Rheologica Acta

, Volume 54, Issue 3, pp 169–183 | Cite as

Analytic slip-link expressions for universal dynamic modulus predictions of linear monodisperse polymer melts

  • Maria Katzarova
  • Ling Yang
  • Marat Andreev
  • Andrés Córdoba
  • Jay D. Schieber
Original Contribution

Abstract

The discrete slip-link model (DSM) is a robust mesoscopic theory that has great success predicting the rheology of flexible entangled polymer liquids and gels. In the most coarse-grained version of the DSM, we exploit heavily the universality observed in the shape of the relaxation modulus of linear monodisperse melts. For this type of polymer, we present here analytic expressions for the relaxation modulus. The high-frequency dynamics which are typically coarse-grained out from the DSM are added back into these expressions by using a Rouse chain with fixed ends to represent the fast motion of Kuhn steps between entanglements. We find consistency in the friction used for both fast and slow modes. We test these expressions against experimental data for three chemistries and molecular weights with good agreement. Using these analytic expressions, the polymer density, the molecular weight of a Kuhn step, MK, and the low-frequency cross-over between the storage and loss moduli, \(G^{\prime }\) and \(G^{\prime \prime }\), it is now straightforward to estimate model parameter values and obtain predictions over the experimentally accessible frequency range without performing expensive numerical calculations.

Keywords

Polymer melt Dynamic moduli Rouse theory Relaxation modulus Modelling Rheology 

Notes

Acknowledgments

Support of this work by Army Research Office Grants W911NF-08-2-0058 and W911NF-09-2-0071 are gratefully acknowledged. We would like to thank Dr. S. Coppola for the PB204 data in Fig. 9 and Dr. D. Auhl for the PI data in Fig. 10.

Supplementary material

397_2015_836_MOESM1_ESM.nb (266 kb)
(NB 265 KB)

