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

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 (Formula presented.) of the vector space (Formula presented.) of (Formula presented.) real symmetric traceless matrices, or (Formula presented.)-tensors. According to Longa and Trebin (1989), a biaxial nematic system is called soft biaxial if the tensor order parameter (Formula presented.) satisfies the constraint (Formula presented.). After the introduction of a (Formula presented.)-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 (Formula presented.) takes values in a four-dimensional sphere (Formula presented.) of radius (Formula presented.) in the five-dimensional space (Formula presented.) with inner product (Formula presented.). The rotation group (Formula presented.) acts orthogonally on (Formula presented.) by conjugation and hence induces an action on (Formula presented.). This action has generic orbits of codimension one that are diffeomorphic to an eightfold quotient (Formula presented.) of the unit three-sphere (Formula presented.), where (Formula presented.) is the quaternion group, and has two degenerate orbits of codimension two that are diffeomorphic to the projective plane (Formula presented.). 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 (Formula presented.) 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 (Formula presented.) of diagonal (Formula presented.)-tensors of fixed norm (Formula presented.) is a (geodesic) great circle in (Formula presented.) which meets every orbit of (Formula presented.) orthogonally and is then a section for (Formula presented.) in the sense of the general theory of canonical forms. We compute necessary and sufficient coercivity conditions for the elastic energy by exploiting the (Formula presented.)-invariance of the elastic energy (frame-indifference), the existence of the section (Formula presented.) for (Formula presented.), and the geometry of the model, which allow us to reduce to a suitable invariant problem on (an arc of) (Formula presented.). Our approach can ultimately be seen as an application of the general method of reduction of variables, or cohomogeneity method.