Analytic Structure of the SCFT Energy Functional of Multicomponent Block Copolymers (Communications in Computational Physics, 2018)
This paper concerns the analytic structure of the self-consistent field theory(SCFT) energy functional of multicomponent block copolymer systems which containmore than two chemically distinct blocks. The SCFT has enjoyed considered successand 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 attentionover the years. A recent work by Fredrickson and collaborators [see the monographby Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, (2006), pp.203–209] has analysed the mathematical structure of the field energy functional forpolymeric systems, and clarified the index-1 saddle point nature of the problem causedby the incompressible constraint. In this paper, our goals are to draw further attentionto multicomponent block copolymers utilizing the Hubbard-Stratonovich transformationused by Fredrickson and co-workers. We firstly show that the saddle point characterof the SCFT energy functional of multicomponent block copolymer systems may behigh index, not only produced by the incompressible constraint, but also by the Flory-Hugginsinteraction parameters. Our analysis will be beneficial to many theoreticalstudies, such as the nucleation theory of ordered phases, the mesoscopic dynamics.As an application, we utilize the discovery to develop the gradient-based iterativeschemes to solve the SCFT equations, and illustrate its performance through severalnumerical experiments taking ABC star triblock copolymers as an example.