The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy

Abstract

This is the second in a series of papers in which we derive a Γ-expansion for the two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion known as the Ohta–Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In our previous paper, we computed the Γ-limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order Γ-limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg–Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Thus, without appealing to the Euler–Lagrange equation, we establish for all configurations which have “almost minimal energy” the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of Γ-equivalence, the obtained results also yield an expansion of the minimal energy and a characterization of the zero super-level sets of the minimizers for the original Ohta–Kawasaki energy. This leads to the expectation of seeing triangular lattices of droplets as energy minimizers.