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

Abstract

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.

A toroidal tube solution to a problem involving mean curvature and Newtonian potential

“…In R 2 there exists a ring shaped solution of the form {x ∈ R 2 : 0 < R 1 < |x| < R 2 }, if γ is greater than a threshold value γ 0 ; see Kang and Ren [14]. The inner and outer radii R 1 and R 2 of the ring solution depend on γ .…”

The Impact of the Domain Boundary on an Inhibitory System: Existence and Location of a Stationary Half Disc

“…(1.4); the intersection condition (1.5) is not needed. Fortunately there are many stationary sets whose interfaces do not meet the domain boundary [12,13,17,[19][20][21][22].…”

Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

“…All solutions to (1.10) in one dimension are known to be local minimizers of J B [26]. Many solutions in two and three dimensions have been found that match the morphological phases in diblock copolymers [24,30,29,31,32,15,16,33,35,39]. Global minimizers of J B are studied in [2,37,19,5,18,17,11] for various parameter ranges.…”