Skip to main content
SpringerLink
Log in

An Efficient Algorithm for Self-consistent Field Theory Calculations of Complex Self-assembled Structures of Block Copolymer Melts

  • Article
  • Published:
Chinese Journal of Polymer Science Aims and scope Submit manuscript

Abstract

Self-consistent field theory (SCFT), as a state-of-the-art technique for studying the self-assembly of block copolymers, is attracting continuous efforts to improve its accuracy and efficiency. Here we present a fourth-order exponential time differencing Runge-Kutta algorithm (ETDRK4) to solve the modified diffusion equation (MDE) which is the most time-consuming part of a SCFT calculation. By making a careful comparison with currently most efficient and popular algorithms, we demonstrate that the ETDRK4 algorithm significantly reduces the number of chain contour steps in solving the MDE, resulting in a boost of the overall computation efficiency, while it shares the same spatial accuracy with other algorithms. In addition, to demonstrate the power of our ETDRK4 algorithm, we apply it to compute the phase boundaries of the bicontinuous gyroid phase in the strong segregation regime and to verify the existence of the triple point of the O70 phase, the lamellar phase and the cylindrical phase.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Edwards, S. F. Statistical mechanics of polymer with excluded volume. Proc. Phys. Soc. London 1965, 85 (546P), 613–624.

    Article  CAS  Google Scholar 

  2. de Gennes, P. G. "Scaling concepts in polymer physics", Cornell University Press, Ithaca 1969.

    Google Scholar 

  3. Helfand, E. Block copolymer theory. 3. Statistical-mechanics of microdomain structure. Macromolecules 1975, 8(4), 552–556.

    Article  Google Scholar 

  4. Hong, K. M.; Noolandi, J. Theory of inhomogeneous multicomponent polymer systems. Macromolecules 1981, 14(3), 727–736.

    Article  CAS  Google Scholar 

  5. Fredrickson, G. H. "The equilibrium theory of inhomogeneous polymers", Oxford University Press, New York 2006.

    Google Scholar 

  6. Matsen, M. W. Undulation instability in block-copolymer lamellae subjected to a perpendicular electric field. Soft Matter 2006, 2(12), 1048–1056.

    Article  CAS  Google Scholar 

  7. Bates, F. S.; Hillmyer, M. A.; Lodge, T. P.; Bates, C. M.; Delaney, K. T.; Fredrickson, G. H. Multiblock polymers: panacea or pandora’s box? Science 2012, 336(6080), 434–440.

    Article  CAS  Google Scholar 

  8. Matsen, M. W.; Thompson, R. B. Equilibrium behavior of symmetric ABA triblock copolymer melts. J. Chem. Phys. 1999, 111(15), 7139–7146.

    Article  CAS  Google Scholar 

  9. Tang, P.; Qiu, F.; Zhang, H. D.; Yang, Y. L. Morphology and phase diagram of complex block copolymers: ABC star triblock copolymers. J. Phys. Chem. B 2004, 108(24), 8434–8438.

    Article  CAS  Google Scholar 

  10. Xie, N.; Liu, M. J.; Deng, H. L.; Li, W. H.; Qiu, F.; Shi, A. C. Macromolecular metallurgy of binary mesocrystals via designed multiblock terpolymers. J. Am. Chem. Soc. 2014, 136(8), 2974–2977.

    Article  CAS  Google Scholar 

  11. Duchs, D.; Sullivan, D. E. Entropy-induced smectic phases in rod-coil copolymers. J. Phys. Condens. Matter 2002, 14(46), 12189–12202.

    Article  CAS  Google Scholar 

  12. Matsen, M. W. Thin films of block copolymer. J. Chem. Phys. 1997, 106(18), 7781–7791.

    Article  CAS  Google Scholar 

  13. Leibler, L. Theory of microphase separation in block co-polymers. Macromolecules 1980, 13(6), 1602–1617.

    Article  CAS  Google Scholar 

  14. Semenov, A. N. Contribution to the theory of microphase layering in block-copolymer melts. Zh. Eksp. Teor. Fiz. 1985, 88(4), 1242–1256.

    CAS  Google Scholar 

  15. Matsen, M. W.; Schick, M. Stable and unstable phases of a diblock copolymer melt. Phys. Rev. Lett. 1994, 72(16), 2660–2663.

    Article  CAS  Google Scholar 

  16. Drolet, F.; Fredrickson, G. H. Combinatorial screening of complex block copolymer assembly with self-consistent field theory. Phys. Rev. Lett. 1999, 83(21), 4317–4320.

    Article  CAS  Google Scholar 

  17. Rasmussen, K. O.; Kalosakas, G. Improved numerical algorithm for exploring block copolymer mesophases. J. Polym. Sci., Part B: Polym. Phys. 2002, 40(16), 1777–1783.

