We review constitutive modeling of solutions and melts of linear polymers, focusing on changes in rheological behavior in shear and extensional flow as the concentration increases from unentangled dilute, to entangled, to dense melt. The rheological changes are captured by constitutive equations, prototypes of which are the FENE-P model for unentangled solutions and the DEMG model for entangled solutions and melts. From these equations, and supporting experimental data, for dilute solutions, the extensional viscosity increases with the strain rate from the low–strain rate to the high–strain rate asymptote, but in the densely entangled state, the high–strain rate viscosity is lower than the low–shear rate value, especially when orientation-dependent friction is accounted for. In shearing flow, shear thinning increases dramatically as the entanglement density increases, which can eventually lead to a shear-banding inhomogeneity. Recent improvements in constitutive modeling are paving the way for robust and accurate numerical simulations of polymer fluid mechanics and industrial processing of polymers.


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Literature Cited

  1. Acharya MV, Bhattacharjee PK, Nguyen DA, Sridhar T. 2008. Are entangled polymeric solutions different from melts?. AIP Conf. Proc. 1027:391–93 [Google Scholar]
  2. Adams JM, Fielding SM, Olmsted PD. 2008. The interplay between boundary conditions and flow geometries in shear banding: hysteresis, band configurations, and surface transitions. J. Non-Newton. Fluid Mech. 151:101–18 [Google Scholar]
  3. Adams JM, Fielding SM, Olmsted PD. 2011. Transient shear banding in entangled polymers: a study using the Rolie-Poly model. J. Rheol. 55:1007–32 [Google Scholar]
  4. Adams JM, Olmsted PD. 2009. Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions. Phys. Rev. Lett. 103:067801 [Google Scholar]
  5. Archer LA, Larson RG, Chen Y-L. 1995. Direct measurements of slip in sheared polymer solutions. J. Fluid Mech. 301:133–51 [Google Scholar]
  6. Armstrong RC, Gupta SK, Basaran O. 1980. Conformational changes of macromolecules in transient elongational flow. Polym. Eng. Sci. 20:466–72 [Google Scholar]
  7. Baaijens FPT. 1998. Mixed finite element methods for viscoelastic flow analysis: a review. J. Non-Newton. Fluid Mech. 79:361–85 [Google Scholar]
  8. Bach A, Almdal K, Hassager O, Rasmussen HK. 2003. Elongational viscosity of narrow molar mass distribution polystyrene. Macromolecules 36:5174–79 [Google Scholar]
  9. Bhattacharjee PK, Oberhauser J, McKinley GH, Leal LG, Sridhar T. 2002. Extensional rheometry of entangled solutions. Macromolecules 35:10131–48 [Google Scholar]
  10. Bird RB, Curtiss CR, Armstrong RC, Hassager O. 1987. Dynamics of Polymeric Liquids 2 Kinetic Theory New York: Wiley, 2nd ed..
  11. Boukany PE, Wang SQ. 2007. A correlation between velocity profile and molecular weight distribution in sheared entangled polymer solutions. J. Rheol. 51:217–33 [Google Scholar]
  12. Boukany PE, Wang SQ. 2009. Shear banding or not in entangled DNA solutions depending on the level of entanglement. J. Rheol. 53:73–83 [Google Scholar]
  13. Byars JA, Binnington RJ, Boger DV. 1997. Entry flow and constitutive modelling of fluid S1. J. Non-Newton. Fluid Mech. 72:219–35 [Google Scholar]
  14. Cohen A. 1991. A Padé approximant to the inverse Langevin function. Rheol. Acta 30:270–73 [Google Scholar]
  15. Dalal IS, Hoda N, Larson RG. 2012. Multiple regimes of deformation in shearing flow of isolated polymers. J. Rheol. 56:305–32 [Google Scholar]
  16. Dealy JM, Larson RG. 2006. Structure and Rheology of Molten Polymers: From Structure to Flow Behavior and Back Again. Munich: Hanser [Google Scholar]
  17. Denn MM. 1990. Issues in viscoelastic fluid mechanics. Annu. Rev. Fluid Mech. 22:13–24 [Google Scholar]
  18. Desai P, Larson RG. 2014. Constitutive equation that shows extension thickening for entangled solutions and extension thinning for melts. J. Rheol. 58:255–79 [Google Scholar]
  19. Doi M, Edwards SF. 1986. The Theory of Polymer Dynamics Oxford, UK: Clarendon
  20. Doyle PS, Shaqfeh ESG, Gast AP. 1997. Dynamic simulation of freely draining flexible polymers in steady linear flows. J. Fluid Mech. 334:251–91 [Google Scholar]
  21. Ferry JD. 1980. Viscoelastic Properties of Polymers New York: Wiley
  22. Flory PJ. 1953. Principles of Polymer Chemistry Ithaca, NY: Cornell Univ. Press
  23. Fukuda M, Osaki K, Kurata M. 1975. Nonlinear viscoelasticity of polystyrene solutions. 1. Strain-dependent relaxation modulus. J. Polym. Sci. B Polym. Phys. 13:1563–76 [Google Scholar]
  24. Ganvir V, Lele A, Thaokar R, Gautham BR. 2009. Prediction of extrudate swell in polymer melt extrusion using an Arbitrary Lagrangian Eulerian (ALE) based finite element method. J. Non-Newton. Fluid Mech. 156:21–28 [Google Scholar]
