Multiscale Analysis of a Landau--de Gennes Model for Nematic thin-film composites

Abstract

We study the simultaneous limits of homogenisation $(\varepsilon\to0)$ and dimension reduction $(h\to0)$ for thin heterogeneous nematic liquid crystal films in the Landau--de Gennes $Q$-tensor framework. The elastic energy density is governed by a tensor $\mathbf{A}(x'/\varepsilon,x_3/h)$ that is periodic in the in-plane fast variable and measurable in the normalised thickness variable. Surface anchoring on the top and bottom faces is modelled by a weak anchoring energy of strength $h^\gamma$, $\gamma\geq0$, in a general set-valued framework covering the principal classical anchoring geometries. The simultaneous limit reveals two principal features. First, the anchoring scaling yields a hierarchy of effective behaviours depending on $\gamma$: a hard constraint $\mathcal{Q}\in H^1(\omega;\mathscr{A})$ for $0\leq\gamma<1$, a finite surface density contribution for $\gamma=1$, and vanishing anchoring for $\gamma>1$. For $0\leq \gamma<1$ with uniaxial anchoring, the homogenised energy reduces on the constrained class to anisotropic Oseen--Frank and Ericksen energies. Second, the effective elastic response depends on the scale ratio $\rho=\lim h/\varepsilon\in[0,\infty]$: each regime has a distinct corrector structure and yields a two-dimensional LdG energy with homogenised tensor $\mathbf{A}^{\mathrm{hom}}_\rho$, characterised by a regime-dependent cell problem. To the best of our knowledge, this is the first rigorous treatment of the simultaneous homogenisation and dimension reduction limit for the Landau--de Gennes energy, and the first in which the anchoring strength enters as a scaling parameter, producing a hierarchy of qualitatively distinct effective models.