Double Fourier Sphere Methods with Low Rank Approximation for Block Copolymer Systems on Sphere

TL;DR

This paper develops spectral methods using the Double Fourier Sphere approach combined with low-rank approximation to efficiently simulate pattern formation in block copolymer systems on spherical domains, revealing patterns similar to biological membranes.

Contribution

It introduces a novel spectral discretization method with low-rank approximation for the OK and NO models on spheres, enabling efficient and accurate simulation of complex pattern formation.

Findings

Patterns resemble experimental biomembrane structures

Confirmed the two-thirds law relating bubbles to repulsive strength

Demonstrated the effectiveness of the spectral method for copolymer systems

Abstract

We introduce spectral methods for the Ohta-Kawasaki (OK) and Nakazawa-Ohta (NO) models on a spherical domain, examining their coarsening dynamics and equilibrium pattern formations. We employed the Double Fourier Sphere (DFS) method for spatial discretization and the second-order Backward Differentiation Formula (BDF2) scheme for time evolution, resulting in an efficient energy-stable scheme to simulate the OK and NO models on the unit sphere. Our numerical experiments reveal various self-assembled patterns, such as single-bubble assemblies in binary systems and double-bubble and mixed-bubble assemblies in ternary systems. These patterns closely resemble experimental biomembrane patterns, demonstrating the effectiveness of the OK model in real-world applications. Additionally, our study explores the relationship between repulsive strength and the number of bubbles in assemblies, confirming the two-thirds law in the OK model. This provides quantitative evidence of how self-assembled patterns depend on system parameters in copolymer systems.