Droplet Minimizers of an Isoperimetric Problem with Long‐Range Interactions

Abstract

Abstract. We give a detailed description of the geometry of single droplet patterns in a nonlocal isoperimetric problem. In particular we focus on the sharp interface limit of the OhtaKawasaki free energy for diblock copolymers, regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long-range interactions. Exploiting fine properties of the regularity theory for minimal surfaces, we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns. IntroductionIn several physical systems competing short-range attractive and long-range repulsive interactions often lead to the formation of mesoscopic scale patterns. Roughly speaking, the short-range interactions favor phase-separation on a microscopic scale, while the long-range ones frustrate such an ordering on the scale of the whole sample. When these systems can be described in terms of a free energy, such a phenomenon is usually referred to as an energy-driven pattern formation. Examples of energy-driven patterns are ubiquitous in physics: among the others we recall ferromagnetic and polymeric systems, type-I superconductor films and Langmuir layers. Even if these systems are driven by different physical laws, they exhibit remarkable similarities in the overall geometry of the formed patterns (see [33] and [46]).Our principal interest is the description of the geometry of patterns. For this reason, we leave for further studies the detailed analysis of more realistic systems and we focus here on an energy model which encodes only the main features of pattern-formation. More specifically, in what follows we are interested in the minimization of the following energy functional:where u is the order parameter of a two-phases system confined in Ω ⊂ R n , and γ and m are two nonnegative numerical parameters. The two terms in the energy mimic attractive short-range and repulsive long-range energies between the phases. More precisely the first term is local, it favors minimal interface area and drives the system toward a partition into few pure phases while the second term, involving a Coulomb-like kernel G, is non-local and favor a fine mixing of the phases. A detailed description of the energy is given in § 1. The competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns, whose geometry strongly depends on the choice of the parameters γ and m (e.g. spherical spots, cylinders, gyroids and lamellae). 0.1. The Ohta-Kawasaki functional for diblock copolymers. The model we consider arises as a simplification of a Ginzburg-Landau functional proposed by Ohta and Kawasaki in their pioneering paper [41] as a possible description of a diblock copolymers' (DBC) system. Even though it is questionable whether such an energy actually describes DBCs (see Choksi and Ren [15], Muratov [39] and Niethammer and Oshita [40]), nevertheless it is a first, and mathematically non-trivial, attempt to capture some...

Low Density Phases in a Uniformly Charged Liquid

“…Importantly, the model in (1.1) is a paradigm for the energy-driven pattern forming systems in which spatial patterns (global or local energy minimizers) form as a result of the competition of short-range attractive and long-range repulsive forces. This is why this model and its generalizations attracted considerable attention of mathematicians in recent years (see, e.g., [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], this list is certainly not exhaustive). In particular, the volume-constrained global minimization problem for (1.1) in the whole space with no neutralizing background, which we will also refer to as the "self-energy problem", has been investigated in [34,37,45,49].…”

On global minimizers for a variational problem with long-range interactions

“…Recently Cicalese and Spadaro [16] have given a rather detailed asymptotic description of the energy in (NLIP) in all space dimensions. They focus on regimes associated to a small volume fraction wherein the isoperimetric term is stronger than the nonlocal one and give a detailed description of the geometry of minimizers, focusing on a single droplet.…”

Rigidity of critical points for a nonlocal Ohta–Kawasaki energy

“…Such problem first appeared in the liquid drop model of the atomic nuclei proposed by Gamow in 1928 [13] and then developed by other researchers [3,9], and it is also relevant in some models of diblock copolymer melts [5,17]. In [15,16] (see also [6]) the authors showed that global minimizers of (1) exist if the volume m is small, and do not exist if the volume is large enough. They also showed that minimizers are necessarily balls if the volume is small enough.…”