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Rheologica Acta

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26-04-2024 | Original Contribution

Variable entanglement density constitutive rheological model for polymeric fluids

Author: Pavlos S. Stephanou

Published in: Rheologica Acta | Issue 5/2024

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Abstract

A variable-entanglement density constitutive model is developed for the description of the rheological properties of entangled polymer melts and concentrated polymer solutions using non-equilibrium thermodynamics (NET). It proposes two evolution equations: one for the average number of entanglements per chain and one for the orientation of entanglement strands. Direct comparison with non-equilibrium molecular dynamics simulation data shows that the model can accurately describe the loss of entanglements due to the applied flow for three molecular weights by using the same value for the convective constraint release (CCR) parameter. The CCR relaxation time depends on the trace of the inverse of the orientation tensor instead of an explicit dependency on the velocity gradient. Finally, the stress tensor contains an additional contribution inspired by the Curtiss-Bird or tumbling snake model. Overall, the model proposed here carefully derives via NET and builds upon the work of Ianniruberto-Marrucci when stretching is not considered.

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Baig C, Mavrantzas VG, Kröger M (2010a) Flow effects on melt structure and entanglement network of linear polymers: results from a nonequilibrium molecular dynamics simulation study of a polyethylene melt in steady shear. Macromolecules 43:6886–6902. https://doi.org/10.1021/ma100826uCrossRef

Baig C, Stephanou PS, Tsolou G et al (2010b) Understanding dynamics in binary mixtures of entangled cis- 1,4-polybutadiene melts at the level of primitive path segments by mapping atomistic simulation data onto the tube model. Macromolecules 43:8239–8250. https://doi.org/10.1021/ma101211bCrossRef

Beris AN, Edwards BJ (1994) Thermodynamics of flowing systems: with internal microstructure. Oxford University Press, New YorkCrossRef

Bird RB, Curtiss FC, Amstrong CR, Ole H (1987) Dynamics of polymeric liquids, second edition volume 2: kinetic theory. Wiley-Interscience

Booij HC (1984) The energy storage in the Rouse model in an arbitrary flow field. J Chem Phys 80:4571–4572. https://doi.org/10.1063/1.447210CrossRef

Boudaghi M, Nafar Seddashti MH, Edwards BJ, Khomami B (2022) Elucidating the role of network topology dynamics on the coil-stretch transition hysteresis in extensional flow of entangled polymer melts. J Rheol 66:551–569. https://doi.org/10.1122/8.0000422/16617021/551_1_ACCEPTED_MANUSCRIPT.PDFCrossRef

Chappa VC, Morse DC, Zippelius A, Müller M (2012) Translationally invariant slip-spring model for entangled polymer dynamics. Phys Rev Lett 109:148302. https://doi.org/10.1103/PhysRevLett.109.148302CrossRefPubMed

Curtiss CF, Byron Bird R (1980a) A kinetic theory for polymer melts. I. The equation for the single-link orientational distribution function. J Chem Phys 74:2016–2025. https://doi.org/10.1063/1.441246CrossRef

Curtiss CF, Byron Bird R (1980b) A kinetic theory for polymer melts. II. The stress tensor and the rheological equation of state. J Chem Phys 74:2026–2033. https://doi.org/10.1063/1.441247CrossRef

De Gennes PG (1971) Reptation of a polymer chain in the presence of fixed obstacles. J Chem Phys 55:572–579. https://doi.org/10.1063/1.1675789CrossRef

Doi M, Edwards SF (1988) The theory of polymer dynamics. Clarendon Press, Oxford

Doi M, Takimoto JI, De Gennes PG et al (2003) Molecular modelling of entanglement. Philos Trans R Soc A Math Phys Eng Sci 361:641–650. https://doi.org/10.1098/rsta.2002.1168CrossRef

Dolata BE, Olmsted PD (2023) A thermodynamically consistent constitutive equation describing polymer disentanglement under flow. J Rheol 67:269–292. https://doi.org/10.1122/8.0000476CrossRef

Dressler M, Edwards BJ, Öttinger HC (1999) Macroscopic thermodynamics of flowing polymeric liquids. Rheol Acta 38:117–136. https://doi.org/10.1007/s003970050162CrossRef

