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

Jun-Qing Song; Yi-Xin Liu; Hong-Dong Zhang

doi:10.1007/s10118-018-2037-7

Your Location：

Home >

Browse articles >

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

Articles | Updated：2021-02-18

- An Efficient Algorithm for Self-consistent Field Theory Calculations of Complex Self-assembled Structures of Block Copolymer Melts
- Chinese Journal of Polymer Science Vol. 36, Issue 4, Pages: 488-496(2018)
- Affiliations：
  
  State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
- Author bio：
  
  Yi-Xin Liu, E-mail lyx@fudan.edu.cn
- Funds：
- DOI：10.1007/s10118-018-2037-7
  CLC：
- Received：02 July 2017，
  
  Accepted：08 September 2017，
  
  Online First：29 December 2017，
  
  Published：05 April 2018
- Accepted：
Scan QR Code
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：

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.

摘要

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 O

phase

the lamellar phase and the cylindrical phase.

关键词

Keywords

references

S. F. Edwards . Statistical mechanics of polymer with excluded volume . Proc. Phys. Soc. London , 1965 . 85 ( 546P ): 613 - 624 . http://iopscience.iop.org/article/10.1088/0370-1328/85/4/301 http://iopscience.iop.org/article/10.1088/0370-1328/85/4/301 , .

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

E. Helfand . Block copolymer theory. 3. Statistical-mechanics of microdomain structure . Macromolecules , 1975 . 8 ( 4 ): 552 - 556 . DOI: 10.1021/ma60046a032 http://doi.org/10.1021/ma60046a032 .

K. M. Hong , J. Noolandi . Theory of inhomogeneous multicomponent polymer systems . Macromolecules , 1981 . 14 ( 3 ): 727 - 736 . DOI: 10.1021/ma50004a051 http://doi.org/10.1021/ma50004a051 .

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

M. W. Matsen . Undulation instability in block-copolymer lamellae subjected to a perpendicular electric field . Soft Matter , 2006 . 2 ( 12 ): 1048 - 1056 . DOI: 10.1039/b611064a http://doi.org/10.1039/b611064a .

F. S. Bates , M. A. Hillmyer , T. P. Lodge , C. M. Bates , K. T. Delaney , G. H. Fredrickson . Multiblock polymers:panacea or pandora's box? . Science , 2012 . 336 ( 6080 ): 434 - 440 . DOI: 10.1126/science.1215368 http://doi.org/10.1126/science.1215368 .

M. W. Matsen , R. B. Thompson . Equilibrium behavior of symmetric ABA triblock copolymer melts . J. Chem. Phys. , 1999 . 111 ( 15 ): 7139 - 7146 . DOI: 10.1063/1.480006 http://doi.org/10.1063/1.480006 .

P. Tang , F. Qiu , H. D. Zhang , Y. L. Yang . Morphology and phase diagram of complex block copolymers:ABC star triblock copolymers . J. Phys. Chem. B , 2004 . 108 ( 24 ): 8434 - 8438 . DOI: 10.1021/jp037911q http://doi.org/10.1021/jp037911q .

N. Xie , M. J. Liu , H. L. Deng , W. H. Li , F. Qiu , A. C. Shi . Macromolecular metallurgy of binary mesocrystals via designed multiblock terpolymers . J. Am. Chem. Soc. , 2014 . 136 ( 8 ): 2974 - 2977 . DOI: 10.1021/ja412760k http://doi.org/10.1021/ja412760k .

D. Duchs , D. E. Sullivan . Entropy-induced smectic phases in rod-coil copolymers . J. Phys. Condens. Matter , 2002 . 14 ( 46 ): 12189 - 12202 . DOI: 10.1088/0953-8984/14/46/321 http://doi.org/10.1088/0953-8984/14/46/321 .

M. W. Matsen . Thin films of block copolymer . J. Chem. Phys. , 1997 . 106 ( 18 ): 7781 - 7791 . DOI: 10.1063/1.473778 http://doi.org/10.1063/1.473778 .

L. Leibler . Theory of microphase separation in block co-polymers . Macromolecules , 1980 . 13 ( 6 ): 1602 - 1617 . DOI: 10.1021/ma60078a047 http://doi.org/10.1021/ma60078a047 .

A. N. Semenov . Contribution to the theory of microphase layering in block-copolymer melts . Zh. Eksp. Teor. Fiz. , 1985 . 88 ( 4 ): 1242 - 1256 . https://www.researchgate.net/publication/267765749_Theory_of_Microphase_Layering_in_Block-Copolymer_Melts https://www.researchgate.net/publication/267765749_Theory_of_Microphase_Layering_in_Block-Copolymer_Melts , .

M. W. Matsen , M. Schick . Stable and unstable phases of a diblock copolymer melt . Phys. Rev. Lett. , 1994 . 72 ( 16 ): 2660 - 2663 . DOI: 10.1103/PhysRevLett.72.2660 http://doi.org/10.1103/PhysRevLett.72.2660 .

F. Drolet , G. H. Fredrickson . Combinatorial screening of complex block copolymer assembly with self-consistent field theory . Phys. Rev. Lett. , 1999 . 83 ( 21 ): 4317 - 4320 . DOI: 10.1103/PhysRevLett.83.4317 http://doi.org/10.1103/PhysRevLett.83.4317 .

K. O. Rasmussen , G. Kalosakas . Improved numerical algorithm for exploring block copolymer mesophases . J. Polym. Sci., Part B:Polym. Phys. , 2002 . 40 ( 16 ): 1777 - 1783 . DOI: 10.1002/(ISSN)1099-0488 http://doi.org/10.1002/(ISSN)1099-0488 .

