Alfrey’s rule

Alternatively, one can use various approximations (see Ferry, 1980) to determine the relaxation spectra directly from modulus data or a mathematical function fit to the data. A method that works well is known as Alfrey s rule, in which the exponential function in the integral in Eq, 7,42 is 0 at small x s and 1 at large x s and is thus replaced by a step function H(t-T). With this simplification, Eq, 7,42 can be differentiated to obtain... 

Application of Alfrey s rule to the polycarbonate data of Fig, 7,20 results in a continuous and smooth relaxation spectra as shown in Fig, 7,24. The shape and magnitude of the spectra obtained in this fashion corresponds to the discrete Prony elements of Fig, 7,21 when smoothed so as to eliminate the oscillations obtained in the least squares fitting process. While the spectrum in Fig. 7.20 still exhibits slight non-smoothness, manipulation of the Prony elements and/or use of a more accurate method to determine H(t) could provide a smoother curve. For most purposes, the spectra shown is adequately smooth. 

Fig. 7.24 Relaxation spectra of polycarbonate calculated from the Prony series elements in Fig. 7.21 via Alfrey s rule.

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