Modulus shift

Fig. 9.10 (A) Frequency shift factors aj and (B) modulus shift factor bj as a function of temperature. Reprinted from [40],...

Fig. 25 Plot of activation energies calculated from storage modulus shift factors versus measured UPy content for linear random copolymers (plus signs) and for covalently cross-linked networks (circles). Reprinted with permission from [143], Copyright 2011 American Chemical Society...

Bahrami-Samani, M., Cook, C.D., Madden, J.D., et al. (2008) Quartz crystal microbalance study of volume changes and modulus shift in electrochemicaUy switched polypyrrole. Thin Solid Films, 516, 9, 2800-7. 

Time-temperature superposition was performed in the following sequence first, tan 8 data were plotted against log to. The superposition involves the log to axis only, because any modulus-shift in G and G" cancels out. From this procedure the time-shift factor, aj, was evaluated. Next, log IG" I data were plotted against log (Oa. From these plots... 

When the modulus shift [32], Equation 6.28, is applied, master curves are formed for CBll, CB22, and CB23. Figure 6.27 shows the result for CB22. 

With the emulsion-polymerised rubbers, the modulus shift is required only when an extensive amount of macrogel is present [22]. However, there is no gel in the samples referred to in this work. This is the only time when strain hardening was observed for the sample containing no gel. It implies that the long branches in these BRs are very long. 

Figure 6.28 shows the modulus shift, r(a), for the samples of CBll, CB22 and CB23 as... 

Figure 6.28 Modulus shift factor as a function of extension ratio.

When the modulus shift. Equation 6.29 is applied, and master curves are formed for all rubbers, as in Figure 6.42 for VCR412. 

Figure 6.43 shows the modulus shift Ffa) as a function of the extension ratio. 

The temperature shift factors found in shifting the observed curves of G and G are given in Table 6.8. For the accuracy of superposition, the curves of G and G are shifted together. The time shift factor, aj and modulus shift factor, Pj are both based on 25 °C, Tq = 298 K. The WLF constants [6], Cj and C2 are also given in Table 6.8 ... 

The modulus shift, Pj, is relatively small and subject to a proportionally large error. However, the shift is required for the superposition. This is in contrast to previous work with EPM and EPDM elastomers where such a shift was not required. The value of Pj decreases with increasing temperature the extent of decrease is similar to what is expected from the theory, PqTq/pT. However, this subject is not pursued any further, since it is somewhat outside the scope of the present study. 

Another importance of linear viscoelasticity is that it provides a reference for non-linear behaviour that is, the latter is expressed as a deviation from the former with use of appropriate parameters. First is the universal parameter, the elongation ratio, a, which reduces the time-scale of nonlinear behaviour to that of linear behaviour. Next is the modulus shift factor, T(a), which indicates the degree of strain-hardening or strainsoftening. Finally, comparison of linearised elongation data with that of shear data indicates, if they disagree, the presence of strain-induced crystallisation or strain-induced association. All these deviations from linearity are related to the structure of rubber. 

The lines connect the data at the same strain and parallel lines at different strains were obtained. Master curves after application of the modulus shift, Equation. 6.29, are shown in Figures 7.27 and 7.28. 

With all compounds the master curves were obtained after the time and the modulus shift. As shown in Figures 7.29-7.31, the direction of the modulus shift was that of strain softening for all compounds. 

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