Time-temperature shift factors

Figure 12.4 Temperature-time shift factor explanation in the text. (After [36].)...

The 150 °C results were obtained in dnplicate, which were in good agreement with each other. The time-temperature correspondence in the viscoelastic behaviour of elastomers may be represented with the WLF equation [17]. For the present elastomer, i.e., butadiene-acrylonitrile copolymer having 33% acrylonitrile, the temperature time shift factor, aj was previously given by [18]... 

PPG (at higher temperatures) behaves like a typical pseudoplastic non-Newtonian fluid. The activation energy of the viscosity in dependence of shear rate (284-2846 Hz) and Mn was detected using a capillary rheometer in the temperature range of 150-180°C at 3.0-5.5 kJ/mol (28,900 Da) and 12-13 kJ/mol (117,700 Da) [15]. The temperature-dependent viscosity for a PPG of 46 kDa between 70 and 170°G was also determined by DMA (torsion mode). A master curve was constructed using the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data). A plateau for G was not observed for this molecular weight. The temperature-dependent shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

For linear thermorheologically simple materials a single temperature-dependent shift factor, aT T), can be used to predict the transient thermal response [20]. The mechanical response is history dependent and involves the use of reduced times, ( ) and (t), which can be found from the shift factor as... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

The pattern can be obtained from the polymer temperature or concentration variations in addition to the change of G°N. The relaxation function may be too complicated a mathematical expression ever to be calculated, nonetheless, it obeys a property of invariance which allows the superposition of all normalised relaxation curves to one another by adjusting a suitable factor to the time scale of each curve. The time shift factor is found to obey the equation... 

As we have seen, the shift factor is the relative change in time (f/fQ) needed to simulate a certain property (which is known at a reference temperature (T0) and a reference time (fQ)) at a changed temperature. The shift factor proves to be the relative time shortening to... 

Curves of stress (divided by absolute temperature) versus log time-to-break at various temperatures can be made to coincide by introducing the temperature-dependent shift factor flT. Application of the same shift factor causes the curves of the elongation at the break br versus the logarithm of time-to-break at various temperatures to coincide. A direct consequence is that all tensile strengths (divided by absolute temperature), when plotted against elongation at break, fall on a common failure envelope, independent of the temperature of testing. Fig. 13.84 shows the behaviour of Viton B elastomer. 

Table I shows the values of these activation parameters for the materials tested. A time—temperature superposition shift factor (A) can be calculated from Equation 2 as follows ...

Figure 12.24 Double logarithmic plot of the aging time shift factors versus aging time, tg, for an epoxy glass aged at different temperatures below its Tg. T — Tg. (O) 30°C, (X) 24°C, ( ) 20.8°C (O) 10.3°C ( ) 6.TC. (From Ref. 30.)...

FIGURE 5.14 (a) Volume as a function of elapsed time after the second step of two-step temperature histories, (b) aging time shift factors corresponding to the changing specific volume of part (a). (After Struik [1978], with permission.)... 

Figure 2.15 Construction of the time-temperature superposition and derivation of temperature dependent shift factor.

The third step is to shift the short-term creep curves measured at different temperatures along the log (time) axis until they superpose one onto another, even partially. This operation defines, for each testing temperature, a shift factor Oj, and produces a master curve (Figure 2.15b). The shift factor is defined as follows ... 

Fig. 14 Time-temperature superposition shift factor in rheoiogy, ax. as a ftuiction of temperature, normalised to Tg, for the supramolecuiar poiymer blend 7 and 8 (diamonds) and linear polystyrene (squares). (Reprinted with permission from [75], copyright 2009 RSC)...

It can be observed that the term of (1 -x) describes the contribution of Tf. Here, Tg is the glass transition temperature. denotes the Vogel temperature, defined as (Tg - 50) (°C), tq corresponds to the reference relaxation time at Tg, and B is the loeal slope at Tg of the trace of the time-temperature superposition shift factor in the global WiUiam-Landel-Ferry (WLF) equation [53]. 

Figure 12.8 Creep compliance master curves (a) and stress-time shift factor (b). The master curve obtained using the stress-time superposition is indicated by short dashes the point symbols are the same as in Figure 12.7. The master curve obtained by using the temperature-time superposition is shown by long dashes (b cf. Figure 12.3) after [36].

Fig. 6.11 Illustration of time-temperature superposition principle for the stress relaxation of polymers. The right-hand-side master curve at a constant temperature is obtained by the parallel shift of the left-hand-side curves at various temperatures. The shift factor used to construct the master curve follows the WLF equation...

Shift factor n. The amount by which the logarithm of the modulus (or comphance) of a plastic, measured at temperature T (K) must be shifted along the time axis to bring it onto a single curve with the modulus measured at Tg, the glass-transition temperature the shift factor relationships is... 

To calculate the time-temperature superposition shift factor [2], use... 

To calculate the time-temperature superposition shift factor by using the WLF (Williams-Landel-Ferry) equation for polymers at temperatures less than 100°C (232 F) above their Tg [2], use... 

Figure 10.14 (36-38) illustrates the time-temperature superposition principle using polyisobutylene data. The reference temperature of the master curve is 25°C. The reference temperature is the temperature to which all the data are converted by shifting the curves to overlap the original 25°C curve. Other equivalent curves can be made at other temperatures. The shift factor shown in the inset corresponds to the WLF shift factor. Thus the quantitative shift of the data in the range Tg to Tg x 50°C is governed by the WLF equation, and... 

Fig. 2.27. Recoverable-compliance, Ji(t), data of PPMS 5000 at temperatures -32.2°C ( ), -35.0°C U),-38.6°C (t),-40.0°C ( ), -41.1 °C (x), -42.6 °C n, -44.5 °C ( ), -45.2 °C (V), -46.9 °C (A), and -50 °C (o). The data taken at different temperatures have been shifted horizontally along the log t axis by a temperature-dependent shift factor log ut in order to superpose the curves at the short-time end with the data for -35.0 °C. The inset shows the retardation spectrum, L, as a function of the reduced retardation time X with reference temperature To = -35.0 °C, which was obtained numerically from J lf) data.

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