Edwards BJ, Beris AN, Grmela M (1990) Generalized constitutive equation for polymeric liquid crystals Part 1. Model formulation using the Hamiltonian (poisson bracket) formulation. J Nonnewton Fluid Mech 35:51–72. https://doi.org/10.1016/0377-0257(90)85072-7CrossRef

Edwards BJ, Beris AN, Grmela M (1991) The dynamical behavior of liquid crystals: a continuum description through generalized brackets. Mol Cryst Liq Cryst 201:51–86. https://doi.org/10.1080/00268949108038637CrossRef

Edwards BJ, Dressler M, Grmela M, Ait-Kadi A (2003) Rheological models with microstructural constraints. Rheol Acta 42:64–72. https://doi.org/10.1007/s00397-002-0256-9CrossRef

Edwards BJ, Nafar Sefiddashti MH, Khomami B (2022) Atomistic simulation of shear flow of linear alkane and polyethylene liquids: a 50-year retrospective. J Rheol 66:415. https://doi.org/10.1122/8.0000365CrossRef

Everaers R (2012) Topological versus rheological entanglement length in primitive-path analysis protocols, tube models, and slip-link models. Phys Rev E - Stat Nonlinear, Soft Matter Phys 86:1–5. https://doi.org/10.1103/PhysRevE.86.022801CrossRef

Fang J, Kröger M, Öttinger HC (2000) A thermodynamically admissible reptation model for fast flows of entangled polymers. II. Model predictions for shear and extensional flows. J Rheol 44:1293–1317. https://doi.org/10.1122/1.1308522CrossRef

Fetters LJ, Lohse DJ, Milner ST, Graessley WW (1999) Packing length influence in linear polymer melts on the entanglement, critical, and reptation molecular weights. Macromolecules 32:6847–6851. https://doi.org/10.1021/MA990620O/ASSET/IMAGES/LARGE/MA990620OF00006.JPEGCrossRef

Foteinopoulou K, Karayiannis NC, Mavrantzas VG, Kröger M (2006) Primitive path identification and entanglement statistics in polymer melts: results from direct topological analysis on atomistic polyethylene models. Macromolecules 39:4207–4216. https://doi.org/10.1021/MA060306B/ASSET/IMAGES/MEDIUM/MA060306BN00001.GIFCrossRef

Galvani Cunha MA (2021) Nonequilibrium dynamics in polymers near interfaces and fluids under flow. Johns Hopkins University

Graham RS, Likhtman AE, McLeish TCB, Milner ST (2003) Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J Rheol 47:1171–1200. https://doi.org/10.1122/1.1595099CrossRef

Grmela M, Öttinger HC (1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys Rev E - Stat Physics Plasmas Fluids Relat Interdiscip Top 56:6620–6632. https://doi.org/10.1103/PhysRevE.56.6620CrossRef

Hawke LGD, Huang Q, Hassager O, Read DJ (2015) Modifying the pom-pom model for extensional viscosity overshoots. J Rheol 59:995–1017. https://doi.org/10.1122/1.4922060CrossRef

Hua CC, Schieber JD (1998) Segment connectivity, chain-length breathing, segmental stretch, and constraint release in reptation models. I. Theory and single-step strain predictions. J Chem Phys 109:10018–10027. https://doi.org/10.1063/1.477670CrossRef

Ianniruberto G (2015) Quantitative appraisal of a new CCR model for entangled linear polymers. J Rheol 59:211–235. https://doi.org/10.1122/1.4903495CrossRef

Ianniruberto G, Marrucci G (1996) On compatibility of the Cox-Merz rule with the model of Doi and Edwards. J Nonnewton Fluid Mech 65:241–246. https://doi.org/10.1016/0377-0257(96)01433-4CrossRef

Ianniruberto G, Marrucci G (2014) Convective constraint release (CCR) revisited. J Rheol 58:89–102. https://doi.org/10.1122/1.4843957CrossRef

Jeong SH, Kim JM, Yoon J et al (2016) Influence of molecular architecture on the entanglement network: topological analysis of linear, long- and short-chain branched polyethylene melts via Monte Carlo simulations. Soft Matter 12:3770–3786. https://doi.org/10.1039/C5SM03016ACrossRefPubMed

