  In this paper we present a generalization of Berreman's model for the elastic
contribution to the surface free-energy density of a nematic liquid crystal in
presence of a sawtooth substrate which favours homeotropic anchoring, as a
function of the wavenumber of the surface structure $q$, the tilt angle
$\alpha$ and the surface anchoring strength $w$. In addition to the previously
reported non-analytic contribution proportional to $q\ln q$, due to the
nucleation of disclination lines at the wedge bottoms and apexes of the
substrate, the next-to-leading contribution is proportional to $q$ for a given
substrate roughness, in agreement with Berreman's predictions. We characterise
this term, finding that it has two contributions: the deviations of the nematic
director field with respect to the corresponding to the isolated disclination
lines, and their associated core free energies. Comparison with the results
obtained from the Landau-de Gennes model shows that our model is quite accurate
in the limit $wL>1$, when strong anchoring conditions are effectively achieved.

Soft Condensed Matter