A globally convergent modified Newton method for the direct minimization of the Ohta-Kawasaki energy with application to the directed self-assembly of diblock copolymers

TL;DR

This paper introduces a fast, robust, and globally convergent modified Newton method for minimizing the Ohta-Kawasaki energy, significantly improving efficiency in modeling diblock copolymer microphase separation and self-assembly.

Contribution

It develops a novel, mass-conservative, energy-descending Newton scheme with line search for direct energy minimization, outperforming gradient flow methods in efficiency.

Findings

The method is asymptotically quadratically convergent.

It is three orders of magnitude more efficient than gradient flow approaches.

Numerical results demonstrate effective modeling of diblock copolymer self-assembly.

Abstract

We propose a fast and robust scheme for the direct minimization of the Ohta-Kawasaki energy that characterizes the microphase separation of diblock copolymer melts. The scheme employs a globally convergent modified Newton method with line search which is shown to be mass-conservative, energy-descending, asymptotically quadratically convergent, and three orders of magnitude more efficient than the commonly-used gradient flow approach. The regularity and the first-order condition of minimizers are analyzed. A numerical study of the chemical substrate guided directed self-assembly of diblock copolymer melts, based on a novel polymer-substrate interaction model and the proposed scheme, is provided.