蠕虫状链模型在高分子物理研究中的应用

蒋滢; 陈征宇

doi:10.7498/aps.65.178201

摘要

蠕虫状链模型可以更好地描述非柔性高分子的空间链构象统计，因此被公认为是更加接近真实高分子的粗粒化高分子链模型。本文从蠕虫状链模型的物理特点出发，简单回顾了该模型在自洽场理论方法中的发展历程，着重从三个研究方向总结了近年来蠕虫状链模型在高分子物理研究中的应用：高分子液晶相结构及其转变的研究；几何表面对高分子体系的影响；蠕虫状嵌段共聚物自组装。最后，针对现有理论的发展现状，对未来基于该模型的场论模拟方法的发展方向提出了展望。

关键词:

Abstract

The continuum version of the wormlike chain model (WLC), which was initially developed by Saito, Takahashi and Yunoki in 1967, is particularly suitable for description of polymer conformational properties affected by the chain rigidity. The WLC model is capable of covering an extensive range of chain rigidity, from the flexible chains to the rigid chains, by tuning the persistence length directly. It is widely accepted as a coarse-grained model that can be used to capture the physical properties, such as conformation and structures, of a larger class of real polymers than the Gaussian chain (GSC). Recently, the WLC model attracts increasing interests because of its advantages in studying a variety of polymeric systems, including liquid crystalline polymers and conjugated polymers. This review article focuses on applications of the WLC model, incorporated in the framework of self-consistent field theory, which is an effective method in theoretical exploration of phase separation in polymer systems. The article also pays particular attention to the developments of the numerical schemes to solve the modified diffusion equation governing the probability distribution of polymers. In addition, we summarize recent applications of the self-consistent field theories based on WLC model in the following three areas: phase transitions in liquid-crystalline polymers; the influence of surface curvature on polymeric systems involving the chain orientation effects; self-assembly of wormlike block copolymers. These studies are beyond the scope of self-consistent field theories based on a GSC model, which have been utilized in a large number of theoretical studies in recent years. Finally, we propose the perspectives of theoretical developments in field-theory simulations based on the WLC model for future work. In the polymer literature, it is generally appreciated that chain-rigidity is an important factor that influences the properties of structural stabilities on the meso-scale. The theoretical studies indentify the key physical mechanisms that play crucial roles in many experimental systems with attractively promising applications in practice, for systems such as liquid crystalline polymers and organic solar cell based on the conjugated polymers.

Keywords:

作者及机构信息

蒋滢¹,
陈征宇²

1.
北京航空航天大学软物质物理研究中心, 化学与环境学院, 北京 100191;

2.
Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1

通信作者: 蒋滢, jiangy.uwaterloo@gmail.com;jeffchen@uwaterloo.ca ; 陈征宇, jiangy.uwaterloo@gmail.com;jeffchen@uwaterloo.ca

基金项目: 国家自然科学基金（批准号：21204067，21574006），中央高校基础科研业务费和 Natural Sciences and Engineering Research Council（Canada）资助的课题.

Authors and contacts

Jiang Ying¹,
Chen Jeff Z. Y.²

1.
School of Chemistry and Environment, Center of Soft Matter Physics and its Applications, Beihang University, Beijing 100191, China;

2.
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Corresponding author: Jiang Ying, jiangy.uwaterloo@gmail.com;jeffchen@uwaterloo.ca ; Chen Jeff Z. Y., jiangy.uwaterloo@gmail.com;jeffchen@uwaterloo.ca

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 21204067, 21574006), the Fundamental Research Funds for the Central Universities, China and Natural Sciences and Engineering Research Council (Canada).

