Origin of epitaxies between ordered phases of block copolymers

The origin of epitaxial relationships between different ordered phases of diblock copolymers is studied by examining the nucleation barrier of an arbitrarily oriented nucleus. It is found that the orientation-dependence of the nucleation barrier exhibits a very corrugated landscape with a few deep minima, corresponding to preferred crystallographic orientations of the nuclei. These minima are identified as the epitaxies between the ordered phases. Examination of the nucleation pathways reveals that the origin of the epitaxial relationships can be attributed to the matching of the dominant Fourier components of the density distributions. The predicted epitaxial relationships among lamellae, cylinders, spheres and gyroid phases are in excellent agreement with previous experimental and theoretical observations. Furthermore, new expitaxial relationships between the Fddd and lamellar phases are revealed by the theory.