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Rheologica Acta

, Volume 57, Issue 4, pp 339–347 | Cite as

Regularization methods for finding the relaxation time spectra of linear polydisperse polymer melts

Original Contribution

Abstract

The calculation of discrete or continuous relaxation time spectra from rheometric measurables of polydisperse polymers is an ill-posed problem. In this paper, a curve fitting method for solving this problem is presented and compared to selected models from the literature. It is shown that the new method is capable of correctly predicting the molecular mass distributions of linear polydisperse polymer melts as well as their relaxation time spectra.

Keywords

Polymer melts Relaxation time spectrum Polydisperse III-posed problems Regularization 

Notes

Acknowledgements

The author wishes to acknowledge I. Teasdale from the Institute of Polymer Chemistry at the Johannes Kepler University Linz, Austria for the help in conducting and evaluating HT-GPC measurements and M. Gall for the fruitful discussions. This research is funded by the European Union within the Horizon 2020 project under the DiStruc Marie Skłodowska Curie innovative training network; grant agreement no. 641839.

References

  1. Baumgaertel M, Schausberger A, Winter HH (1990) The relaxation of polymers with linear flexible chains of uniform length. Rheol Acta 31:400–408CrossRefGoogle Scholar
  2. Carrot C, Guillet J (1997) From dynamic moduli to molecular weight distribution: a study of various polydisperse linear polymers. J Rheol 41:1203CrossRefGoogle Scholar
  3. Cole KS, Cole RH (1942) Dispersion and absorption in dielectrics ii. Direct current characteristics. J Chem Phys 10:98CrossRefGoogle Scholar
  4. deGennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, LondonGoogle Scholar
  5. des Cloizeaux J (1988) Double repattion vs. simple reptation in polymer melts. Europhys Lett 5:437–442CrossRefGoogle Scholar
  6. de L. Kronig R (1926) On the theory of dispersion of x-rays. J Opt Soc Am 12:547–557Google Scholar
  7. Doi M, Edwards SF (1986) The theory of polymer dynamics. OxfordGoogle Scholar
  8. Eder G, Janeschitz-Kriegel H, Liedauer S, Schausberger A, Stadlauer W, Schindlauer G (1989) The influence of molar mass distribution on the complex moduli of polymer melts. J Rheol 33:805–820CrossRefGoogle Scholar
  9. Friedrich C, Braun H (1992) Generalized Cole-Cole behavior and its rheological relevance. Rheol Acta 31:309–322CrossRefGoogle Scholar
  10. Friedrich C, Loy RJ, Anderssen RS (2009) Relaxation time spectrum molecular weight distribution relationships. Rheol Acta 48:151–162CrossRefGoogle Scholar
  11. Hadamard J (1902) Sur les problemes aux derive espartielles et leur signification physique. Priceton University Bulletin 13:49–52Google Scholar
  12. Kramers HA (1927) La diffusion de la lumiere par les atomes. Atti Cong Intern Fisici (Transactions of Volta Centenary Congress) Como 2:545557Google Scholar
  13. Lang C (2015) MMD of polydisperse linear polymers from rheological data. Akademiker Verlag, SaarbrückenGoogle Scholar
  14. Lang C (2017) A Laplace transform method for molecular mass distribution calculation from rheometric data. J Rheol 61:947–954CrossRefGoogle Scholar
  15. Maier D, Eckstein A, Friedrich C, Honerkamp J (1998) Evaluation of models combining rheological data with molecular weight distribution. J Rheol 42:1153–1173CrossRefGoogle Scholar
  16. Mead DW (1994) Determination of molecular weight distributions of linear flexible polymers from linear viscoelastic material functions. J Rheol 38:1769–1795CrossRefGoogle Scholar
  17. Orbey N, Dealy JM (1991) Determination of the relaxation time spectra from oscillatory shear data. J Rheol 35:1035–1049CrossRefGoogle Scholar
  18. Roths T, Maier D, Freidrich C, Marth M, Honerkamp J (2000) Determination of the relaxation time spectra from dynamic moduli using an edge preserving regularization method. Rheol Acta 39:163–173CrossRefGoogle Scholar
  19. Rouse PE (1953) Constrained inversion of rheological data to molecular weight distribution for polymer melts. J Chem Phys 21:1271–1280CrossRefGoogle Scholar
  20. Schausberger A (1986) On the stability of molecular weight distributions as computed from the flow curves of polymer melts. Rheol Acta 25:596–605CrossRefGoogle Scholar
  21. Temme NM (1987) Incomplete laplace integrals: uniform asymptotic expansion with application to the incomplete beta function. SIAM J Math Anal 18:1638–1662CrossRefGoogle Scholar
  22. Thimm W, Friedrich C, Marth M, Honerkamp J (1999) An analytical relation between relaxation time spectrum and molecular weight distribution. J Rheol 43:1663–1672CrossRefGoogle Scholar
  23. Thimm W, Friedrich C, Marth M, Honerkamp J (2000) An analytical relation between relaxation time spectrum and molecular weight distribution. J Rheol 44:429–438CrossRefGoogle Scholar
  24. Tikhonov A (1963) Solution of incorrectly formulated problems and the regularization method. Soviet Math Dokl 5:1035–1038Google Scholar
  25. Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior-an introduction. Springer, BerlinCrossRefGoogle Scholar
  26. Tsenoglou C (1991) Molecular weight polydispersity effects on the viscoelasticity of entangled linear polymers. Macromolecules 24:1762–1767CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Forschungszentrum JülichICS 3JülichGermany
  2. 2.Laboratory for Soft Matter and BiophysicsKU LeuvenLeuvenBelgium

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