参考文献

[1]	Rubinstein M, Colby R H 2003 Polymer Physics (Oxford University Press)
[2]	Doi M, Edwards S F 1986 The Theory of Polymer Dynamics (Clarendon Press)
[3]	Yamakawa H 1997 Helical Wormlike Chains in Polymer Solutions (New York: Springer-Verlag)
[4]	de Gennes P G, Prost J 1993 The Physics of Liquid Crystals (New York: Oxford University Press)
[5]	Kratky O, Porod G 1949 Recl. Trav. Chim. 68 1106
[6]	Saito N, Takahashi K, Yunoki Y 1967 J. Phys. Soc. Jpn. 22 219
[7]	Tagamil Y 1969 Macromolecules 2 8
[8]	Harris R A, Hearst J E 1966 J. Chem. Phys. 44 2595
[9]	Freed K F 1972 Adv. Chem. Phys. 22 1
[10]	Fredrickson G H, 2006 The Equilibrium Theory of Inhomogeneous Polymer (Oxford: Clarendon Press)
[11]	Vroege G J, Odijk T 1988 Macromolecules 21 2848
[12]	Chen Z Y 1993 Macromolecules 26 3419
[13]	Chen J Z Y, Sullivan D E, Yuan X Q 2007 Macromolecules 40 1187
[14]	Morse D C, Fredrickson G H 1994 Phys. Rev. Lett. 73 3235
[15]	Schmid F, Mller M 1995 Macromolecules 28 863
[16]	Dchs D, Sullivan D E 2002 J. Phys.: Condens. Matter 14 12189
[17]	Ganesan V, Khounlavong L, Pryamitsyn V 2008 Phys. Rev. E 78 051804
[18]	Song W D, Tang P, Zhang H D, Yang Y L, Shi A C 2009 Macromolecules 42 6300
[19]	Gao J, Song W D, Tang P, Yang Y L 2011 Soft Matter 7 5208
[20]	Gao J, Tang P, Yang Y L. 2013 Soft Matter 9 69
[21]	Jiang Y, Chen J Z Y 2013 Phys. Rev. Lett. 110 138305
[22]	Jiang Y, Chen J Z Y 2013 Phys. Rev. E 88 042603
[23]	Cui S M, Akcakir O, Chen Z Y 1995 Phys. Rev. E 51 4548
[24]	Matsen M W 1996 J. Chem. Phys. 104 7758
[25]	Deng M G, Jiang Y, Liang H J, Chen J Z Y 2010 Macromolecules 43 3455
[26]	Jiang Y, Chen J Z Y 2010 Macromolecules 43 10668
[27]	Jiang Y, Zhang W Y, Chen J Z Y 2011 Phys. Rev. E 84 041803
[28]	Song W D, Tang P, Qiu F, Yang Y L, Shi A C 2011 Soft Matter 7 929
[29]	Liu A J, Fredrickson G H 1993 Macromolecules 26 2817
[30]	Drovetsky B Y, Liu A J, Mak C H 1999 J. Chem. Phys. 111 4334
[31]	Puech N, Grelet E, Poulin P, Blanc C, van der Schoot P 2010 Phys. Rev. E 82 020702
[32]	Yang G, Tang P, Yang Y 2012 Macromolecules 45 3590
[33]	Shah M, Pryamitsyn V, Ganesan V 2008 Macromolecules 41 218
[34]	Li S, Jiang Y, Chen J Z Y 2014 Soft Matter 10 8932
[35]	Netz R R, Andelman D 2003 Phys. Rep. 380 1
[36]	de Gennes P G 1969 Rep. Prog. Phys. 32 187
[37]	Semenov A N 2002 Eur. Phys. J. E 9 353
[38]	Deng M G, Jiang Y, Liang H J, Chen J Z Y 2010 J. Chem. Phys. 133 034902
[39]	Ivanov V A, Martemyanoova J A, Mller M, Paul W, Binder K 2009 J. Phys. Chem. B 113 3653
[40]	Netz R R, Joanny J F 1999 Macromolecules 32 9026
[41]	Chen Y L, Schweizer K S 2002 J. Chem. Phys. 117 1351
[42]	Cao D. P, Jiang T, Wu J. Z. 2006 J. Chem. Phys. 124 164904
[43]	Daoulas K, Therodorou D N, Harmandaris V A, Karayiannis N, Mavrantzas V G 2005 Macromolecules 38 7134
