## Y. Wang - G. Canevari - A. Majumdar

# Order Reconstruction for Nematics on Squares with Isotropic Inclusions:
A Landau-de Gennes Study

created by canevari on 29 Jan 2021

[

BibTeX]

*preprint*

**Inserted:** 29 jan 2021

**Year:** 2018

**Abstract:**

We prove the existence of a well order reconstruction solution (WORS) type
Landau-de Gennes critical point on a square domain with an isotropic concentric
square inclusion, with tangent boundary conditions on the outer square edges.
There are two geometrical parameters - the outer square edge length $\lambda$,
and the aspect ratio $\rho$, which is the ratio of the inner and outer square
edge lengths. The WORS exists for all geometrical parameters and for all
temperatures; we prove that the WORS is globally stable for either $\lambda$
small enough or for $\rho$ sufficiently close to unity. We study three
different types of critical points in this model setting: critical points with
the minimal two degrees of freedom consistent with the imposed boundary
conditions, critical points with three degrees of freedom and critical points
with five degrees of freedom. In the two-dimensional case, we use
$\Gamma$-convergence techniques to identify the energy-minimizing competitors.
We decompose the second variation of the Landau-de Gennes energy into three
separate components to study the effects of different types of perturbations on
the WORS solution and find that it is most susceptible to in-plane
perturbations. In the three-dimensional setting, we numerically find up to $28$
critical points for moderately large values of $\rho$ and we find two critical
points with the full five degrees of freedom for very small values of $\rho$,
with an escaped profile around the isotropic square inclusion.