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Rheologica Acta

, Volume 49, Issue 10, pp 1041–1057 | Cite as

Effect of incomplete datasets on the calculation of continuous relaxation spectra from dynamic-mechanical data

  • Florian J. Stadler
Original Contribution

Abstract

In this article, the effect of an incomplete frequency range on relaxation spectra calculated with the new spline-based method (Stadler and Bailly, Rheol Acta 48(1):33–49, 2009) presented before is discussed. The range, in which the spectrum can be determined, is limited by the range of the input data, but not directly by the inverse frequency. The actual limits depend on the range of input data. Depending on the shape of the spectrum the relaxation spectrum can be determined from the input data in a range up to three decades larger than the input data. This can be explained by the influence of the modes outside the inverse frequency range. For this purpose, a new concept, the relevance factor analysis, was introduced, which allows for a determination of the limits of spectrum calculation. The characteristic relaxation times are discussed in comparison for to the calculation of \(J_e^{\rm 0}\) and η 0 from the spectrum.

Keywords

Relaxation spectrum Continuous spectrum Relevance range Error minimization Hermite spline 

Notes

Acknowledgements

The author wants to acknowledge the financial aid from Communauté Française de Belgique. FJS would like to thank Prof. Dr. Christian Bailly (Université catholique de Louvain (UCL), Belgium), Prof. em. Dr. F. R. Schwarzl and Dr. J. Kaschta (University Erlangen-Nürnberg), and Prof. H. H. Winter (University of Massachusetts, Amherst) for stimulating discussions about this topic.

Supplementary material

397_2010_479_MOESM1_ESM.pdf (398 kb)
(PDF 398 KB)

