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

Abstract

Block copolymers provide a wonderful platform in the study of 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 precision, 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. More significantly, the proposed numerical framework can be applied to study the phase behaviors of block copolymers on general surfaces accurately.