Consistent numerical evaluation of the anchoring energy of a grooved surface

We evaluate the azimuthal anchoring energy of a grooved surface by calculating numerically the Frank elastic energy of a nematic cell composed of the grooved surface and a flat one with rigid azimuthal anchoring, where the director is fixed along the $\varphi$ direction. We pay attention to the surface anchoring induced by elastic distortions of the director due to its contact with a nonflat surface, which impose local planar degenerate anchoring. Surface anchoring of this kind was analyzed analytically for shallow grooves by Berreman [Phys. Rev. Lett. 28, 1683 (1972)] and critically reexamined by the present authors [Phys. Rev. Lett. 98, 187803; 99, 139902(E) (2007)]. We consider two types of surface. one is a surface with one-dimensional sinusoidal parallel grooves, and the other is a surface with two-dimensional square patterns whose surface height is given by a sum of two sinusoidal functions with orthogonal wave vectors. The total energy is the sum of the anchoring energy and the twist energy in the bulk. For the calculation of the twist energy to be eliminated and the evaluation of the azimuthal-angle dependence of the anchoring energy, the “average” azimuthal angle at the bottom, $\phi \left(0\right)$, must be determined. We adopt two methods to determine $\phi \left(0\right)$. One is a simple extrapolation of the twist deformation in the bulk. The other relates $\phi \left(0\right)$ to the variation of the total Frank elastic energy with respect to $\varphi$. Our calculations indicate that both methods give essentially the same results, which indicates the consistency of those two methods. We also show that, for a surface with square patterns, the agreement between theory and numerical calculations is quite good even when the maximum of the surface slope is around 0.4, which theory assumes is much smaller than unity. When the surface slope is of order unity, the deviation of numerical results from theory crucially depends on the the surface elastic constant $K_{24}$.

Figure 1

(Color online) Rescaled anchoring energy ${\tilde{\mathcal{F}}}_{\mathrm{a}}$ for one-dimensional parallel grooves as a function of the azimuthal angle at the bottom, $\phi \left(0\right)$ for $qA=\pi ∕160\simeq 0.0196$. The surface elastic constant is $K_s∕K_3=0.6$ (a) and 1 (b). The solid lines represent the rescaled analytic anchoring energy ${\mathcal{F}}_{\mathrm{a}}$ for small $qA$.

Figure 2

(Color online) Same as Fig. 1 for $qA=\pi ∕16\simeq 0.196$.

Figure 3

(Color online) Same as Fig. 1 for $qA=3\pi ∕16\simeq 0.589$.

Figure 4

(Color online) Azimuthal angle of the director at the bottom one-dimensionally grooved surface, $\phi \left(0\right)$, as a function of the (fixed) azimuthal angle at the top flat surface $\varphi$ for $qA=\pi ∕160\simeq 0.0196$. The surface elastic constant is $K_s∕K_3=0.6$ (a) and 1 (b). The solid line corresponds to $\phi \left(0\right)=\varphi$, which implies the absence of twist deformations in the bulk.

Figure 5

(Color online) Same as Fig. 4 for $qA=\pi ∕16\simeq 0.196$.

Figure 6

(Color online) Same as Fig. 4 for $qA=3\pi ∕16\simeq 0.589$.

Figure 7

(Color online) Plots of the rescaled anchoring energy ${\tilde{\mathcal{F}}}_{\mathrm{a}}$ for a surface with square pattern as a function of the azimuthal angle at the bottom, $\phi \left(0\right)$ for $qA=\pi ∕160\simeq 0.0196$. The surface elastic constant is $K_s∕K_3=0.6$ for (a) and $K_s∕K_3=1$ for (b). The solid lines represent the rescaled analytic anchoring energy ${\mathcal{F}}_{\mathrm{a}}$ for small $qA$.

Figure 8

(Color online) Same as Fig. 7 for $qA=\pi ∕16\simeq 0.196$.

Figure 9

(Color online) Same as Fig. 7 for $qA=3\pi ∕16\simeq 0.589$.

Figure 10

(Color online) Azimuthal angle of the director at the bottom grooved surface with square patterns, $\phi \left(0\right)$ as a function of the (fixed) azimuthal angle at the top flat surface $\varphi$ for $qA=\pi ∕160\simeq 0.0196$. The surface elastic constant is $K_s∕K_3=0.6$ (a) and 1 (b). The solid line corresponds to $\phi \left(0\right)=\varphi$, which implies the absence of twist deformations in the bulk.

Figure 11

(Color online) Same as Fig. 10 for $qA=\pi ∕16\simeq 0.196$.

Figure 12

(Color online) Same as Fig. 10 for $qA=3\pi ∕16\simeq 0.589$.

Figure 13

(Color online) Distribution of the azimuth ${\phi }_{\mathit{n}}$ of the director with the variation of $z∕L_z$ in the case of $qA=3\pi ∕16$ and $\varphi =10°$ for $K_{\mathrm{s}}∕K_3=\left(\mathrm{a}\right)$ 0.6 and (b) 1. Solid lines in each curve specify the region in which ${\phi }_{\mathit{n}}\left(z\right)$ is expected to reside in a naive theoretical argument (see text).

Figure 14

Schematic illustration as to how to solve Eq. (9).

Figure 15

Schematic illustration to explain why $\phi \left(0\right)⩽\phi {\left(0\right)}_{\max }<90°$.