Droplet phase in a nonlocal isoperimetric problem under confinement

Stan Alama; Lia Bronsard; Rustum Choksi; Ihsan Topaloglu

doi:10.3934/cpaa.2020010

Article Contents

2020, Volume 19, Issue 1: 175-202. Doi: 10.3934/cpaa.2020010

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Droplet phase in a nonlocal isoperimetric problem under confinement

1.
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada
2.
Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada
3.
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA

Received: July 31, 2018

Revised: February 28, 2019

Published: July 2019

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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 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.

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Mathematics Subject Classification: 35Q70, 49Q20, 49S05, 74N15, 82D60.

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Figure 1. The attraction to the origin and scaling at the rate $ \delta = \eta^{1/3} $

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Figure 2. Minimizing configurations of the second-order energy $ \mathsf{F}_{m^1,\dots,m^n} $ with equal mass $ m^i = 1/100 $ for 100 particles with varying powers $ q $ of degenerate penalization $ \rho(x)-\rho_0\sim |x|^q $. Minimizing configurations are obtained as steady-states of the gradient flow of the energy $ \mathsf{F}_{m^1,\dots,m^n} $

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