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

Full Finite Element Scheme for Reaction-Diffusion Systems on Embedded Curved Surfaces in ℝ 3

“…In many of the previous references, the system of equations is solved numerically using the finite difference method; in some cases, the finite element is used but with automatic meshers. Recently, there are some proposals for extensions of finite element methods to solve this kind of systems [19,20].…”

An adaptive high-order surface finite element method for the self-consistent field theory on general curved surfaces

“…Geometric constraints drastically affect the formation of ordered structures, under which some traditional ordered phases of block copolymers are rearranged to form novel patterns [1,2,3,4,5,6]. Theories play an important role in understanding and predicting the phase behavior of block copolymers under geometrical confinements [7,8,9,10]. Among these theories, self-consistent field theory (SCFT) [11] is one of the most powerful tools in studying the self-assembly behaviors of inhomogeneous polymers and related soft-matter systems.…”

Self-consistent field theory of inhomogeneous homopolymers and grafted polymers

“…Motivated by these factors, some researchers developed real space formulations of the SCFT problem using the finite element method. [155][156][157] Even so, there are several aspects in which state-of-the-art SCFT combined with FEM (SCFFEM) literature is lacking:…”

“…Solving this problem analytically requires rough approximations, compelling analytic solutions to be quite limited in applicability, because they demand the system to have very particular properties. 155 In most cases of interest, numerical methods are required to solve the set of integro-differential equations.…”

“…They investigated the details of the implementation on block copolymer systems of self-assembled structures in complex geometries. Huayi Wei et al 155 proposed a linear surface FEM to solve the SCFT model and studied the self-assembly behaviors of block copolymers on general curved surfaces. An essential point missing from both works is that they mainly illustrate confinement effects for specific geometries, without being concerned about how representative of the real world the system is.…”