On relaxation-spectrum estimation for decades of data: accuracy and sampling-localization considerations

In the inversion of data using the relaxation-spectrum model of electrical and rheological phenomena and processes, an important goal is to obtain a c omprehensive and accurate approximation of the distribution of relaxation t imes (DRT) associated with the phenomenon or process under investigation. T he estimation of such a DRT poses theoretical as well as experimental chall enges. In the latter, it is often necessary to measure many decades of freq uency-response data before sufficient information has been collected to all ow an adequate estimate of the structure of the required spectrum to be cal culated. At the theoretical level, among other things, one must solve the i nverse problem of the relaxation-spectrum estimation, along with the freque nt need to work with many decades of data. Some consequences of these two a spects of the problem are examined in this payer. In order to address this problem, accurate (or noisy), wide-range frequency -response data sets derived from the Kohlrausch-Williams-Watts (KWW) contin uous DRT function (with a value of its beta (0) parameter of 0.5) are inver ted numerically using a complex nonlinear least-squares method with variabl e tau free parameters and variable quadrature weighting. a method superior to Tikhonov regularization for data of the present type when inversion accu racy need not be sacrificed for increased resolution. The relative errors i n the resulting DRT point estimates are investigated for various frequency ranges for both frequency-response data derived from the usual KWW DRT and from such a DRT which is abruptly cut off at its low tau end, Although the inversions are ill-posed, DRT errors were found to be very small over appre ciable tau ranges, but the effects of cutoff of the range of the DRT and of a limited frequency range led to rapid relative-error increases at the end s of the DRT tau range, and allowed separation and quantification of the ef fects of these two limitations. Important resolution differences are illustrated between the results of inv ersions of accurate and of noisy data by the present method and by Tikhonov regularization. Finally, the temporal response was calculated very simply from frequency-response inversion DRT estimates and compared with exact str etched-exponential response function values associated with the KWW dispers ion model. Extremely small relative errors were found, but they nevertheles s showed the effects of the above limitations at short and long times. Tran sformation to the time domain of the kind illustrated here for wide-range d ata is far simpler, more accurate, and more convenient than Fourier transfo rmation.