Time-temperature superposition viscosity data

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the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data).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shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

The time-temperature superposition method can be also applied to viscosity data (Ferry, 1980). For any viscoelastic parameter, exact matching of the adjacent curves is an important criterion for the applicability of the method. In addition, when possible, the same values of oy must superpose all the viscoelastic parameters and the temperature dependence of ar should have a reasonable form based on experience. One advantage of the method is that the range of frequencies are extended beyond those available experimentally. The time-temperature method has been also referred to as thermorheological simplicity (Plazek, 1996). 

Figure 8.10 The time-temperature superposition data at the reference temperature of 345 C for Thermx LNOOOl. Complex viscosity ( ), G (A), and G" ( ) [5].

The data in Figure 3.2.1 cover about five decades of time. Three to five decades are all that are practical for a typical single viscoelastic experiment. However, to fully describe G t) we often need a wider range. This can be accomplished by collecting data at different temperatures and shifting it to one reference temperature. This idea of time-temperature superposition has been described, eq. 2.6.9, with respect to viscosity and is also illustrated in Chapter 11.6. Equation 2.6.9 holds for any of the viscoelastic functions, for example, G(t). 

A cone and plate rheometer was used in order to measure the complex melt viscosity n (tt)) as a function of frequency (o [1]. Rgure 13.1 illustrates the master curve of t ((0, T) data measured at various temperatures T for SPS using the time-temperature superposition principle [1]. The reference temperature 7 is 290 °C. All the data are converted by shifting the curves to overlap the original 290 °C curve. 

This chapter covers some of the methods and instruments used to determine the mechanical properties of polymers. Examples of instrument designs and typical data generated in these measurements will be introduced. In particular, automated axial tensiometers (to find elastic modulus, yield stress, and ultimate stress), dynamic mechanical analyzers (to determine storage and loss moduli), and rheometers (to measure flow viscosity) will be introduced. This chapter considers the principles behind the devices used to establish and measure the properties of viscometric flows. One of the common techniques used to determine viscous flow properties, PoisueiUe (laminar) flow in cylindrical tubes, is also important in technical applications, as polymer melts and solutions are often transported and processed in this manner. The time-temperature superposition principle is also covered as a way to predict polymer behavior over long timescales by testing materials across a range of temperatures. 

Another type of Cole-Cole representation of rheological data is a plot of the imaginary versus the real parts of the complex viscosity. Such a plot for a monodisperse polystyrene is shown in Fig. 5.29 [139]. Note that the viscosities are strong functions of temperature, and data taken at various temperatures therefore do not superpose. However, if time-temperature superposition is obeyed, a temperature-independent plot can be obtained by use of reduced complex viscosity components as shown below. 

Like engine oil, polymer melts show reduced viscosity with increasing temperature. In the absence of experimental data defining these curves, which is mostly the case, this can be determined by using the Williams, Landel, and Ferry relafionship for time-temperature superposition [13]. This relationship can be applied to modulus values and to viscosity as well. Applying the relationship to viscosity we get Equation 16C.5 ... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

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