Published Paper

Inserted: 11 oct 2016
Last Updated: 4 may 2017

Journal: J. Nonlinear Sci.
Year: 2017
Doi: 10.1007/s00332-017-9383-4
Notes:

Published online: 02 May 2017

Abstract:

In the Landau-de Gennes theory of liquid crystals, the propensities for alignments of molecules are represented at each point $x$ of the fluid by an element $Q(x)$ of the vector space $\mathcal S_0$ of $3\times 3$ real symmetric traceless matrices, or $Q$-tensors. % According to Longa and Trebin {Phys. Rev. A}, {39} (1989), 2160--2168, a biaxial nematic phase is called \textit{soft biaxial} if the tensor order parameter $Q$ satisfies the constraint $tr (Q^2) = \text{const}$. After the introduction of a $Q$-tensor model for soft biaxial nematic systems and the description of its geometric structure, we address the question of coercivity for the most common four-elastic-constant form of the Landau-de Gennes elastic free-energy in this model. For a soft biaxial nematic system, the tensor field $Q$ takes values in a four-dimensional sphere $S^4_\rho$ of radius $\rho\leq\sqrt{2/3}$ in the five-dimensional space $\mathcal S_0$ with inner product $\langle Q, \mathbf P \rangle = tr (Q \mathbf P)$. The symmetry group of the system is the rotation group $SO(3)$ which acts orthogonally on $\mathcal S_0$ by conjugation and hence induces an action on $S^4_\rho \subset \mathcal S_0$. This action has generic orbits of codimension one that are diffeomorphic to an eightfold quotient $S^3/\mathcal H$ of the unit 3-sphere $S^3$, where $\mathcal H$ is the quaternion group, and has two degenerate orbits of codimension two that are diffeomorphic to the projective plane $\mathbb RP^2$. Each generic orbit can be interpreted as the order parameter space of a constrained biaxial nematic system and each singular orbit as the order parameter space of a constrained uniaxial nematic system. It turns out that $S^4_\rho$ is a cohomogeneity one manifold, i.e., a manifold with a group action whose orbit space is one-dimensional. Another important geometric feature of the model is that the set $\Sigma_\rho$ of diagonal $Q$-tensors of fixed norm $\rho$ is a (geodesic) great circle in $S^4_\rho$ which meets every orbit of $S^4_\rho$ orthogonally and is then a \textit{section} for $S^4_\rho$ in the sense of the general theory of canonical forms. % We compute necessary and sufficient coercivity conditions for the elastic energy by exploiting the $SO(3)$-invariance of the elastic energy (frame-indifference), the existence of the section $\Sigma_\rho$ for $S_\rho$, and the geometry of the model, which allow us to reduce to a suitable invariant problem on (an arc of) $\Sigma_\rho$. Our approach can ultimately be seen as an application of the general method of reduction of variables, or cohomogeneity method.

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