The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics

TL;DR

This paper investigates the asymptotic behavior of critical points in a two-dimensional ferronematic model, revealing how energy concentrates along singular sets of different dimensions and characterizing their geometric properties.

Contribution

It provides a detailed analysis of the singular structures formed in ferronematics, including the concentration of energy on points and lines, and the geometric properties of these singular sets.

Findings

Energy concentrates at finite points for the Q-tensor component.

Magnetisation energy concentrates along a rectifiable set.

Curvature of the magnetisation singular set is concentrated at finite points.

Abstract

We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\mathbf{Q}$-tensor for the liquid crystal component and a magnetisation vector field~$\mathbf{M}$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\mathbf{Q}$ and~$\mathbf{M}$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\varepsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~$\mathbf{Q}$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~$\mathbf{M}$-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component.