Kim JM, Keffer DJ, Kröger M, Edwards BJ (2008) Rheological and entanglement characteristics of linear-chain polyethylene liquids in planar Couette and planar elongational flows. J Nonnewton Fluid Mech 152:168–183. https://doi.org/10.1016/j.jnnfm.2007.03.005CrossRef

Konstantinou PC, Stephanou PS (2023) Predicting high-density polyethylene melt rheology using a multimode tube model derived using non-equilibrium thermodynamics. Polymers (Basel) 15:3322. https://doi.org/10.3390/polym15153322CrossRefPubMed

Kröger M (2005) Shortest multiple disconnected path for the analysis of entanglements in two- and three-dimensional polymeric systems. Comput Phys Commun 168:209–232. https://doi.org/10.1016/J.CPC.2005.01.020CrossRef

Kröger M, Dietz JD, Hoy RS, Luap C (2023) The Z1+ package: shortest multiple disconnected path for the analysis of entanglements in macromolecular systems. Comput Phys Commun 283:108567. https://doi.org/10.1016/j.cpc.2022.108567CrossRef

Langeloth M, Masubuchi Y, Böhm MC, Müller-Plathe F (2013) Recovering the reptation dynamics of polymer melts in dissipative particle dynamics simulations via slip-springs. J Chem Phys 138:104907. https://doi.org/10.1063/1.4794156/192810CrossRefPubMed

Larson RG, Sridhar T, Leal LG et al (2003) Definitions of entanglement spacing and time constants in the tube model. J Rheol 47:809–818. https://doi.org/10.1122/1.1567750CrossRef

Li Y, Abberton BC, Kröger M, Liu WK (2013) Challenges in multiscale modeling of polymer dynamics. Polymers (Basel) 5:751–832. https://doi.org/10.3390/polym5020751CrossRef

Likhtman AE (2005) Single-chain slip-link model of entangled polymers: simultaneous description of neutron spin-echo, rheology, and diffusion. Macromolecules 38:6128–6139. https://doi.org/10.1021/ma050399hCrossRef

Likhtman AE, Graham RS (2003) Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation. J Nonnewton Fluid Mech 114:1–12. https://doi.org/10.1016/S0377-0257(03)00114-9CrossRef

Marrucci G (1996) Dynamics of entanglements: a nonlinear model consistent with the Cox-Merz rule. J Nonnewton Fluid Mech 62:279–289. https://doi.org/10.1016/0377-0257(95)01407-1CrossRef

Marrucci G, Ianniruberto G (2003) Flow-induced orientation and stretching of entangled polymers. Philos Trans R Soc London Ser A Math Phys Eng Sci 361:677–688. https://doi.org/10.1098/RSTA.2002.1156CrossRef

Marrucci G, Titomanlio G, Sarti GC (1973) Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts. Rheol Acta 12:269–275. https://doi.org/10.1007/BF01635115CrossRef

Masubuchi Y, Takimoto JI, Koyama K et al (2001) Brownian simulations of a network of reptating primitive chains. J Chem Phys 115:4387–4394. https://doi.org/10.1063/1.1389858CrossRef

Masubuchi Y, Ianniruberto G, Greco F, Marrucci G (2003) Entanglement molecular weight and frequency response of sliplink networks. J Chem Phys 119:6925–6930. https://doi.org/10.1063/1.1605382CrossRef

Masubuchi Y, Langeloth M, Böhm MC et al (2016) A multichain slip-spring dissipative particle dynamics simulation method for entangled polymer solutions. Macromolecules 49:9186–9191. https://doi.org/10.1021/acs.macromol.6b01971CrossRef

McIlroy C, Olmsted PD (2017) Deformation of an amorphous polymer during the fused-filament-fabrication method for additive manufacturing. J Rheol 61:379–397. https://doi.org/10.1122/1.4976839CrossRef

McLeish TCB (2002) Tube theory of entangled polymer dynamics. Adv Phys 51:1379–1527. https://doi.org/10.1080/00018730210153216CrossRef

