Citation: Jun-Qing Song, Yi-Xin Liu, Hong-Dong Zhang. An Efficient Algorithm for Self-consistent Field Theory Calculations of Complex Self-assembled Structures of Block Copolymer Melts[J]. Chinese Journal of Polymer Science, ;2018, 36(4): 488-496. doi: 10.1007/s10118-018-2037-7 shu

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

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

Corresponding author: Yi-Xin Liu, lyx@fudan.edu.cn
Received Date: 2 July 2017
Accepted Date: 8 September 2017
Available Online: 29 December 2017

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.
- Block copolymer,
- Self-consistent field theory,
- Algorithm,
- Pseudo-spectral,
- Phase structure
1. [1]
  Edwards S. F.. Statistical mechanics of polymer with excluded volume[J]. Proc. Phys. Soc. London, 1965,85(546P):613-624.
2. [2]
  de Gennes, P. G. "Scaling concepts in polymer physics", Cornell University Press, Ithaca 1969.
3. [3]
  Helfand E.. Block copolymer theory. 3. Statistical-mechanics of microdomain structure[J]. Macromolecules, 1975,8(4):552-556. doi: 10.1021/ma60046a032
4. [4]
  Hong K. M., Noolandi J.. Theory of inhomogeneous multicomponent polymer systems[J]. Macromolecules, 1981,14(3):727-736. doi: 10.1021/ma50004a051
5. [5]
  Fredrickson, G. H. "The equilibrium theory of inhomogeneous polymers", Oxford University Press, New York 2006.
6. [6]
  Matsen M. W.. Undulation instability in block-copolymer lamellae subjected to a perpendicular electric field[J]. Soft Matter, 2006,2(12):1048-1056. doi: 10.1039/b611064a
7. [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?[J]. Science, 2012,336(6080):434-440. doi: 10.1126/science.1215368
8. [8]
  Matsen M. W., Thompson R. B.. Equilibrium behavior of symmetric ABA triblock copolymer melts[J]. J. Chem. Phys., 1999,111(15):7139-7146. doi: 10.1063/1.480006
9. [9]
  Tang P., Qiu F., Zhang H. D., Yang Y. L.. Morphology and phase diagram of complex block copolymers:ABC star triblock copolymers[J]. J. Phys. Chem. B, 2004,108(24):8434-8438. doi: 10.1021/jp037911q
10. [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]. J. Am. Chem. Soc., 2014,136(8):2974-2977. doi: 10.1021/ja412760k
11. [11]
  Duchs D., Sullivan D. E.. Entropy-induced smectic phases in rod-coil copolymers[J]. J. Phys. Condens. Matter, 2002,14(46):12189-12202. doi: 10.1088/0953-8984/14/46/321
12. [12]
  Matsen M. W.. Thin films of block copolymer[J]. J. Chem. Phys., 1997,106(18):7781-7791. doi: 10.1063/1.473778
13. [13]
  Leibler L.. Theory of microphase separation in block co-polymers[J]. Macromolecules, 1980,13(6):1602-1617. doi: 10.1021/ma60078a047
14. [14]
  Semenov A. N.. Contribution to the theory of microphase layering in block-copolymer melts[J]. Zh. Eksp. Teor. Fiz., 1985,88(4):1242-1256.
15. [15]
  Matsen M. W., Schick M.. Stable and unstable phases of a diblock copolymer melt[J]. Phys. Rev. Lett., 1994,72(16):2660-2663. doi: 10.1103/PhysRevLett.72.2660
16. [16]
  Drolet F., Fredrickson G. H.. Combinatorial screening of complex block copolymer assembly with self-consistent field theory[J]. Phys. Rev. Lett., 1999,83(21):4317-4320. doi: 10.1103/PhysRevLett.83.4317
17. [17]
  Rasmussen K. O., Kalosakas G.. Improved numerical algorithm for exploring block copolymer mesophases[J]. J. Polym. Sci., Part B:Polym. Phys., 2002,40(16):1777-1783. doi: 10.1002/(ISSN)1099-0488
18. [18]
  Cochran E. W., Garcia-Cervera C. J., Fredrickson G. H.. Stability of the gyroid phase in diblock copolymers at strong segregation[J]. Macromolecules, 2006,39(7):2449-2451. doi: 10.1021/ma0527707
19. [19]
  Ranjan A., Qin J., Morse D. C.. Linear response and stability of ordered phases of block copolymer melts[J]. Macromolecules, 2008,41(3):942-954. doi: 10.1021/ma0714316
20. [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]. J. Phys. Chem. B, 2011,115(39):11307-11317. doi: 10.1021/jp204904b
21. [21]
  Stasiak P., Matsen M. W.. Efficiency of pseudo-spectral algorithms with Anderson mixing for the SCFT of periodic block-copolymer phases[J]. Eur. Phys. J. E, 2011,34(10)110. doi: 10.1140/epje/i2011-11110-0
22. [22]
  Liu Y. X., Zhang H. D.. Exponential time differencing methods with Chebyshev collocation for polymers confined by interacting surfaces[J]. J. Chem. Phys., 2014,140(22)224101. doi: 10.1063/1.4881516
23. [23]
  Cox S. M., Matthews P. C.. Exponential time differencing for stiff systems[J]. J. Comput. Phys., 2002,176(2):430-455. doi: 10.1006/jcph.2002.6995
24. [24]
  Kassam A. K., Trefethen L. N.. Fourth-order time-stepping for stiff PDEs[J]. SIAM J. Sci. Comput.,, 2005,26(4):1214-1233. doi: 10.1137/S1064827502410633
25. [25]
  Krogstad, S. "Topics in numerical Lie group integration", Ph. D. thesis, The University of Bergen, 2003.
26. [26]
  Matsen M. W., Bates F. S.. Block copolymer microstructures in the intermediate-segregation regime[J]. J. Chem. Phys., 1997,106(6):2436-2448. doi: 10.1063/1.473153
27. [27]
  Oberkampf, W. L.; Roy, C. J. "Verification and validation for scientific computing", Cambridge University Press, New York, 2010.
28. [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[J]. Macromolecules, 1986,19(8):2197-2202. doi: 10.1021/ma00162a016
29. [29]
  Matsen M. W., Bates F. S.. Origins of complex self-assembly in block copolymers[J]. Macromolecules, 1996,29(23):7641-7644. doi: 10.1021/ma960744q
30. [30]
  Matsen M. W.. Fast and accurate SCFT calculations for periodic block-copolymer morphologies using the spectral method with Anderson mixing[J]. Eur. Phys. J. E,, 2009,30(4):361-369. doi: 10.1140/epje/i2009-10534-3
31. [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[J]. Macromolecules, 2004,37(22):8325-8341. doi: 10.1021/ma048762s
32. [32]
  Bailey T. S., Hardy C. M., Epps T. H., Bates F. S.. A noncubic triply periodic network morphology in poly(isoprene-b-styrene-b-ethylene oxide) triblock copolymers[J]. Macromolecules, 2002,35(18):7007-7017. doi: 10.1021/ma011716x
33. [33]
  Tyler C. A., Morse D. C.. Orthorhombic Fddd network in triblock and diblock copolymer melts[J]. Phys. Rev. Lett., 2005,94(20)208302. doi: 10.1103/PhysRevLett.94.208302
34. [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.
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