Progress on Rheological Constitutive Models for Branched Polymer Melts

XIONG Zhong-qiang; YU Wei

doi:10.14028/j.cnki.1003-3726.2024.24.271

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Progress on Rheological Constitutive Models for Branched Polymer Melts

更新时间：2025-01-23

- Progress on Rheological Constitutive Models for Branched Polymer Melts
- Polymer Bulletin Vol. 38, Issue 2, Pages: 265-283(2025)
- 作者机构：
  
  1.中国科学院大学温州研究院，温州 325000
  2.上海交通大学化学化工学院，流变学研究所，上海 200240
- 作者简介：
- 基金信息：
- DOI：10.14028/j.cnki.1003-3726.2024.24.271
  CLC：
- Received：16 September 2024，
  
  Accepted：07 November 2024，
  
  Published Online：18 December 2024，
  
  Published：20 February 2025
- 稿件说明：
移动端阅览
熊钟强, 俞炜. 支化高分子熔体流变本构模型的研究进展. 高分子通报, 2025, 38(2), 265–283

Xiong, Z. Q.; Yu, W. Progress on rheological constitutive models for branched polymer melts. Polym. Bull. (in Chinese), 2025, 38(2), 265–283
熊钟强, 俞炜. 支化高分子熔体流变本构模型的研究进展. 高分子通报, 2025, 38(2), 265–283 DOI： 10.14028/j.cnki.1003-3726.2024.24.271.

Xiong, Z. Q.; Yu, W. Progress on rheological constitutive models for branched polymer melts. Polym. Bull. (in Chinese), 2025, 38(2), 265–283 DOI： 10.14028/j.cnki.1003-3726.2024.24.271.

摘要

分子本构模型的构建是理解微结构动力学和宏观流变行为关系的重要途径。从非平衡态统计物理的角度出发，流变学本构模型的研究已经取得显著的进展。计算的分子结构包括但不限于线型链、星型链、H型链、绒球链、梳型链、瓶刷链、树型链、随机支化链等。本文综述的高分子流变本构模型，虽然由于历史原因，模型之间的命名并不相关，但其解决的问题是一脉相承，由简入繁的。我们将从Onsager变分原理的角度开始，介绍每个线性黏弹性模型针对的问题和假设，使得模型发展的路径呈现得更加清晰。线性黏弹性模型的成功，使我们能够深刻地认识结构性能之间的关系，为非线性黏弹性模型的构建打下坚实的基础。通过结合线性黏弹性谱和非线性松弛机理，构建的非线性模型能够在实验中的振荡剪切、启动剪切/拉伸、阶跃剪切等流变测试模式下得到很好的验证。同时也由于分子间动态相互作用和非线性耦合的复杂性等问题，对于链吸附、动态反应、滑环滑动等结构的动态性，模型的构建目前仍然存在很大的挑战。

Abstract

The construction of molecular constitutive models is an important way to understand the relationship between microstructure dynamics and macroscopic rheological behavior. From the perspective of non-equilibrium statistical physics

significant p

rogress has been made in the study of rheological constitutive models. The calculated molecular structures include but are not limited to linear chains

star chains

H chains

pom-pom chains

comb chains

bottle-brush chains

tree-like chains

and randomly branched chains

etc

. The rheological constitutive models of polymers reviewed in this article

although the names of these models are not related due to historical reasons

models themselves are developed from simplicity to complexity. We will start from the perspective of the Onsager variational principle to introduce the problems and assumptions addressed by each linear viscoelastic model

in order to present the development path of the model more clearly. The success of linear viscoelastic models enables us to deeply understand how structure affects performance

laying a solid foundation for the construction of nonlinear viscoelastic models. By combining linear viscoelastic spectra with nonlinear relaxation mechanism

the constructed nonlinear model can be well validated in rheological testing

such as oscillatory shear

start-up shear/extension

and step shear in experiments. However

due to the complexity of dynamic intermolecular interactions and nonlinear coupling

there are still significant challenges in constructing models for the dynamics of structures

such as chain adsorption

dynamic reactions

and slide-ring sliding.

关键词

Keywords

references

Berry, G. C. ; Fox, T. G . The viscosity of polymers and their concentrated solutions . Advances in Polymer Science . Berlin/Heidelberg : Springer-Verlag , 1968 , 261 − 357 .

