Order Reconstruction for Nematics on Squares and Hexagons: A Landau-de Gennes Study

We construct an order reconstruction (OR)-type Landau-de Gennes critical point on a square domain of edge length 2λ, motivated by the well order reconstruction solution numerically reported in [1]. The OR critical point is distinguished by an uniaxial cross with negative scalar order parameter along the square diagonals. The OR critical point is defined in terms of a saddle-type critical point of an associated scalar variational problem. The OR-type critical point is globally stable for small λ and undergoes a supercritical pitchfork bifurcation in the associated scalar variational setting. We consider generalizations of the OR-type critical point to a regular hexagon, accompanied by numerical estimates of stability criteria of such critical points on both a square and a hexagon in terms of material-dependent constants.