Rheologica Acta

, Volume 48, Issue 1, pp 33–49 | Cite as

A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data

Original Contribution


We present a new method for the determination of highly precise continuous relaxation spectra. The method is based on the use of piecewise cubic Hermite splines, which are fairly easy to tabulate by using their knots. The Hermite splines method allows a continuous description of the spectrum by a series of polynomial functions. The numerical instabilities of the spectrum calculation are minimized by limiting the slope of the spectrum to physically meaningful values. The reproducibility of the spectrum calculation is within an error margin of about ±10% in the physically relevant relaxation time range. This method is able to retrieve the spectrum based on data calculated from a benchmark with high accuracy and precision.


Relaxation spectrum Continuous spectrum Discrete spectrum Linear viscoelasticity Hermite spline 



The authors want to acknowledge the financial support from Communauté Française de Belgique. FJS would like to thank Dr. J. Kaschta and Prof. em. Dr. F. R. Schwarzl (University Erlangen-Nürnberg) and Prof. Dr. H. H. Winter (University of Massachusetts, Amherst) for the stimulating discussions about this topic.

Supplementary material

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  1. Baumgärtel M, Winter HH (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol Acta 28:511–519CrossRefGoogle Scholar
  2. Baumgärtel M, Winter HH (1992) Interrelation between continuous and discrete relaxation time spectra. J Non-Newton Fluid Mech 44:15–36CrossRefGoogle Scholar
  3. Baumgärtel M, Schausberger A, Winter HH (1990) The relaxation of polymers with linear flexible chains of uniform length. Rheol Acta 29(5):400–408CrossRefGoogle Scholar
  4. Boltzmann L (1874) Sitzb Kgl Akad Wiss Wien 2(Abt 70):725Google Scholar
  5. Brabec CJ, Schausberger A (1995) An improved algorithm for calculating relaxation-time spectra from material functions of polymers with monodisperse and bimodal molar-mass distributions. Rheol Acta 34(4):397–405CrossRefGoogle Scholar
  6. Davies AR, Anderssen RS (1997) Sampling localization in determining the relaxation spectrum. J Non-Newton Fluid Mech 73(1–2):163–179CrossRefGoogle Scholar
  7. Dealy J, Larson RG (2006) Structure and rheology of molten polymers—from structure to flow behavior and back again. Hanser, MunichGoogle Scholar
  8. Elster C, Honerkamp J, Weese J (1992) Using regularization methods for the determination of relaxation and retardation spectra of polymeric liquids. Rheol Acta 31(2):161–174CrossRefGoogle Scholar
  9. Emri I, Tschoegl NW (1993) Generating line spectra from experimental responses. Part I: relaxation modulus and creep compliance. Rheol Acta 32(3):311–321CrossRefGoogle Scholar
  10. Emri I, Tschoegl NW (1994) Generating line spectra from experimental responses. 4. Application to experimental-data. Rheol Acta 33(1):60–70CrossRefGoogle Scholar
  11. Ferry JD (1980) Viscoelastic properties of polymers. Wiley, New YorkGoogle Scholar
  12. Ferry JD, Williams ML (1952) Second approximation methods for determining the relaxation-time spectrum of a viscoelastic material. J Colloid Sci 7:347–353CrossRefGoogle Scholar
  13. Gabriel C, Münstedt H (1999) Creep recovery behavior of metallocene linear low-density polyethylenes. Rheol Acta 38(5):393–403CrossRefGoogle Scholar
  14. Gabriel C, Münstedt H (2002) Influence of long-chain branches in polyethylenes on linear viscoelastic flow properties in shear. Rheol Acta 41(3):232–244CrossRefGoogle Scholar
  15. 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
  16. Hansen S (2007) Estimation of the relaxation spectrum from dynamic experiments using Bayesian analysis and a new regularization constraint. Rheol Acta 47:169–178. doi:10.1007/s00397-007-0225-4 CrossRefGoogle Scholar