Pearson, D. S. ; Helfand, E . Viscoelastic properties of star-shaped polymers . Macromolecules , 1984 , 17 ( 4 ), 888 − 895 .

Holzapfel, G . Nonlinear solid mechanics: a continuum approach for engineering science . Meccanica , 2000 , 37 , 489 − 490 .

Beris, A. N. ; Edwards, B. J . Thermodynamics of Flowing Systems: with Internal Microstructure . New York : Oxford University Press , 1994 .

Milner, S. T. ; McLeish, T. C. B . Parameter-free theory for stress relaxation in star polymer melts . Macromolecules , 1997 , 30 ( 7 ), 2159 − 2166 .

McLeish, T. C. B. ; Larson, R. G . Molecular constitutive equations for a class of branched polymers: the pom-pom polymer . J. Rheol. , 1998 , 42 ( 1 ), 81 − 110 .

Lentzakis, H. ; Das, C. ; Vlassopoulos, D. ; Read, D. J . Pom-pom-like constitutive equations for comb polymers . J. Rheol. , 2014 , 58 ( 6 ), 1855 − 1875 .

Xiong, Z. Q. ; Yu, W . An orientation-stretch coupled model for entangled comb polymers . J. Rheol. , 2021 , 65 ( 2 ), 113 − 128 .

Dalsin, S. J. ; Hillmyer, M. A. ; Bates, F. S . Linear rheology of polyolefin-based bottlebrush polymers . Macromolecules , 2015 , 48 ( 13 ), 4680 − 4691 .

Blackwell, R. J. ; Harlen, O. G. ; McLeish, T. C. B . Theoretical linear and nonlinear rheology of symmetric treelike polymer melts . Macromolecules , 2001 , 34 ( 8 ), 2579 − 2596 .

Xiong, Z. Q. ; Yu, W . A nonlinear constitutive model for entangled symmetric dendrimers . J. Rheol. , 2022 , 66 ( 5 ), 907 − 923 .

Park, S. J. ; Shanbhag, S. ; Larson, R. G . A hierarchical algorithm for predicting the linear viscoelastic properties of polymer melts with long-chain branching . Rheol. Acta , 2005 , 44 ( 3 ), 319 − 330 .

Das, C. ; Inkson, N. J. ; Read, D. J. ; Kelmanson, M. A. ; McLeish, T. C. B . Computational linear rheology of general branch-on-branch polymers . J. Rheol. , 2006 , 50 ( 2 ), 207 − 234 .

Read, D. J. ; Auhl, D. ; Das, C. ; den Doelder, J. ; Kapnistos, M. ; Vittorias, I. ; McLeish, T. C. B . Linking models of polymerization and dynamics to predict branched polymer structure and flow . Science , 2011 , 333 ( 6051 ), 1871 − 1874 .

Wojtecki, R. J. ; Jones, G. O. ; Yuen, A. Y. ; Chin, W. ; Boday, D. J. ; Nelson, A. ; García, J. M. ; Yang, Y. Y. ; Hedrick, J. L . Developments in dynamic covalent chemistries from the reaction of thiols with hexahydrotriazines . J. Am. Chem. Soc. , 2015 , 137 ( 45 ), 14248 − 14251 .

Yasuda, Y. ; Hidaka, Y. ; Mayumi, K. ; Yamada, T. ; Fujimoto, K. ; Okazaki, S. ; Yokoyama, H. ; Ito, K . Molecular dynamics of polyrotaxane in solution investigated by quasi-elastic neutron scattering and molecular dynamics simulation: sliding motion of rings on polymer . J. Am. Chem. Soc. , 2019 , 141 ( 24 ), 9655 − 9663 .

Xiong, Z. Q. ; Yu, W . Sliding dynamics of slide-ring polymers based on the bead-spring model . Chinese. J. Polym. Sci. , 2023 , 41 ( 9 ), 1410 − 1424 .

Doi, M. ; Edwards, S. F . Dynamics of concentrated polymer systems. Part 3.—The constitutive equation . J. Chem. Soc., Faraday Trans. 2 , 1978 , 74 , 1818 − 1832 .

Doi, M. ; Edwards, S. F . Dynamics of concentrated polymer systems. Part 1.—Brownian motion in the equilibrium state . J. Chem. Soc., Faraday Trans. 2 , 1978 , 74 , 1789 − 1801 .

