Orientable and non-orientable director fields for liquid crystals

John Ball

Oxford University
ABSTRACT:

Uniaxial nematic liquid crystals are often modelled using the
Oseen-Frank theory, in which the mean orientation of the rod-like
molecules is modelled 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 S^2, it should take values in the
sphere with opposite points identified, i.e. in the real projective
plane RP^2. The de Gennes theory respects this symmetry by working with
the tensor Q=s(n otimes n- 1/3I). In the case of a non-zero constant
scalar order parameter, s, the 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
crystals, 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.

This is joint work with Arghir Zarnescu.