Saddle-splay term induced orientational instability in nematic liquid crystal cells and director fluctuations at substrates

We analyze stability of the planar orientational structure in a nematic liquid crystal (NLC) cell with planar anchoring conditions at both substrates. Specifically, we study the instabilities of the ground state caused by surface elasticity at large saddle-splay elastic constant, $K_{24}$. For relatively small $K_{24}$ violating the Ericksen inequalities the theory predicts that the critical fluctuation mode of the wavelength, ${\lambda }_c$, will render the structure unstable when the thickness of the cell is below its critical value, $d_c$. The parity of the critical mode changes as the twist-splay ratio $K_2\mathrm{/}{K}_1$ is passing through the unity. Further increase of $K_{24}$ beyond the second threshold value, $4K_1{K}_2\mathrm{/}(K_1+K_2)$, leads to the instability with respect to the short wavelength fluctuations regardless of the cell thickness. We compute the critical thickness and the critical wavelength as functions of $K_{24}$, the twist-splay ratio and the azimuthal anchoring strength.