Molecular organization of nematic liquid crystals between concentric cylinders: Role of the elastic anisotropy

The orientational order in a nematic liquid crystal sample confined to an annular region between two concentric cylinders is investigated by means of lattice Monte Carlo simulations. Strong anchoring and homeotropic orientations, parallel to the radial direction, are implemented at the confining surfaces. The elastic anisotropy is taken into account in the bulk interactions by using the pair potential introduced by Gruhn and Hess [T. Gruhn and S. Hess, Z. Naturforsch. A 51, 1 (1996)] and parametrized by Romano and Luckhurst [S. Romano, Int. J. Mod. Phys. B 12, 2305 (1998); Phys. Lett. A 302, 203 (2002); G. R. Luckhurst and S. Romano, Liq. Cryst. 26, 871 (1999)], i.e., the so-called GHRL potential. In the case of equal elastic constants, a small but appreciable deformation along the cylinder axis direction is observed, whereas when the values of $K_{11}/K_{33}$ if $K_{22}=K_{33}$ are low enough, all the spins in the bulk follow the orientation imposed by the surfaces. For larger values of $K_{11}/K_{33}$, spontaneous deformations, perpendicular to the polar plane, increase significantly. Our findings indicate that the onset of these deformations also depends on the ratio $K_{22}/K_{33}$ and on the radius of the cylindrical surfaces. Although expected from the elastic theory, no tangential component of the deformations was observed in the simulations for the set of parameters analyzed.

Figure 1

Profiles for equal elastic constants (LL model) of (a) $P_{2R}$, (b) $P_{2Z}$, (c) $P_{2\theta }$, and (d) $P_2$ as a function of the distance from the cylinder axis for various inner cylinder radii $r_1$ boundaries. The maximum observed in $P_{2Z}$ has a minimum correspondent in $P_{2R}$, but no change in $P_{2\theta }$ is detected. The $P_2$ profiles present values very close to 0.25. Profiles for different elastic constants (GHLR model) of $P_{2Z}$ as a function of the distance from the cylinder axis for the sets (e) $K_{11}/K_{33}=0.5$ and $K_{22}/K_{33}=1.0$ and (f) $K_{11}/K_{33}=3.0$ and $K_{22}/K_{33}=1.0$. In the latter case, a more evident maximum is observed for $P_{2Z}$ for a larger value of the scaled splay elastic constant.

Figure 2

Snapshots of the configuration of the spins at the end of the simulations for $r_1=4,K_{22}/K_{33}=1.0$, and (a) $K_{11}/K_{33}=1$, (b) $K_{11}/K_{33}=2.0$, (c) $K_{11}/K_{33}=3$, and (d) $K_{11}/K_{33}=3.5$. We can verify that there are significant deformations only for high values of the ratio $K_{11}/K_{33}$. These deformations are more concentrated near the inner cylinder and their size increase as $K_{11}/K_{33}$ increases.

Figure 3

Microscopic texture between linear crossed polarizers for $K_{11}/K_{33}=3.0$ and $K_{22}/K_{33}=1.0$ for a few values of inner radius (a) $r_1=2$, (b) $r_1=4$, (c) $r_1=8$, and (d) $r_1=12$. The circular dark rings indicate the presence of a deformation in the $\mathbf{Z}$ direction.

Figure 4

Profiles of (a) $P_{2R}$, (b) $P_{2Z}$, (c) $P_{2\theta }$, and (d) $P_2$ as a function of the distance from the cylinder axis for $K_{11}/K_{33}=2.0,r_1=4$, and a few values of $K_{22}/K_{33}$. The scaled twist elastic constant seems to induce some relevant deformation parallel to the $\mathbf{Z}$ axis.

Figure 5

(a) Maximum amplitude of $P_{2Z}$ $\left(P_{2Z\text{−}\max }\right)$ as a function of the scaled splay elastic constant. A critical behavior is observed for a given ratio $K_{11}/K_{33}$, which varies according to the values of the scaled twist elastic constant. (b) $P_{2Z\text{−}\max }$ as a function of the radius of the inner cylinder for a few values of $K_{11}/K_{33}$ and $K_{22}/K_{33}=1.0$. (c) $P_{2Z\text{−}\max }$ versus the radius of the inner cylinder when $K_{11}/K_{33}=2.5$ for several values of $K_{22}/K_{33}$. Both cases indicate that the inner surface can induce the deformation. (d) $P_{2Z\text{−}\max }$ as a function of $r_1/r_2$ for two sets of the elastic constant and two values of the radius of the external cylinder. This plot suggests that, in contrast to predictions from the elastic theory, the Fréedericksz-like transition may depend on the absolute values of the radius of the surfaces.