Thermodynamically Consistent Algorithms for Models of Diblock Copolymer Solutions Interacting with Electric and Magnetic Fields

TL;DR

This paper develops thermodynamically consistent models for diblock copolymer solutions interacting with electric and magnetic fields, along with energy-preserving numerical schemes and numerical validation of pattern formation and field effects.

Contribution

It introduces thermodynamically consistent models coupled with electric and magnetic fields and designs energy-preserving numerical schemes for these complex systems.

Findings

Numerical schemes preserve energy dissipation rates.

Electric and magnetic fields influence pattern formation.

Hysteresis effects observed under external fields.

Abstract

We derive thermodynamically consistent models for diblock copolymer solutions coupled with the electric and magnetic field, respectively. These models satisfy the second law of thermodynamics and therefore are therefore thermodynamically consistent. We then design a set of 2nd order, linear, semi-discrete schemes for the models using the energy quadratization method and the supplementary variable method, respectively, which preserve energy dissipation rates of the models. The spatial discretization is carried out subsequently using 2nd order finite difference methods, leading to fully discrete algorithms that preserve discrete energy-dissipation-rates of the models so that the resulting fully discrete models are thermodynamically consistent. Convergence rates are numerically confirmed through mesh refinement tests and several numerical examples are given to demonstrate the role of the mobility in pattern formation, defect removing effect of both electric and magnetic fields as well as the hysteresis effect for applied external fields in copolymer solutions.