  We present numerical calculations of lamellar phases of di-block copolymers
(BCP) confined between two surfaces, where the top surface is flat and the
bottom one is corrugated. The corrugated substrate is assumed to have a single
$q$-mode of lateral undulations with a wavenumber q_s and amplitude R. We focus
on the effects of substrate roughness, parameterized by the dimensionless
quantity, q_sR, on the relative stability between parallel and perpendicular
orientations of the lamellar phase. The competition between film confinement,
energy cost of elastic deformation and gain in surface energy induces a
parallel-to-perpendicular transition of the BCP lamellae. Employing
self-consistent field theory (SCFT), we study the critical substrate roughness
value corresponding to this transition. The critical value increases as
function of the surface preference towards one of the two BCP components, and
as function of film thickness. But, it decreases with increasing values of the
Flory-Huggins parameter. Our findings are equivalent to stating that the
critical value decreases as the BCP molecular weight or the natural BCP
periodicity increases. We further show that the rough substrate can overcome
the formation of parallel lamellae in cases where the top surface has a
preference towards one of the two BCP components. Our results are in good
agreement with previous experiments, and highlight the physical conditions
behind the perpendicular orientation of lamellar phases, as is desired in
nanolithography and other industrial applications.
