Influence of the extrinsic curvature on two-dimensional nematic films

Nematic films are thin fluid structures, ideally two dimensional, endowed with an in-plane degenerate nematic order. In this paper we examine a generalization of the classical Plateau problem to an axisymmetric nematic film bounded by two coaxial parallel rings. At equilibrium, the shape of the nematic film results from the competition between surface tension, which favors the minimization of the area, and the nematic elasticity, which instead promotes the alignment of the molecules along a common direction. We find two classes of equilibrium solutions in which the molecules are uniformly aligned along the meridians or parallels. Depending on two dimensionless parameters, one related to the geometry of the film and the other to the constitutive moduli, the Gaussian curvature of the equilibrium shape may be everywhere negative, vanishing, or positive. The stability of these equilibrium configurations is investigated.

Figure 1

Schematic representation of an axisymmetric 2D nematic film. At any point $P$ we define both the Darboux frame $\{\mathbf{n},\mathbf{t},\mathbf{\nu }\}$ and the orthonormal basis $\{{\mathbf{e}}_p,{\mathbf{e}}_m,\mathbf{\nu }\}$, with ${\mathbf{e}}_p$ and ${\mathbf{e}}_m$ being the principal directions.

Figure 2

Solutions to (24) for different values of $c$. Solid lines correspond to stable equilibrium configurations in the case of in-plane strong anchoring (see Sec. 4b). The dashed lines correspond instead to unstable equilibria.

Figure 3

Equilibrium configurations with the molecules of the nematics oriented along parallels for different values of $c$.

Figure 4

Equilibrium configurations with the molecules of the nematics oriented along the meridians for different values of $c$.

Figure 5

Critical threshold for the stability of $(S_p,{\alpha }_p)$ as a function of $c$. The equilibrium configuration $(S_p,{\alpha }_p)$ is stable if and only if $\xi \le {\xi }_{\mathrm{cr}}\left(c\right)$. Beyond this critical value, at a stable equilibrium configuration, the alignment of the molecules must be inhomogeneous.