Anchoring energy for nematic liquid crystals: Contribution from the spatial variation of the elastic constants

The influence of the spatial variation of the elastic constants of a nematic liquid crystal on the detectable surface anchoring energy is analyzed. It is shown that, in the framework of the usual elastic approach, the apparent surface free energy connected with the spatial variation of the elastic constants is proportional to the difference between the average value of the elastic constant in the surface layer and the bulk value. The related extrapolation length is estimated to be of the order of the interaction range of the forces responsible for the nematic phase. When the spatial variation of the elastic constants is large and the usual elastic approach does not work, the same analysis is performed in the framework of the second-order elasticity. In this case a nonlocal form for the surface energy is used. By considering simple situations the equivalent surface anchoring energy is deduced. Our results generalize the conclusions obtained by other researchers for a similar problem.