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Analytic Structure of the SCFT Energy Functional of Multicomponent Block Copolymers

Published online by Cambridge University Press:  03 June 2015

Kai Jiang ,
Weiquan Xu  and
Pingwen Zhang
Kai Jiang
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P.R. China LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Weiquan Xu
Affiliation:
LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Pingwen Zhang*
Affiliation:
LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
*
*Corresponding author. Email addresses: kaijiang@xtu.edu.cn (K. Jiang), weiquanxu1986@gmail.com (W. Xu), pzhang@pku.edu.cn (P. Zhang)
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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 caused by the incompressible 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 firstly 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 incompressible 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 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.

Keywords

82.35.JK82.20.Wt83.80.Uv83.80.SgMulticomponent block copolymerssaddle point approximationSCFT energy functionalgradient-based iterative methodsABC star triblock copolymers
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 5 , May 2015 , pp. 1360 - 1387
Copyright
Copyright © Global-Science Press 2015 

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