A Double Bubble Assembly as a New Phase of a Ternary Inhibitory System

Abstract

A ternary inhibitory system is a three component system characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In the meantime a longer ranging confinement mechanism prevents unlimited spreading. Together they lead to a locally self-enhancing and self-organizing process. The model considered here is a planar nonlocal geometric problem derived from the triblock copolymer theory. An assembly of perturbed double bubbles is mathematically constructed as a stable stationary point of the free energy functional. Triple junction, a phenomenon in which the three components meet at a single point, is a key issue addressed in the construction. Coarsening, an undesirable scenario of excessive growth, is prevented by a lower bound on the long range interaction term in the free energy. The proof involves several ideas: perturbation of double bubbles in a restricted class; use of internal variables to remove nonlinear constraints, local minimization in a restricted class formulated as a nonlinear problem on a Hilbert space; and reduction to finite dimensional minimization. This existence theorem predicts a new morphological phase of a double bubble assembly.