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- Title
On the Landau-de Gennes Elastic Energy of a Q-Tensor Model for Soft Biaxial Nematics.
- Authors
Mucci, Domenico; Nicolodi, Lorenzo
- Abstract
In the Landau-de Gennes theory of liquid crystals, the propensities for alignments of molecules are represented at each point of the fluid by an element $$\mathbf{Q}$$ of the vector space $${\mathcal {S}}_0$$ of $$3\times 3$$ real symmetric traceless matrices, or $$\mathbf{Q}$$ -tensors. According to Longa and Trebin (1989), a biaxial nematic system is called soft biaxial if the tensor order parameter $$\mathbf{Q}$$ satisfies the constraint $$\mathrm{tr}(\mathbf{Q}^2) = \text {const}$$ . After the introduction of a $$\mathbf{Q}$$ -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 (Iyer et al. 2015) in this model. For a soft biaxial nematic system, the tensor field $$\mathbf{Q}$$ takes values in a four-dimensional sphere $${{\mathbb {S}}}^4_\rho $$ of radius $$\rho \le \sqrt{2/3}$$ in the five-dimensional space $${\mathcal {S}}_0$$ with inner product $$\langle \mathbf{Q}, {\mathbf {P}} \rangle = \mathrm{tr}(\mathbf{Q}{\mathbf {P}})$$ . The rotation group $$\textit{SO}(3)$$ acts orthogonally on $${\mathcal {S}}_0$$ by conjugation and hence induces an action on $${{\mathbb {S}}}^4_\rho \subset {\mathcal {S}}_0$$ . This action has generic orbits of codimension one that are diffeomorphic to an eightfold quotient $${{\mathbb {S}}}^3/{\mathcal {H}}$$ of the unit three-sphere $${{\mathbb {S}}}^3$$ , where $${{\mathcal {H}}}=\{\pm 1, \pm \mathsf{i}, \pm \mathsf{j}, \pm \mathsf{k}\}$$ is the quaternion group, and has two degenerate orbits of codimension two that are diffeomorphic to the projective plane $${\mathbb {R}}P^2$$ . 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 $${{\mathbb {S}}}^4_\rho $$ 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 $$\mathbf{Q}$$ -tensors of fixed norm $$\rho $$ is a (geodesic) great circle in $${{\mathbb {S}}}^4_\rho $$ which meets every orbit of $${{\mathbb {S}}}^4_\rho $$ orthogonally and is then a section for $${{\mathbb {S}}}^4_\rho $$ in the sense of the general theory of canonical forms. We compute necessary and sufficient coercivity conditions for the elastic energy by exploiting the $$\textit{SO}(3)$$ -invariance of the elastic energy (frame-indifference), the existence of the section $$\Sigma _\rho $$ for $${{\mathbb {S}}}^4_\rho $$ , 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.
- Subjects
TENSOR algebra; MANIFOLDS (Mathematics); QUANTUM mechanics; FREE energy (Thermodynamics); EQUATIONS of motion; MATHEMATICAL symmetry
- Publication
Journal of Nonlinear Science, 2017, Vol 27, Issue 6, p1687
- ISSN
0938-8974
- Publication type
Article
- DOI
10.1007/s00332-017-9383-4