Low Density Phases in a Uniformly Charged Liquid

Abstract

This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase.Under periodic boundary conditions, we prove that the considered energy Γ-converges to an energy functional of the limit "homogenized" measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplets per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

An Old Problem Resurfaces Nonlocally: Gamow's Liquid Drops Inspire Today's Research and Applications

“…While one may conjecture that it is possible to choose 0 = 1 = 2 , based on physical evidence, to date it remains open to prove or disprove whether any (or all) of the constants , = 0, 1, 2, may be pairwise equal. One result in that direction was obtained by Knüpfer, Muratov, and Novaga [4] in 2016, who show that the set ℐ = { > 0 ∶ ( ) is attained} is a closed (hence compact) subset of (0, ∞).…”

“…The energy in (5) was introduced in 1986 by Ohta and Kawasaki as a simple model for self-assembly of diblock copolymer melts, even if its nuclear physics analog goes back all the way to the classical 1935 paper of von Weizsäcker. It can also be considered as a tool for numerical studies of (4).…”

“…This separation has been exploited in a number of papers by Ren and Wei (2007,2008,2009) and by Cristoferi (2016) for constructing critical points of variants of (4) perturbatively. To gain further insights into this separation, an asymptotic analysis was presented by Choksi and Peletier [2], [3] and by Knüpfer, Muratov, and Novaga [4] for the droplet regimes wherein → 0. Here, Gamow's liquid drop model (on all ℝ 3 ) emerged as the leading-order energy describing the shape of a single droplet, ignoring the next-order effects of the neighboring droplets.…”

“…Finally, referring back to the third purpose of the liquid drop model, existence of the tightest bound nucleus was established in 2016 by Knüpfer, Muratov, and Novaga [4], who show that there exists * ∈ ℐ such that * ∶= inf…”

On Equilibrium Shape of Charged Flat Drops

“…In particular, the non-local isoperimetric problems governed by (1.1), or (1.8), or their generalizations involving Riesz energy, in which the measure µ is fixed to be the uniform measure, was recently treated in [9,23,25,26,[35][36][37]41,44] (see also [2,16,31,32,38] for closely related problems on large bounded domains; these lists are by far not meant to be exhaustive). One of the motivations for those studies comes from the fact that the three-dimensional problem with uniform charge density corresponds to the famous model of the atomic nucleus introduced by Gamow [27] and is used to explain the stability of nuclear matter and nuclear fission [8,52].…”