[44]	Lavrentovich O D 2014 Soft Matter 10 1264
[45]	Bonthuis D J, Meyer C, Stein D, Dekker C 2008 Phys. Rev. Lett. 101 108303
[46]	Kyubong J, Dhingra D M, Odijk T, de Pablo J J, Graham M D, Runnheim R, Forrest D, Schwartz D C 2007 Proc. Natl. Acad. Sci. U.S.A. 104 2673
[47]	Marenduzzo D, Orlandini E, Stasiak A, Sumner D W, Tubiana L, Micheletti C 2009 Proc. Natl. Acad. Sci. U.S.A. 106 22269
[48]	Reisner W, Pedersen J N, Austin R H 2012 Rep. Prog. Phys. 75 106601
[49]	de Gennes P G 1979 Scaling Concepts in Polymer Physics (Ithaca, NY: Cornell University Press)
[50]	Odijk T 1983 Macromolecules 16 1340
[51]	Burkhardt T W, Yang Y, Gompper G 2010 Phys. Rev. E 82 041801
[52]	Chen J Z Y, Sullivan D E 2006 Macromolecules 39 7769
[53]	Odijk T 2008 Phys. Rev. E 77 060901
[54]	Dai L, Ng S Y, Doyle P S, van der Maarel J 2012 ACS Macro Lett 1 1046
[55]	Wang Y, Tree D R, Dorfman K D 2011 Macromolecules 44 6594
[56]	Tree D R, Wang Y, Dorfman K D 2013 Phys. Rev. Lett. 110 208103
[57]	Dai L, van der Maarel J, Doyle P S 2014 Macromolecules 47 2445
[58]	Forrey C, Muthukumar M 2006 Biophys. J. 91 25
[59]	LaMarque J C, Le T L, Harvey S C 2004 Biopolymers 73 348
[60]	Fathizadeh A, Heidari M, Mossallam B E, Ejtehadi M R 2013 J. Chem. Phys. 139 044912
[61]	Petrov A S, Boz M B, Harvey S C 2007 J. Struct. Biol. 160 241
[62]	Ivanov V A, Rodionova A S, Martemyanova J A, Stukan M R, Mller M, Paul W, Binder K 2013 J. Chem. Phys. 138 234903
[63]	Liang Q, Li J F, Zhang P W, Chen J Z Y 2013 J. Chem. Phys. 138 244910
[64]	Chen J Z Y 2013 Macromolecules 46 9837
[65]	Gao J, Tang P, Yang Y L, Chen J Z Y 2014 Soft Matter 10 4674
[66]	Khanna V, Cochran E W, Hexemer A, Stein G E, Fredrickson G H, Kramer E J, Li X, Wang J, Hahn S F 2006 Macromolecules 39 9346
[67]	Yang G, Tang P, Yang Y L, Wang Q 2010 J. Phys. Chem. B 114 14897
[68]	Semenov A N, Vasilenko S V 1986 Sov. Phys. JETP 63 70
[69]	Semenov A N 1991 Molecular Crystals and Liquid Crystals 209 191
[70]	Williams D R M, Fredrickson G H 1992 Macromolecules 25 3561
[71]	Matsen M W, Barrett C 1998 J. Chem. Phys. 109 4108
[72]	Holyst R, Schick M 1992 J. Chem. Phys. 96 730
[73]	Singh C, Goulian M, Liu A. J, Fredrickson G H 1994 Macromolecules 27 2974
[74]	Reenders M, ten Brinke G 2002 Macromolecules 35 3266
[75]	Pryamitsyn V, Ganesan V 2004 J. Chem. Phys. 120 5824
[76]	Chen J Z, Zhang C X, Sun Z Y, Zheng Y S, An L J 2006 J. Chem. Phys. 124 104907
[77]	Kriksin Y A, Khalatur P G 2012 Macromolecular Theory and Simulations 21 382
[78]	Zhang X H, Jiang Y, Miao B, Chen Y L, Yan D D, Chen J Z Y 2014 Soft Matter 10 5405
[79]	Leibler L 1980 Macromolecules 13 1602
[80]	Jiang Y, Zhang X H, Miao B, Yan D D, Chen J Z Y 2016 Soft Matter 12 2481

