DIBLOCK COPOLYMER LAMELLAE AT ROUGH SURFACES

We study theoretically the behavior of a diblock copolymer lamellar layer, in the melt, assembled at a substrate with a small-amplitude periodic roughness, characterized by a wavevector q. The polymer configurations are described using the Alexander-de Gennes model of polymer brushes. For a single lamella of thickness h, at small wavevectors (qh < 1), the profile of the free surface follows closely the roughness of the substrate. There exists a finite value of the wavevector q(c) (q(c)h almost-equal-to 2.6) at which the upper surface is completely flat. At larger wavevectors q > q(c), the undulation of the free surface has a sign opposite to that of the solid surface; the two surfaces of the lamella undulate out of phase. In the limit of infinite wavevector, the midplane of the lamella is flat and the deformations of the two surfaces are symmetric with respect to reflection about the midplane. We also consider thicker films where many lamellar layers form a stack next to the surface. We determine the smectic elasticity of the lamellar phase for small wavevectors. We then discuss the penetration of the undulation induced by the substrate into the lamellar stack. At low wavevector the penetration depth diverges as q-2, as in usual smectics; it vanishes at the wavevector q(c) and then increases with wavevector.