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

TL;DR

This paper develops a comprehensive diffuse interface model for diblock copolymers interacting with electric fields, demonstrating existence, uniqueness, regularization, and convergence to equilibrium of solutions.

Contribution

It introduces a coupled system of equations for copolymer and solvent dynamics with electric interactions, proving well-posedness and long-term behavior.

Findings

Existence of global weak solutions in 2D and 3D

Uniqueness in constant mobility case

Solutions converge to a stationary state with affine-linear permittivity

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.