  We consider the relative stability of parallel and perpendicular lamellar
layers on corrugated surfaces. The model can be applied to smectic phases of
liquid crystals, to lamellar phases of short-chain amphiphiles and to lamellar
phases of long-chain block copolymers. The corrugated surface is modelled by
having a single $q$-mode lateral corrugation of a certain height. The lamellae
deform close to the surface as a result of chemical interaction with it. The
competition between the energetic cost of elastic deformations and the gain in
surface energy determines whether parallel or perpendicular lamellar
orientation (with respect to the surface) is preferred. Our main results are
summarized in two phase diagrams, each exhibiting a transition line from the
parallel to perpendicular orientations. The phase diagrams depend on the three
system parameters: the lamellar natural periodicity, and the periodicity and
amplitude of surface corrugations. For a fixed lamellar periodicity (or polymer
chain length), the parallel orientation is preferred as the amplitude of
surface corrugation decreases and/or its periodicity increases. Namely, for
surfaces having small corrugations centered at long wavelengths. For a fixed
corrugation periodicity, the parallel orientation is preferred for small
corrugation amplitude and/or large lamellae periodicity. Our results are in
agreement with recent experimental results carried out on thin block copolymer
films of PS-PMMA (polystyrene-polymethylmethacrylate) in the lamellar phase,
and in contact with several corrugated surfaces.
