Diblock Copolymer Thin Films on Heterogeneous Striped Surfaces: Commensurate, Incommensurate and Inverted Lamellae

We study a symmetric melt $\mathrm{AB}$ diblock copolymer thin film which is placed on a periodically striped $\mathrm{CD}$ surface. Because of different surface energies each stripe has a different preference for each of the blocks. There is thus a competition between the preferred bulk lamellar spacing, $L_b$, and the width of the stripes, $\lambda$. This system is similar to the Frenkel-Kontorowa model of solid state physics. We consider the case $L_b\ge \lambda$, and assume perpendicular alignment. The periodic striped surface will induce lamellae with unequal spacings. We also show that the striped potential can induce inverted bilayers, i.e., an $AB,AB$ pattern rather than the more usual $AB,BA$.