Equilibrium configurations of nematic liquid crystals on a torus

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy and how new configurations emerge. In particular, our analysis predicts the existence of stable equilibria with complex windings.

Figure 1

Schematic representation of the torus $\mathbb{T}$.

Figure 2

Schematic representation of the constant equilibrium configurations ${\alpha }_m$, ${\alpha }_p$, and ${\alpha }_h$, where the director field $\mathbf{n}$ is aligned along meridians, parallels, and helixes of $\mathbb{T}$, respectively.

Figure 3

Surface energy $W_{\mathrm{NV}}$ (rescaled by $k{\pi }^2$) as a function of the deviation $\alpha$ from ${\mathbf{e}}_{\theta }$, for the one-constant approximation $k:=k_1=k_2=k_3$. The ratio of the radii of the torus $\mu =R/r$ is $\mu =1.1$ (dotted line), $\mu =2/\sqrt{3}$ (dashed-dotted line), $\mu =1.25$ (dashed line), and $\mu =1.6$ (continuous line). The behavior for equal $k_j$'s depends on the fact that the only nonconstant term in the energy $W_{\mathrm{NV}}$ changes sign when $2\mu ={\mu }^2/\sqrt{{\mu }^2-1}$, i.e., when $\mu =2/\sqrt{3}$.

Figure 4

Surface energy $W_{\mathrm{NV}}$ (rescaled by $k_1{\pi }^2$) as a function of the deviation $\alpha$ from ${\mathbf{e}}_{\theta }$, for $\mu =1.25$, $k_1=k_3=1$, and different values of $k_2$. Studying the second derivative of $W_{\mathrm{NV}}$, one finds critical values ${\xi }_1=2\sqrt{{\mu }^2-1}/\mu$ and ${\xi }_2=2-{\xi }_1$, which characterize the cases: ${\alpha }_m$ is the local minimum and ${\alpha }_p$ is the global minimum ($k_2>{\xi }_1$; solid line); ${\alpha }_m$ is the global maximum and ${\alpha }_p$ is the global minimum (${\xi }_1\ge k_2\ge {\xi }_2$; between the dashed and the dashed-dotted lines); and $\pm {\alpha }_h$ are the global minima, ${\alpha }_p$ is the local maximum, and ${\alpha }_m$ is the global maximum ($k_2<{\xi }_2$, dotted line).

Figure 5

Plots of the vector field ${\mathbf{n}}_{\infty }$ corresponding to the limit solution ${\alpha }_{\infty }$ to (10) for ${\alpha }_0=\pi /4$ and different ratios $\mu$. (The color code represents ${\alpha }_{\infty }$.) For $\mu <2$, the configuration ${\alpha }_m$ is preferable in a strip close to the central hole, while for every choice of $\mu$, ${\alpha }_p$ is more convenient around the external equator. As the term $|{\nabla }_s{\alpha |}^2$ in the energy penalizes the transition from ${\alpha }_p$ to ${\alpha }_m$, minimizers exhibit a smooth rotation of the director field along the meridians. The amplitude of the rotation increases as $\mu$ decreases to 1.

Figure 6

Examples of (numerical) local minimizers with different winding numbers, on the torus with $\mu =1.8$. The unoriented segments represent the direction $\pm \mathbf{n}$. The winding number can be easily visualized by fixing a closed meridian or parallel line and counting how many times a complete (and oriented) scale of colors appears on the chosen line. The value $W$ under each figure indicates the value of the energy (rescaled by $k$) for that configuration. The energy value of the ground state ${\alpha }_p$, for this choice of parameters, is $W=9.85$.