Linear relaxation: Distributions, thermal activation, structure, and ambiguity

摘要

The equations governing the smallhyphen;signal response of relaxing, nonresonant systems which may be described by a distribution of relaxation times (DRT) and/or a distribution of activation energies (DAE) are summarized and generalized and their implications discussed for several popular distributions. Much past work, both experimental and theoretical, associated with these distributions is discussed. A distinction is made between physically realistic distributions, which involve finite shortest and longest relaxation times, and the usual mathematical approaches which involve limiting zero and infinite relaxation times. The Leacute;vy DRT, which is of the latter character and which leads to the popular stretched exponential (SE) time and Williamsndash;Watts (WW) frequency responses, is inconsistent with a temperaturehyphen;independent DAE, reducing its range of applicability for a thermally activated situation. The SEhyphen;WW response has been termed universal; it is not, both because of the above facts and also because it does not lead to the often found symmetrical loghyphen;frequency response. Both Gaussian and exponential DAEs can lead to both symmetrical and skewed results, and can involve either temperaturehyphen;dependent or temperaturehyphen;independent DAEs. However, the Gaussian DAE does not yield fractional powerhyphen;law time or frequency response over a finite, nonzero range, behavior found in nearly all distributed data. However, all DAEs involving exponential probability densities do lead to such behavior and provide, as well, an explanation of the temperature dependence of powerhyphen;law exponents. In addition, it appears that the response of systems involving an exponential DAE can fit that of virtually all previous models, including the SEhyphen;WW, and thus can fit all data for thermally activated systems which have been fitted by these models. Problems in data fitting and many sources and types of ambiguity and their resolution are discussed. Special attention is devoted to the distinction between parallel, sequential, and hierarchical microscopichyphen;model structure and response, and the various different, but, surprisingly, equivalent ways the overall response can be represented mathematically or by means of equivalent circuits of different connectivity.