Supported membranes on chemically structured and rough surfaces.

We present a general linear response description of membrane adhesion at rough or chemically structured surfaces. Our method accounts for nonlocal Van der Waals effects and contains the more approximate (and local) Deryagin approach in a simple limit. Specializing to supported membranes we consider the effects of substrate structure on the membrane adhesion energy and configuration. Adhesion is usually less favorable for rough substrates and the membrane shape tends to follow that of the surface contours. Chemical patterning (described by a spatially varying Van der Waals force), however, favors adhesion with the membrane configuration being out of phase with the surface structure. Finally, considering a surface indented with "V"-shaped trenches, we show that our approach is in good agreement with an exact numerical solution.