Time-shift factor

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... 

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.)...

Fig. 11.15 The apparent time-shift factor, /rc, as a function of the molecular weight (MW) of the low MW component (denoted by Y) in the binary blends F80/Y = 75/25 ( ) F80/Y = 50/50 (o) F35/Y = 75/25 (a) and F35/Y = 50/50 (a). The horizontal lines on the vertical axis indicate the theoretical rf, /tc values calculated from Eki- (11.10) as the limiting values for the four series of blend samples.

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 7. Aging time shift factor as a function of aging time.

The second term in Equation (3.69) represents the terminal entanglement relaxation of PI occurring at Ig h (= p, for PI relaxing much faster than PtBS, as explained earlier). As explained for Equation (3.63), this term is approximately expressed in terms of the modulus Gp, bulk of bulk PI, the PI volume fraction ([jpi, the relaxation time shift factor Ah = Xg,h/xg, and the intensity factor Ipi given by Equation (3.65). 

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

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].

When structural-rheological simplicity does not hold and, for example, fi in equation (7) changes with physical aging, exact reduction of the creep or stress relaxation cnrves cannot be accomplished. However, it is estimated by the author that a change in the KWW parameter of approximately 5-10% would result in a deviation of the logarithm of the aging time shift factor (log ate) of 0.05 over two decades in time scale. Scatter in reduced curves of this order of magnitude is often observed because of the scatter in the creep or stress relaxation data (23,39,40). Researchers have not systematically looked for such small deviations in their rednced curves. 

Application of this formulation by Prest to the data of Fig. 13-12 indicated that V = Vi but V2 s vl, at le t for 0.4. The corresponding time shift factors Xi and X2 were functions of Mw and respectively. It is of interest that, if the relaxation spectrum is to be independent of molecular weight in the transition zone where H is proportional to (Section C2 of Chapter 10), the following equation must be fulfilled ... 

FIG. 19-2. Coefficient of friction for acrylonitrile-butadiene rubber sliding on wavy glass, plotted against logarithm of velocity times shift factor ar reduced to 20 C. Temperature dependence of ar corresponds to equation 40 of Chapter 11 with 7, 30 C. Points identify 13 different temperatures from —15 (at right) to 85 C (at left). (Grosch. )... 

The time compression corresponding to a time-temperature shift is then expressed using a time shift factor o,. For any component property... 

When displayed on the logarithmic time scale, the curves merge by shifting toward the x-axis. Here, the degree of shift represents the time compression, which can be described by the logarithm of the time shift factor log (a according to Eq. 1.36. This means that the service life curves are equidistant when the difference between the reciprocal, absolute temperatures is equal [104]. 

The term At is a time shift factor which can be obtained with the application of the Williams-Landel-Ferry (WLF) equation, reported by Ferry (1961). 

The dependence of the time shift factor on temperature is of great interest, and several empirical expressions have been proposed to describe it. The two that are most commonly used are the Arrhenius dependence and the WLF dependence, which are given by Eqs. 4.68 and 4.69, respectively. 

This implies that if the vertical shift factor is neglected, the time shift factor can be obtained as a function of temperature by measuring the zero-shear viscosity at several temperatures. 

The effect of pressure on linear viscoelastic properties can also be accounted for in terms of shift factors. One can define an isothermal time-shift factor flp(P) that accounts for the effect of pressure on the relaxation times at constant temperature, and it has been found that this factor follows the well-known Bams equation ... 

The parameter is made temperature dependent according to the rheological relaxation time shift factor Uj-... 

Schultheisz, C.R., McKenna, G.B., Leterrier, Y., Stefanis, E. A comparison of structure and aging time shift factors from simultaneous volume and mechanical measurements. In Proceedings of 1995 SEM Spring Conference on Experimental Mechanics (1995)... 

Zhang et al. (2003) developed an alternative method for coping with the complex multiple reflections in spiral-wound membrane modules. They defined an amplitude shift factor Qa and an arrival time shift factor Qt that characterize the entire reflected waveform response or selected regions of it as follows ... 

Figure 33.4 Comparison between the permeation flux relative to that for an unfouled membrane (upper data and left ordinate) and the normalized arrival time shift factor (lower data and right ordinate) as a function of operating time for a sequence consisting of pure water feed (0-30 h), fouling with a 1.6-g/L aqueous solution of calcium-sulfate (30-100 h), and cleaning with a pure water feed (100-171 h) in a Koch 2521 spiral-wound membrane module operating at 0.7 MPa and 20°C.

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