  Self-assembly of ordered nanometer-scale patterns is interesting in itself,
but its practical value depends on the ability to predict and control pattern
formation. In this paper we demonstrate theoretically and numerically that
engineering of extrinsic as well as intrinsic substrate geometry may provide
such a controllable ordering mechanism for block copolymers films. We develop
an effective two-dimensional model of thin films of striped-phase diblock
copolymers on general curved substrates. The model is obtained as an expansion
in the film thickness and thus takes the third dimension into account, which
crucially allows us to predict the preferred orientations even in the absence
of intrinsic curvature. We determine the minimum-energy textures on several
curved surfaces and arrive at a general principle for using substrate curvature
as an ordering field, namely that the stripes will tend to align along
directions of maximal curvature.