References

  1. Bates FS, Rosedale JH, Fredrickson GH (1990) Fluctuation effects in a symmetric diblock copolymer near the order-disorder transition. J Chem Phys 92(10):6255–6270CrossRefADSGoogle Scholar
  2. Capodagli J, Lakes R (2008) Isothermal viscoelastic properties of PMMA and LDPE over 11 decades of frequency and time: a test of time-temperature superposition. Rheol Acta 47(7):777–786CrossRefGoogle Scholar
  3. Carella JM, Gotro JT, Graessley WW (1986) Thermorheological effects of long-chain branching in entangled polymer melts. Macromolecules 19(3):659–667CrossRefADSGoogle Scholar
  4. Davies AR, Anderssen RS (1997) Sampling localization in determining the relaxation spectrum. J non-Newton Fluid Mech 73(1–2):163–179CrossRefGoogle Scholar
  5. Dealy J, Larson RG (2006) Structure and rheology of molten polymers—from structure to flow behavior and back again. Hanser, MunichGoogle Scholar
  6. Ferry JD (1980) Viscoelastic properties of polymers. Wiley, New YorkGoogle Scholar
  7. Gabriel C, Kaschta J (1998) Comparison of different shear rheometers with regard to creep and creep recovery measurements. Rheol Acta 37:358–364CrossRefGoogle Scholar
  8. Gabriel C, Münstedt H (1999) Creep recovery behavior of metallocene linear low-density polyethylenes. Rheol Acta 38(5):393–403CrossRefGoogle Scholar
  9. Gabriel C, Kaschta J, Münstedt H (1998) Influence of molecular structure on rheological properties of polyethylenes I. Creep recovery measurements in shear. Rheol Acta 37(1):7–20CrossRefGoogle Scholar
  10. Hartwig G (1994) Polymer properties at room and cryogenic temperatures. Plenum, New YorkGoogle Scholar
  11. Honerkamp J (1989) Ill-posed problems in rheology. Rheol Acta 28(5):363–371zbMATHCrossRefGoogle Scholar
  12. Honerkamp J, Weese J (1989) Determination of the relaxation spectrum by a regularization method. Macromolecules 22(11):4372–4377CrossRefADSGoogle Scholar
  13. Honerkamp J, Weese J (1993) A nonlinear regularization method for the calculation of relaxation spectra. Rheol Acta 32(1):65–73CrossRefGoogle Scholar
  14. Lippits DR, Rastogi S, Talebi S, Bailly C (2006) Formation of entanglements in initially disentangled polymer melts. Macromolecules 39(26):8882–8885CrossRefADSGoogle Scholar
  15. Liu CY, Bailly C, Yao M, Garritano RG, Franck AJ (2007) Instrument compliance effects revisited: linear viscoelasticity measurements (submitted)Google Scholar
  16. Mandelkern L (1993) The crystalline state, 2nd edn., Chap. 4. ACS, Washington DCGoogle Scholar
  17. Piel C, Stadler FJ, Kaschta J, Rulhoff S, Münstedt H, Kaminsky W (2006) Structure-property relationships of linear and long-chain branched metallocene high-density polyethylenes and SEC-MALLS. Macromol Chem Phys 207(1):26–38. doi: 10.1002/macp.200500321 CrossRefGoogle Scholar
  18. Plazek DJ (1966) Viscoelastic behavior of polymers at long times. Mellon Inst., Pittsburgh, PA, USA, 9 ppGoogle Scholar
  19. Plazek DJ (1968) Magnetic-bearing torsional creep apparatus. J Polym Sci, Part A, Gen Pap 6(3):621–638Google Scholar
  20. Plazek DJ, Echeverria I (2000) Don’t cry for me Charlie Brown, or with compliance comes comprehension. J. Rheol 44(4):831–841CrossRefADSGoogle Scholar
  21. Plazek DJ, Raghupathi N, Kratz RF, Miller WR, Jr. (1979) Recoverable compliance behavior of high-density polyethylenes. J Appl Polym Sci 24(5):1305–1320CrossRefGoogle Scholar
  22. Schwarzl FR (1993) Polymermechanik. Springer, HeidelbergGoogle Scholar
  23. Stadler FJ, Münstedt H (2008) Terminal viscous and elastic rheological characterization of ethene-/α-olefin copolymers. J Rheol 52(3):697–712. doi: 610.1122/1121.2892039 CrossRefADSGoogle Scholar
  24. Stadler FJ, Bailly C (2009) A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data. Rheol Acta 48(1):33–49. doi: 10.1007/s00397-00008-00303-00392 CrossRefGoogle Scholar
  25. Stadler FJ, van Ruymbeke E (2010) An improved method to obtain direct rheological evidence of monomer density reequilibration for entangled polymer melts. MacromoleculesGoogle Scholar
  26. Stadler FJ, Kaschta J, Münstedt H (2005) Dynamic-mechanical behavior of polyethylenes and ethene-/α-olefin-copolymers: part I: α’-relaxation. Polymer 46(23):10311–10320. doi: 10310.11016/j.polymer.12005.10307.10099 CrossRefGoogle Scholar
  27. Stadler FJ, Kaschta J, Münstedt H (2008) Thermorheological behavior of long-chain branched metallocene catalyzed polyethylenes. Macromolecules 41(4):1328–1333. doi: 1310.1021/ma702367a CrossRefADSGoogle Scholar
  28. Stadler FJ, Piel C, Kaminsky W, Münstedt H (2006a) Rheological characterization of long-chain branched polyethylenes and comparison with classical analytical methods. Macromol Symp 236(1):209–218. doi: 210.1002/masy.200650426 CrossRefGoogle Scholar
  29. Stadler FJ, Piel C, Klimke K, Kaschta J, Parkinson M, Wilhelm M, Kaminsky W, Münstedt H (2006b) Influence of type and content of very long comonomers on long-chain branching of ethene-/α-olefin copolymers. Macromolecules 39(4):1474–1482. doi: 1410.1021/ma0514018 CrossRefADSGoogle Scholar
  30. Stadler FJ, Schumers J-M, Fustin C-A, Gohy J-F, Pyckhout-Hintzen W, Bailly C (2009) Rheological characterization of telechelic polybutadiene based temporary networks. Macromolecules 42:6181–6192. doi: 6110.1021/ma802488a CrossRefADSGoogle Scholar
  31. Sternstein SS (1983) Transient and dynamic characterization of viscoelastic solids. Adv Chem Ser 203:123–147CrossRefGoogle Scholar
  32. Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior. Springer, New YorkzbMATHGoogle Scholar
  33. Wood-Adams PM, Costeux S (2001) Thermorheological behavior of polyethylene: effects of microstructure and long chain branching. Macromolecules 34(18):6281–6290CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Semiconductor and Chemical EngineeringChonbuk National UniversityJeonjuRepublic of Korea

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