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<p>We give a brief introduction to a divergence penalized Landau-de Gennes functional as a toy model for the study of nematic liquid crystal with colloid inclusion, in the case of unequal elastic constants. We assume that the nematic occupies the exterior of the unit ball, satisfies homeotropic anchoring at the surface of the colloid and approaches a uniform uniaxial state as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|x|\rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We study the “small particle” limit and obtain a representation formula for solutions to the associated Euler-Lagrange equations. We also present a numerical analysis of these equations based on a finite element approach and discuss the effect of the divergence penalization on the “Saturn ring” defects and on the properties of the <i>Q</i>-tensor.</p>

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On a divergence penalized Landau-de Gennes model

  • 期刊论文

  • 发表日期：

    2025年3月10日

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  • Lia Bronsard,
  • Jinqi Chen,
  • Léa Mazzouza,
  • Daniel McDonald,
  • Nathan Singh,
  • Dominik Stantejsky,
  • Lee van Brussel

摘要

We give a brief introduction to a divergence penalized Landau-de Gennes functional as a toy model for the study of nematic liquid crystal with colloid inclusion, in the case of unequal elastic constants. We assume that the nematic occupies the exterior of the unit ball, satisfies homeotropic anchoring at the surface of the colloid and approaches a uniform uniaxial state as \(|x|\rightarrow \infty \) | x | → ∞ . We study the “small particle” limit and obtain a representation formula for solutions to the associated Euler-Lagrange equations. We also present a numerical analysis of these equations based on a finite element approach and discuss the effect of the divergence penalization on the “Saturn ring” defects and on the properties of the Q-tensor.