Droplet phase in a nonlocal isoperimetric problem under confinement

TL;DR

This paper investigates the asymptotic behavior of a nonlocal isoperimetric problem modeling self-assembly of diblock copolymers under confinement, revealing how droplet configurations depend on volume and confinement strength.

Contribution

It introduces a two-stage asymptotic analysis of droplet formation under variable confinement, connecting droplet volume to the existence of solutions and droplet splitting behavior.

Findings

Large droplet volume leads to multiple droplets at intermediate scales.

Small droplet volume results in a single droplet at the confinement maximum.

The analysis links droplet configurations to Gamow's Liquid Drop model.

Abstract

We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter $\eta$ to represent the size of the domains of the minority phase, and study the resulting droplet regime as $\eta\to 0$. By considering confinement densities which are spatially variable and attain a nondegenerate maximum, we present a two-stage asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter $M$ which gives the total volume of the droplets at order $\eta^3$ and its relation to existence and non-existence of Gamow's Liquid Drop model on $\mathbb{R}^3$. For large values of $M$, the minority phase splits into several droplets at an intermediate scale $\eta^{1/3}$, while for small $M$ minimizers form a single droplet converging to the maximum of the confinement density.