施引文献

[1]	Rubinstein M, Colby R H 2003 Polymer Physics (Oxford University Press)
[2]	Doi M, Edwards S F 1986 The Theory of Polymer Dynamics (Clarendon Press)
[3]	Yamakawa H 1997 Helical Wormlike Chains in Polymer Solutions (New York: Springer-Verlag)
[4]	de Gennes P G, Prost J 1993 The Physics of Liquid Crystals (New York: Oxford University Press)
[5]	Kratky O, Porod G 1949 Recl. Trav. Chim. 68 1106
[6]	Saito N, Takahashi K, Yunoki Y 1967 J. Phys. Soc. Jpn. 22 219
[7]	Tagamil Y 1969 Macromolecules 2 8
[8]	Harris R A, Hearst J E 1966 J. Chem. Phys. 44 2595
[9]	Freed K F 1972 Adv. Chem. Phys. 22 1
[10]	Fredrickson G H, 2006 The Equilibrium Theory of Inhomogeneous Polymer (Oxford: Clarendon Press)
[11]	Vroege G J, Odijk T 1988 Macromolecules 21 2848
[12]	Chen Z Y 1993 Macromolecules 26 3419
[13]	Chen J Z Y, Sullivan D E, Yuan X Q 2007 Macromolecules 40 1187
[14]	Morse D C, Fredrickson G H 1994 Phys. Rev. Lett. 73 3235
[15]	Schmid F, Mller M 1995 Macromolecules 28 863
[16]	Dchs D, Sullivan D E 2002 J. Phys.: Condens. Matter 14 12189
[17]	Ganesan V, Khounlavong L, Pryamitsyn V 2008 Phys. Rev. E 78 051804
[18]	Song W D, Tang P, Zhang H D, Yang Y L, Shi A C 2009 Macromolecules 42 6300
[19]	Gao J, Song W D, Tang P, Yang Y L 2011 Soft Matter 7 5208
[20]	Gao J, Tang P, Yang Y L. 2013 Soft Matter 9 69
[21]	Jiang Y, Chen J Z Y 2013 Phys. Rev. Lett. 110 138305
[22]	Jiang Y, Chen J Z Y 2013 Phys. Rev. E 88 042603
[23]	Cui S M, Akcakir O, Chen Z Y 1995 Phys. Rev. E 51 4548
[24]	Matsen M W 1996 J. Chem. Phys. 104 7758
[25]	Deng M G, Jiang Y, Liang H J, Chen J Z Y 2010 Macromolecules 43 3455
[26]	Jiang Y, Chen J Z Y 2010 Macromolecules 43 10668
[27]	Jiang Y, Zhang W Y, Chen J Z Y 2011 Phys. Rev. E 84 041803
[28]	Song W D, Tang P, Qiu F, Yang Y L, Shi A C 2011 Soft Matter 7 929
[29]	Liu A J, Fredrickson G H 1993 Macromolecules 26 2817
[30]	Drovetsky B Y, Liu A J, Mak C H 1999 J. Chem. Phys. 111 4334
[31]	Puech N, Grelet E, Poulin P, Blanc C, van der Schoot P 2010 Phys. Rev. E 82 020702
[32]	Yang G, Tang P, Yang Y 2012 Macromolecules 45 3590
[33]	Shah M, Pryamitsyn V, Ganesan V 2008 Macromolecules 41 218
[34]	Li S, Jiang Y, Chen J Z Y 2014 Soft Matter 10 8932
[35]	Netz R R, Andelman D 2003 Phys. Rep. 380 1
[36]	de Gennes P G 1969 Rep. Prog. Phys. 32 187
[37]	Semenov A N 2002 Eur. Phys. J. E 9 353
[38]	Deng M G, Jiang Y, Liang H J, Chen J Z Y 2010 J. Chem. Phys. 133 034902
[39]	Ivanov V A, Martemyanoova J A, Mller M, Paul W, Binder K 2009 J. Phys. Chem. B 113 3653
[40]	Netz R R, Joanny J F 1999 Macromolecules 32 9026
[41]	Chen Y L, Schweizer K S 2002 J. Chem. Phys. 117 1351
[42]	Cao D. P, Jiang T, Wu J. Z. 2006 J. Chem. Phys. 124 164904
[43]	Daoulas K, Therodorou D N, Harmandaris V A, Karayiannis N, Mavrantzas V G 2005 Macromolecules 38 7134
