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On a diffuse interface model for diblock copolymers interacting with an electric field

Helmut Abels, Andrea Di Primio, Harald Garcke

Abstract

We consider a diffuse interface model describing a ternary system constituted by a conductive diblock copolymer and a homopolymer acting as solvent. The resulting dynamics is modeled by two Cahn--Hilliard--Oono equations for the copolymer blocks, accounting for long-range interactions; a classical Cahn--Hiliard equation for the homopolymer and the Maxwell equation for the electric displacement field. A multiphase singular potential is employed in order to ensure physical consistency. First, we show existence of global weak solutions in two and three dimensions. Uniqueness of weak solutions is established in the constant mobility case, and a conditional result is given in the general case. Instantaneous regularization and long-time behavior are also investigated, the latter in the case of affine-linear electric permittivity, showing in particular that solutions converge to a single stationary state.

On a diffuse interface model for diblock copolymers interacting with an electric field

Abstract

We consider a diffuse interface model describing a ternary system constituted by a conductive diblock copolymer and a homopolymer acting as solvent. The resulting dynamics is modeled by two Cahn--Hilliard--Oono equations for the copolymer blocks, accounting for long-range interactions; a classical Cahn--Hiliard equation for the homopolymer and the Maxwell equation for the electric displacement field. A multiphase singular potential is employed in order to ensure physical consistency. First, we show existence of global weak solutions in two and three dimensions. Uniqueness of weak solutions is established in the constant mobility case, and a conditional result is given in the general case. Instantaneous regularization and long-time behavior are also investigated, the latter in the case of affine-linear electric permittivity, showing in particular that solutions converge to a single stationary state.

Paper Structure

This paper contains 22 sections, 38 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Notation and functional setting
  4. Functional spaces.
  5. The Gibbs simplex.
  6. Weak Laplace-like operators.
  7. Basic assumptions.
  8. Main results.
  9. Proofs of Theorem \ref{['thm:wellposed']} and Theorem \ref{['thm:energyid']}
  10. Regularization of the singular potential.
  11. The semi-implicit time discretization.
  12. Uniform estimates.
  13. Higher integrability properties.
  14. Passage to the limit.
  15. Global solutions.
  16. ...and 7 more sections

Key Result

Lemma 2.1

If every component of $F(\boldsymbol s)$ is differentiable and there exists $C_F > 0$ such that then it holds for all $\boldsymbol \varphi_1,\,\boldsymbol \varphi_2 \in L^4(\Omega;\mathbb{R}^3)$ and $\boldsymbol \eta \in Y$.

Theorems & Definitions (83)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Corollary 2.8
  • ...and 73 more