Ring pattern solutions of a free boundary problem in diblock copolymer morphology

The cross section of a diblock copolymer in the cylindrical phase is made up of a large number of microdomains of small discs with high concentration of the minority monomers. Often several ring like microdomains appear among the discs. We show that a ring like structure may exist as a stable solution of a free boundary problem derived from the Ohta-Kawasaki theory of diblock copolymers. The existence of such a stable, single ring structure explains why rings exist for a long period of time before they eventually disappear or become discs in a diblock copolymer. A variant of Lyapunov-Schmidt reduction process is carried out that rigorously reduces the free boundary problem to a finite dimensional problem. The finite dimensional problem is solved numerically. A stability criterion on the parameters determines whether the ring solution is stable.