Relaxation Spectrum from Storage Modulus

The method of Williams and Ferry provides two formulas depending on whether m, the negative slope of // on a doubly logarithmic plot, is greater or less than 1. Almost invariably, w 1, in which case  

As with equation 4, the calculation is carried out in two stages first A is set equal to unity, and a preliminary calculation is made with each point at a given value of (0 yielding a value of A/ at t = 1 /co. From a tentative graph of H, the value of m is measured at each point, and the appropriate correction factor A is applied. Values of A and A are also given at the end of the chapter. 

The second order approximation formulas of TschoegF a. - are probably more accurate  

Equation 16 is used when the slope of H(t) is positive (m negative) and equation 17 when the slope is negative (m positive). In terms of log-log plots, equation 17 is 

An example of a third-order approximation from the method of Tschoegl is, for m positive, 

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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... 

Figure 29. Storage shear modulus vs. frequency for System 1. The points are experimental, uncorrected for the apparent yield stress, the lines computed from the relaxation spectrum.

For a more complete analysis of this sec for example Wetton [24]. The relaxation prtKX ss in polymers can also be approached from the measurement of a wide range of frequencies at a given temperature, but in practice this is impossible, so approximations have been made where measurements over 3 or 4 decades only are used. Such second-order approximations by Ferry [7] and Schwarzl Staverman [25] show that the relaxation spectrum can be represented from the storage modulus by... 

Fig. 11.9 Comparison of the measured storage-modulus spectrum (o and ) and that calculated from the Rouse theory (solid line) for sample A. The dashed line indicates the separation of the contributions from the G1 and GIO components. The arrow at 1/ti(1) indicates the frequency that is the reciprocal of the relaxation time of the first Rouse mode of the G1 component calculated from Eq. (7.57) with K = x 10, wheresis the arrow at l/ri(2) indicates the same for the GIO component.

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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