Periodic splay-twist Fréedericksz transition for nematics confined between two concentric cylinders

This paper derives theoretical results for the periodic splay-twist Fréedericksz transition in nematic liquid crystals confined between two infinite concentric cylinders. The calculation of Lonberg and Meyer [Phys. Rev. Lett. 55, 718 (1985)], for nematics sandwiched between two infinite planes, is extended to annular domains. The phase transition is triggered by an applied voltage between the outer and the inner delimiting walls. The critical threshold behavior is analyzed via the linearized Euler-Lagrange equations related to the Frank’s free energy. It is found that, the threshold depends on both the ratio between the twist and the splay elastic constants, and the sample radii ratio. Results for planar samples are recovered in the thin cell limit. With respect to the planar geometry, our analysis predicts that for annular geometries the periodic Fréedericksz transition is also allowed for elastic anisotropies $K_2/K_1>0.303$.

Figure 1

Schematic representation of Fréedericksz transitions for a nematic confined between two concentric cylinders, when a radial electric field is applied. The vector field $\mathbf{n}$ is portrayed in a right section of the domain. Top: field-free configuration. Strong planar boundary conditions impose an uniform alignment of the molecules along the cylinder axis. Center: HFT. The director field results distorted: the molecules rotate the plane spanned ${\mathbf{e}}_r$ and ${\mathbf{e}}_z$, maintaining zero their azimuthal component. Bottom: PFT. The molecules undergo a spatial distortion which is periodic in the azimuthal direction. The director is plotted according to our numerical solution (see Fig. 5) and it is worth noticing the close resemblance with the path reported in Fig. 1 of [14].

Figure 2

The determinant $G$ as a function of $\upsilon$.

Figure 3

The critical threshold ${\upsilon }_c$ (top), the critical undulation wave number (center) as a function of $\kappa$ for different values of $\rho$. In the bottom is shown a comparison with the Lonberg and Meyer results (see text).

Figure 4

Critical $\kappa$ as a function of $\rho$. The dashed curve is given by the formula (29)

Figure 5

An example of numerical solution for $\overline{\phi }\left(y\right)$ and $\overline{\gamma }\left(y\right)$ [see Eqs. (17)]. The parameters are $\kappa =0.2$, $\rho =1.5$, $q=7$, and the critical voltage numerically found ${\upsilon }_c\approx 2.965886$.