Dynamics of active nematic defects on the surface of a sphere

TL;DR

This paper develops a particle-based theoretical model for the complex dynamics of topological defects in active nematic liquid crystals on spherical surfaces, combining numerical validation with Onsager's variational principle for clarity.

Contribution

It introduces an effective particle theory for active nematic defects on spheres, deriving their equations of motion and elucidating defect mobility using Onsager's variational principle.

Findings

Defects exhibit periodic or chaotic trajectories depending on activity strength.

Numerical solutions confirm and clarify defect dynamics and global rotation effects.

Theoretical framework aligns with observed defect behaviors in active nematics.

Abstract

A nematic liquid crystal confined to the surface of a sphere exhibits topological defects of total charge $+2$ due to the topological constraint. In equilibrium, the nematic field forms four $+1/2$ defects, located at the corners of a regular tetrahedron inscribed within the sphere, since this minimizes the Frank elastic energy. If additionally the individual nematogens exhibit self-driven directional motion, the resulting active system creates large-scale flow that drives it out of equilibrium. In particular, the defects now follow complex dynamic trajectories which, depending on the strength of the active forcing, can be periodic (for weak forcing) or chaotic (for strong forcing). In this paper we derive an effective particle theory for this system, in which the topological defects are the degrees of freedom, whose exact equations of motion we subsequently determine. Numerical solutions of these equations confirm previously observed characteristics of their dynamics and clarify the role played by the time dependence of their global rotation. We also show that Onsager's variational principle offers an exceptionally transparent way to derive these dynamical equations, and we explain the defect mobility at the hydrodynamics level.