a1 Mathematical Institute, University of Oxford, 24–29 St.Giles, Oxford, OX1 3LB, UK email: firstname.lastname@example.org
We study equilibrium liquid crystal configurations in three-dimensional geometries, within the continuum Landau-de Gennes theory. We obtain explicit bounds for the equilibrium scalar order parameters in terms of the temperature and material-dependent constants. We explicitly quantify the temperature regimes where the Landau-de Gennes predictions match and the temperature regimes where the Landau-de Gennes predictions do not match the probabilistic second-moment definition of the Q-tensor order parameter. The regime of agreement may be interpreted as the regime of validity of the Landau-de Gennes theory since the Landau-de Gennes theory predicts large values of the equilibrium scalar order parameters – larger than unity, in the low-temperature regime. We discuss a modified Landau-de Gennes energy functional which yields physically realistic values of the equilibrium scalar order parameters in all temperature regimes.
(Received May 28 2009)
(Revised December 18 2009)
(Accepted December 21 2009)