Mead DW, Larson RG, Doi M (1998) A molecular theory for fast flows of entangled polymers. Macromolecules 31:7895–7914. https://doi.org/10.1021/ma980127xCrossRef

Mwasame PM, Wagner NJ, Beris AN (2017) On the macroscopic modelling of dilute emulsions under flow. J Fluid Mech 831:433–473. https://doi.org/10.1017/jfm.2017.578CrossRef

Nafar Sefiddashti MH, Edwards BJ, Khomami B (2015) Individual chain dynamics of a polyethylene melt undergoing steady shear flow. J Rheol 59:119–153. https://doi.org/10.1122/1.4903498CrossRef

Nafar Sefiddashti MH, Edwards BJ, Khomami B (2016) Steady shearing flow of a moderately entangled polyethylene liquid. J Rheol 60:1227–1244. https://doi.org/10.1122/1.4963800CrossRef

Nafar Sefiddashti MH, Edwards BJ, Khomami B (2019) Elucidating the molecular rheology of entangled polymeric fluids via comparison of atomistic simulations and model predictions. Macromolecules 52:8124–8143. https://doi.org/10.1021/ACS.MACROMOL.9B01099/ASSET/IMAGES/LARGE/MA9B01099_0010.JPEGCrossRef

Öttinger HC (1999) A thermodynamically admissible reptation model for fast flows of entangled polymers. J Rheol 43:165–185. https://doi.org/10.1122/1.551055CrossRef

Öttinger HC (2005) Beyond equilibrium thermodynamics. WileyCrossRef

Öttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E - Stat Physics Plasmas Fluids Relat Interdiscip Top 56:6633–6655. https://doi.org/10.1103/PhysRevE.56.6633CrossRef

Qin J, Milner ST, Stephanou PS, Mavrantzas VG (2012) Effects of tube persistence length on dynamics of mildly entangled polymers. J Rheol 56:707–723. https://doi.org/10.1122/1.4708594CrossRef

Ramírez-Hernández A, Detcheverry FA, Peters BL et al (2013) Dynamical simulations of coarse grain polymeric systems: rouse and entangled dynamics. Macromolecules 46:6287–6299. https://doi.org/10.1021/MA400526V/ASSET/IMAGES/LARGE/MA-2013-00526V_0019.JPEGCrossRef

Schieber JD, Andreev M (2014) Entangled polymer dynamics in equilibrium and flow modeled through slip links. Annu Rev Chem Biomol Eng 5:367–381. https://doi.org/10.1146/ANNUREV-CHEMBIOENG-060713-040252CrossRefPubMed

Schieber JD, Córdoba A (2021) Nonequilibrium thermodynamics for soft matter made easy(er). Phys Fluids 33:083103. https://doi.org/10.1063/5.0057081CrossRef

Schieber JD, Neergaard J, Gupta S (2003) A full-chain, temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching. J Rheol 47:213–233. https://doi.org/10.1122/1.1530155CrossRef

Sefiddashti MHN, Edwards BJ, Khomami B (2019) Individual molecular dynamics of an entangled polyethylene melt undergoing steady shear flow: steady-state and transient dynamics. Polymers (Basel) 11:476. https://doi.org/10.3390/polym11030476CrossRef

Shanbhag S, Larson RG, Takimoto J, Doi M (2001) Deviations from dynamic dilution in the terminal relaxation of star polymers. Phys Rev Lett 87:195502. https://doi.org/10.1103/PhysRevLett.87.195502CrossRefPubMed

Steenbakkers RJA, Tzoumanekas C, Li Y et al (2014) Primitive-path statistics of entangled polymers: mapping multi-chain simulations onto single-chain mean-field models. New J Phys 16:015027. https://doi.org/10.1088/1367-2630/16/1/015027CrossRef

Stephanou PS, Kröger M (2016) Solution of the complete Curtiss-Bird model for polymeric liquids subjected to simple shear flow. J Chem Phys 144:124905. https://doi.org/10.1063/1.4944674CrossRefPubMed

Stephanou PS, Kröger M (2017) Non-constant link tension coefficient in the tumbling-snake model subjected to simple shear. J Chem Phys 147:174903. https://doi.org/10.1063/1.4991935CrossRefPubMed

