Core shells and double bubbles in a weighted nonlocal isoperimetric problem

TL;DR

This paper investigates the shapes of phase-separated structures in a three-component copolymer model, showing how surface tensions influence the formation of double bubbles and core shells, including effects of nonlocal interactions.

Contribution

It provides a detailed analysis of how surface tension variations affect the geometry of minimizers in a weighted nonlocal isoperimetric problem, including the emergence of core shell structures.

Findings

Symmetric surface tensions lead to double bubble configurations.

Asymmetry in surface tensions results in asymmetric double bubbles.

Degenerate cases produce core shell geometries in nonlocal systems.

Abstract

We consider a sharp-interface model of $ABC$ triblock copolymers, for which the surface tension $\sigma_{ij}$ across the interface separating phase $i$ from phase $j$ may depend on the components. We study global minimizers of the associated ternary local isoperimetric problem in $\mathbb{R}^2$, and show how the geometry of minimizers changes with the surface tensions $\sigma_{ij}$, varying from symmetric double-bubbles for equal surface tensions, through asymmetric double bubbles, to core shells as the values of $\sigma_{ij}$ become more disparate. Then we consider the effect of nonlocal interactions in a droplet scaling regime, in which vanishingly small particles of two phases are distributed in a sea of the third phase. We are particularly interested in a degenerate case of $\sigma_{ij}$ in which minimizers exhibit core shell geometry, as this phase configuration is expected on physical grounds in nonlocal ternary systems.