Parallel and Perpendicular Lamellae on Corrugated Surfaces

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 shortchain 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.