Analytic Structure of the SCFT Energy Functional of Multicomponent Block Copolymers

摘要

This paper concerns the analytic structure of the self-consistent fieldtheory (SCFT) energy functional of multicomponent block copolymer systems whichcontain more than two chemically distinct blocks. The SCFT has enjoyedconsidered success and wide usage in investigation of the complex phasebehavior of block copolymers. It is well-known that the physical solutions ofthe SCFT equations are saddle points, however, the analytic structure of theSCFT energy functional has received little attention over the years. A recentwork by Fredrickson and collaborators [see the monograph by Fredrickson, TheEquilibrium Theory of Inhomogeneous Polymers, (2006), pp. 203-209] has analysedthe mathematical structure of the field energy functional for polymericsystems, and clarified the index-1 saddle point nature of the problem producedby the incompressibility constraint. In this paper, our goals are to drawfurther attention to multicomponent block copolymers utilizing theHubbard-Stratonovich transformation used by Fredrickson and co-workers. Wefirst show that the saddle point character of the SCFT energy functional ofmulticomponent block copolymer systems may be high index, not only produced bythe incompressibility constraint, but also by the Flory-Huggins interactionparameters. Our analysis will be beneficial to many theoretical studies, suchas the nucleation theory of ordered phases, the mesoscopic dynamics. As anapplication, we then utilize the discovery to develop the gradient-basediterative schemes to solve the SCFT equations, and illustrate its performancethrough several numerical experiments taking ABC star triblock copolymers as anexample.