WLF shift factor

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... 

Figure 16 WLF shift factors for various molecular weight poly(a-methylstyrenes). Reference temperature, 2(I4°C. (From Ref. I60.)...

Plazek (183) carried out very accurate creep experiments on natural rubber as a function of cross-linking. He found that data at different temperatures could be superimposed by the usual WLF shift factors which were developed for non-cross-linkcd poiymers (27). Temperature-superposed... 

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. 

Fig. 6.4 Correlation between a, the WLF shift factor and the conductivity for PEO networks.

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.

It has been remarked that time (frequency) - temperature reduced data on carbon black filled rubbers exhibit increased scatter compared to similar data on unfilled polymers. Payne (102) ascribes this to the effects of secondary aggregation. Possibly related to this are the recent observations of Adicoff and Lepie (174) who show that the WLF shift factors of filled rubbers giving the best fit are slightly different for the storage and loss moduli and that they are dependent on strain. Use of different shift factors for the various viscoelastic functions is not justified theoretically and choice of a single, mean ar-funetion is preferred as an approximation. The result, of course, is increased scatter of the experimental points of the master curve. This effect is small for carbon black... 

A typical example of recent studies of time-temperature-modulus relationships may be found in papers by Moehlenpah et al. (1970, 1971), who examined crosslinked epoxy resins filled with glass beads, fibers, or air bubbles. The initial tangent modulus in compression was seen to increase with a decrease in strain rate flexural and tensile moduli were reported to behave in a similar fashion. The WLF shift factor was essentially independent of the type of filler used and of the mode of loading. Kerner s equation was found to hold for the particulate composites in the glassy range. 

In the same studies, Moehlenpah et al (1970, 1971) obtained master curves for the stress relaxation of their epoxy systems, at least into the glass-to-rubber transition region (Figure 12.4), and demonstrated similar behavior of both the stress relaxation modulus and the tensile modulus as a function of strain rate. As with the strain rate studies mentioned, no effect of filler type on the WLF shift factor was observed. All solid fillers increased the modulus of the system, the fibers being more effective than the spheres. The bubbles, as expected (Nielsen, 1967a), decreased the modulus. 

Figure 12.10. Tensile yield stress vs. strain rate for epoxy composites (Tref = 50°C) (Moehlenpah et a/., 1970, 1971). The term oLjy is the WLF shift factor. (x) Continuous transverse (A) particulate-filled ( + ) unfilled (O) foam (J) brittle failure, range of ultimate strengths.

Interpretations of the experiments that have been performed in this area have concentrated on the second aspect. Karim and co-workers (Karim et al. 1990) and Stamm and co-workers (Stamm et al. 1991) both used neutron reflectivity to look at the very early stages of the interdiflfusion of polystyrene and deuterated polystyrene. Figure 4.26 shows the results of Karim et al. The graph depicts the extent of interfacial broadening as a fimction of time and temperature, whereby a WLF shift factor (equation (4.4.9)) has been used to reduce the data for a number of different temperatures on to one curve. At very early times... 

Figure 4.26. Interfacial broadening as a function of time for polyst5n-ene and deuterated polystyrene, both of relative molecular mass 200000, as measured by neutron reflectivity. Data taken at various temperatures have been reduced to a reference temperature of 120 C by using a WLF shift factor. After Karim et al. (1990).

Figure 7.12. The fracture energy versus the effective rate of peeling at —20 °C, with data from a number of different temperatures ( , —40 °C , —20 °C o, 0 °C a, 25 °C , 50 °C A, 80 °C and V, 130 °C) reduced using the WLF shift factors (equation (7.2.2)). The line is an estimate of the low-peeling-rate limit of the fracture energy Gq. After Gent and Lai (1994).

Vogel temperature) for the Williams-Landel-Ferry (WLF) shift factor where... 

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... 

Measurements were made of 0 as a function of crack velocity, the debonding being macroscopically interfacial in all cases. The data obtained are summarized in Figure 7 in the form of a plot of log 0 vs. log ca, where is the WLF shift factor for data obtained at different temperatures. The curve for cohesive tearing of the elastomer (or adhesive) was also included. As will be seen, the predicted parallel behavior (see Section 6 above) was in fact obtained, the curves for cohesive failure and for adhesive detachment from different substrates all being superimposable by vertical shifting. 

Figure 5.17. WLF shift factor curve for the data of Fig. 5.15 reference temperature Ti (the inflection temperature) is equivalent to Tg. Note that Q and C2 here are not the universal values but are specific for the polycarbonate polymer. [From Mercier et al. (1965) reprinted with permission of John Wiley and Sons, Inc.]...

Barquins and Maugis (6) have approached this problem from the view point of fracture mechanics for the special case that viscoelastic losses are localized at the crack tip. They completely ignore effects of creep. Viscoelastic response is assumed to be proportional to the thermodynamic work of adhesion in analogy with the results of peel experiments. Specifically, F -w=w0(flj.v) where T is the strain release rate and (t> ajV) is assumed to be a universal function of crack speed V for a viscoelastic material and Ct is the WLF shift factor (). For macroscopic glass-pol3mrethane contacts (a = 50-250 pm), Maugis and Barquins showed experimentally that with n = 0.6 for 10 m/s < V < 10 m/s and a T) . Most SFM experiments are at V much smaller... 

Another shift factor that has been used is based on the work of Williams, Landel, and Ferry (WLF), and referred to as the WLF shift factor... 

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