Ordering on a Thin Crystalline Substrate

We have so far considered the substrate to be infinitely extended both in the x- and in the y-axis directions. Actually, polymers form thin lamellae and the crystallization takes place on the narrow side surface of the lamellae. 

Since the interaction energy /s with the finite substrate is not easy to handle due to the loss of two dimensional periodicity assumed in the derivation of Eq. 5, we simplify the substrate interaction. We considered that the substrate was a sandwich of the amorphous and the crystalline layers, and the attractive potential Uo(z) works at any point (x,y), while the translational barrier Ui(z) cos(2nx/k) only works on the crystalline substrate. The implicit assumption is that the atomic densities of the crystal and the amorphous are not so different. 

---