Mechanical shift factors, dynamic

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

Fig. 4.11 Temperature dependence of the shift factors as reported in the literature for atactic polypropylene 1 dynamic mechanical measurements [140], 2 NMR data of Pschorn et al. [141], 3 photon correlation spectroscopy [142], 4 from NMR measurements of Moe...

Fig. 5.16 Q-dependence of the characteristic times of the KWW functions describing the PIB dynamic structure factor at 335 K filled circle), 365 K empty square) and 390 K filled triangle), a Shows the values obtained for each temperature. Taking 365 K as reference temperature, the application of the rheological shift factor to the times gives b and a shift factor corresponding to an activation energy of 0.43 eV delivers c. The arrows in a show the interpolated mechanical susceptibility relaxation times at the temperatures indicated. (Reprinted with permission from [147]. Copyright 2002 The American Physical Society)...

Mijovic et al. analyzed the annealed blends from melts using dynamic mechanical thermal analysis and achieved similar results after an adjustment for shifting factors, AT s, as shown in Figure 7.3. The results were extended to include blends having a PVDF concentration greater than 80 wt %. It can be observed that the glass transition temperatures of the annealed blends reduce rapidly when the PVDF concentrations are above 80 wt %. 

Detailed analysis of the isothermal dynamic mechanical data obtained as a function of frequency on the Rheometrics apparatus lends strong support to the tentative conclusions outlined above. It is important to note that heterophase (21) polymer systems are now known to be thermo-rheologically complex (22,23,24,25), resulting in the inapplicability of traditional time-temperature superposition (26) to isothermal sets of viscoelastic data limitations on the time or frequency range of the data may lead to the appearance of successful superposition in some ranges of temperature (25), but the approximate shift factors (26) thus obtained show clearly the transfer viscoelastic response... 

In the present case, all of our dynamic mechanical data could be reduced successfully into master curves using conventional shifting procedures. As an example, Figure 7 shows storage and loss-modulus master curves and demonstrates the good superposition obtained. In all cases, the shifting was not carried out empirically in order to obtain the best possible superposition instead the appropriate shift factors were calculated from the WLF equation (26) ... 

A final comment seems to be pertinent. In most cases actual measurements are not made at the frequencies of interest. However, one can estimate the corresponding property at the desired frequency by using the time (fre-quency)-temperature superposition techniques of extrapolation. When different apparatuses are used to measure dynamic mechanical properties, we note that the final comparison depends not only on the instrument but also on how the data are analyzed. This implies that shifting procedures must be carried out in a consistent manner to avoid inaccuracies in the master curves. In particular, the shape of the adjacent curves at different frequencies must match exactly, and the shift factor must be the same for all the viscoelastic functions. Kramers-Kronig relationships provide a useful tool for checking the consistency of the results obtained. 

FIGURE 6.7 Atactic polypropylene segmental relaxation times (solid symbols) from mechanical spectroscopy, dynamic light scattering, dielectric relaxation, and NMR, along with the global time-temperature shift factors (open symbols) from dynamic mechanical spectroscopy, creep compliance, and viscosity. Vertical shifts were applied to superpose the data (Roland et al., 2001). 

Vibration forces applying a dynamic stress load to viscoelastic materials results in a phase shift by the phase angle 8 between stress a and elongation e. The tangent of 8 is called the mechanical loss factor d or mechanical damping. Damping is thus a measure of the heat produced by application of dynamic loads as a result of internal friction (dissipatiOTi) (Fig. 24). 

We have explained the correspondence principles in Section 24.1.3, including the time— frequency correspondence. We were not able to apply this particular principle before becoming familiar with dynamic mechanical experiments. We need to provide at least an example of the application of the correspondence in the frequency domain. In Fig. 24.24 we show results from [58] pertaining to HDPE. The shift factors used to obtain that diagram have been calculated from equations in Section 24.1.3. More examples can be found for instance in the same paper [58]. 

