Nematic liquid crystals in planar and cylindrical hybrid cells: Role of elastic anisotropy on the director deformations

Nematic samples filling a flat cell or the annular region between two concentric cylinders with hybrid anchoring conditions at the boundaries are investigated by setting up and minimizing their Frank elastic free energy. The coupling with the surfaces is taken to be strong on one side and weak on the other. The equations are numerically solved and the conditions for which the molecular organization inside the cell becomes uniform are analyzed. The classical calculation performed by G. Barbero and R. Barberi [J. Phys. 44, 609 (1983)] is reproduced and investigated from a different point of view, in order to compare the results of planar and cylindrical geometries. The results suggest that the cylindrical cell presents some unusual features deserving a more complete investigation. Although most part of the transitional phenomena are found for $K_{11}>K_{33}$, a case not common for ordinary (lyotropic and thermotropic) liquid crystals, it is possible to find a completely uniform cell even for $K_{11}<K_{33}$ in both the geometries considered here.

Figure 1

Schematic representation of the planar sample with the surfaces placed at a distance $d$.

Figure 2

The first three plates (a,b,c) refer to the ZwXs case. Anchoring angle at the surface located at $z=-d/2$ as a function of the scaled extrapolation length for a few values of the elastic constant parameter (a) and as function of the elastic anisotropy parameter for some values of the scaled extrapolation length (b); the dependence of the critical value of the elastic parameter on the scaled extrapolation length (c). The plate (d) shows the director angle close to the surface located in $z=-d/2$ versus the scaled extrapolation length for the XwZs case.

Figure 3

A schematic representation of the annular region confined between two concentric cylinders and a sketch of the coordinates $\mathbf{r},\mathbf{\phi }$, and $\mathbf{z}$. The surfaces are placed at distances $r_1$ and $r_2$ from the center, with $r_2-r_1=d$.

Figure 4

RsZw case. Profile of ${\varphi }_{s1}-\pi /2$ as function of $L_1$, for $r_1=14$ and a few values of the elastic parameters (a). ${\varphi }_{s1}-\pi /2$ versus $r_1$ for $L=10.0$ and a few values of the elastic parameter (b). The behavior of ${\varphi }_{s1}-\pi /2$ as function of $k$ for $r_1=14$ and some values of the extrapolation length (c). Critical values of the elastic parameter versus the extrapolation length, for some values of internal radio (d).

Figure 5

RwZs case. The Profile of ${\varphi }_{s2}$ is shown as function of $L_2$ (a), $k$ (b), and $r_1$ (c). In (a) and (c) we have used $r_1=10$ and some values of the elastic parameter and extrapolation length, respectively, as indicated in the figure. In (b) the numerical calculation was performed for $L_2=3.0$ and a few values of the elastic parameter. The profiles of critical values of elastic parameter as function of the extrapolation length, for some values of the internal radius (d).

Figure 6

ZsRw case. Profile of ${\varphi }_{s1}$ as a function of $L_1$, for $r_1=10$, with some values of the elastic parameters (a), versus $r_1$ for $L_1=1.5$ and a few values of the elastic parameter (in which the line styles are the same used in (a)) (b) and as a function of $k$ for $r_1=10$ and some values of extrapolation length (c). (d) As in Fig. 4.

Figure 7

ZwRs case. Results of the numerical analysis of ${\varphi }_{s2}-\pi /2$ as function of $L_2$, for some values of the elastic parameters (a). Profile of ${\varphi }_{s2}-\pi /2$ versus $r_1$ for $L_2=20.0$ and a few values of the elastic parameter (b). (c) ${\varphi }_{s2}-\pi /2$ versus $k$ for a few values of extrapolation length. In both cases (a) and (c) we have used $r_1=14$. (d) As described in Fig. 4.