The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. I. Droplet Density

Dorian Goldman, Cyrill B. Muratov, Sylvia Serfaty

Research output: Contribution to journalArticle

Abstract

This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on 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 this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.

Original languageEnglish (US)
Pages (from-to)581-613
Number of pages33
JournalArchive for Rational Mechanics and Analysis
Volume210
Issue number2
DOIs
StatePublished - Nov 2013

Fingerprint

Droplet
Charge density
Energy
Minimizer
Diffuse Interface
Energy Functional
Charge
Block copolymers
Volume fraction
Copolymer
Ginzburg-Landau
Rescaling
Weak Convergence
Volume Fraction
Model
Upper and Lower Bounds
Radius
kernel
Converge
Series

ASJC Scopus subject areas

  • Analysis
  • Mechanical Engineering
  • Mathematics (miscellaneous)

Cite this

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

In: Archive for Rational Mechanics and Analysis, Vol. 210, No. 2, 11.2013, p. 581-613.

Research output: Contribution to journalArticle

Goldman, Dorian ; Muratov, Cyrill B. ; Serfaty, Sylvia. / The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. I. Droplet Density. In: Archive for Rational Mechanics and Analysis. 2013 ; Vol. 210, No. 2. pp. 581-613.
@article{d12d755b6d7943c0b7e4590b23c5cd4c,
title = "The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. I. Droplet Density",
abstract = "This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on 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 this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.",
author = "Dorian Goldman and Muratov, {Cyrill B.} and Sylvia Serfaty",
year = "2013",
month = "11",
doi = "10.1007/s00205-013-0657-1",
language = "English (US)",
volume = "210",
pages = "581--613",
journal = "Archive for Rational Mechanics and Analysis",
issn = "0003-9527",
publisher = "Springer New York",
number = "2",

}

TY - JOUR

T1 - The Γ-Limit of the Two-Dimensional Ohta-Kawasaki Energy. I. Droplet Density

AU - Goldman, Dorian

AU - Muratov, Cyrill B.

AU - Serfaty, Sylvia

PY - 2013/11

Y1 - 2013/11

N2 - This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on 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 this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.

AB - This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion, also known as the Ohta-Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on 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 this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta-Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.

UR - http://www.scopus.com/inward/record.url?scp=84884162891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884162891&partnerID=8YFLogxK

U2 - 10.1007/s00205-013-0657-1

DO - 10.1007/s00205-013-0657-1

M3 - Article

AN - SCOPUS:84884162891

VL - 210

SP - 581

EP - 613

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -