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.
1. [1]
  Yiwen Lin , Yijie Chen , Chunhui Deng , Nianrong Sun . Integration of resol/block-copolymer carbonization and machine learning: A convenient approach for precise monitoring of glycan-associated disorders. Chinese Chemical Letters, 2024, 35(12): 109813-. doi: 10.1016/j.cclet.2024.109813
2. [2]
  Shudi Yu , Jie Li , Jiongting Yin , Wanyu Liang , Yangping Zhang , Tianpeng Liu , Mengyun Hu , Yong Wang , Zhengying Wu , Yuefan Zhang , Yukou Du . Built-in electric field and core-shell structure of the reconstructed sulfide heterojunction accelerated water splitting. Chinese Chemical Letters, 2024, 35(12): 110068-. doi: 10.1016/j.cclet.2024.110068
3. [3]
  Xin Dong , Tianqi Chen , Jing Liang , Lei Wang , Huajie Wu , Zhijin Xu , Junhua Luo , Li-Na Li . Structure design of lead-free chiral-polar perovskites for sensitive self-powered X-ray detection. Chinese Journal of Structural Chemistry, 2024, 43(6): 100256-100256. doi: 10.1016/j.cjsc.2024.100256
4. [4]
  Changhui Yu , Peng Shang , Huihui Hu , Yuening Zhang , Xujin Qin , Linyu Han , Caihe Liu , Xiaohan Liu , Minghua Liu , Yuan Guo , Zhen Zhang . Evolution of template-assisted two-dimensional porphyrin chiral grating structure by directed self-assembly using chiral second harmonic generation microscopy. Chinese Chemical Letters, 2024, 35(10): 109805-. doi: 10.1016/j.cclet.2024.109805
5. [5]
  Xue Zhao , Rui Zhao , Qian Liu , Henghui Chen , Jing Wang , Yongfeng Hu , Yan Li , Qiuming Peng , John S Tse . A p-d block synergistic effect enables robust electrocatalytic oxygen evolution. Chinese Chemical Letters, 2024, 35(11): 109496-. doi: 10.1016/j.cclet.2024.109496
6. [6]
  Kai An , Qinglong Qiao , Lovelesh , Syed Ali Abbas Abedi , Xiaogang Liu , Zhaochao Xu . "Superimposed" spectral characteristics of fluorophores arising from cross-conjugation hybridization. Chinese Chemical Letters, 2025, 36(1): 109786-. doi: 10.1016/j.cclet.2024.109786
7. [7]
  Qian Wang , Ting Gao , Xiwen Lu , Hangchao Wang , Minggui Xu , Longtao Ren , Zheng Chang , Wen Liu . Nanophase separated, grafted alternate copolymer styrene-maleic anhydride as an efficient room temperature solid state lithium ion conductor. Chinese Chemical Letters, 2024, 35(7): 108887-. doi: 10.1016/j.cclet.2023.108887
8. [8]
  Pengcheng Su , Shizheng Chen , Zhihong Yang , Ningning Zhong , Chenzi Jiang , Wanbin Li . Vapor-phase postsynthetic amination of hypercrosslinked polymers for efficient iodine capture. Chinese Chemical Letters, 2024, 35(9): 109357-. doi: 10.1016/j.cclet.2023.109357
9. [9]
  Ce Liang , Qiuhui Sun , Adel Al-Salihy , Mengxin Chen , Ping Xu . Recent advances in crystal phase induced surface-enhanced Raman scattering. Chinese Chemical Letters, 2024, 35(9): 109306-. doi: 10.1016/j.cclet.2023.109306
10. [10]
  Mengjia Luo , Yi Qiu , Zhengyang Zhou . Exploring temperature-driven phase dynamics of phosphate: The periodic to incommensurately modulated long-range ordered phase transition in CsCdPO4. Chinese Journal of Structural Chemistry, 2025, 44(1): 100446-100446. doi: 10.1016/j.cjsc.2024.100446
11. [11]
