High-accurate and efficient numerical algorithms for the self-consistent field theory of liquid-crystalline polymers

TL;DR

This paper develops high-order, efficient numerical algorithms to solve the complex six-dimensional PDEs in self-consistent field theory for liquid crystalline polymers, enabling detailed simulations of their self-assembly behaviors.

Contribution

It introduces advanced PDE solvers, an improved Anderson iteration, and domain adjustment techniques specifically for SCFT of liquid crystalline polymers, enhancing computational efficiency and accuracy.

Findings

Efficient algorithms significantly reduce computation time.

Simulations reveal complex 3D self-assembly structures.

Methods enable exploration of high-dimensional polymer behaviors.

Abstract

Self-consistent field theory (SCFT) is one of the most widely-used framework in studying the equilibrium phase behaviors of inhomogenous polymers. For liquid crystalline polymeric systems, the main numerical challenges of solving SCFT encompass efficiently solving plenty of six dimensional partial differential equations (PDEs), precisely determining the subtle energy difference among self-assembled structures, and developing effective iterative methods for nonlinear SCF iteration. To address these challenges, this work introduces a suite of high-order and efficient numerical methods tailored for SCFT of liquid-crystalline polymers. These methods include various advaced PDE solvers, an improved Anderson iteration algorithm to accelerate SCFT calculations, and an optimization technique of adjusting the computational domain during the SCF iterations. Extensive numerical tests demonstrate the efficiency of the proposed methods. Based on these algorithms, we further explore the self-assembly behavior of liquid crystalline polymers through simulations in four, five, and six dimensions, uncovering intricate three-dimensional spatial structures.