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

TL;DR

The paper addresses the inadequacy of strict Dirichlet anchoring in confined nematics by incorporating a finite Rapini–Papoular surface energy within a reduced 2D Landau–de Gennes model and analyzing the combined small-domain ($ε o 0$) and Oseen–Frank ($ε o ∞$) limits via matched asymptotics. The leading-order small-domain state is fixed by the boundary data average $q^{(0)} = rac{1}{|oundary ext{Ω}|} igoint_{oundary ext{Ω}} q_b \, dl$, potentially yielding isotropic melting when geometrical frustration makes this average vanish; in the large-domain limit, the surface energy produces an $O(1/ε)$ correction to the director near boundaries, a feature absent in rigid Dirichlet models. The analysis reveals that interior and boundary defect cores are smoother under Robin-type boundary conditions, and numerical experiments in square and circular wells illustrate significant differences in defect morphology compared with Dirichlet boundary conditions. The framework provides a physically consistent description of anchoring energetics crucial for accurate predictions in nanoscale nematic systems and guides design of micro- and nano-patterned devices. Key contributions include (i) a distinguished-limit treatment of strong anchoring with finite surface energy, (ii) a rigorous small-domain expansion with boundary-average dominance, (iii) a first-order $O(1/ε)$ correction to the director field in the Oseen–Frank regime, and (iv) demonstrations that Robin-type boundaries yield more realistic defect structures than rigid Dirichlet conditions.

Abstract

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 ($ε= h/ξ\to 0$; $h$ is the domain size) and Oseen-Frank $(ε\to \infty$) 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 $O(1/ε)$ correction to the director field near boundaries, in contrast to the $O(1/ε^2)$ 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. The results demonstrate that a consistent treatment of anchoring energetics is essential for accurate modelling of nematic equilibria in micro- and nano-scale confined geometries.

Figures (4)

• Figure 1: Contours of $\varphi_1$ for the diagonal state in a square well, calculated with $\gamma=1$. The director field pertains to $\mathbf{n}_0$.
• Figure 2: The leading-order scalar order parameter $S_0(\eta)$ for three values of the defect charge $|m|$.
• Figure 3: Equilibria in a square well. In each pair of figures, the left plots correspond to our mixed boundary conditions \ref{['BCs']}, whereas the right ones correspond to the Dirichlet boundary conditions \ref{['BCDcs']}. The colour contour represent the scalar order parameter $s$. The top row corresponds to the diagonal (D) states, the first pair in the bottom row to the cross-defect (X) states, and the last pair to the boundary-defect (BD) states.
• Figure 4: Equilibria in a circular well. In each pair of figures, the left plots correspond to our mixed boundary conditions \ref{['BCs']}, whereas the right ones correspond to the Dirichlet boundary conditions \ref{['BCDcs']}. The colour contour represent the scalar order parameter $s$. The top row corresponds to the state with two $+\tfrac{1}{2}$-defects, whereas the bottom row corresponds to the the state with one $+1$-defect (a vortex or ring solution).