Séminaire EDP-Analyse ICJ

# Droplet Phase for a Ternary Nonlocal Isoperimetric Problem

## by Stan Alama (McMaster University)

Europe/Paris
Fokko du Cloux (ICJ)

### Fokko du Cloux

#### ICJ

Description

We study the Nakazawa-Ohta ternary inhibitory system, which describes domain morphologies in a triblock copolymer as a nonlocal isoperimetric problem for three interacting phase domains. The free energy consists of two parts: the local interface energy measures the total perimeter of the phase boundaries, while a longer-range Coulomb interaction energy reflects the connectivity of the polymer chains and promotes splitting into micro-domains. We consider global minimizers on the two-dimensional torus, in a limit in which two of the species have vanishingly small mass but the interaction strength is correspondingly large. In this limit there is splitting of the masses, and each vanishing component rescales to a minimizer of an isoperimetric problem for clusters in $\mathbb{R}^2$. Depending on the relative strengths of the coefficients of the interaction terms we may see different structures for the global minimizers, ranging from a lattice of isolated simple droplets of each minority species to double-bubbles, and coexistence of droplets and double-bubbles. This represents work with L. Bronsard, X. Lu, and C. Wang.