E. W. Cochran , C. J. Garcia-Cervera , G. H. Fredrickson . Stability of the gyroid phase in diblock copolymers at strong segregation . Macromolecules , 2006 . 39 ( 7 ): 2449 - 2451 . DOI: 10.1021/ma0527707 http://doi.org/10.1021/ma0527707 .

A. Ranjan , J. Qin , D. C. Morse . Linear response and stability of ordered phases of block copolymer melts . Macromolecules , 2008 . 41 ( 3 ): 942 - 954 . DOI: 10.1021/ma0714316 http://doi.org/10.1021/ma0714316 .

C. H. Tong , Y. J. Zhu , H. D. Zhang , F. Qiu , P. Tang , Y. L. Yang . 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 . DOI: 10.1021/jp204904b http://doi.org/10.1021/jp204904b .

P. Stasiak , M. W. Matsen . Efficiency of pseudo-spectral algorithms with Anderson mixing for the SCFT of periodic block-copolymer phases . Eur. Phys. J. E , 2011 . 34 ( 10 ): 110 DOI: 10.1140/epje/i2011-11110-0 http://doi.org/10.1140/epje/i2011-11110-0 .

Y. X. Liu , H. D. Zhang . Exponential time differencing methods with Chebyshev collocation for polymers confined by interacting surfaces . J. Chem. Phys. , 2014 . 140 ( 22 ): 224101 DOI: 10.1063/1.4881516 http://doi.org/10.1063/1.4881516 .

S. M. Cox , P. C. Matthews . Exponential time differencing for stiff systems . J. Comput. Phys. , 2002 . 176 ( 2 ): 430 - 455 . DOI: 10.1006/jcph.2002.6995 http://doi.org/10.1006/jcph.2002.6995 .

A. K. Kassam , L. N. Trefethen . Fourth-order time-stepping for stiff PDEs . SIAM J. Sci. Comput., , 2005 . 26 ( 4 ): 1214 - 1233 . DOI: 10.1137/S1064827502410633 http://doi.org/10.1137/S1064827502410633 .

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

M. W. Matsen , F. S. Bates . Block copolymer microstructures in the intermediate-segregation regime . J. Chem. Phys. , 1997 . 106 ( 6 ): 2436 - 2448 . DOI: 10.1063/1.473153 http://doi.org/10.1063/1.473153 .

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

E. L. Thomas , D. B. Alward , D. J. Kinning , D. C. Martin , D. L. Handlin , L. J. Fetters . Ordered bicontinuous doublediamond structure of star block copolymers-a new equilibrium microdomain morphology . Macromolecules , 1986 . 19 ( 8 ): 2197 - 2202 . DOI: 10.1021/ma00162a016 http://doi.org/10.1021/ma00162a016 .

M. W. Matsen , F. S. Bates . Origins of complex self-assembly in block copolymers . Macromolecules , 1996 . 29 ( 23 ): 7641 - 7644 . DOI: 10.1021/ma960744q http://doi.org/10.1021/ma960744q .

M. W. Matsen . 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 . DOI: 10.1140/epje/i2009-10534-3 http://doi.org/10.1140/epje/i2009-10534-3 .

T. H. Epps , E. W. Cochran , T. S. Bailey , R. S. Waletzko , C. M. Hardy , F. S. Bates . Ordered network phases in linear poly(isoprene- b -styrene- b -ethylene oxide) triblock copolymers . Macromolecules , 2004 . 37 ( 22 ): 8325 - 8341 . DOI: 10.1021/ma048762s http://doi.org/10.1021/ma048762s .

T. S. Bailey , C. M. Hardy , T. H. Epps , F. S. Bates . A noncubic triply periodic network morphology in poly(isoprene- b -styrene- b -ethylene oxide) triblock copolymers . Macromolecules , 2002 . 35 ( 18 ): 7007 - 7017 . DOI: 10.1021/ma011716x http://doi.org/10.1021/ma011716x .

C. A. Tyler , D. C. Morse . Orthorhombic Fddd network in triblock and diblock copolymer melts . Phys. Rev. Lett. , 2005 . 94 ( 20 ): 208302 DOI: 10.1103/PhysRevLett.94.208302 http://doi.org/10.1103/PhysRevLett.94.208302 .

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

Views

411

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

AI-driven Automated Construction of Block Copolymer Phase Diagrams

Supramolecular Self-assembly Behaviors of Asymmetric Diblock Copolymer Blends with Hydrogen Bonding Interactions between Shorter Blocks Modelled by Yukawa Potentials

Enlarged Stable Phase Region of Hybrid Lamella-sphere Phase Enabled by A1B1A2(B2)m Branched Tetrablock Copolymer

Core Structure Modulation of 2D Nanosheets by Selective Crystallization-driven Self-assembly of Polyhedral Oligomeric Silsesquioxane-containing Block Copolymers

Manipulating Hierarchical Aggregation Structure via Film-forming Kinetics Enables Organic Photovoltaics over 20% Efficiency

Related Author

Qing-Shu Dong

Qing-Liang Song

Kun Tian

Wei-Hua Li

Xu Zhang

Jialiang Chen

Lin Xu

Tianxi Liu

Related Institution

State Key Laboratory of Molecular Engineering of Polymers, Research Center of AI for Polymer Science, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University

Innovation Center for Textile Science and Technology, State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, College of Materials Science and Engineering, Donghua University

National Garment and Accessories Quality Supervision Testing Center (Fujian), Fujian Provincial Key Laboratory of Textiles Inspection Technology, Fujian Fiber Inspection Center

Key Laboratory of Synthetic and Biological Colloids, Ministry of Education, School of Chemical and Material Engineering, Jiangnan University

Department of Physics, Taizhou University

⁰