# Orientation of topological defects in 2D nematic liquid crystals

## TL;DR

This paper introduces a tensor-based formalism to describe the orientation of topological defects in 2D nematic liquid crystals, providing new insights into defect interactions and dynamics.

## Contribution

It proposes an alternative tensor formalism for defect orientation, improving upon previous vector-based models, and applies it to analyze defect interactions and motion.

## Key findings

- Tensor formalism effectively describes defect orientations.
- Orientation influences defect interaction energies.
- Numerical simulations show defect motion depends on orientation.

## Abstract

Topological defects are an essential part of the structure and dynamics of all liquid crystals, and they are particularly important in experiments and simulations on active liquid crystals. In a recent paper, Vromans and Giomi [Soft Matter, 2016, 12, 6490] pointed out that topological defects are not point-like objects but actually have orientational properties, which strongly affect the energetics and motion of the defects. That paper developed a mathematical formalism which describes the orientational properties as vectors. Here, we agree with the basic concept of defect orientation, but we suggest an alternative mathematical formalism. We represent the defect orientation by a tensor, with a rank that depends on the topological charge: rank 1 for a charge of +1/2, rank 3 for a charge of -1/2. Using this tensor formalism, we calculate the orientation-dependent interaction between defects, and we present numerical simulations of defect motion.