Time-Temperature Correspondence Principle

6 Differentiation of Regions in the Master Curves of the Viscoelastic Functions 

8 Effects of Cross-Linking on the Viscoelastic Functions Problem Sets 

A fundamental characteristic of the so-called thermorheologically simple systems is that consecutive isotherms have similar habits, so they overlie each other when they are shifted horizontally along the log t axis. In other words, the time-temperature correspondence principle holds. This property in creep experiments can be expressed by the relation (2,3) 

The viscosity increases as the temperature decreases, this increase being sharp in the vicinity of the glass transition temperature. According to Eq. (8.1), the value of r can be determined by means of the equation 

The viscosity of viscoelastic liquids at temperatures slightly above Tg can be determined by the procedures outUned above. Thus a step shear stress is imposed on the viscoelastic liquid at temperatures well above Tg, and once steady state is reached the sample is cooled to the temperature of interest then the straight line of J t) vs. t is recorded, and the viscosity is determined from the reciprocal of the slope of the straight line. By assuming that the elastic and viscous mechanisms have the same temperature dependence, the shift factor can be written in terms of the viscosity as (2,5) 

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The procedure described above is an application of the time-temperature correspondence principle. By shifting a set of plots of modulus (or compliance) versus time (or frequency) at any temperature (subscript 1) along the log t axis, we obtain the value of that mechanical property at another time and temperature (subscript 2). Using the shear modulus as an example, the time-temperature correspondence principle states... 

By applying the time-temperature correspondence principle to /"(co) and / (co) at low frequencies, one obtains... 

If the time-temperature correspondence principle holds, Eq. (8.35) suggests that the temperature dependence of the shift factor in the time (or frequency) domain can be written as... 

The time-temperature correspondence principle holds not only for the viscosity but also for the normal stresses. In the latter case, however, the... 

In thermorheological simple systems, the time-temperature correspondence principle holds. Chapter 8 gives examples of isotherms for compliance functions and relaxation moduli. The shift factors are expressed in terms of terminal viscoelastic parameters, and the temperature dependence of the shift factors is interpreted in terms of the free volume and the WLF equation. The chapter outlines methods for determining the molecular weight between entanglements, and analyzes the influence of diluents and plasticizers on the viscoelastic functions. 

The time-temperature correspondence principle states that there are two methods to use to determine the polymer s behavior at longer (or shorter) times than those covered by a stress-relaxation experiment run at 7j. First, one may improve the experiment to measure directly the response at longer (shorter) times. For the longer times, however, this procedure rapidly becomes prohibitively time-consuming because the change is so slow (note that Figure 4-5 is plotted on a log scale). (For the shorter times, the limitations are equipment related, e.g., transducer response time, problems with instrument and sample inertia, etc.) An alternative, according to the time-temperature... 

The time-temperature correspondence principle is known to be widely applicable to polymers other than PLCs, e.g. [37] or [38]. It was therefore of interest to test the applicability of the principle to our PLC. For this purpose, the experimental creep data are presented in Figure 12.2 in logarithmic coordinates. It can be seen that the shape of these curves suggests that a master curve may be constructed by shifting experimental curves along the log time axis until they all superimpose on one curve for the chosen reference temperature. Figure 12.3 presents such a master curve for the reference temperature of 20 C. As can be seen in... 

We have discussed above the applicability of the time-temperature correspondence principle to pure PET/0.6PHB. Now of course the question of the applicability of the principle to PLC-containing blends... 

We conclude that the addition of the PLC to PP strengthens the material in a predictable and quantifiable way. Surveying the above creep as well as stress relaxation, we find that the time-temperature correspondence principle is applicable here in spite of the multiphase character of the PLC. Hence the predictive capabilities, noted above for various specific instances, are quite extensive. It remains to be seen whether other PLC-containing blends behave similarly, although there is no a priori reason why they should behave differently. 

In Section 24.1.3 we have discussed among others the time-temperature correspondence principle. An example of application of that principle is shown in Fig. 24.10. The results pertain to high density polyethylene (HOPE) subjected to different levels of predrawing [58]. The draw ratio is defined as... 

Important here of course is whether the shift factor a-i values calculated from Eq. (24.13) agree with the experimental ones. These results are displayed in Fig. 24.15. The continuous line is calculated from our Eq. (24.13). The dotted line is from an equation proposed in 1955 by Williams, Landel, and Ferry (WLF) [27], a pioneering aj T) formula at that time. We see that the WLF equation works well in a certain temperature range—this seems the reason it is still in use— but fails miserably outside of that range. Nobody else but Ferry [1] stated that range of application of WLF amounts to 50 K or so, not more. If one makes a primitive and unfounded assumption in our Eq. (24.13), one gets from it the WLF equation as a special case [6]. The problem is when people use the WLF equation blindly in wide temperature ranges, obtain bad results, and draw a false conclusion that the time—temperature correspondence principle does not work. 

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