Determination of the discrete relaxation spectrum for polybutadiene and polystyrene by a non-linear regression method

The relaxation spectrum is an important tool for investigating the behaviour of viscoelastic polymer melts. The most popular procedure for determining the relaxation spectrum is the use of a small-amplitude oscillatory shear experiment to determine the parameters in a multi-mode Maxwell model. However, this is an ill-posed problem and its numerical solution is along with difficulties. The determination of the discrete relaxation spectrum by a linear regression approach is widespread in practice and in the literature. In this work, a non-linear regression technique, based on Marquardt-Levenberg procedure, is applied in which the minimization is performed with respect to both the discrete relaxation times and the elastic moduli. Using this non-linear optimization, the spectrum parameters of a multi-mode Maxwell model were obtained for polybutadiene and polystyrene. It has been shown that in comparison with the other methods, the present approach is more efficient and it was found to give a very good fit with the fewest possible parameters.