Possibility of a deformed ground state in free standing nematic films

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Abstract
The possibility of existence of distorted configurations in free standing nematic liquid crystal films is considered. In the framework of Frank elastic theory the stable configurations are always the undistorted ones, compatible with boundary conditions. When the mixed splay-bend elastic term with elastic constant k13 is considered and the Nehring-Saupe free energy density used to analyse the stability of the nematic configurations, it seems that a splayed or bent structure could be more stable than the undistorted one, if k13 satisfies a suitable relation. This result, which is independent of the sample thickness, seems justified by some experimental data. But we show that this strange result is due to an over-simplification of the elastic theory, in which terms depending on the square of the second order derivatives of the nematic molecular average orientation are neglected. In this paper we consider these usally neglected terms. We show that the distorted configurations can still be stable with respect to the undistorted one only if the elastic constant k13 is large enough, and the sample thickness is smaller than a critical value. Furthermore the order of magnitude of the ratio between the new elastic constant, connected to the elastic term depending on the second order derivative, and the usual one is determined by using a simple quasi-microscopic theory