Frequency-dependent master curve

Figure 11. Frequency dependent master curve of completely cured adhesive resulting from time-temperature superposition.

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. 

The shape of the maser curve not only depends on the rubber compound, but also on the surface on which it slides. On dry, clean polished glass the friction master curve for gum rubbers rises from very small values at low log ajv to a maximum which may reach friction coefficients of more than 3 and falls at high log ajv to values which are normally associated with hard materials, i.e., 0.3 shown for an ABR gum compound in Figure 26.2. If the position of the maximum on the log a-fV axis for different gum rubbers is compared with that of their maximum log E frequency curves, a constant length A = 6 X 10 m results which is of molecular dimension, indicating that this is an adhesion process [10]. 

In Fig. 16, the values of the exponent a are plotted against the barrier frequency this is clearly a kind of master curve that summarizes the essence of much of the work reported here. When the rate is calculated from Grote-Hynes theory, this curve depends only weakly on temperature. 

For a quantitative analysis of the percolation behavior of R, considered in Eq. (20), the permittivity data shown in Fig. 30b have been fitted to an empirical Cole-Cole function [131,132]. The fits are quite good, yielding relatively small broadness parameters between 0.36 and 0.5. The obtained relaxation times rR= R 1 are depicted in Fig. 31 in dependence of filler concentration O. Furthermore, the cross-over times T =a)fl and the limiting low frequency plateau values of the conductivity o ( —>0), obtained from the data in Fig. 30a, are represented in Fig. 31. They have been evaluated by a simple shifting procedure for constructing a conductivity master curve, as... 

Conventionally storage modulus versus frequency and temperature results are presented by extrapolated isotherms called master curves. These are plotted by shifting frequency data with a temperature dependent shift factor, a. Most published results for are based on manual graphic methods. Computer work first aimed at systemizing and automating determinations of shift factor was initiated by a desire to efficiently process the hundreds of data points from each FTMA run. 

Figure 3.12 (a) Storage compliance ot poIy(n-octyl methacrylate) versus frequency measured at 24 temperatures, given in degrees Celsius, (b) Master curve of J versus reduced frequency (Mj obtained from the data of (a) by time temperature shifting, fc) Temperature-dependence of the shift... 

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. 

It has been remarked that time (frequency) - temperature reduced data on carbon black filled rubbers exhibit increased scatter compared to similar data on unfilled polymers. Payne (102) ascribes this to the effects of secondary aggregation. Possibly related to this are the recent observations of Adicoff and Lepie (174) who show that the WLF shift factors of filled rubbers giving the best fit are slightly different for the storage and loss moduli and that they are dependent on strain. Use of different shift factors for the various viscoelastic functions is not justified theoretically and choice of a single, mean ar-funetion is preferred as an approximation. The result, of course, is increased scatter of the experimental points of the master curve. This effect is small for carbon black... 

The principle can also be applied to dielectric data, which can be shifted either along the temperature or the frequency axis. An example of the latter type of shift is shown in Figure 13.26, where instead of time-dependence measurements the frequency dependence of the 3-relaxation in poly(vinyl acetate) has been studied at fixed temperature in the range 212 to 266 K. A master curve can be constructed for this relaxation region by plotting (e Vemax) against logio ( / max) where the max subscript refers to the peak maximum at each experimental temperature. 

FIGURE 13.26 (a) Frequency dependence of e" for the P-relaxation in poly(vinyl acetate) measured at different temperatures, (b) Master dielectric loss curve for the data in (a) (O) compared with similar data for the P-relaxation of poly(vinyl benzoate) ( ). (From Ishida, Y. et al., Roll. Z. 180, 108, 1962. With permission from Dr. Dietrich Steinkopff Verlag, Darmstadt.)... 

Partial master curves of 10 g.dL"l solutions of a,o)-alkaline earth dicarboxylato PBD in xylene at 297 K are reported in Figure 10, and result from a good frequency-temperature superposition of the experimental data.l7 Only the G" master curve of the solution of Be-based HTP is ill-defined due to the poor accuracy in the determination of the very small values of G". The shift factors support an apparent Arrhenius-type of dependence (Figure 11), from which the activation energy of the observed secondary ionic relaxation process was calculated and found to decrease as the radius of the alkaline earth cations increases (Figure 12). One also observes that the relaxation spectrum calculated by the first order approximation of Ninomiya and Ferry S is displaced along the time scale in relation with the cation size (Figure 13). The dynamic behavior of the 10 g.dL solution is obviously... 

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