Coarse-graining analysis of the Berreman anchoring

By means of a coarse-graining technique, we derive the effective anchoring energy of a nematic liquid crystal in contact with a macroscopically corrugated surface imposing a weak degenerate planar anchoring. For coarse-grained nematic director's profiles that vary only in the direction perpendicular to the average surface's plane, our results generalize those already known by including the anisotropy of the elastic constants. In the general case, on the contrary, we show that extra surface gradient terms appear that generally are not negligible.