Droplet phase in a nonlocal isoperimetric problem under confinement

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 unique nondegenerate maximum, we present a two-scale 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.