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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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References

[1]Hamley, I. W. and Wiley, J., Developments in block copolymer science and technology, Wiley Online Library, 2004.Google Scholar
[2]Matsen, M. W., The standard Gaussian model for block copolymer melts, Journal of Physics: Condensed Matter, 14:R21–R47, 2002.Google Scholar
[3]Fredrickson, G. H., The equilibrium theory of inhomogeneous polymers, Oxford University Press, USA, 2006.Google Scholar
[4]Jiang, K.Huang, Y., and Zhang, P., Spectral method for exploring patterns of diblock copolymers, Journal of Computational Physics, 229:7796–7805, 2010.Google Scholar
[5]Jiang, K.Wang, C.Huang, Y., and Zhang, P., Discovery of new metastable patterns in diblock copolymers, Communications in Computational Physics, 14:443–460, 2013.Google Scholar
[6]Xu, W.Jiang, K.Zhang, P., and Shi, A. C., A strategy to explore stable and metastable ordered phases of block copolymers, The Journal of Physical Chemistry B, 117:5296–5305, 2013.Google Scholar
[7] J. Vavasour, D. and Whitmore, M. D., Self-consistent mean field theory of the microphases of diblock copolymers, Macromolecules, 25:5477–5486, 1992.Google Scholar
[8]Matsen, M. and Schick, W.M., Stable and unstable phases of a diblock copolymer melt, Physical Review Letters, 72:2660–2663, 1994.Google Scholar
[9]Drolet, F. and Fredrickson, G. H., Combinatorial screening of complex block copolymer assembly with self-consistent field theory, oPhysical Review Letters, 83:4317–4320, 1999.Google Scholar
[10]Rasmussen, K. Ø. and Kalosakas, G., Improved numerical algorithm for exploring block copolymer mesophases, Journal of Polymer Science Part B: Polymer Physics, 40:1777–1783, 2002.Google Scholar
[11]Guo, Z.Zhang, G.Qiu, F.Zhang, H.Yang, Y., and Shi, A. C., Discovering ordered phases of block copolymers: New results from a generic Fourier-space approach, Physical Review Letters, 101:28301, 2008.CrossRefGoogle Scholar
[12]Cochran, E. W.Garcia-Cervera, C. J., and Fredrickson, G. H., Stability of the gyroid phase in diblock copolymers at strong segregation, Macromolecules, 39:2449–2451, 2006.CrossRefGoogle Scholar
[13]Ranjan, A.Qin, J., and Morse, D. C., Linear response and stability of ordered phases of block copolymer melts, Macromolecules, 41:942–954, 2008.Google Scholar
[14]Thompson, R. B.Rasmussen, K.Ø., and Lookman, T., Improved convergence in block copolymer self-consistent field theory by Anderson mixing, The Journal of Chemical Physics, 120:31–34, 2004.Google Scholar
[15]Ceniceros, H. D. and Fredrickson, G. H., Numerical solution of polymer self-consistent field theory, Multiscale Modeling and Simulation, 2:452–474, 2004.Google Scholar
[16]Liang, Q.Jiang, K., and Zhang, P., Efficient numerical schemes for solving self-consistent field equations of flexible-semiflexible diblock copolymers, Accepted by Mathematical Methods in the Applied Sciences, 2013.Google Scholar
[17]Ganesan, V. and Fredrickson, G. H., Field-theoretic polymer simulations, Europhysics Letters, 55:814–820, 2001.Google Scholar
[18]Fredrickson, G. H.Ganesan, V., and Drolet, F., Field-theoretic computer simulation methods for polymers and complex fluids, Macromolecules, 35:16–39, 2002.Google Scholar
[19]Chaikin, P. and Lubensky, M.T. C., Principles of condensed matter physics, Cambridge University Press, 1995.CrossRefGoogle Scholar
[20]Reister, E.Müller, M., and Binder, K., Spinodal decomposition in a binary polymer mixture: dynamic self-consistent-field theory and Monte Carlo simulations, Physical Review E, 64:041804, 2001.Google Scholar
[21]Düchs, D., Ganesan, V., Fredrickson, G.H., and Schmid, F., Fluctuation effects in ternary AB+ A+B polymeric emulsions, Macromolecules, 36:9237–9248, 2003.Google Scholar
