[Paper Review] Boojums in Liquid Crystals Around a Colloid

Yuchen Huang, Yong Yu|arXiv (Cornell University)|Jan 17, 2026

Liquid Crystal Research Advancements0 citations

TL;DR

The paper analyzes the Landau-de Gennes energy for a nematic liquid crystal around a spherical colloid in the one-constant limit, proving that energy-minimizing configurations develop boojum disclinations at the two poles under axially symmetric and Lyuksyutov constraints. It provides a detailed regularity and asymptotic analysis showing boojum-type surface defects at the poles.

ABSTRACT

We study the Landau-de Gennes theory in the one constant limit. The bulk domain is the exterior of a spherical colloid. A Rapini-Papoular surface potential is imposed on the colloid surface, supplemented by a homogeneous far-field condition at spatial infinity. Under the axially symmetric ansatz and the Lyuksyutov constraint, we show that energy minimizers exhibit boojum disclinations at the two poles of the colloid. The local structure of these boojum disclinations is also characterized.

Motivation & Objective

• Motivate understanding of how colloid-induced defects arise in nematic liquid crystals under the Landau-de Gennes framework.
• Investigate the regularity and singularity structure of minimizers with surface anchoring and far-field conditions.
• Characterize the local structure of boojum disclinations at the colloid poles under axial symmetry and Lyuksyutov constraints.

Proposed method

• Formulate the Landau-de Gennes energy with bulk, elastic, and Rapini-Papoular surface terms in the one-constant limit.
• Apply an axially symmetric ansatz and Lyuksyutov constraint to reduce to a reduced energy functional.
• Transform to a boundary-flattened coordinate system near poles and develop a split-family Campanato/Morrey analysis for regularity.
• Prove Hölder continuity of minimizers up to the boundary away from the axis and obtain boundary regularity at poles.
• Derive and analyze the Euler–Lagrange equations and the director field to identify boojum-type singularities at the poles.

Experimental results

Research questions

• RQ1 Do energy-minimizing configurations exhibit boojum disclinations at the poles under the given boundary and far-field conditions?
• RQ2 What is the regularity of minimizers near the colloid poles, and how do boundary conditions affect this regularity?
• RQ3 What is the local structure of the director field near the poles, and can boojum-type defects be rigorously characterized within this framework?

Key findings

• Minimizers wν are regular away from the z-axis and regular near the north and south poles in a small neighborhood.
• At the poles, minimizers satisfy wν(N)=wν(S)=(0,0,−1,0,0)ᵀ.
• Along the symmetry axis, an odd number of singularities occur above and below the colloid.
• The director field is singular at the poles and converges to the radial direction eρ approaching the poles, identifying boojum-type surface defects.
• A novel split-family argument is developed to handle weak-anchoring boundary regularity in the boundary-containing region.