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Self-assembly of ordered nanometer-scale patterns is interesting in itself,but its practical value depends on the ability to predict and control patternformation. In this paper we demonstrate theoretically and numerically thatengineering of extrinsic as well as intrinsic substrate geometry may providesuch a controllable ordering mechanism for block copolymers films. We developan effective two-dimensional model of thin films of striped-phase diblockcopolymers on general curved substrates. The model is obtained as an expansionin the film thickness and thus takes the third dimension into account, whichcrucially allows us to predict the preferred orientations even in the absenceof intrinsic curvature. We determine the minimum-energy textures on severalcurved surfaces and arrive at a general principle for using substrate curvatureas an ordering field, namely that the stripes will tend to align alongdirections of maximal curvature.

Substrate curvature governs texture orientation in thin films of smectic block copolymers