An improved algorithm for the Fourier integral of the KWW function and its application to neutron scattering and dielectric data

摘要

The Kohlrausch-Williams-Watts (KWW) function is widely used to describe relaxation behavior in glass-forming polymers and other condensed systems. The application of this time domain function to frequency domain spectroscopy is, in principle, possible through its Fourier transform. Unfortunately analytical forms for the transform exist only in limited cases, and a number of methods, which circumvent this limitation, are described in the literature. Here we have revisited the problem of evaluating the Fourier integral in the general case, but with the specific aim of producing a fast and accurate algorithm suitable for use within common spreadsheet applications available on personal computers. Two methods were examined, a corrected discrete Fourier transform and, more successfully, a two-series expansion representation. A previous problem associated with the use of complementary series, that they may not adequately cover the whole range of evaluation of the transform, was overcome by the substitution of a Pade approximant for one of the series. This was found to provide the basis for a robust and effective algorithm whose accuracy was tested against both symbolic mathematical expressions and standard tabulated values. The use of this algorithm in iterative nonlinear least squares curve fitting routines is illustrated by reference to quasi-elastic neutron scattering data and dielectric relaxation spectra. The former involves a convolution of the real part of the integral with the instrumental resolution function; very satisfactory fits to scattering spectra for poly(vinyl acetate) and poly(isobutylene), at temperatures above their glass transitions, was achieved and the KWW characteristic relaxation parameters established. The dielectric spectrum due to the a-relaxation of poly(vinyl acetate) is similarly well-described, here using both the real and the imaginary parts of the integral. The KWW description of the PVAc relaxation is compared with that using the Havriliak-Negami equation and was found to provide an equally acceptable description of the data, and by using one less adjustable parameter.