Kinematics and dynamics of disclination lines in three-dimensional nematics

TL;DR

This paper derives an exact kinematic law for disclination line motion in 3D nematic liquid crystals using the tensor order parameter, providing analytical results validated by numerical simulations.

Contribution

It introduces a novel exact expression relating disclination velocity to the tensor order parameter derivatives, enabling precise analysis of defect dynamics in nematics.

Findings

Analytical solutions for line interactions and motion

Agreement between theory and 3D numerical simulations

Insights into defect behavior under external fields and shear flows

Abstract

An exact kinematic law for the motion of disclination lines in nematic liquid crystals as a function of the tensor order parameter $\mathbf{Q}$ is derived. Unlike other order parameter fields that become singular at their respective defect cores, the tensor order parameter remains regular. Following earlier experimental and theoretical work, the disclination core is defined to be the line where the uniaxial and biaxial order parameters are equal, or equivalently, where the two largest eigenvalues of $\mathbf{Q}$ cross. This allows an exact expression relating the velocity of the line to spatial and temporal derivatives of $\mathbf{Q}$ on the line, to be specified by a dynamical model for the evolution of the nematic. By introducing a linear core approximation for $\mathbf{Q}$, analytical results are given for several prototypical configurations, including line interactions and motion, loop annihilation, and the response to external fields and shear flows. Behaviour that follows from topological constraints or defect geometry is highlighted. The analytic results are shown to be in agreement with three dimensional numerical calculations based on a singular Maier-Saupe free energy that allows for anisotropic elasticity.