Mathematical Aspects of Microphase Separation of Diblock Copolymers

Diblock copolymer melts, dubbed “designer materials”, have the remarkable ability to self-Bemble into various ordered structures. These structures are key to the many properties that make diblock copolymers of great technological interest. The density functional theory of Ohta and Kawasaki leads to anonlocal variational problem, and presents an excellent setting for the analysis of microphases. In this note we will first discuss the origins and derivation of this theory, presenting it in connection with the self-consistent mean field theory. Then, focusing on what is know as the strong segregation regime, we will discuss some analytical techniques which provide insight on the scales of minimizing structures (phases). These techniques have the advantage that they are ansatz-free, that is they are not based upon any preassigned bias for the phase geometry. In particular, we will derive ascaling law for the minimum energy in three space dimensions, and will address properties of optimal structures achieving this scaling law