Analytic Structure of the SCFT Energy Functional of Multicomponent Block Copolymers

TL;DR

This paper investigates the complex analytic structure of the SCFT energy functional for multicomponent block copolymers, revealing high-index saddle points influenced by interactions and constraints, and develops gradient-based numerical methods for solving these equations.

Contribution

It uncovers the high-index saddle point nature of the SCFT energy functional in multicomponent systems and introduces gradient-based iterative schemes for solving the equations.

Findings

High-index saddle points identified in the SCFT energy functional.

Gradient-based iterative schemes effectively solve SCFT equations.

Numerical experiments demonstrate the methods on ABC star triblock copolymers.

Abstract

This paper concerns the analytic structure of the self-consistent field theory (SCFT) energy functional of multicomponent block copolymer systems which contain more than two chemically distinct blocks. The SCFT has enjoyed considered success and wide usage in investigation of the complex phase behavior of block copolymers. It is well-known that the physical solutions of the SCFT equations are saddle points, however, the analytic structure of the SCFT energy functional has received little attention over the years. A recent work by Fredrickson and collaborators [see the monograph by Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, (2006), pp. 203-209] has analysed the mathematical structure of the field energy functional for polymeric systems, and clarified the index-1 saddle point nature of the problem produced by the incompressibility constraint. In this paper, our goals are to draw further attention to multicomponent block copolymers utilizing the Hubbard-Stratonovich transformation used by Fredrickson and co-workers. We first show that the saddle point character of the SCFT energy functional of multicomponent block copolymer systems may be high index, not only produced by the incompressibility constraint, but also by the Flory-Huggins interaction parameters. Our analysis will be beneficial to many theoretical studies, such as the nucleation theory of ordered phases, the mesoscopic dynamics. As an application, we then utilize the discovery to develop the gradient-based iterative schemes to solve the SCFT equations, and illustrate its performance through several numerical experiments taking ABC star triblock copolymers as an example.