Droplet phase in a nonlocal isoperimetric problem under confinement
Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloğlu · Communications on Pure & Applied Analysis · 2019
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 \begin{document}$ \eta $\end{document} to represent the size of the domains of the minority phase, and study the resulting droplet regime as \begin{document}$ \eta\to 0 $\end{document} . By considering confinement densities which are spatially variable and attain a unique nondegenerate maximum, we present a two-scale asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter \begin{document}$ M $\end{document} which gives the total volume of the droplets at order \begin{document}$ \eta^3 $\end{document} and its relation to existence and non-existence of Gamow's Liquid Drop model on \begin{document}$ \mathbb{R}^3 $\end{document} . For large values of \begin{document}$ M $\end{document} , the minority phase splits into several droplets at an intermediate scale \begin{document}$ \eta^{1/3} $\end{document} , while for small \begin{document}$ M $\end{document} minimizers form a single droplet converging to the maximum of the confinement density.