Calculus of Variations and Geometric Measure Theory

D. Mucci - L. Nicolodi

On the elastic energy density of constrained $Q$-tensor models for biaxial nematics

created by mucci on 26 Oct 2011
modified on 06 Nov 2012


Published Paper

Inserted: 26 oct 2011
Last Updated: 6 nov 2012

Journal: Arch. Rational Mech. Anal.
Volume: 206
Pages: 853--884
Year: 2012


Within the Landau--de~Gennes theory, the order parameter describing a biaxial nematic liquid crystal assigns a symmetric traceless $3\times 3$ matrix $Q$ with three distinct eigenvalues to every point of the region $\Omega$ occupied by the system. In the constrained case of matrices $Q$ with constant eigenvalues, the order parameter space is diffeomorphic to the eightfold quotient $S^3/\mathcal H$ of the 3-sphere $S^3$, where $\mathcal H$ is the quaternion group, and a configuration of a biaxial nematic liquid crystal is described by a map from $\Omega$ to $S^3/\mathcal H$. We express the (simplest form of the) Landau--de~Gennes elastic free-energy density as a density defined on maps $q: \Omega\to S^3$, whose functional dependence is restricted by the requirements that (1) it is well defined on the class of configuration maps from $\Omega $ to $S^3/\mathcal H$ (residual symmetry) and (2) it is independent of arbitrary superposed rigid rotations (frame indifference). As an application of this representation, we then discuss some properties of the corresponding energy functional, including coercivity, lower semicontinuity and strong density of smooth maps. Other invariance properties are also considered. In the discussion, we take advantage of the identification of $S^3$ with the Lie group of unit quaternions $Sp(1) \cong SU(2)$ and of the relations between quaternions and rotations in $\mathbb R^3$ and $\mathbb R^4$.