Orientational transition in a nematic liquid crystal at a patterned surface

We consider a semi-infinite nematic in contact with a periodic patterned surface with alternate planar and homeotropic stripes. Extending the work of Barbero et al., we find the free energy (assuming $K_1=K_3$) for the situations where the easy direction on the planar stripe is either perpendicular or parallel to the length of the stripes. We find the bulk free energy difference between the structures to be proportional to $\sqrt{K_2∕K_1}$ and so we consider the possibility of a spontaneous transition between the two states if the azimuthal anchoring energy is sufficiently weak and $K_1\ne K_2$. We compute the critical azimuthal anchoring energy for such a transition in terms of the relative width of the stripes and the period of the pattern and find it to be $\sim {10}^{-6}\mathrm{J}{\mathrm{m}}^{-2}$, comparable to experimental values.

Figure 1

Schematic of the director field $\widehat{\mathbf{n}}(x,z)$ for a nematic with $K_1=K_2=K_3$ in contact with a surface patterned with period $\lambda$ and relative homeotropic stripe width $a$. (a) “Splay-Bend” surface; (b) “Twist” Surface.

Figure 2

Critical azimuthal anchoring energy as a function of $a$ for $\lambda =10\mathrm{\mu }\mathrm{m}$, $K_1=1\times {10}^{-11}\mathrm{N}$, and with rigid polar anchoring. Solid lines were computed with $r_c={10}^{-8}\mathrm{m}$ and dashed lines with $r_c=5\times {10}^{-8}\mathrm{m}$. Note that a transition from splay-bend to twist is feasible if $W_{\varphi }<W_{\varphi critical}$.

Figure 3

Critical azimuthal anchoring energy as a function of $\lambda$ with rigid anchoring for $a=0.5$, $r_c={10}^{-8}\mathrm{m}$, and $K_1=1\times {10}^{-11}\mathrm{N}$.

Figure 4

Critical azimuthal anchoring energy as a function of the polar anchoring energy $W_{\theta }$ for $a=0.5$, $\lambda =10\mathrm{\mu }\mathrm{m}$, and $K_1=1\times {10}^{-11}\mathrm{N}$.