Reduced-time master curves

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

Extensive tests have shown that if the final creep strain is not large then a graph of Fractional Recovery against Reduced Time is a master curve which... 

Figure 1. Time-temperature superimposed master curves of storage and loss moduli as functions of reduced frequency. The four networks belong to the D family. Numbers indicate wt. % dicumyl peroxide.

The scaling the functional shape hardly depends on temperature. Curves corresponding to different temperatures superimpose in a single master curve when they are represented against a reduced time variable that includes a T-dependent shift factor. 

Some applications require the material to remain under constant stress for years, yet it is often not reasonable to conduct such extended time measurements. One approach which circumvents this employs time-temperature superposition. Measurements are obtained over a shorter time span at differing temperatures. A master curve of C as a function of a reduced time tl a where a is a shift factor, is generated, and this allows the results to be extended to longer times. The shift factor is obtained by employing the Williams, Landel, and Ferry (WLF) relationship... 

The creep of a viscoelastic body or the stress relaxation of an elasacoviscous one is employed in the evaluation of T] and G. In such studies, the long-time behavior of a material at low temperatures resembles the short-time response at high temperatures. A means of superimposing data over a wide range of temperatures has resulted which permits the mechanical behavior of viscoelastic materials to be expressed as a master curve over a reduced time scale covering as much as twenty decades (powers of ten). 

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... 

Figure 13.20 Master curves of G Tq/T ) versus reduced frequency ajco for the PS-PI block copolymer described in Fig. 13-15 in the high-temperature disordered state and in the low-temperature ordered state flow-aligned into either the parallel or perpendicular directions. The arrows show the time-temperature-shifted frequencies at which the parallel (sideways arrow) and perpendicular (up arrow) orientations are achieved by large-amplitude shearing. The frequency ajcoc 300 secri is the reduced frequency below which G in the disordered and quenched states no longer superpose. (Reprinted with permission from Patel et al., Macromolecules 28 4313. Copyright 1995, American Chemical Society.)...

Figure 19, Time-temperature superposed master curves for Naflon-H, Naflon-Na, and Naflon-Ba reduced to a reference temperature of 50 °C. Reproduced with permission from Ref. 47. Copyright 1984 John Wiley Sons, Inc.

Abstract The present study demonstrates, by means of broadband dielectric measurements, that the primary a- and the secondary Johari-Goldstein (JG) /3-processes are strongly correlated, in contrast with the widespread opinion of statistical independence of these processes. This occurs for different glassforming systems, over a wide temperature and pressure range. In fact, we found that the ratio of the a- and P- relaxation times is invariant when calculated at different combinations of P and T that maintain either the primary or the JG relaxation times constant. The a-P interdependence is quantitatively confirmed by the clear dynamic scenario of two master curves (one for a-, one for P-relaxation) obtained when different isothermal and isobaric data are plotted together versus the reduced variable Tg(P)/T, where Tg is the glass transition temperature. Additionally, the a-P mutual dependence is confirmed by the overall superposition of spectra measured at different T-P combinations but with an invariant a-relaxation time. 

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... 

Under-water stress relaxation of the Nafion acid and Nafion-Na samples with an equivalent weight of 1200 was carried out by Kyu and Eisenberg (40). The degree of neutralization of the Nafion-Na was measured to be ca. 80%. Time-temperature superposed master curves of the two systems, reduced to a reference temperature of 50°C, are shown in Figure 6. The under-water stress relaxation behavior of Nafion acid resembles that of Nafion-Na, except for the fact that the elastic modulus is somewhat lower in the acid. This latter feature may be due to the difference in the degree of water absorption of the acid and salt samples (26,31). The swelling of Nafion acid is greater than that of Nafion-Na, which yields a material of lower modulus. 

Figure 6. Time-temperature superposed master curves from underwater stress relaxation of the Nation acid (0) and Nafion-Na ( j, reduced to a reference temperature of 50° C.

The effect of overall molecular weight or the number of blocks on rheological properties for the samples from the second fractionation can be illustrated as a plot of reduced viscosity vs. a function proportional to the principal molecular relaxation time (Figure 2). This function includes the variables of zero shear viscosity, shear rate, y, and absolute temperature, T, in addition to molecular weight, and allows the data to be expressed as a single master curve (10). All but one of the fractions from the copolymer containing 50% polystyrene fall on this... 

Data was collected over a temperatrrre range of 30 °C to 80 °C, and reduced to a master-curve via the principle of time-temperatrrre superposition, at the reference temperature of 30°C. 

Figure 1. Linear dynamic oscillatory shear response of the 50K PBA based Si02 hybrid sample. The data collected at temperatures between 30 and 80 °C were reduced to a single master curve using the principle of time-temperature superpositioning. The horizontal frequency shift factors (af were similar to that for the pure PBA homopolymer.

Figure 4. Stress relaxation response at 30 as a function of strain and the reduced time - strain master curve.

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