References

  1. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions, vol 1. Dover New York, p 15Google Scholar
  2. Andreev M, Khaliullin RN, Steenbakkers RJ, Schieber JD (2013) Approximations of the discrete slip-link model and their effect on nonlinear rheology predictions. J Rheol 57:535–557Google Scholar
  3. Andreev M, Feng H, Yang L, Schieber JD (2014) Universality and speedup in equilibrium and nonlinear rheology predictions of the fixed slip-link model. J Rheol 58:723–736Google Scholar
  4. Auhl D, Ramirez J, Likhtman AE, Chambon P, Fernyhough C (2008) Linear and nonlinear shear flow behavior of monodisperse polyisoprene melts with a large range of molecular weights. J Rheol (1978-present) 52:801–835Google Scholar
  5. Bach A, Almdal K, Rasmussen HK, Hassager O (2003) Elongational viscosity of narrow molar mass distribution polystyrene. Macromolecules 36:5174–5179CrossRefGoogle Scholar
  6. Baumgaertel M, Schausberger A, Winter H (1990) The relaxation of polymers with linear flexible chains of uniform length. Rheol Acta 29:400–408CrossRefGoogle Scholar
  7. Bernabei M, Moreno AJ, Zaccarelli E, Sciortino F, Colmenero J (2011) Chain dynamics in nonentangled polymer melts: a first-principle approach for the role of intramolecular barriers. Soft Matter 71:364–1368Google Scholar
  8. Berry GC, Fox TG (1968) The viscosity of polymers and their concentrated solutions. SpringerGoogle Scholar
  9. Bhattacharjee PK, Oberhauser JP, McKinley GH, Leal LG, Sridhar T (2002) Extensional rheometry of entangled solutions. Macromolecules 35:10131–10148CrossRefGoogle Scholar
  10. Carella JM, Graessley WW, Fetters LJ (1984) Effects of chain microstructure on the viscoelastic properties of linear polymer melts: polybutadienes and hydrogenated polybutadienes. Macromolecules 17:2775–2786CrossRefGoogle Scholar
  11. Córdoba A, Schieber JD, Indei T (2015) The role of filament length, finite-extensibility and motor force dispersity in stress relaxation and buckling mechanisms in non-sarcomeric active gels. Soft Matter 11:38–57CrossRefGoogle Scholar
  12. Doi Masao (1988) The theory of polymer dynamics, vol 73. oxford university pressGoogle Scholar
  13. Doi M, Edwards SF (1978a) Dynamics of concentrated polymer systems. Part 2. Molecular motion under flow. Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics 74:1802–1817CrossRefGoogle Scholar
  14. Doi M, Edwards SF (1978b) Dynamics of concentrated polymer systems. Part 1. Brownian motion in the equilibrium state. Journal of the Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics 74:1789–1801CrossRefGoogle Scholar
  15. Ferry JD (1980) Viscoelastic properties of polymers. WileyGoogle Scholar
  16. 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 32:6847–6851CrossRefGoogle Scholar
  17. Gardiner CW (2009) Handbook ofstochastic methods: for the natural and social sciences. SpringerGoogle Scholar
  18. Gestoso P, Nicol E , Doxastakis M , Theodorou DN (2003) Atomistic Monte Carlo simulation of polybutadiene isomers: cis-1, 4-polybutadiene and 1, 2-polybutadiene. Macromolecules 36:6925–6938CrossRefGoogle Scholar
  19. Huang Q, Mednova O, Rasmussen HK, Alvarez NJ, Skov AL, Almdal K, Hassager O (2013) Concentrated polymer solutions are different from melts: Role of entanglement molecular weight. Macromolecules 46:5026–5035CrossRefGoogle Scholar
  20. Jensen MK, Khaliullin R, Schieber JD (2012) Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model. Rheol Acta 51:21–35CrossRefGoogle Scholar
  21. Karatasos K, Adolf DB (2000) Slow modes in local polymer dynamics. J Chem Phys 112:8225–8228CrossRefGoogle Scholar
  22. Katzarova M, Andreev M, Sliozberg YR, Mrozek RA, Lenhart JL, Andzelm JW, Schieber JD (2014) Rheological predictions of network systems swollen with entangled solvent. AIChE J 60:1372–1380CrossRefGoogle Scholar
  23. Kavassalis TA, Noolandi J (1987) New view of entanglements in dense polymer systems. Phys Rev lett 59:2674CrossRefGoogle Scholar
  24. Kavassalis TA, Noolandi J (1988) A new theory of entanglements and dynamics in dense polymer systems. Macromolecules 21:2869–2879CrossRefGoogle Scholar
  25. Khaliullin RN, Schieber JD (2009) Self-consistent modeling of constraint release in a single-chain mean-field slip-link model. Macromolecules 42:7504–7517CrossRefGoogle Scholar
  26. Khaliullin RN, Schieber JD (2010) Application of the slip-link model to bidisperse systems. Macromolecules 43:6202–6212CrossRefGoogle Scholar
  27. Koslover EF, Spakowitz AJ (2014) Multiscale dynamics of semiflexible polymers from a universal coarse-graining procedure. Phys Rev E 90:013304CrossRefGoogle Scholar
  28. Kubo R (1966) The fluctuation-dissipation theorem. Rep Prog Phys 29:255CrossRefGoogle Scholar
