Determination of continuous relaxation spectrum based on the Fuoss-Kirkwood relation and logarithmic orthogonal power-series approximation

Abstract

In this study, we suggest a new algorithm for inferring continuous spectrum from dynamic moduli data. The algorithm is based on the Fuoss-Kirkwood relation (Fuoss and Kirkwood, 1941) and logarithmic powerseries approximation. The Fuoss-Kirkwood relation denotes the existence of the uniqueness of continuous spectrum. If we know the exact equation of dynamic moduli, then continuous spectrum can be inferred uniquely. We used the Chebyshev polynomials of the first kind to approximate dynamic moduli data in double-logarithmic scale. After the approximation, a spectrum equation can be derived by use of the complex decomposition method and the Fuoss-Kirkwood relation. We tested our algorithm to both simulated and experimental data of dynamic moduli and compared our result with those obtained from other methods such as the fixed-point iteration (Cho and Park, 2013) and cubic Hermite spline (Stadler and Bailly, 2009).