Bulk and surface biaxiality in nematic liquid crystals

TL;DR

This paper demonstrates that spatial variations in the director field of nematic liquid crystals naturally induce biaxiality, especially near defects or curved surfaces, with the eigenvalues of a specific tensor serving as a measure.

Contribution

It provides a theoretical framework linking director gradients to biaxiality, highlighting how surface curvature influences the order tensor's symmetry breaking.

Findings

Biaxiality arises when the tensor =( abla n)( abla n)^T has two distinct nonzero eigenvalues.

Eigenvalue differences serve as a measure of biaxiality.

Biaxial order is induced along principal directions on curved surfaces with homeotropic anchoring.

Abstract

Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. In this paper we show that any spatially-varying director pattern may be a source of biaxiality. We prove that biaxiality arises naturally whenever the symmetric tensor $\Sb=(\grad \nn)(\grad \nn)^T$ possesses two distinct nonzero eigenvalues. The eigenvalue difference may be used as a measure of the expected biaxiality. Furthermore, the corresponding eigenvectors indicate the directions in which the order tensor \QQ is induced to break the uniaxial symmetry about the director \nn. We apply our general considerations to some examples. In particular we show that, when we enforce homeotropic anchoring on a curved surface, the order tensor become biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.