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

Abstract

We study the simultaneous limits of homogenisation $(\epsilon \to 0)$ and dimension reduction $(h\to 0)$ for thin heterogeneous nematic liquid crystal films in the Landau--de Gennes $Q$-tensor framework. The elastic energy density is governed by a tensor $\mathbf{A}(x^{\prime }/\epsilon ,x_3/h)$ 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 $h^{\gamma }$, $\gamma \ge 0$, 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 $\mathcal{Q}\in H^1(\omega ;\mathcal{A})$ for $0\le \gamma <1$, a finite surface density contribution for $\gamma =1$, and vanishing anchoring for $\gamma >1$. For $0\le \gamma <1$ 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 $\rho =\lim h/\epsilon \in [0,\infty ]$: each regime has a distinct corrector structure and yields a two-dimensional LdG energy with homogenised tensor ${\mathbf{A}}_{\rho }^{\mathrm{hom}}$, 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.

45 pages, 1 figure