Theory of Anisotropic Fluctuations in Ordered Block Copolymer Phases

A general theoretical framework for the study of anisotropic composition fluctuations about an ordered block copolymer phase is developed. The approach is based on the idea that, in order to study the effects of fluctuations around an ordered broken symmetry phase, the theory must be formulated as a self-consistent expansion around the mean-field solution of this ordered state. A random phase approximation treatment of the theory leads to anisotropic correlation functions for the system. It is shown that the calculation of the polymer correlation functions in an ordered phase is equivalent to the calculation of the energy bands and eigenfunctions for an electron in a periodic potential. This general method is applied to the lamellar phase of block copolymers. The calculated anisotropic scattering intensity captures the main features observed experimentally, including the secondary peaks due to fluctuations with hexagonal symmetry. The origin of the anisotropic fluctuations can be traced to the formation of “energy” bands, similar to electronic states in solids.