Using two order tensors, we propose a mean-field model to describe the uniaxial and biaxial phases of nematogenic molecules presenting a shape dispersion of their biaxial dielectric susceptibility. We recover the classical isotropic-uniaxial-biaxial sequence of phases. The phase diagram exhibits a tricritical point, a feature that cannot be retraced in the other mean-field models established for molecules without shape dispersion.

We study the biaxial structure of both line and point defects in a nematic liquid crystal confined within a capillary tube whose lateral boundary enforces homeotropic anchoring. According to Landau–de Gennes theory the local order in the material is described by a second-order tensor $\mathbf{Q},$ which encompasses both uniaxial and biaxial states. Our study is both analytical and numerical. We show that the core of a line defect with topological charge $M=1\mathrm{}$ is uniaxial in the axial direction. At the lateral boundary, the uniaxial ordering along the radial direction is reached in two qualitatively different ways, depending on the sign of the order parameter on the axis. The point defects with charge $M=\pm 1$ exhibit a uniaxial ring in the plane orthogonal to the cylinder axis. This ring is in turn surrounded by a torus on which the degree of biaxiality attains its maximum. The typical lengths that characterize the structure of these defects depend both on the cylinder radius and the biaxial correlation length. It seems that the core of the point defect does not depend on the far nematic director field in the bulk limit.

Within a two-dimensional Landau–de Gennes tensorial formalism we study the equilibrium configurations of nematic shells, thin films of nematic liquid crystal deposited on the boundary of colloidal particles of arbitrary shape enforcing a degenerate tangential anchoring on the nematic molecules. In this formalism, defects appear wherever a scalar order parameter vanishes. Their total number is the colloidal valence, as this is the number of molecular bridges that can bind every colloid to its peers. We show how the defect organization on a colloidal particle is affected by the Gaussian curvature of the colloidal boundary, to the point of changing its valence.