  Shan Jiang , Lingchen Meng , Wenyue Ma , Qingkai Qi , Wei Zhang , Bin Xu , Leijing Liu , Wenjing Tian . Corrigendum to 'Morphology controllable conjugated network polymers based on AIE-active building block for TNP detection' [Chin. Chem. Lett. 32 (2021) 1037-1040]. Chinese Chemical Letters, 2024, 35(12): 108998-. doi: 10.1016/j.cclet.2023.108998
12. [12]
  Wenjia Wang , Xingyue He , Xiaojie Wang , Tiantian Zhao , Osamu Muraoka , Genzoh Tanabe , Weijia Xie , Tianjiao Zhou , Lei Xing , Qingri Jin , Hulin Jiang . Glutathione-depleted cyclodextrin pseudo-polyrotaxane nanoparticles for anti-inflammatory oxaliplatin (Ⅳ) prodrug delivery and enhanced colorectal cancer therapy. Chinese Chemical Letters, 2024, 35(4): 108656-. doi: 10.1016/j.cclet.2023.108656
13. [13]
  Shengyu Zhao , Qinhao Shi , Wuliang Feng , Yang Liu , Xinxin Yang , Xingli Zou , Xionggang Lu , Yufeng Zhao . Suppression of multistep phase transitions of O3-type cathode for sodium-ion batteries. Chinese Chemical Letters, 2024, 35(5): 108606-. doi: 10.1016/j.cclet.2023.108606
14. [14]
  Xue Xin , Qiming Qu , Islam E. Khalil , Yuting Huang , Mo Wei , Jie Chen , Weina Zhang , Fengwei Huo , Wenjing Liu . Hetero-phase zirconia encapsulated with Au nanoparticles for boosting electrocatalytic nitrogen reduction. Chinese Chemical Letters, 2024, 35(5): 108654-. doi: 10.1016/j.cclet.2023.108654
15. [15]
  Tian Yang , Yi Liu , Lina Hua , Yaoyao Chen , Wuqian Guo , Haojie Xu , Xi Zeng , Changhao Gao , Wenjing Li , Junhua Luo , Zhihua Sun . Lead-free hybrid two-dimensional double perovskite with switchable dielectric phase transition. Chinese Chemical Letters, 2024, 35(6): 108707-. doi: 10.1016/j.cclet.2023.108707
16. [16]
  Shu Lin , Kezhen Qi . Phase-dependent lithium-alloying reactions for lithium-metal batteries. Chinese Chemical Letters, 2024, 35(4): 109431-. doi: 10.1016/j.cclet.2023.109431
17. [17]
  Wangyan Hu , Ke Li , Xiangnan Dou , Ning Li , Xiayan Wang . Nano-sized stationary phase packings retained by single-particle frit for microchip liquid chromatography. Chinese Chemical Letters, 2024, 35(4): 108806-. doi: 10.1016/j.cclet.2023.108806
18. [18]
  Zhaohong Chen , Mengzhen Li , Jinfei Lan , Shengqian Hu , Xiaogang Chen . Organic ferroelastic enantiomers with high T_c and large dielectric switching ratio triggered by order-disorder and displacive phase transition. Chinese Chemical Letters, 2024, 35(10): 109548-. doi: 10.1016/j.cclet.2024.109548
19. [19]
  Zhi-Yuan Yue , Hua-Kai Li , Na Wang , Shan-Shan Liu , Le-Ping Miao , Heng-Yun Ye , Chao Shi . Dehydration-triggered structural phase transition-associated ferroelectricity in a hybrid perovskite-type crystal. Chinese Chemical Letters, 2024, 35(10): 109355-. doi: 10.1016/j.cclet.2023.109355
20. [20]
  Zhuoer Cai , Yinan Zhang , Xiu-Ni Hua , Baiwang Sun . Phase transition arising from order-disorder motion in stable layered two-dimensional perovskite. Chinese Journal of Structural Chemistry, 2024, 43(11): 100426-100426. doi: 10.1016/j.cjsc.2024.100426

Get Citation

PDF

XML

Metrics

PDF Downloads(0)
Abstract views(802)
HTML views(32)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142