  We study block copolymers (BCP) on patterned substrates, where the top
polymer film surface is not constrained but is a free interface that adapts its
shape self-consistently. In particular, we investigate the combined effect of
the free interface undulations with the wetting of the BCP film as induced by
nano-patterned substrates. For a finite volume of BCP material, we find
equilibrium droplets composed of coexisting perpendicular and parallel lamellar
domains. The self-assembly of BCP on topographic patterned substrates was also
investigated and it was found that the free interface induces mixed
morphologies of parallel and perpendicular domains coupled with a non-flat free
interface. In both cases, the free interface relaxes the strong constraints
that would otherwise be imposed by a fixed top boundary (which is commonly used
in simulations), and affects strongly the BCP ordering. Our study has some
interesting consequences for experimental setups of graphoepitaxy and
nanoimprint lithography.