[22]Fraaije, J., Dynamic density functional theory for microphase separation kinetics of block copolymer melts, The Journal of Chemical Physics, 99:9202–9212, 1993.Google Scholar
[23]Fraaije, J.Van Vlimmeren, B. A. C.Maurits, N. M.Postma, M.Evers, O. A.Hoffmann, C.Altevogt, P., and Goldbeck-Wood, G., The dynamic mean-field density functional method and its application to the mesoscopic dynamics of quenched block copolymer melts. The Journal of Chemical Physics, 106:4260–4269, 1997.CrossRefGoogle Scholar
[24]Cheng, X.Lin, L.ZhangW. E, P. W. E, P., and Shi, A. C., Nucleation of ordered phases in block copolymers, Physical Review Letters, 104:148301, 2010.Google Scholar
[25]Bates, F. S.Hillmyer, M. A.Lodge, T. P.Bates, C. M.Delaney, K. T., and Fredrickson, G. H., Multiblock polymers: panacea or pandora’s box? Science, 336:434–440, 2012.Google Scholar
[26]Masuda, J.Takano, A.Nagata, Y.Noro, A., and Matsushita, Y., Nanophase-separated synchronizing structure with parallel double periodicity from an undecablock terpolymer, Physical Review Letters, 97:98301, 2006.Google Scholar
[27]Epps, T. H. III, Cochran, E. W., Bailey, T. S., Waletzko, R. S., Hardy, C. M., and Bates, F. S., Ordered network phases in linear poly (isoprene-b-styrene-b-ethylene oxide) triblock copolymers, Macromolecules, 37:8325–8341, 2004.Google Scholar
[28]Tang, P.Qiu, F.Zhang, H., and Yang, Y., Morphology and phase diagram of complex block copolymers: ABC star triblock copolymers, The Journal of Physical Chemistry B, 108:8434–8438, 2004.Google Scholar
[29]Rubinstein, M. and Colby, R., Polymer physics (chemistry), Oxford University Press, USA, 2003.Google Scholar
[30]Laplace, P. S., Memoir on the probability of the causes of events, Statistical Science, 1:364–378, 1986.Google Scholar
[31]Miller, P. D., Applied asymptotic analysis, American Mathematical Society, 2006.Google Scholar
[32]Bender, C.M. and Orszag, S. A.Advanced mathematical methods for scientists and engineers: Asymptotic methods and perturbation theory, Springer Verlag, 1999.Google Scholar
[33]Ellis, R. S. and Rosen, J. S., Asymptotic analysis of gaussian integrals, I: Isolated minimum points, Transactions of the American Mathematical Society, 273:447–481, 1982.Google Scholar
[34]Ellis, R. S. and Rosen, J. S., Asymptotic analysis of gaussian integrals, II: Manifold of minimum points, Communications in Mathematical Physics, 82:153–181, 1981.CrossRefGoogle Scholar
[35]Ellis, R. S. and Rosen, J. S., Laplace’s method for gaussian integrals with an application to statistical mechanics, The Annals of Probability, 10:47–66, 1982.Google Scholar
[36]Press, W. H., Numerical recipes: the art of scientific computing, Cambridge University Press, 2007.Google Scholar
[37]Frigo, M. and Johnson, S., Fftw: An adaptive software architecture for the FFT, ICASSP Conf. Proc., 3:1381–1384, 1998.Google Scholar
[38]Tang, P.Qiu, F.Zhang, H., and Yang, Y., Morphology and phase diagram of complex block copolymers: ABC linear triblock copolymers, Physical Review E, 69:31803, 2004.Google Scholar
[39]Tyler, C. A.Qin, J.Bates, F. S., and Morse, D. C., SCFT study of nonfrustrated ABC triblock copolymer melts, Macromolecules, 40:4654–4668, 2007.Google Scholar
[40]Liu, M.Li, W.Qiu, F., and Shi, A. C., Theoretical study of phase behavior of frustrated ABC linear triblock copolymers, Macromolecules, 45:9522–9530, 2012.Google Scholar
[41]Li, W. and Shi, A. C.Theory of hierarchical lamellar Structures from A(BC)nBA Multiblock Copolymers, Macromolecules, 42:811–819, 2009.Google Scholar
[42]Walker, H. F. and Ni, P., Anderson acceleration for fixed-point iterations, SIAM Journal on Numerical Analysis, 49:1715–1735, 2011.CrossRefGoogle Scholar