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

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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.

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Notes

  1. Another reason for the limitation of TTS for extending the frequency regime is the accessible temperature range, which is limited by crystallization, glass transition, and degradation of the polymer.

  2. The contributions were calculated by calculating the ratio of G ′(ω) and G ″(ω) from the whole range to G ′(ω) and G ″(ω) calculated from the reduced relaxation time range. The contributions given in the text refer to the maximum value of this ratio.

  3. The calculation of the relevance factors is very much straightforward, but requires a lot of computations for complex spectra (and, therefore, anybody should be warned to use arrays with significantly more than 1,000 frequencies by 1,000 relaxation times for numerical reasons). The Matlab®-script written by the author for that purpose can be obtained upon request. To use it, a full Matlab®-installation is required.

  4. Tests on PBd and telechelic PBd on the TA Instruments ARES (Stadler and van Ruymbeke 2010; Stadler et al. 2009) indicate that the limit for the correction probably is somewhat higher than one decade in modulus.

References

  • Bates FS, Rosedale JH, Fredrickson GH (1990) Fluctuation effects in a symmetric diblock copolymer near the order-disorder transition. J Chem Phys 92(10):6255–6270

    Article  CAS  ADS  Google Scholar 

  • 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):777–786

    Article  CAS  Google Scholar 

  • Carella JM, Gotro JT, Graessley WW (1986) Thermorheological effects of long-chain branching in entangled polymer melts. Macromolecules 19(3):659–667

    Article  CAS  ADS  Google Scholar 

  • Davies AR, Anderssen RS (1997) Sampling localization in determining the relaxation spectrum. J non-Newton Fluid Mech 73(1–2):163–179

    Article  CAS  Google Scholar 

  • Dealy J, Larson RG (2006) Structure and rheology of molten polymers—from structure to flow behavior and back again. Hanser, Munich

    Google Scholar 

  • Ferry JD (1980) Viscoelastic properties of polymers. Wiley, New York

    Google Scholar 

  • Gabriel C, Kaschta J (1998) Comparison of different shear rheometers with regard to creep and creep recovery measurements. Rheol Acta 37:358–364

    Article  CAS  Google Scholar 

  • Gabriel C, Münstedt H (1999) Creep recovery behavior of metallocene linear low-density polyethylenes. Rheol Acta 38(5):393–403

    Article  CAS  Google Scholar 

  • 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):7–20

    Article  CAS  Google Scholar 

  • Hartwig G (1994) Polymer properties at room and cryogenic temperatures. Plenum, New York

    Google Scholar 

  • Honerkamp J (1989) Ill-posed problems in rheology. Rheol Acta 28(5):363–371

    Article  MATH  CAS  Google Scholar 

  • Honerkamp J, Weese J (1989) Determination of the relaxation spectrum by a regularization method. Macromolecules 22(11):4372–4377

    Article  CAS  ADS  Google Scholar 

  • Honerkamp J, Weese J (1993) A nonlinear regularization method for the calculation of relaxation spectra. Rheol Acta 32(1):65–73

    Article  CAS  Google Scholar 

  • Lippits DR, Rastogi S, Talebi S, Bailly C (2006) Formation of entanglements in initially disentangled polymer melts. Macromolecules 39(26):8882–8885

    Article  CAS  ADS  Google Scholar 

  • Liu CY, Bailly C, Yao M, Garritano RG, Franck AJ (2007) Instrument compliance effects revisited: linear viscoelasticity measurements (submitted)

  • Mandelkern L (1993) The crystalline state, 2nd edn., Chap. 4. ACS, Washington DC

    Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Plazek DJ (1966) Viscoelastic behavior of polymers at long times. Mellon Inst., Pittsburgh, PA, USA, 9 pp

  • Plazek DJ (1968) Magnetic-bearing torsional creep apparatus. J Polym Sci, Part A, Gen Pap 6(3):621–638

    CAS  Google Scholar 

  • Plazek DJ, Echeverria I (2000) Don’t cry for me Charlie Brown, or with compliance comes comprehension. J. Rheol 44(4):831–841

    Article  CAS  ADS  Google Scholar 

  • Plazek DJ, Raghupathi N, Kratz RF, Miller WR, Jr. (1979) Recoverable compliance behavior of high-density polyethylenes. J Appl Polym Sci 24(5):1305–1320

    Article  CAS  Google Scholar 

  • Schwarzl FR (1993) Polymermechanik. Springer, Heidelberg

    Google Scholar 

  • 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

    Article  CAS  ADS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • Stadler FJ, van Ruymbeke E (2010) An improved method to obtain direct rheological evidence of monomer density reequilibration for entangled polymer melts. Macromolecules

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  ADS  Google Scholar 

  • 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

    Article  CAS  Google Scholar 

  • 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

    Article  CAS  ADS  Google Scholar 

  • 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

    Article  CAS  ADS  Google Scholar 

  • Sternstein SS (1983) Transient and dynamic characterization of viscoelastic solids. Adv Chem Ser 203:123–147

    Article  CAS  Google Scholar 

  • Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior. Springer, New York

    MATH  Google Scholar 

  • Wood-Adams PM, Costeux S (2001) Thermorheological behavior of polyethylene: effects of microstructure and long chain branching. Macromolecules 34(18):6281–6290

    Article  CAS  ADS  Google Scholar 

Download references

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.

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  1. School of Semiconductor and Chemical Engineering, Chonbuk National University, 664-14, 1-ga Deokjin-dong, Deokjin-gu, Jeonju, Jeonbuk, 561-756, Republic of Korea

    Florian J. Stadler

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Correspondence to Florian J. Stadler.

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Stadler, F.J. Effect of incomplete datasets on the calculation of continuous relaxation spectra from dynamic-mechanical data. Rheol Acta 49, 1041–1057 (2010). https://doi.org/10.1007/s00397-010-0479-0

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