Anchoring of a nematic liquid crystal induced by surface grooves: A numerical study

To examine the anchoring energy of a surface with one-dimensional grooves of sinusoidal shape, we carry out numerical calculation of the Frank elastic energy of a nematic cell composed of such a grooved surface and a flat surface. We evaluate the anchoring energy of the grooved surface by carefully eliminating the contribution from a uniform twist deformation in the bulk. When $qA\lesssim 0.2$, with $q$ and $A$ being the wave number and the amplitude of the surface groove, we find that the azimuthal-angle dependence of the calculated anchoring energy agrees perfectly with our previous analytical result under the assumption of $qA⪡1$ [Fukuda et al., Phys. Rev. Lett. 98, 187803 (2007); 99, 139902(E) (2007)]. Even when $qA\simeq 0.6$ or 1, we observe an unexpectedly good agreement between the calculated and the analytical anchoring energies, indicating the wide applicability of the analytical anchoring energy in spite of the assumption of $qA⪡1$ in its derivation.

Figure 1

(Color online) Calculated anchoring energy ${\tilde{\mathcal{F}}}_{\mathrm{a}}$ as a function $\phi \left(0\right)$ (in degrees) in the case of (a) $qA=\pi ∕160\simeq 0.0196⪡1$ and (b) $qA=\pi ∕16\simeq 0.196$. The symbols × and + are the results for $K_{\mathrm{s}}∕K_3=0.6$ and $K_{\mathrm{s}}∕K_3=1$, respectively. The upper and lower curves are ${\mathcal{F}}_{\mathrm{a}}$ for $K_{\mathrm{s}}∕K_3=0.6$ and $K_{\mathrm{s}}∕K_3=1$, respectively.

Figure 2

(Color online) Calculated anchoring energy ${\tilde{\mathcal{F}}}_{\mathrm{a}}$ as a function $\phi \left(0\right)$ (in degrees) in the case of (a) $qA=3\pi ∕16\simeq 0.589$ and (b) $qA=1$. The symbols × and + are the results for $K_{\mathrm{s}}∕K_3=0.6$ and $K_{\mathrm{s}}∕K_3=1$, respectively. The upper and lower curves are ${\mathcal{F}}_{\mathrm{a}}$ for $K_{\mathrm{s}}∕K_3=0.6$ and $K_{\mathrm{s}}∕K_3=1$, respectively.

Figure 3

(Color online) Distribution of the azimuth of the director ${\phi }_{\mathit{n}}$ with the variation of $z∕L_z$ in the case of $qA=3\pi ∕16$ and $\varphi =90°$ for (a) $K_{\mathrm{s}}∕K_3=0.6$ and (b) $K_{\mathrm{s}}∕K_3=1$. Two curves in each graph specify the region in which ${\phi }_{\mathit{n}}\left(z\right)$ is expected to reside in a naive theoretical argument (see text).