Orientable and non-orientable director fields for liquid crystals

Uniaxial nematic liquid crystals are often modelled using the Oseen-Frank theory, in which the mean orientation of the rodlike 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 − ⅓Id). 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 Sobolev space to which Q belongs. We also analyze the circumstances in which the non-orientable configurations are energetically favoured over the orientable ones. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)