Intermediate periodic "saddle-splay" nematic phase in the vicinity of a nematic-smectic-A transition

We consider possible spontaneous modulations of the nematic director induced by the elastic saddle-splay K24 term when the value of the elastic constant K24 does not satisfy the Ericksen stability condition for the homogeneous ground state. According to the standard formula expressing K24 in terms of the twist elastic constant K22, this can be expected close to the nematic-smectic-A transition where K22 becomes very large. It is predicted that in a planar nematic layer (or, more generally, if the surface director alignment is sufficiently close to a planar one), a modulated phase with observable long wavelength period can occur in samples considerably thicker than the anchoring extrapolation length. The modulated nematic phase is expected to persist into the smectic phase so that its temperature of the transition to smectic phase has to be lower than that for the homogeneous nematic liquid crystal. Low amplitude short wavelength modulations are predicted for any thickness if the surface director is sufficiently far from a pure homeotropic alignment. At the expense of this mode the temperature of a nematic-smectic-A transition in a planar cell with isotropic surfaces has to be lower than that for a homeotropic cell even if the periodic structure is not accessible for the direct observation.