  29. Kubo R (1957) Statistical-mechanical theory of irreversible processes. I. general theory and simple applications to magnetic and conduction problems. J Phys Soc Jpn 12:570–586CrossRefGoogle Scholar
  30. Larson RG, Sridhar T, Leal LG, McKinley GH, Likhtman AE, McLeish TCB (2003) Definitions of entanglement spacing and time constants in the tube model. J Rheol (1978-present) 47:809–818Google Scholar
  31. Likhtman AE (2005) Single-chain slip-link model of entangled polymers: simultaneous description of neutron spin-echo, rheology, and diffusion. Macromolecules 38:6128–6139CrossRefGoogle Scholar
  32. Likhtman AE, McLeish TC (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332–6343CrossRefGoogle Scholar
  33. Liu C, He J, van Ruymbeke E, Keunings R, Bailly C (2006) Evaluation of different methods for the determination of the plateau modulus and the entanglement molecular weight. Polymer 47:4461–4479CrossRefGoogle Scholar
  34. Marciano Y, Brochard-Wyart F (1995) Normal modes of stretched polymer chains. Macromolecules 28:985–990CrossRefGoogle Scholar
  35. Mark JE (1999) Polymer data handbook. Oxford University, England, pp 804–805Google Scholar
  36. Pilyugina E, Andreev M, Schieber JD (2012) Dielectric relaxation as an independent examination of relaxation mechanisms in entangled polymers using the discrete slip-link model. Macromolecules 45:5728–5743CrossRefGoogle Scholar
  37. Qiu X, Ediger MD (2000) Local and global dynamics of unentangled polyethylene melts by 13c nmr. Macromolecules 33:490–498CrossRefGoogle Scholar
  38. Quake SR, Babcock H, Chu S (1997) The dynamics of partially extended single molecules of dna. Nature 388:151–154CrossRefGoogle Scholar
  39. Roland C (2006) Mechanical behavior of rubber at high strain rates. Rubber Chem Technol 79:429–459CrossRefGoogle Scholar
  40. Rouse PE (1953) A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J Chem Phys 21:1272–1280CrossRefGoogle Scholar
  41. Rubinstein M, Colby RH (2003) Polymer physics. Oxford University PressGoogle Scholar
  42. van Ruymbeke E, Coppola S, Balacca L, Righi S, Vlassopoulos D (2010) Decoding the viscoelastic response of polydisperse star/linear polymer blends. J Rheol (1978-present) 54:507–538Google Scholar
  43. Saphiannikova M, Toshchevikov V , Gazuz I , Petry F , Westermann S , Heinrich G (2014) Multiscale approach to dynamic-mechanical analysis of unfilled rubbers. Macromolecules 47(14):4813– 4823CrossRefGoogle Scholar
  44. Schausberger A, Schindlauer G , Janeschitz-Kriegl H (1985) Linear elastico-viscous properties of molten standard polystyrenes. Rheol Acta 24:220–227CrossRefGoogle Scholar
  45. Schieber JD (2003) Fluctuations in entanglements of polymer liquids. J Chem Phys 118:5162–5166CrossRefGoogle Scholar
  46. Schieber JD, Neergaard J, Gupta S (2002) A full-chain, temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching. J Rheol 47:213–233CrossRefGoogle Scholar
  47. Schieber JD, Andreev M (2014) Entangled polymer dynamics in equilibrium and flow modeled through slip links. Ann Rev Chem Biomol EngGoogle Scholar
  48. Schieber JD, Indei T, Steenbakkers RJ (2013) Fluctuating entanglements in single-chain mean-field models. Polymers 5:643–678CrossRefGoogle Scholar
  49. Steenbakkers RJ, Schieber JD (2012) Derivation of free energy expressions for tube models from coarse-grained slip-link models. J Chem Phys 137:034901CrossRefGoogle Scholar
  50. Steenbakkers RJ, Tzoumanekas C, Li Y, Liu WK, Kröger M, Schieber JD (2014) Primitive-path statistics of entangled polymers: mapping multi-chain simulations onto single-chain mean-field models. New J Phys 16:015027CrossRefGoogle Scholar
  51. Van Kampen NG (1992) Stochastic processes in physics and chemistry, vol 1. ElsevierGoogle Scholar
  52. Vandoolaeghe WL, Terentjev EM (2007) A Rouse-tube model of dynamic rubber viscoelasticity. J Phys A: Math Theor 40:14725Google Scholar
  53. Wang S, Wang S-Q, Halasa A, Hsu W-L (2003) Relaxation dynamics in mixtures of long and short chains: tube dilation and impeded curvilinear diffusion. Macromolecules 36:5355–5371CrossRefGoogle Scholar
  54. Whitley DM, Adolf DB (2012) Local segmental dynamics of cis-1, 4-polybutadiene, polypropylene and polyethylene terephthalate via molecular dynamics simulations. Mol Simul 38:119–123CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Maria Katzarova
    • 1
  • Ling Yang
    • 1
  • Marat Andreev
    • 2
    • 3
  • Andrés Córdoba
    • 1
  • Jay D. Schieber
    • 1
    • 2
  1. 1.Department of Chemical and Biological Engineering and Center for Molecular Study of Condensed Soft MatterIllinois Institute of TechnologyChicagoUSA
  2. 2.Department of Physics IllinoisInstitute of TechnologyChicagoUSA
  3. 3.Center for Molecular Study of Condensed Soft MatterIllinois Institute of TechnologyChicagoUSA

Personalised recommendations