Stephanou PS, Mavrantzas VG (2013) Quantitative predictions of the linear viscoelastic properties of entangled polyethylene and polybutadiene melts via modified versions of modern tube models on the basis of atomistic simulation data. J Nonnewton Fluid Mech 200:111–130. https://doi.org/10.1016/j.jnnfm.2013.04.003CrossRef

Stephanou PS, Mavrantzas VG (2014) Accurate prediction of the linear viscoelastic properties of highly entangled mono and bidisperse polymer melts. J Chem Phys 140:214903. https://doi.org/10.1063/1.4878500CrossRefPubMed

Stephanou PS, Baig C, Tsolou G et al (2010) Quantifying chain reptation in entangled polymer melts: topological and dynamical mapping of atomistic simulation results onto the tube model. J Chem Phys 132:124904. https://doi.org/10.1063/1.3361674CrossRefPubMed

Stephanou PS, Baig C, Mavrantzas VG (2011a) Projection of atomistic simulation data for the dynamics of entangled polymers onto the tube theory: calculation of the segment survival probability function and comparison with modern tube models. Soft Matter 7:380–395. https://doi.org/10.1039/c0sm00327aCrossRef

Stephanou PS, Baig C, Mavrantzas VG (2011b) Toward an improved description of constraint release and contour length fluctuations in tube models for entangled polymer melts guided by atomistic simulations. Macromol Theory Simul 20:752–768. https://doi.org/10.1002/mats.201100052CrossRef

Stephanou PS, Tsimouri IC, Mavrantzas VG (2016) Flow-induced orientation and stretching of entangled polymers in the framework of nonequilibrium thermodynamics. Macromolecules 49:3161–3173. https://doi.org/10.1021/acs.macromol.5b02805CrossRef

Stephanou PS, Schweizer T, Kröger M (2017) Communication: Appearance of undershoots in start-up shear: experimental findings captured by tumbling-snake dynamics. J Chem Phys 146:161101. https://doi.org/10.1063/1.4982228CrossRefPubMed

Stephanou PS, Tsimouri IC, Mavrantzas VG (2020) Simple, accurate and user-friendly differential constitutive model for the rheology of entangled polymer melts and solutions from nonequilibrium thermodynamics. Materials (Basel) 13:2867. https://doi.org/10.3390/ma13122867CrossRef

Tzoumanekas C, Theodorou DN (2006) Topological analysis of linear polymer melts: a statistical approach. Macromolecules 39:4592–4604. https://doi.org/10.1021/MA0607057/SUPPL_FILE/MA0607057SI20060329_065346.AVICrossRef

Uneyama T, Masubuchi Y (2012) Multi-chain slip-spring model for entangled polymer dynamics. J Chem Phys 137:154902. https://doi.org/10.1063/1.4758320CrossRefPubMed

Van Ruymbeke E, Liu C-Y, Bailly C et al (2007) Quantitative tube model predictions for the linear viscoelasticity of linear polymers. Rheol Rev 53–134. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=22e3529664ea417aca3cf75f484e5e8115a9e85a

van Wiechen PH, Booij HC (1971) A general solution to the necklace model problem in the rheology of macromolecules. J Eng Math 5:89–98. https://doi.org/10.1007/BF01535441/METRICSCrossRef

Watanabe H (1999) Viscoelasticity and dynamics of entangled polymers. Prog Polym Sci 24:1253–1403. https://doi.org/10.1016/S0079-6700(99)00029-5CrossRef

Yaoita T, Isaki T, Masubuchi Y et al (2008) Statics, linear, and nonlinear dynamics of entangled polystyrene melts simulated through the primitive chain network model. J Chem Phys 128:154901. https://doi.org/10.1063/1.2899653CrossRefPubMed

Title: Variable entanglement density constitutive rheological model for polymeric fluids
Author: Pavlos S. Stephanou
Publication date: 26-04-2024
Publisher: Springer Berlin Heidelberg
Published in: Rheologica Acta / Issue 5/2024
Print ISSN: 0035-4511
Electronic ISSN: 1435-1528
DOI: https://doi.org/10.1007/s00397-024-01444-0

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