    Article  CAS  Google Scholar 

  18. Cochran, E. W.; Garcia-Cervera, C. J.; Fredrickson, G. H. Stability of the gyroid phase in diblock copolymers at strong segregation. Macromolecules 2006, 39(7), 2449–2451.

    Article  CAS  Google Scholar 

  19. Ranjan, A.; Qin, J.; Morse, D. C. Linear response and stability of ordered phases of block copolymer melts. Macromolecules 2008, 41(3), 942–954.

    Article  CAS  Google Scholar 

  20. Tong, C. H.; Zhu, Y. J.; Zhang, H. D.; Qiu, F.; Tang, P.; Yang, Y. L. The self-consistent field study of the adsorption of flexible polyelectrolytes onto two charged nano-objects. J. Phys. Chem. B 2011, 115(39), 11307–11317.

    Article  CAS  Google Scholar 

  21. Stasiak, P.; Matsen, M. W. Efficiency of pseudo-spectral algorithms with Anderson mixing for the SCFT of periodic block-copolymer phases. Eur. Phys. J. E 2011, 34(10), DOI: 10.1140/epje/i2011-11110-0

    Google Scholar 

  22. Liu, Y. X.; Zhang, H. D. Exponential time differencing methods with Chebyshev collocation for polymers confined by interacting surfaces. J. Chem. Phys. 2014, 140(22), DOI: 10.1063/1.4881516

    Google Scholar 

  23. Cox, S. M.; Matthews, P. C. Exponential time differencing for stiff systems. J. Comput. Phys. 2002, 176(2), 430–455.

    Article  CAS  Google Scholar 

  24. Kassam, A. K.; Trefethen, L. N. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput., 2005, 26(4), 1214–1233

    Article  Google Scholar 

  25. Krogstad, S. "Topics in numerical Lie group integration", Ph.D. thesis, The University of Bergen, 2003.

    Google Scholar 

  26. Matsen, M. W.; Bates, F. S. Block copolymer microstructures in the intermediate-segregation regime. J. Chem. Phys. 1997, 106(6), 2436–2448.

    Article  CAS  Google Scholar 

  27. Oberkampf, W. L.; Roy, C. J. "Verification and validation for scientific computing", Cambridge University Press, New York, 2010.

    Book  Google Scholar 

  28. Thomas, E. L.; Alward, D. B.; Kinning, D. J.; Martin, D. C.; Handlin, D. L.; Fetters, L. J. Ordered bicontinuous doublediamond structure of star block copolymers-a new equilibrium microdomain morphology. Macromolecules 1986, 19(8), 2197–2202.

    Article  CAS  Google Scholar 

  29. Matsen, M. W.; Bates, F. S. Origins of complex self-assembly in block copolymers. Macromolecules 1996, 29(23), 7641–7644.

    Article  CAS  Google Scholar 

  30. Matsen, M. W. Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing. Eur. Phys. J. E, 2009, 30(4), 361–369.

    Article  CAS  Google Scholar 

  31. Epps, T. H.; Cochran, E. W.; Bailey, T. S.; Waletzko, R. S.; Hardy, C. M.; Bates, F. S. Ordered network phases in linear poly(isoprene-b-styrene-b-ethylene oxide) triblock copolymers. Macromolecules 2004, 37(22), 8325–8341.

    Article  CAS  Google Scholar 

  32. Bailey, T. S.; Hardy, C. M.; Epps, T. H.; Bates, F. S. A noncubic triply periodic network morphology in poly(isoprene-bstyrene-b-ethylene oxide) triblock copolymers. Macromolecules 2002, 35(18), 7007–7017.

    Article  CAS  Google Scholar 

  33. Tyler, C. A.; Morse, D. C. Orthorhombic Fddd network in triblock and diblock copolymer melts. Phys. Rev. Lett. 2005, 94(20), DOI: 10.1103/PhysRevLett.94.208302

    Google Scholar 

  34. Press, W. H., Teukolsky, S. A. Vetterling, W. T. and Flannery, B. P., "Numerical recipes 3rd edition: The art of scientific computing", Cambridge University Press, New York 2007.

    Google Scholar 

Download references

Acknowledgments

This work was financially supported by the China Scholarship Council (No. 201406105018), the National Natural Science Foundation of China (No. 21004013) and the National Basic Research Program of China (No. 2011CB605701).

Author information

Authors and Affiliations

  1. State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai, 200433, China

    Jun-Qing Song, Yi-Xin Liu & Hong-Dong Zhang

Authors
  1. Jun-Qing Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi-Xin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong-Dong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi-Xin Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, JQ., Liu, YX. & Zhang, HD. An Efficient Algorithm for Self-consistent Field Theory Calculations of Complex Self-assembled Structures of Block Copolymer Melts. Chin J Polym Sci 36, 488–496 (2018). https://doi.org/10.1007/s10118-018-2037-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10118-018-2037-7

Keywords

Search

Navigation