  17. Honerkamp J (1989) Ill-posed problems in rheology. Rheol Acta 28(5):363–371CrossRefGoogle Scholar
  18. Honerkamp J, Weese J (1989) Determination of the relaxation spectrum by a regularization method. Macromolecules 22(11):4372–4377CrossRefGoogle Scholar
  19. Honerkamp J, Weese J (1993) A nonlinear regularization method for the calculation of relaxation spectra. Rheol Acta 32(1):65–73CrossRefGoogle Scholar
  20. Kaschta J, Schwarzl FR (1994a) Calculation of discrete retardation spectra from creep data: 1. Method. Rheol Acta 33(6):517–529CrossRefGoogle Scholar
  21. Kaschta J, Schwarzl FR (1994b) Calculation of discrete retardation spectra from creep data: 2. Analysis of measured creep curves. Rheol Acta 33(6):530–541CrossRefGoogle Scholar
  22. Kaschta J, Stadler FJ (2008) Avoiding waviness in the calculation of relaxation spectra. Rheol Acta (accepted)Google Scholar
  23. Laun HM, Wagner MH, Janeschitz-Kriegl H (1979) Model analysis of nonlinear viscoelastic behavior by use of a single integral constitutive equation: stresses and birefringence of a polystyrene melt in intermittent shear flows. Rheol Acta 18(5):615–622CrossRefGoogle Scholar
  24. Mandelkern L (1993) The crystalline state, 2nd edn. Chap. 4. Washington DC, ACSGoogle Scholar
  25. Mead DW (1994) Numerical interconversion of linear viscoelastic material functions. J Rheol 38(6):1769–1795CrossRefGoogle Scholar
  26. 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):26–38CrossRefGoogle Scholar
  27. Plazek DJ, Echeverria I (2000) Don’t cry for me Charlie Brown, or with compliance comes comprehension. J Rheol 44(4):831–841CrossRefGoogle Scholar
  28. Schwarzl FR (1993) Polymermechanik. Springer, HeidelbergGoogle Scholar
  29. Schwarzl F, Staverman AJ (1952) Higher approximations of relaxation spectra. Physica (The Hague) 18:791–798CrossRefGoogle Scholar
  30. Stadler FJ, Bailly C (2008) Effect of incomplete datasets on the calculation of continuous relaxation spectra from dynamic-mechanical data (submitted)Google Scholar
  31. 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 CrossRefGoogle Scholar
  32. Stadler FJ, Kaschta J, Münstedt H (2005) Dynamic-mechanical behavior of polyethylenes and ethene-/α-olefin-copolymers: part I: α -Relaxation. Polymer 46(23):10311–10320CrossRefGoogle Scholar
  33. 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):209–218CrossRefGoogle Scholar
  34. 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):1474–1482CrossRefGoogle Scholar
  35. Stadler FJ, Gabriel C, Münstedt H (2007) Influence of short-chain branching of polyethylenes on the temperature dependence of rheological properties in shear. Macromol Chem Phys 208(22):2449–2454CrossRefGoogle Scholar
  36. 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 CrossRefGoogle Scholar
  37. Tschoegl NW (1989) The phenomenological theory of linear viscoelastic behavior. Springer, BerlinGoogle Scholar
  38. Tschoegl NW, Emri I (1992) Generating line spectra from experimental responses. III. Interconversion between relaxation and retardation behavior. Int J Polym Mater 18(1–2):117–127CrossRefGoogle Scholar
  39. Tschoegl NW, Emri I (1993) Generating line spectra from experimental responses. Part II: storage and loss functions. Rheol Acta 32(3):322–327CrossRefGoogle Scholar
  40. van Ruymbeke E, Orfanou K, Kapnistos M, Iatrou H, Pitsikalis M, Hadjichristidis N, Lohse DJ, Vlassopoulos D (2007) Entangled dendritic polymers and beyond: rheology of symmetric Cayley-tree polymers and macromolecular self-assemblies. Macromolecules 40(16):5941–5952CrossRefGoogle Scholar
  41. Winter HH, Mours M (2005) IRIS-handbook. IRIS Development, Amherst, USAGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Unité de Physique et de Chimie des Hauts PolymèresUniversité catholique de LouvainLouvain-la-NeuveBelgium

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