Rheologica Acta

, Volume 47, Issue 7, pp 689–700 | Cite as

Determining polymer molecular weight distributions from rheological properties using the dual-constraint model

  • Cattaleeya Pattamaprom
  • Ronald G. Larson
  • Anuvat Sirivat
Original Contribution


Although multiple models now exist for predicting the linear viscoelasticity of a polydisperse linear polymer from its molecular weight distribution (MWD) and for inverting this process by predicting the MWD from the linear rheology, such inverse predictions do not yet exist for long-chain branched polymers. Here, we develop and test a method of inverting the dual-constraint model (Pattamaprom et al., Rheol Acta 39:517–531, 2000; Pattamaprom and Larson, Macromolecules 34:5229–5237, 2001), a model that is able to predict the linear rheology of polydisperse linear and star-branched polymers. As a first step, we apply this method only to polydisperse linear polymers, by comparing the inverse predictions of the dual-constraint model to experimental GPC traces. We show that these predictions are usually at least as good, or better than, the inverse predictions obtained from the Doi–Edwards double-reptation model (Tsenoglou, ACS Polym Prepr 28:185–186, 1987; des Cloizeaux, J Europhys Lett 5:437–442, 1988; Mead, J Rheol 38:1797–1827, 1994), which we take as a “benchmark”—an acceptable invertible model for polydisperse linear polymers. By changing the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, the predictions of the dual-constraint model are slightly improved. These results suggest that models that are complex enough to predict branched polymer rheology can be inverted, at least for linear polymers, to obtain molecular weight distribution. Further work will be required to invert such models to allow prediction of the molecular weight distribution of branched polymers.


Molecular weight distribution Rheology Dual-constraint model Double-reptation model 



The authors are grateful for the financial support from Thailand research fund grant number TRG4580064. Evelyn van Ruymbeke and BASF research laboratory, especially Dr. Martin Laun, are deeply appreciated for their kind consideration in sharing experimental data. R.G. Larson was supported by the NSF, grant numbers DMR-0096688 and DMR-0604965. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).


