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

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Abstract

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.

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Notes

  1. Or of a retardation time and strength.

  2. Such a low error means that the largest deviation between input and fitted data is in the range of 0.1% (in this case around ω = 10 − 5 s − 1) and that that deviation is rather the consequence of numerical limitations of double precision numbers than that of a mismatch between the real and the fitted spectrum. Double precision numbers are absolutely required to correctly calculate spectra; otherwise, a very strange result might occur as a consequence of numerical errors.

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Acknowledgements

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.

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Authors and Affiliations

  1. Unité de Physique et de Chimie des Hauts Polymères, Université catholique de Louvain, Croix du Sud, 1, 1348, Louvain-la-Neuve, Belgium

    Florian J. Stadler & Christian Bailly

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  1. Florian J. Stadler
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  2. Christian Bailly
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Correspondence to Christian Bailly.

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Stadler, F.J., Bailly, C. A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data. Rheol Acta 48, 33–49 (2009). https://doi.org/10.1007/s00397-008-0303-2

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