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.
Bibliographical noteKAUST Repository Item: Exported on 2021-09-16
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: A. Majumdar was supported by a Royal Commission for the Exhibition of 1851 Research Fellowship till October 2008. This publication is now based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) to the Oxford Centre for Collaborative Applied Mathematics. The author gratefully acknowledges helpful comments and suggestions made by John Ball, Giovanni De Matteis, Geoffrey Luckhurst, Carlos Mora-Corral, Tim Sluckin and the three referees.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
ASJC Scopus subject areas
- Applied Mathematics