The shift factors, ar,s, of the softening dispersion with G ranging from about 10 < G < 10 dyne/cm and temperature Tfrom -27.5 °C to 70 °C. ar,s of the entire softening dispersion, from dynamic mechanical measurement of J (f), 10[Pg.457]

Poly(vlnyl acetate) PVAc 349 8.86 101.6 305 [67,68] aj s of the entire softening dispersion from dynamic mechanical J f) with 10equilibrium with ambient moisture and the water absorbed lower the sample s Tg. Its T-dependence Is considerably weaker than either of the shift factors obtained from creep compliance In dried samples given Immediately below. 

PVC commercial sample Solvay Cie, type Solvic 229) 346.5 11.2 34.6 338.7 from dilatometry at a cooling rate of 3K/h) [56] aj s of the softening dispersion from creep, J(f), data. It has a much weaker temperature dependence compared with ar, from dynamic mechanical and dielectric relaxation given above. This discrepancy between the shift factors of the mechanical data of Schwarzl with the other sets of data may be due to the much lower Tg of the sample used. 

Poly(methyl acrylate) PMA 324 8.86 101.6 276 value seems too low) [79,80] This shift factor was obtained by combining the dynamic mechanical data of the entire softening dispersion (25<7<90 °Cand30dielectric relaxation data of Mead and Fuoss [79] in a comparable frequency range. Its temperature dependence is weaker than that of the shift factors ar s and ar,s to be described below. 

The shift factor aj could be measured by dynamic mechanical or dielectric relaxation spectroscopy. 

Fig. 19. Temperature dependence of the shift factors of the viscosity (T), terminal dispersion ( ), and softening dispersion (0) of app from Ref. 73. The temperature dependence of the local segmental relaxation time determined by dynamic light scattering ( ) (30) and by dynamic mechanical relaxation (o) (74). The two solid lines are separate fits to the terminal shift factor and local segmental relaxation by the Vogel-Fulcher-Tammann-Hesse equation. The uppermost dashed line is the global relaxation time tr, deduced from nmr relaxation data (75). The dashed curve in the middle is tr after a vertical shift indicated by the arrow to line up with the shift factor of viscosity (73). The lowest dashed curve is the local segmental relaxation time tgeg deduced from nmr relaxation data (75).

Figure 21 Segmental relaxation times and shift factors for global chain dynamics of atactic polypropylene. Segmental relaxation i>, NMR v, dielectric spectroscopy , light scattering o, mechanical dynamic spectroscopy 0, creep compliance measurements. Shift factors for global chain dynamics

Time-temperature equivalence in its simplest form implies that the viscoelastic behaviour at one temperature can be related to that at another temperature by a change in the timescale only. Consider the idealised double logarithmic plots of creep compliance versus time shown in Figure 7.7(a). The compliances at temperatures Ti and T2 can be superimposed exactly by a horizontal displacement log a, where a, is called the shift factor. Similarly (Figure 7.7(b)), in dynamic mechanical experiments, double logarithmic plots of tan 5 versus frequency show an equivalent shift with temperature. 

The linear viscoelastic master curve of a material serves as an important fingerprint for its mechanical behavior and the fine features of these master curves correlate with the particular materials molecular details. For these reasons, master curves are widely generated in practice. Below, we illustrate an example [38] of master curve generation where the original data were taken under dynamic testing. Fig. 2 shows the data. Fig. 3 shows the master curve obtained by means of the shift factor calculated from the data in Fig. 2. Finally, Fig. 4 shows a plot of the shift factor that is seen to display WLF-type behavior. 

T vo additional aspects enhance the utility of the time-temperature superposition concept. First, the same temperature shift factors apply to a particular polymer regardless of the nature of the mechanical response, that is, the shift factors as determined in stress relaxation are apphcable to the prediction of the time-temperature behavior in creep or dynamic testing. Second, if the polymer s glass transition temperature is chosen as the... 

Ferry went to Harvard University in 1937 and worked there in a variety of posts, including as a Junior Fellow, until he joined the University of Wisconsin in 1946. He was promoted to Full Professor in 1947 His extensive measurements of the temperature dependence of the dynamic mechanical properties of polymers led to the concept of reduced variables in rheology. His demonstration that time-temperature superposition applied to many systems is the basis for the rational description of polymer rheology. He measured the dynamic response over a very wide range of frequency. One of the fruits of this work is the Williams-Landel-Ferry (WLF) equation for time-temperature shift factors. 

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