Onsager, L . Reciprocal relations in irreversible processes. I . Phys. Rev. , 1931 , 37 ( 4 ), 405 − 426 .

Onsager, L . Reciprocal relations in irreversible processes. II . Phys. Rev. , 1931 , 38 ( 12 ), 2265 − 2279 .

Doi, M . Soft Matter Physics . Oxford : Oxford University Press, USA , 2013 .

Mohar, B . Some applications of Laplace eigenvalues of graphs . Graph Symmetry . Dordrecht : Springer Netherlands , 1997 , 225 − 275 .

Doi, M. ; Edwards, S. F . The Theory of Polymer Dynamics . Oxford University Press , 1986 , 1 − 380 .

Rouse, P. E . A theory of the linear viscoelastic properties of dilute solutions of coiling polymers . J. Chem. Phys. , 1953 , 21 ( 7 ), 1272 − 1280 .

Bird, R. ; Curtiss, C. F. ; Armstrong, R. ; Hassager, O . Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory , 2 nd edition , 1987 .

Zimm, B. H . Dynamics of polymer molecules in dilute solution: viscoelasticity, flow birefringence and dielectric loss . 1956 , 24 ( 2 ), 269 − 278 .

Kratky, O. ; Porod, G . Röntgenuntersuchung gelöster fadenmoleküle . Recl. Trav. Chim. Pays-Bas , 1949 , 68 ( 12 ), 1106 − 1122 .

Harris, R. ; Hearst, J . On polymer dynamics . J. Chem. Phys. , 1966 , 44 , 2595 − 2602 .

Hearst, J. ; Harris, R. ; Beals, E . On polymer dynamics. II . J. Chem. Phys. , 1966 , 45 , 3106 − 3113 .

Morse, D. C . Viscoelasticity of concentrated isotropic solutions of semiflexible polymers. 1. Model and stress tensor . Macromolecules , 1998 , 31 ( 20 ), 7030 − 7043 .

Gittes, F. ; MacKintosh, F. C . Dynamic shear modulus of a semiflexible polymer network . Phys. Rev. E , 1998 , 58 ( 2 ), R1241 − R1244 .

Shankar, V. ; Pasquali, M. ; Morse, D. C . Theory of linear viscoelasticity of semiflexible rods in dilute solution . J. Rheol. , 2002 , 46 ( 5 ), 1111 − 1154 .

Xiong, Z. Q. ; Seto, R. ; Doi, M . Bending-rotation coupling in the viscoelasticity of semiflexible polymers─rigorous perturbation analysis from the rod limit . Macromolecules , 2024 , 57 ( 11 ), 5289 − 5299 .

Amblard, F. ; Maggs, A. C. ; Yurke, B. ; Pargellis, A. N. ; Leibler, S . Subdiffusion and anomalous local viscoelasticity in actin networks . Phys. Rev. Lett. , 1996 , 77 ( 21 ), 4470 − 4473 .

Sheiko, S. S. ; Dobrynin, A. V . Architectural code for rubber elasticity: from supersoft to superfirm materials . Macromolecules , 2019 , 52 ( 20 ), 7531 − 7546 .

Ham, J. S . Viscoelastic theory of branched and cross—linked polymers . J. Chem. Phys. , 1957 , 26 ( 3 ), 625 − 633 .

Forsman, W. C . Graph theory and the statistics and dynamics of polymer chains . J. Chem. Phys. , 1976 , 65 ( 10 ), 4111 − 4115 .

Gurtovenko, A. A. ; Blumen, A . Generalized Gaussian structures: models for polymer systems with complex topologies . Advances in Polymer Science . Berlin, Heidelberg : Springer Berlin Heidelberg , 2005 , 171 − 282 .

Jurjiu, A. ; Gomes Maia Júnior, D. ; Galiceanu, M . Relaxation dynamics of generalized scale-free polymer networks . Sci. Rep. , 2018 , 8 , 3731 .

Oliveira, E. S. ; Galiceanu, A. C. A. M. ; Jurjiu, A. ; Galiceanu, M . Relaxation dynamics of semiflexible treelike small-world polymer networks . Phys. Rev. E , 2019 , 100 ( 2-1 ), 022501 .

von Ferber, C. ; Blumen, A . Dynamics of dendrimers and of randomly built branched polymers . 2002 , 116 ( 19 ), 8616 − 8624 .

