Some mathematical aspects of the micro-phase separation in diblock copolymers

A free energy functional of nonlocal type is considered that was originally introduced to describe the micro-phase separation of diblock copolymer. A mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology. The scaling law is equivalent to finding a nice rescaling in order to have a well-defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer, and the associated stationary problem becomes a morphology equation that governs the configuration of final patterns. Although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nature, a possible rigorous approach is presented to stability and morphological selection problems for them.