Strong anchoring boundary conditions in nematic liquid crystals: Higher-order corrections to the Oseen-Frank limit and a revised small-domain theory

Strong anchoring boundary conditions are conventionally modelled by imposing Dirichlet conditions on the order parameter in Landau-de Gennes theory, neglecting the finite surface energy of realistic anchoring. This work revisits the strong anchoring limit for nematic liquid crystals in confined two-dimensional domains. By explicitly retaining a Rapini-Papoular surface energy and adopting a scaling where the extrapolation length $l_{ex}$ is comparable to the coherence length $ξ$, we analyse both the small-domain ($\ep = h/ξ\to 0$; $h$ is the domain size) and Oseen-Frank $(\ep \to \infty$) asymptotic regimes. In the small-domain limit, the leading-order equilibrium solution is given by the average of the boundary data, which can vanish in symmetrically frustrated geometries, leading to isotropic melting. In the large-domain limit, matched asymptotic expansions reveal that surface anchoring introduces an $O(1/\ep)$ correction to the director field, in contrast to the $O(1/\ep^2)$ correction predicted by Dirichlet conditions. The analysis captures the detailed structure of interior and boundary defects, showing that mixed (Robin-type) boundary conditions yield smoother defect cores and more physical predictions than rigid Dirichlet conditions. Numerical solutions for square and circular wells with tangential anchoring illustrate the differences between the two boundary condition treatments, particularly in defect morphology. These results demonstrate that a consistent treatment of anchoring energetics, together with stability considerations, is essential for accurate modelling of nematic equilibria in micro- and nano-scale confined geometries.

💡 Research Summary

This paper revisits the modeling of strong anchoring boundary conditions for nematic liquid crystals confined in two‑dimensional domains, explicitly retaining a Rapini‑Papoular surface energy rather than imposing the usual Dirichlet condition on the Landau‑de Gennes order‑parameter tensor. The authors adopt a scaling in which the extrapolation length (l_{\mathrm{ex}}=L/W) is comparable to the nematic coherence length (\xi) (i.e. (\gamma=\xi/l_{\mathrm{ex}}=O(1))). This regime is relevant for micro‑ and nano‑scale wells where the domain size (h) may be of the same order as (\xi) or (l_{\mathrm{ex}}).

After reducing the full three‑dimensional Landau‑de Gennes model to a two‑dimensional setting (where the tensor reduces to two scalar fields (q_{1},q_{2}) or equivalently a scalar order parameter (s) and director angle (\phi)), the free‑energy functional is nondimensionalised, introducing the single parameter (\varepsilon=h/\xi). The surface term then scales as (\varepsilon\gamma), which is strong for large domains and weak for small ones.

Small‑domain limit ((\varepsilon\to0)).
A regular perturbation expansion yields at leading order a Laplace equation with homogeneous Neumann boundary conditions, forcing the solution to be spatially constant. The constant is not arbitrary; the solvability condition fixes it to the average of the prescribed boundary tensor: \