On the Landau–de Gennes Elastic Energy of a Q-Tensor Model for Soft Biaxial Nematics

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.