SOME MATHEMATICAL ASPECTS OF THE MICROPHASE SEPARATION IN DIBLOCK COPOLYMERS

A free energy functional of nonlocal type is considered that was origi nally introduced to describe the micro-phase separation of diblock cop olymer. A mathematical framework is given to the issues of the scaling law for stationary states and the governing equation of morphology. T he scaling law is equivalent to finding a nice rescaling in order to h ave a well-defined limiting interfacial problem which is free from the interfacial thickness and the total chain length of copolymer, and th e associated stationary problem becomes a morphology equation that gov erns the configuration of final patterns. Although the steady patterns become finer and finer in our scaling limit due to the mesoscopic nat ure, a possible rigorous approach is presented to stability and morpho logical selection problems for them.