Master curves shifting

The master curves of J (O) and J" %) for the polymers with various dryness are shown in Fig. 4 [182]. As the volume fi iction of the polymer V2 increases, the master curve shifts towards lower ftequency without... 

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. 

Williams and Ferryf measured the dynamic compliance of poly(methyl acrylate) at a number of temperatures. Curves measured at various temperatures were shifted to construct a master curve at 25°C, and the following shift factors were obtained ... 

Master curves can also be constmcted for crystalline polymers, but the shift factor is usually not the same as the one calculated from the WLF equation. An additional vertical shift factor is usually required. This factor is a function of temperature, partly because the modulus changes as the degree of crystaHiuity changes with temperature. Because crystaHiuity is dependent on aging and thermal history, vertical factors and crystalline polymer master curves tend to have poor reproducibiUty. 

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

Thus all the different temperature related data in Fig. 2.58 could be shifted to a single master curve at the reference temperature (7 ). Alternatively if the properties are known at Tref then it is possible to determine the property at any desired temperature. It is important to note that the shift factor cannot be applied to a single value of modulus. This is because the shift factor is on the horizontal time-scale, not the vertical, modulus scale. If a single value of modulus 7, is known as well as the shift factor ar it is not possible to... 

FIGURE 1.7 Construction of a master curve of dynamic modulus /a versus log (frequency) by lateral shifting of experimental results made over a small frequency range but at several different temperatures. 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

Figure 8 WLF time-temperature superposition applied to stress-relaxation data obtained at several temperatures to obtain a master curve. The master curve, made by shifting the data along the horizontal axis by amounts shown in the insert for r> is shown with circles on a line.

Master curves can often be made for crystalline as well as for amorphous polymers (33-38). The horizontal shift factor, however, will generally not correspond to a WLF shift factor. In addition, a vertical shift factor is generally required which, has a strong dependence on temperature (36-38). At least part of the vertical shift factor results from the change in... 

An example of experimental stress-relaxation data is shown in Figure 14 (160). Master stress-relaxation curves made from the experimental data on different molecular weight materials are shown in Figure 15. The temperature-shift factors used in making the master curves are shown in Figure 16. Note that the shift factors a, are the same for all molecular weights... 

Several attempts have been made to superimpose creep and stress-relaxation data obtained at different temperatures on styrcne-butadiene-styrene block polymers. Shen and Kaelble (258) found that Williams-Landel-Ferry (WLF) (27) shift factors held around each of the glass transition temperatures of the polystyrene and the poly butadiene, but at intermediate temperatures a different type of shift factor had to be used to make a master curve. However, on very similar block polymers, Lim et ai. (25 )) found that a WLF shift factor held only below 15°C in the region between the glass transitions, and at higher temperatures an Arrhenius type of shift factor held. The reason for this difference in the shift factors is not known. Master curves have been made from creep and stress-relaxation data on partially miscible graft polymers of poly(ethyl acrylate) and poly(mcthyl methacrylate) (260). WLF shift factors held approximately, but the master curves covered 20 to 25 decades of time rather than the 10 to 15 decades for normal one-phase polymers. 

An alternative to constructing the Arrhenius plot (log(X) against 1/T) is to shift the plots of parameter against time along the time axis to construct a master curve. 

To obtain an estimate of lifetime, use the WLF equation to determine the shift factor from the reference temperature to the temperature of interest. Apply that shift factor to each of the points on the master curve to obtain the required property/time curve and read the time to reach the threshold value. 

Burnay [14] has developed a predictive model, which is based on the use of the superposition technique to determine thermal and dose rate shift factors relative to a master curve of compression set of a rubber seal versus time. The relation between the shift factors and environmental parameters of temperature and dose rate are given by ... 

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.

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. 

Fig. 4.4 Scaling representation of the spin-echo data at Q ax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve. The shift factors are taken from macroscopic viscosity measurements, a Polyisobutylene at Qmax l-0 A [125] measured on INI 1 (viscosity data from [34]). b Atactic polypropylene at Qmax=l-ll (viscosity data from [131]). (b Reprinted with permission from [126]. Copyright 2001 Elsevier)...

Fig. 4.13 Momentum transfer dependence of the characteristic time associated to the self-motion of protons in the a-relaxation regime Master curve (time exponentiated to p) constructed with results from six polymers polyisoprene (340 K, p=0.57) (filled square) [9] polybutadiene (280 K, p=0Al) (filled circle) [146] polyisobutylene (390 K, p=0.55) (empty circle) [147] poly (vinyl methyl ether) (375 K, f=0A4) (filled triangle) [148] phenoxy (480 K, p=0A0) (filled diamond) [148] and poly(vinyl ethylene) (340 K, p=0A3) (empty diamond) [ 146]. The data have been shifted by a polymer dependent factor Tp to obtain superposition. The solid line displays a Q -dependence corresponding to the Gaussian approximation (Eq. 4.11). (Reprinted with permission from [149]. Copyright 2003 Institute of Physics)...

Finally we compare the temperature dependencies reported for the structural relaxation and the self-motion of hydrogens studied by NSE. For PI, the shift factors used for the construction of the master curve on Q,T) (Fig. 4.17) are identical to those observed for the structural relaxation time [8]. This temperature dependence also agrees with DS and rheological studies. The case of PIB is more complex [ 147]. The shift factors obtained from the study of Teif(Q>T) (Fig. 4.14b) reveal an apparent activation energy close to that reported from NMR results (-0.4 eV) [136]. This temperature dependence is substantially weaker than that observed for the structural relaxation time (=0.7 eV, coinciding with rheological measurements) in the same temperature range (see Fig. 4.20). 

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). 

Figure 5.65 Modulus-time master curve based on WLF-shift factors using data from Figure 5.63 with a reference temperature of 114°C. Reprinted, by permission, from F. Rodriguez, Principles of Polymer Systems, 2nd ed., p. 217. Copyright 1982 by Hemisphere Publishing Corporation.

The time-temperature superposition principle has practical applications. Stress relaxation experiments are practical on a time scale of 10 to 10 seconds (10 to 10 hours), but stress relaxation data over much larger time periods, including fractions of a second for impacts and decades for creep, are necessary. Temperature is easily varied in stress relaxation experiments and, when used to shift experimental data over shorter time intervals, can provide a master curve over relatively large time intervals, as shown in Figure 5.65. The master curves for several crystalline and amorphous polymers are shown in Figure 5.66. 

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