# A finite element method of the self-consistent field theory on general curved surfaces

## TL;DR

This paper introduces a finite element method for accurately simulating self-assembled block copolymer structures on arbitrary curved surfaces using self-consistent field theory, expanding the scope of numerical analysis in soft matter physics.

## Contribution

It develops a rigorous surface finite element method and an adaptive surface optimization approach for studying block copolymer phases on general curved surfaces.

## Key findings

- Efficient numerical method for complex curved surfaces.
- Ordered structures consistent with known results.
- Capable of analyzing phase behavior on arbitrary surfaces.

## Abstract

Block copolymers provide a wonderful platform in studying the soft condensed matter systems. Many fascinating ordered structures have been discovered in bulk and confined systems. Among various theories, the self-consistent field theory (SCFT) has been proven to be a powerful tool for studying the equilibrium ordered structures. Many numerical methods have been developed to solve the SCFT model. However, most of these focus on the bulk systems, and little work on the confined systems, especially on general curved surfaces. In this work, we developed a linear surface finite element method, which has a rigorous mathematical theory to guarantee numerical precsion, to study the self-assembled phases of block copolymers on general curved surfaces based on the SCFT. Furthermore, to capture the consistent surface for a given self-assembled pattern, an adaptive approach to optimize the size of the general curved surface has been proposed. To demonstrate the power of this approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces, including five closed surfaces and an unclosed surface. Numerical results illustrate the efficiency of the proposed method. The obtained ordered structures are consistent with the previous results on standard surfaces, such as sphere and torus. Certainly, the proposed numerical framework has the capability of studying the phase behaviors on general surfaces precisely.