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 cell with planar anchoring conditions at both substrates. Specifically, we study the instabilities of the ground state caused by surface elasticity with the saddle-splay elastic constant K-24 violating the Ericksen inequalities. We express the surface part of static correlation functions as a functional integral over the fluctuation field induced by director fluctuations at confining walls and derive the stability conditions for the planar structure with respect to the fluctuation modes characterized by the in-plane wave numbers and by the parity. These conditions are analyzed in the cell thickness-fluctuation wavelength plane through the parametrization for the boundary curve of the instability region. For relatively small K-24, the fluctuation mode of the critical wavelength is found to render the structure unstable when the thickness of the cell is below its critical value. The parity of the critical mode changes as the twist-splay ratio K-2/K-1 is passing through unity. Further increase of K-24 beyond the second threshold value, 4K(1)K(2)/(K-1+K-2), leads to the instability with respect to short wavelength fluctuations regardless of the cell thickness. We compute the critical thickness and the critical wavelength as a function of K-24, the twist-splay ratio, and the azimuthal anchoring strength.