  25. Graessley WW. 2008. Polymeric Liquids and Networks: Dynamics and Rheology New York: Garland Sci.
  26. Graham RS, Likhtman AE, Milner ST, McLeish TCB. 2003. Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J. Rheol. 47:1171–200 [Google Scholar]
  27. Hayes KA, Buckley MR, Cohen I, Archer LA. 2008. High resolution shear profile measurements in entangled polymers. Phys. Rev. Lett. 101:218301 [Google Scholar]
  28. Hinch EJ. 1993. The flow of an Oldroyd fluid around a sharp corner. J. Non-Newton. Fluid Mech. 50:161–71 [Google Scholar]
  29. Huang Q, Alvarez NJ, Matsumiya Y, Rasmussen HK, Watanabe H, Hassager O. 2013a. Extensional rheology of entangled polystyrene solutions suggest importance of nematic interactions. ACS Macro Lett. 2:741–44 [Google Scholar]
  30. Huang Q, Mednova O, Rasmussen HK, Alvarez NJ, Skov AL. et al. 2013b. Concentrated polymer solutions are different from melts: role of entanglement molecular weight. Macromolecules 46:5026–35 [Google Scholar]
  31. Hur JS, Shaqfeh ESG, Larson RG. 2000. Brownian dynamics simulations of single DNA molecules in shear flow. J. Rheol. 44:713–42 [Google Scholar]
  32. Ianniruberto G, Marrucci G. 2001. A simple constitutive equation for entangled polymers with chain stretch. J. Rheol. 45:1305–18 [Google Scholar]
  33. Kaye A. 1963. A bouncing liquid stream. Nature 197:1001–2 [Google Scholar]
  34. Keunings R. 1986. On the high Weissenberg number problem. J. Non-Newton. Fluid Mech. 20:Suppl. 1209–26 [Google Scholar]
  35. Kolkka RW, Malkus DS, Hansen MG, Ierley GR, Worthing RA. 1988. Spurt phenomena of the Johnson-Segalman fluid and related models. J. Non-Newton. Fluid Mech. 29:303–35 [Google Scholar]
  36. Larson RG. 1988. Constitutive Equations for Polymer Solutions and Melts Boston, MA: Butterworths
  37. Larson RG. 1992. Instabilities in viscoelastic flows. Rheol. Acta 31:213–63 [Google Scholar]
  38. Larson RG. 1999. The Structure and Rheology of Complex Fluids New York: Oxford Univ. Press
  39. Larson RG. 2005. The rheology of dilute solutions of flexible polymers: progress and problems. J. Rheol. 39:1–70 [Google Scholar]
  40. Larson RG, Muller SJ, Shaqfeh ESG. 1990. A purely elastic instability in Taylor-Couette flow. J. Fluid Mech. 218:573–600 [Google Scholar]
  41. Lee CS, Tripp BC, Magda JJ. 1992. Does N1 or N2 control the onset of edge fracture?. Rheol. Acta 31:306–8 [Google Scholar]
  42. Likhtman AE. 2009. Whither tube theory: from believing to measuring. J. Non-Newton. Fluid Mech. 157:158–61 [Google Scholar]
  43. Likhtman AE, Graham RS. 2003. Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation. J. Non-Newton. Fluid Mech. 114:1–12 [Google Scholar]
  44. Likhtman AE, McLeish TCB. 2002. Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332–43 [Google Scholar]
  45. Luo XL. 1999. Numerical simulation of Weissenberg phenomena: the rod-climbing of viscoelastic fluids. Comput. Methods Appl. Mech. Eng. 180:393–412 [Google Scholar]
  46. Marrucci G. 1996. Dynamics of entanglements: a nonlinear model consistent with the Cox-Merz rule. J. Non-Newton. Fluid Mech. 62:279–89 [Google Scholar]
  47. Marrucci G, Grizzuti N. 1988. Fast flows of concentrated polymers: predictions of the tube model of chain stretching. Gazz. Chim. Ital. 118:179–85 [Google Scholar]
  48. Masubuchi Y, Yaoita T, Matsumiya Y, Watanabe H, Ianniruberto G, Marrucci G. 2013. Stretch/orientation induced acceleration in stress relaxation in coarse-grained molecular dynamics simulations. J. Soc. Rheol. Jpn. 41:35–37 [Google Scholar]
  49. McKinley GH, Hassager O. 1999. The Considère condition and rapid stretching of linear and branched polymer melts. J. Rheol. 43:1195–212 [Google Scholar]
  50. McKinley GH, Sridhar T. 2002. Filament-stretching rheometry of complex fluids. Annu. Rev. Fluid Mech. 34:375–415 [Google Scholar]
  51. McLeish TCB, Ball RC. 1986. A molecular approach to the spurt effect in polymer melt flow. J. Polym. Sci. B Polym. Phys. 24:1735–45 [Google Scholar]