Dolgushev, M. ; Blumen, A . Dynamics of semiflexible treelike polymeric networks . J. Chem. Phys. , 2009 , 131 ( 4 ), 044905 .

Dolgushev, M. ; Blumen, A . Dynamics of chains and dendrimers with heterogeneous semiflexibility . J. Chem. Phys. , 2010 , 132 ( 12 ), 124905 .

Dolgushev, M. ; Berezovska, G. ; Blumen, A . Maximum entropy principle applied to semiflexible ring polymers . J. Chem. Phys. , 2011 , 135 ( 9 ), 094901 .

Xiong, Z. Q. ; Yu, W . Molecular constitutive equation for unentangled branch copolymers . Rheol. Acta , 2021 , 60 ( 8 ), 439 − 455 .

Kumar, A. ; Biswas, P . Dynamics of semiflexible dendrimers in dilute solutions . Macromolecules , 2010 , 43 ( 17 ), 7378 − 7385 .

Swan, J. W. ; Wang, G . Rapid calculation of hydrodynamic and transport properties in concentrated solutions of colloidal particles and macromolecules . Phys. Fluids , 2016 , 28 ( 1 ), 011902 .

Milner, S. T. ; McLeish, T. C. B . Reptation and contour-length fluctuations in melts of linear polymers . Phys. Rev. Lett. , 1998 , 81 , 725 − 728 .

Roovers, J. ; Toporowski, P. M . Relaxation by constraint release in combs and star-combs . Macromolecules , 1987 , 20 ( 9 ), 2300 − 2306 .

Marrucci, G . Relaxation by reptation and tube enlargement: a model for polydisperse polymers . J. Polym. Sci. B Polym. Phys. , 1985 , 23 ( 1 ), 159 − 177 .

Ball, R. C. ; McLeish, T. C. B . Dynamic dilution and the viscosity of star-polymer melts . Macromolecules , 1989 , 22 ( 4 ), 1911 − 1913 .

Likhtman, A. E. ; McLeish, T. C. B . Quantitative theory for linear dynamics of linear entangled polymers . Macromolecules , 2002 , 35 ( 16 ), 6332 − 6343 .

Des Cloizeaux, J . Double reptation versus simple reptati on in polymer melts . EPL Europhys. Lett. , 1988 , 5 , 437 .

Tsenoglou, C . Molecular weight polydispersity effects on the viscoelasticity of entangled linear polymers . Macromolecules , 1991 , 24 ( 8 ), 1762 − 1767 .

Onogi, S. ; Masuda, T. ; Kitagawa, K . Rheological properties of anionic polystyrenes. I. Dynamic viscoelasticity of narrow-distribution polystyrenes . Macromolecules , 1970 , 3 ( 2 ), 109 − 116 .

Gennes, P . Reptation of stars . J. Phys. France , 1975 , 36 , 1199 − 1203 .

Wang, Z. W. ; Chen, X. ; Larson, R. G . Comparing tube models for predicting the linear rheology of branched polymer melts . J. Rheol. , 2010 , 54 ( 2 ), 223 − 260 .

Sato, T. ; Kwon, Y. ; Matsumiya, Y. ; Watanabe, H . A constitutive equation for Rouse model modified for variations of spring stiffness, bead friction, and Brownian force intensity under flow . Phys. Fluids , 2021 , 33 ( 6 ), 063106 .

Matsumiya, Y. ; Watanabe, H . Non-universal features in uniaxially extensional rheology of linear polymer melts and concentrated solutions: a review . Prog. Polym. Sci. , 2021 , 112 , 101325 .

Likhtman, A. E. ; Graham, R. S . Simple constitutive equation for linear polymer melts derived from molecular theory: rolie-poly equation . J. Non Newton. Fluid Mech. , 2003 , 114 ( 1 ), 1 − 12 .

Graham, R. S. ; Likhtman, A. E. ; McLeish, T. C. B. ; Milner, S. T . Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release . J. Rheol. , 2003 , 47 ( 5 ), 1171 − 1200 .

Boudara, V. A. H. ; Peterson, J. D. ; Leal, L. G. ; Read, D. J . Nonlinear rheology of polydisperse blends of entangled linear polymers: rolie-double-poly models . J. Rheol. , 2019 , 63 ( 1 ), 71 − 91 .

Marrucci, G. ; Ianniruberto, G . Flow-induced orientation and stretching of entangled polymers . Phil. Trans. R. Soc. A. , 2003 , 361 ( 1805 ), 677 − 688 .

