Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

Abstract

We consider a nematic liquid crystal occupying the exterior region in R 3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.

Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

“…where C ⊂ Ω are the colloidal particles included in the NLC environment and f s (Q, ν) is a surface energy quantifying the interaction between the NLC and the colloids (here ν stands for the exterior derivative). The equations corresponding to critical points of F LDGs are the same as (3.1) but on Ω\C with boundary conditions on ∂C of Robin type (coming out of the surface energy term, see for instance [93][94][95]). There exist a number of works in this direction studying either the case of a single coloidal particle and the qualitative structure of defects around it [ [95,[97][98][99].…”

Topics in the mathematical design of materials

“…Liquid crystal colloids are a subject of intense research in the current physics literature [169] while being largely untouched from a mathematical point of view. The existing mathematical studies focus on either the presence and qualitative properties of a defect-type structure around a single particle [170][171][172] or many particles and the homogenization effects [173][174][175][176].…”

Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation

“…Let P ⊆ R 3 be a closed, convex set that satisfies (H 5 ). Let p ∈ [2,4]. Then, there exists C = C(P, φ, p) > 0 such that, for any a > 0, b ≥ 2a and any u ∈ H 1 (bP\aP), there holdŝ…”