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

Dorian Goldman, Cyrill B. Muratov, Sylvia Serfaty

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)445-501
Number of pages57
JournalArchive for Rational Mechanics and Analysis
Volume212
Issue number2
DOIs
StatePublished - May 2014

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Droplet
Arrangement
Energy
Minimizer
Block copolymers
Minimal Energy
Volume fraction
Vortex flow
Roundness
Ginzburg-Landau Model
Ergodic Theorem
Copolymer
Euler-Lagrange Equations
Ginzburg-Landau
Triangular Lattice
Volume Fraction
Level Set
Vortex
Express
Limiting

ASJC Scopus subject areas

  • Analysis
  • Mechanical Engineering
  • Mathematics (miscellaneous)

Cite this

The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. Droplet Arrangement via the Renormalized Energy. / Goldman, Dorian; Muratov, Cyrill B.; Serfaty, Sylvia.

In: Archive for Rational Mechanics and Analysis, Vol. 212, No. 2, 05.2014, p. 445-501.

Research output: Contribution to journalArticle

Goldman, Dorian ; Muratov, Cyrill B. ; Serfaty, Sylvia. / The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. Droplet Arrangement via the Renormalized Energy. In: Archive for Rational Mechanics and Analysis. 2014 ; Vol. 212, No. 2. pp. 445-501.
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