Determination of Relaxation Time Spectra from Rheological Data Using an Edge Preserving Regularization Method

1. INTRODUCTION In order to describe linear viscoelastic properties of polymers one is interested in the relaxation time spectrum h, as it supplies information about the material properties. Moreover given h it is easy to convert one material function into another one. Additionally many microscopic theories are based on h and provide predictions about its character. Unfortunately the relaxation time spectrum is not directly accessible. It is only possible to estimate it from measured data, e.g. the dynamic moduli G' and G", which is an ill-posed inverse problem. Ill-posed problems are solved with regularization methods: Apart from the constraint from the data additional information about the spectrum independent from the data (a priori knowledge) is imposed upon the spectrum. A typical a priori knowledge is the smoothness constraint used in the algorithm proposed by Honerkamp and Weese (1993), which will be denoted the common method: The spectrum and its first derivative are assumed to be continuous. In contrast to this assumption the models due to spectra that are piecewise smooth in double logarithmic representation and exhibit edges. Unfortunately the common method is not able to reconstruct these spectra. Therefore we introduced a new edge preserving regularization method. By using the a priori knowledge " piecewise smoothness " this method is able to reconstruct even piecewise smooth spectra. 2. MODEL 2.1 Linear viscoelastic model The dependence of the dynamic moduli G' and G" on the relaxation time spectrum h is given by the linear viscoelastic model: ω ω τ ω τ τ ε σ ω ω τ ω τ τ ε σ τ τ 2 2 2 1 0 1 0 (1) denote the noisy data. The errors ε σ (,) 0 i due to measurement are assumed to be relative. 2.2 Theoretical models for the spectrum h For polymer solutions Rouse (1953) and Zimm (1956) predicted up to the Rouse time τ R a linearly decreasing spectrum in double logarithmic scaling with a gradient of-1/2 and-2/3 respectively. The material specific Rouse time is independent of the molecular weight and varies from 10-4 s to 10-3 s. The BSW spectrum due to Baumgaertel et al. (1990) predicts another linear increase with a gradient of 0-¼ in the time interval from τ R to the terminal relaxation time τ τ 0 R >. In particular the spectrum does not show any peak, i.e. any characteristic relaxation time in this interval. The transition …