  52. McLeish TCB, Larson RG. 1998. Molecular constitutive equations for a class of branched polymers: the pom-pom polymers. J. Rheol. 42:81–110 [Google Scholar]
  53. Mead DW, Larson RG, Doi M. 1998. A molecular theory for fast flows of entangled polymers. Macromolecules 31:7895–914 [Google Scholar]
  54. Menasveta MJ, Hoagland DA. 1991. Light-scattering from dilute poly(styrene) solutions in uniaxial elongational flow. Macromolecules 24:3427–33 [Google Scholar]
  55. Menezes EV, Graessley WW. 1982. Non-linear rheological behavior of polymer systems in several shear-flow histories. J. Polym Sci. B Polym. Phys. 20:1817–33 [Google Scholar]
  56. Milner ST. 2005. Predicting the tube diameter in melts and solutions. Macromolecules 38:4929–39 [Google Scholar]
  57. Minegishi A, Nishioka A, Takahashi T, Masubuchi Y, Takimoto J, Koyama K. 2001. Uniaxial elongational viscosity of PS/a small amount of UHMW-PS blends. Rheol. Acta 40:329–38 [Google Scholar]
  58. Morrison FA, Larson RG. 1992. A study of shear-stress relaxation anomalies in binary mixtures of monodisperse polystyrenes. J. Polym. Sci. B Polym. Phys. 30:943–50 [Google Scholar]
  59. Mu Y, Zhao GQ, Wu XH, Zhai JQ. 2012. Modeling and simulation of three-dimensional planar contraction flow of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive equations. Appl. Math. Comput. 218:8429–43 [Google Scholar]
  60. Nielsen JK, Rasmussen HK, Hassager O, McKinley GH. 2006. Elongational viscosity of monodisperse and bidisperse polystyrene melts. J. Rheol. 50:453–76 [Google Scholar]
  61. Owen RG, Phillips TN. 2006. Computational Rheology London: Imperial Coll. Press
  62. Pattamaprom C, Larson RG. 2001. Constraint release effects in monodisperse and bidisperse polystyrenes in fast transient shear flows. Macromolecules 34:5229–37 [Google Scholar]
  63. Pearson DS, Herbolzheimer E, Grizzuti N, Marrucci G. 1991. Transient behavior of entangled polymers at high shear rates. J. Polym. Sci. B Polym. Phys. 29:1589–97 [Google Scholar]
  64. Renardy M. 1995. A matched solution for corner flow of the upper convected Maxwell fluid. J. Non-Newton. Fluid Mech. 58:83–89 [Google Scholar]
  65. Renardy M. 1997. Re-entrant corner behavior of the PTT fluid. J. Non-Newton. Fluid Mech. 69:99–104 [Google Scholar]
  66. Rodd LE, Scott TP, Cooper-White JJ, McKinley GH. 2005. Capillary break-up rheometry of low-viscosity elastic fluids. Appl. Rheol. 14:12–27 [Google Scholar]
  67. Rosenberg JR, Keunings R. 1988. Further results on the flow of a Maxwell fluid through an abrupt contraction. J. Non-Newton. Fluid Mech. 29:295–302 [Google Scholar]
  68. Schweizer T, Meerveld JV, Öttinger HC. 2004. Nonlinear shear rheology of polystyrene melt with narrow molecular weight distribution: experiment and theory. J. Rheol. 48:1345–63 [Google Scholar]
  69. Smith DE, Babcock HP, Chu S. 1999. Single-polymer dynamics in steady shear flow. Science 283:1724–27 [Google Scholar]
  70. Solomon MJ, Muller SJ. 1996. Study of mixed solvent quality in a polystyrene-dioctyl phthalate-polystyrene system. J. Polym. Sci. B Polym. Phys. 34:181–92 [Google Scholar]
  71. Tanner RI. 1985. Engineering Rheology New York: Oxford Univ. Press
  72. Tanner RI, Keentok M. 1983. Shear fracture in cone plate rheometry. J. Rheol. 27:47–57 [Google Scholar]
  73. Tapadia P, Wang SQ. 2006. Direct visualization of continuous simple shear in non-Newtonian polymeric fluids. Phys. Rev. Lett. 96:016001 [Google Scholar]
  74. Thomas DG, Khomami B, Surrshkumar R. 2009. Nonlinear dynamics of viscoelastic Taylor-Couette flow: effect of elasticity on pattern selection, molecular conformation and drag. J. Fluid Mech. 620:353–82 [Google Scholar]
  75. Yaoita T, Isaki T, Masubuchi Y, Watanabe H, Ianniruberto G, Marrucci G. 2012. Primitive chain network simulation of elongational flows of entangled linear chains: stretch/orientation-induced reduction of monomeric friction. Macromolecules 45:2773–82 [Google Scholar]

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