  1. Anderssen RS, Mead DW, Driscoll JJ (1997) On the recovery of molecular weight functionals from the double reptation model. J Non-Newtonian Fluid Mech 68:291–301CrossRefGoogle Scholar
  2. Ball RC, McLeish TCB (1989) Dynamic dilution and the viscosity of star polymer melts. Macromolecules 22:1911–1913CrossRefGoogle Scholar
  3. Baumgaertel M, De Rosa ME, Machado J, Masse M, Winter HH (1992) The relaxation time spectrum of nearly monodisperse polybutadiene melts. Rheol Acta 31:75–82CrossRefGoogle Scholar
  4. Carrot C, Guillet J (1997) From dynamic moduli to molecular weight distribution: a study of various polydisperse linear polymers. J Rheol 41:1203–1220CrossRefGoogle Scholar
  5. Carrot C, Revenu P, Guillet J (1996) Rheological behavior of degraded polypropylene melts: from MWD to dynamic moduli. J Appl Polym Sci 61:1887–1897CrossRefGoogle Scholar
  6. Cocchini F, Nobile MR (2003) Constrained inversion of rheological data to molecular weight distribution for polymer melts. Rheol Acta 42:232–242Google Scholar
  7. Das C, Inkson NJ, Read DJ, Kelmanson MA, McLeish TCB (2006) Computational linear rheology of general branch-on-branch polymers. J Rheol 50:207–234CrossRefGoogle Scholar
  8. Dealy JM, Wissbrun KF (1990) Melt rheology and its role in plastics processing. Van Nostrand, New YorkGoogle Scholar
  9. des Cloizeaux J (1988) Double reptation vs. simple reptation in polymer melts. J Europhys Lett 5:437–442CrossRefGoogle Scholar
  10. des Cloizeaux J (1990) Relaxation and viscosity anomaly of melts made of entangled polymers. Time dependent reptation. Macromolecules 23:4678–4687CrossRefGoogle Scholar
  11. Doi M, Edwards SF (1978) Dynamics of concentrated polymer systems, part 1–3. J Chem Soc, Faraday Trans II 74:1789–1832CrossRefGoogle Scholar
  12. Doi M, Edwards SF (1979) Dynamics of concentrated polymer systems, part 4—rheological properties. J Chem Soc, Faraday Trans II 75:38–54CrossRefGoogle Scholar
  13. Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon, OxfordGoogle Scholar
  14. Fetters LJ, Lohse DJ, Richter D, Witten TA, Zirkel A (1994) Connection between polymer molecular weight, density, chain dimensions, and melt viscoelastic properties. Macormolecules 27:4639–4647CrossRefGoogle Scholar
  15. Gloor WE (1978) The numerical evaluation of parameters in distribution functions polymer from their molecular weight distribution. J Appl Poly Sci 22:1177–1182CrossRefGoogle Scholar
  16. Gloor WE (1983) Extending the continuum of molecular weight distribution based on the generalized exponential (GEX) distribution. J Appl Poly Sci 22:1177–1182CrossRefGoogle Scholar
  17. Larson RG, Sridhar T, Leal LG, McKinley GH (2003) Definitions of entanglement spacing and time constants in the tube model. J Rheol 47(3):809–818CrossRefGoogle Scholar
  18. Léonardi F, Majesté JC, Allal A, Marin GJ (2000) A critical review of rheological models based on the double reptation concept: the effect of a polydisperse environment. J Rheol 44:675CrossRefGoogle Scholar
  19. Léonardi F, Allal A, Marin G (2002) Molecular weight distribution from viscoelastic data: the importance of tube renewal and rouse mode. J Rheol 46(1):209–224CrossRefGoogle Scholar
  20. Likhtman AE, McLeish TCB (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332–6343CrossRefGoogle Scholar
  21. Maier D, Eckstein A, Friedrich C, Honerkamp J (1998) Evaluation of models combining rheological data with the molecular weight distribution. J Rheol 42:1153–1173CrossRefGoogle Scholar
  22. Mead DW (1994) Determination of molecular weight distributions of linear flexible polymers from linear viscoelastic material functions. J Rheol 38:1797–1827CrossRefGoogle Scholar
  23. Milner ST, McLeish TCB (1997) Parameter-free theory for stress relaxation in star polymer melts. Macromolecules 30:2159–2166CrossRefGoogle Scholar
  24. Milner ST, McLeish TCB, Young RN, Hakiki A, Johnson JM (1998) Dynamic dilution, constraint-release, and star-linear blends. Macromolecules 31:9345–9353CrossRefGoogle Scholar
  25. Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313Google Scholar
  26. Nobile MR, Cocchini F, Lawler JV (1996) On the stability of molecular weight distributions as computed from the flow curves of polymer melts. J Rheol 40:363–382CrossRefGoogle Scholar
  27. Park SJ, Shanbhag S, Larson RG (2005) A hierarchical algorithm for predicting the linear viscoelastic properties of polymer melts with long-chain branching. Rheol Acta 44:319–330CrossRefGoogle Scholar
  28. Pattamaprom C, Larson RG (2001) Constraint release effects in monodisperse and bidisperse polystyrenes in fast transient shearing flows. Macromolecules 34:5229–5237CrossRefGoogle Scholar
  29. Pattamaprom C, Larson RG, van Dykes TJ (2000) Quantitative predictions of linear viscoelastic rheological properties of entangled polymers. Rheol Acta 39:517–531CrossRefGoogle Scholar
  30. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran 77, 2nd edn. Cambridge University Press, USAGoogle Scholar
  31. Struglinski MJ, Graessley GG, Fetters L (1985) Effects of polydispersity on the linear viscoelastic properties of entangled polymers. 1. Experimental observations for binary mixtures of linear polybutadiene. Macromolecules 18:2630–2643CrossRefGoogle Scholar
  32. Thimm W, Friedrich C, Marth M, Honerkamp J (2000) On the Rouse spectrum and the determination of the molecular weight distribution from rheological data. J Rheol 44:429–438CrossRefGoogle Scholar
  33. Tsenoglou C (1987) Viscoelasticity of binary homopolymer blends. ACS Polym Prepr 28:185–186Google Scholar
  34. Tuminello WH (1986) Molecular weight and molecular weight distribution from dynamic measurements of polymer melts. Pol Eng Sci 26:1339–1347CrossRefGoogle Scholar
  35. van Ruymbeke E, Keunings R, Stéphenne V, Hagenaars A, Bailly C (2002a) Evaluation of reptation models for predicting the linear viscoelastic properties of entangled linear polymers. Macromolecules 35:2689–2699CrossRefGoogle Scholar
  36. van Ruymbeke E, Keunings R, Bailly C (2002b) Determination of the molecular weight distribution of entangled linear polymers from linear viscoelasticity data. J Non-Newtonian Fluid Mech 105:153–175CrossRefGoogle Scholar
  37. Viovy JL, Rubinstein M, Colby RH (1991) Constraint release in polymer melts: tube reorganization versus tube dilution. Macromolecules 24:3587–3596CrossRefGoogle Scholar
  38. Wasserman SH (1995) Calculating the molecular weight distribution from linear viscoelastic response of polymer melts. J Rheol 39:601–625CrossRefGoogle Scholar
  39. Wasserman SH, Graessley WW (1992) Effects of polydispersity on linear viscoelasticity in entangled polymer melts. J Rheol 36:543–572CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Cattaleeya Pattamaprom
    • 1
  • Ronald G. Larson
    • 2
  • Anuvat Sirivat
    • 3
  1. 1.Department of Chemical EngineeringThammasat UniversityPatumthaniThailand
  2. 2.Department of Chemical EngineeringThe University of MichiganAnn ArborUSA
  3. 3.The Petroleum and Petrochemical CollegeChulalongkorn UniversityBangkokThailand

Personalised recommendations