McLeish, T. C. B. ; Allgaier, J. ; Bick, D. K. ; Bishko, G. ; Biswas, P. ; Blackwell, R. ; Blottière, B. ; Clarke, N. ; Gibbs, B. ; Groves, D. J. ; Hakiki, A. ; Heenan, R. K. ; Johnson, J. M. ; Kant, R. ; Read, D. J. ; Young, R. N . Dynamics of entangled H-polymers: theory, rheology, and neutron-scattering . Macromolecules , 1999 , 32 ( 20 ), 6734 − 6758 .

Giesekus, H . A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility . J. Non Newton. Fluid Mech. , 1982 , 11 ( 1-2 ), 69 − 109 .

Verbeeten, W. M. H. ; Peters, G. W. M. ; Baaijens, F. P. T . Differential constitutive equations for polymer melts: the extended pom-pom model . J. Rheol. , 2001 , 45 ( 4 ), 823 − 843 .

Das, C. ; Read, D. J. ; Auhl, D. ; Kapnistos, M. ; den Doelder, J. ; Vittorias, I. ; McLeish, T. C. B . Numerical prediction of nonlinear rheology of branched polymer melts . J. Rheol. , 2014 , 58 ( 3 ), 737 − 757 .

Bishko, G. ; McLeish, T. C. B. ; Harlen, O. G. ; Larson, R. G . Theoretical molecular rheology of branched polymers in simple and complex flows: the pom-pom model . Phys. Rev. Lett. , 1997 , 79 ( 12 ), 2352 − 2355 .

Inkson, N. J. ; McLeish, T. C. B. ; Harlen, O. G. ; Groves, D. J . Predicting low density polyethylene melt rheology in elongational and shear flows with “pom-pom” constitutive equations . J. Rheol. , 1999 , 43 ( 4 ), 873 − 896 .

Stephanou, P. S. ; Tsimouri, I. C. ; Mavrantzas, V. G . Flow-induced orientation and stretching of entangled polymers in the framework of nonequilibrium thermod-ynamics . Macromolecules , 2016 , 49 ( 8 ), 3161 − 3173 .

Hyun, K. ; Wilhelm, M . Establishing a new mechanical nonlinear coefficient Q from FT-rheology: first investigation of entangled linear and comb polymer model systems . Macromolecules , 2009 , 42 ( 1 ), 411 − 422 .

Matsumiya, Y. ; Inoue, T. ; Oishi, Y. ; Watanabe, H . Orientational anisotropy of bead-spring star chains during creep process . J. Polym. Sci. Polym. Phys. , 2006 , 44 , 3501 − 3517 .

Lee, J. H. ; Orfanou, K. ; Driva, P. ; Iatrou, H. ; Hadjichristidis, N. ; Lohse, D. J . Linear and nonlinear rheology of dendritic star polymers: experiment . Macromolecules , 2008 , 41 ( 23 ), 9165 − 9178 .

Cai, L. H. ; Panyukov, S. ; Rubinstein, M . Hopping diffusion of nanoparticles in polymer matrices . Macromolecules , 2015 , 48 ( 3 ), 847 − 862 .

Xu, Z. Y. ; Dai, X. B. ; Bu, X. Y. ; Yang, Y. ; Zhang, X. Y. ; Man, X. K. ; Zhang, X. H. ; Doi, M. ; Yan, L. T . Enhanced heterogeneous diffusion of nanoparticles in semiflexible networks . ACS Nano , 2021 , 15 ( 3 ), 4608 − 4616 .

Boudara, V. A. H. ; Read, D. J. ; Ramírez, J . Reptate rheology software: toolkit for the analysis of theories and experiments . J. Rheol. , 2020 , 64 ( 3 ), 709 − 722 .

Risken, H . The Fokker-Planck Equation: Methods of Solution and Applications . 2 ed . Berlin , Springer-Verlag , 1996 , 107 − 108 .

Brady, J. F. ; Bossis, G . Stokesian dynamics . Annu. Rev. Fluid Mech. , 1988 , 20 , 111 − 157 .

Harvey, S. ; Garcia de la Torre, J . Coordinate systems for modeling the hydrodynamic resistance and diffusion coefficients of irregularly shaped rigid macromolecules . Macromolecules , 1980 , 13 ( 4 ), 960 − 964 .

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