[44]	Lavrentovich O D 2014 Soft Matter 10 1264
[45]	Bonthuis D J, Meyer C, Stein D, Dekker C 2008 Phys. Rev. Lett. 101 108303
[46]	Kyubong J, Dhingra D M, Odijk T, de Pablo J J, Graham M D, Runnheim R, Forrest D, Schwartz D C 2007 Proc. Natl. Acad. Sci. U.S.A. 104 2673
[47]	Marenduzzo D, Orlandini E, Stasiak A, Sumner D W, Tubiana L, Micheletti C 2009 Proc. Natl. Acad. Sci. U.S.A. 106 22269
[48]	Reisner W, Pedersen J N, Austin R H 2012 Rep. Prog. Phys. 75 106601
[49]	de Gennes P G 1979 Scaling Concepts in Polymer Physics (Ithaca, NY: Cornell University Press)
[50]	Odijk T 1983 Macromolecules 16 1340
[51]	Burkhardt T W, Yang Y, Gompper G 2010 Phys. Rev. E 82 041801
[52]	Chen J Z Y, Sullivan D E 2006 Macromolecules 39 7769
[53]	Odijk T 2008 Phys. Rev. E 77 060901
[54]	Dai L, Ng S Y, Doyle P S, van der Maarel J 2012 ACS Macro Lett 1 1046
[55]	Wang Y, Tree D R, Dorfman K D 2011 Macromolecules 44 6594
[56]	Tree D R, Wang Y, Dorfman K D 2013 Phys. Rev. Lett. 110 208103
[57]	Dai L, van der Maarel J, Doyle P S 2014 Macromolecules 47 2445
[58]	Forrey C, Muthukumar M 2006 Biophys. J. 91 25
[59]	LaMarque J C, Le T L, Harvey S C 2004 Biopolymers 73 348
[60]	Fathizadeh A, Heidari M, Mossallam B E, Ejtehadi M R 2013 J. Chem. Phys. 139 044912
[61]	Petrov A S, Boz M B, Harvey S C 2007 J. Struct. Biol. 160 241
[62]	Ivanov V A, Rodionova A S, Martemyanova J A, Stukan M R, Mller M, Paul W, Binder K 2013 J. Chem. Phys. 138 234903
[63]	Liang Q, Li J F, Zhang P W, Chen J Z Y 2013 J. Chem. Phys. 138 244910
[64]	Chen J Z Y 2013 Macromolecules 46 9837
[65]	Gao J, Tang P, Yang Y L, Chen J Z Y 2014 Soft Matter 10 4674
[66]	Khanna V, Cochran E W, Hexemer A, Stein G E, Fredrickson G H, Kramer E J, Li X, Wang J, Hahn S F 2006 Macromolecules 39 9346
[67]	Yang G, Tang P, Yang Y L, Wang Q 2010 J. Phys. Chem. B 114 14897
[68]	Semenov A N, Vasilenko S V 1986 Sov. Phys. JETP 63 70
[69]	Semenov A N 1991 Molecular Crystals and Liquid Crystals 209 191
[70]	Williams D R M, Fredrickson G H 1992 Macromolecules 25 3561
[71]	Matsen M W, Barrett C 1998 J. Chem. Phys. 109 4108
[72]	Holyst R, Schick M 1992 J. Chem. Phys. 96 730
[73]	Singh C, Goulian M, Liu A. J, Fredrickson G H 1994 Macromolecules 27 2974
[74]	Reenders M, ten Brinke G 2002 Macromolecules 35 3266
[75]	Pryamitsyn V, Ganesan V 2004 J. Chem. Phys. 120 5824
[76]	Chen J Z, Zhang C X, Sun Z Y, Zheng Y S, An L J 2006 J. Chem. Phys. 124 104907
[77]	Kriksin Y A, Khalatur P G 2012 Macromolecular Theory and Simulations 21 382
[78]	Zhang X H, Jiang Y, Miao B, Chen Y L, Yan D D, Chen J Z Y 2014 Soft Matter 10 5405
[79]	Leibler L 1980 Macromolecules 13 1602
[80]	Jiang Y, Zhang X H, Miao B, Yan D D, Chen J Z Y 2016 Soft Matter 12 2481

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