Relaxation Spectrum from Loss Modulus

In zero approximation, the loss modulus at co has sometimes been directly taken as the relaxation spectrum at t = 1 /co, but there is no need to use such a crude approximation when very simple methods are available to improve it. 

The method of Williams and Ferry again provides two alternative formulas depending on whether m is positive or negative, but these can be readily synthesized by using the absolute values of m and of the slope d log C/d log co  

The calculation proceeds in two steps as in the other Williams-Ferry methods values of B from equation 27 are given at the end of the chapter. 

Equation 29 is used when the slope of f(r) is positive and equation 30 when it is negative. 

An equation of Schwarzl and Staverman uses only the second derivative  

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Abstract In the present study a nonlinear regression with regularization and inverse Fourier transformation methods were developed to determine the relaxation spectrum from the frequency-dependent storage and loss modulus data. The spectra obtained were used for the determination of the molecular mass distribution in a calculation process... 

The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

From this equation the relaxation spectrum is approximately obtained in terms of the loss relaxation modulus,... 

Since the Gross frequency relaxation spectrum can be computed from r , i.e., from the loss modulus, G = T co, the agreement between the computed and measured G values provides good means of verifying both the computational and experimental procedures. It has been found that Eqs 7.83 and 7.84 are useful to evaluate the rheological performance of systems that obey the linear viscoelastic principles. 

Fig. 13 Storage modulus G and loss modulus G" of unfilled LDPE and highly filled LDPE/LDH nanocomposite (20 wt%) in the frequency sweep experiment. Also shown are the fits of G and G" obtained from conversion of the LDPE relaxation spectrum (see Table 2)...

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