WLF rate—temperature equivalence

For simple C-C crosslinked elastomers (Gent and Lai, 1994), the reduction factors, ar, used to transform tear energy results at different temperatures as in Figure 10.12 to yield a master curve as in Figure 10.13 are found to correspond closely to the universal form of the WLF rate-temperature equivalence relation (Ferry, 1970) ... 

The rate of peel and test temperature at which the abrupt transition occurs are directly dependent upon the rate of Brownian motion of molecular segments. Simple viscoelastic adhesives therefore obey the WLF rate-temperature equivalence, Eq. (30), as shown in Fig. 35. The peel strength above the critical rate depends upon factors discussed previously interfacial attractions and dissipative processes within the adherends. Below the critical rate, the peel strength is primarily a measure of the work of... 

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. 

In fact, the same increase in joint strength that is obtained with a simple viscoelastic adhesive on increasing the rate of debonding, can be achieved by a suitable reduction in test temperature. This is referred to as the principle of rate-temperature equivalence. For amorphous glass-forming liquids above their glass transition temperature Tg, Williams, Landel, and Ferry (WLF) proposed a universal relationship for the ratio of corresponding test rates at temperatures Tand Tgi ... 

It is well established that between Tg and about Tg + 50 K, the relaxation kinetics obeys the WLF law (Williams et al., 1955). If Pr is a property depending on the macromolecular mobility (relaxation modulus, complex modulus, viscosity, diffusion rate, etc.), the time-temperature equivalence principle may be formulated as... 

For rubber-like materials, the viscoelastic losses vary with the strain rate and the temperature. The principle of time-temperature equivalence propose in 1955 by Williams et al (75) allows us to superimpose the experimental curves obtained at different temperatures through the known translation factor ar of the WLF transformation. As a consequence, at fixed geometry, the adherence forces provoking crack extensions at different speeds V can be studied as a function of the reduced parameter ajV. 

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.

Fracture energy master curves were determind as a function of nearly equivalent ranges of reduced crack velocity (King and Andrews), and extension rate (Swetlin). In both cases, T was used as the reference temperature. King and Andrews master curves were obtained using the WLF Equation and the universal constants, while Swetlin s master curves were determined via numerical best-fit shifting. 

Fig. 3.4. With a multi-frequency measurement, frequencies beyond the measurable range of the DMA can be achieved by using the superposition method. Employing the Williams-Landel-Ferry (WLF) equation, and with a treatment of the data, designated as the method of reduced variables or time-temperature superposition (TTS) it is possible to overcome the difficulty of extrapolating limited laboratory tests at shorter times to longer-term, more real world conditions. The underlying bases for TTS are that the processes involved in molecular relaxation or rearrangements in viscoelastic materials occur at accelerated rates at higher temperatures and that there is a direct equivalency between time (the frequency of the measurement) and temperature.

Equations [7] and [9] are mathematically equivalent. TRef is a reference temperature and v(TRef) the value of the relaxation rate at this temperature. Ci and C2 = TRef 7o are so-called WLF parameters. It was argued that these parameters should have universal values that are independent of the polymer investigated if the glass transition temperature Tg is chosen as reference temperature However, it was found that these... 

According to Williams-Landel-Ferry (WLF) principle longer time is equivalent to higher temperature. Thus at higher rates the modulus is shifted to the direction of lower temperature and vice versa [210]. 

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