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Boojums in Liquid Crystals Around a Colloid

Yuchen Huang, Yong Yu

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.

Boojums in Liquid Crystals Around a Colloid

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.
Paper Structure (27 theorems, 244 equations, 5 figures)

This paper contains 27 theorems, 244 equations, 5 figures.

Key Result

Theorem 1.5

For any normalized anchoring strength $\nu>0$, let $w_{\nu}$ be a minimizer of the functional $F_{\nu}$ within the configuration space $\mathcal{F}_{\nu}$. Then:

Figures (5)

  • Figure 1: Boojum type defects of the director on poles
  • Figure 2: Four types of Campanato balls and covering schemes
  • Figure 3: Comparison map, Case I
  • Figure 4: Waxing moon covering scheme
  • Figure 5: 2D coordinates transformation

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3: Campanato Criteria
  • proof
  • ...and 48 more