically favoured over the orientable ones.