Williams, landel and Ferry, WLF

When the test temperature is raised, the rate of Brownian motion increases by a certain factor, denoted Ox. and it would therefore be necessary to raise the frequency of oscillation by the same factor flx to obtain the same physical response, as shown in Figure 1.6. The dependence of Uj upon the temperature difference T—Tg follows a characteristic equation, given by Williams, Landel, and Ferry (WLF) [11] ... 

According to the more widely used Williams, Landel, and Ferry (WLF) equations, all linear, amorphous polymers have similar viscoelastic properties at Tg and at specific temperatures above Tg, such as Tg + 25 K, and the constants Ci and C2 related to holes or free volume, the following relationship holds ... 

In any case, the Arrhenius equation is not particularly useful at temperatures above Tg + 100 K. The overall temperature-dependence of polymer flexibility at temperatures of Tt to T% + 100 K can be expressed by the empirical Williams, Landel, and Ferry (WLF) equation... 

Some applications require the material to remain under constant stress for years, yet it is often not reasonable to conduct such extended time measurements. One approach which circumvents this employs time-temperature superposition. Measurements are obtained over a shorter time span at differing temperatures. A master curve of C as a function of a reduced time tl a where a is a shift factor, is generated, and this allows the results to be extended to longer times. The shift factor is obtained by employing the Williams, Landel, and Ferry (WLF) relationship... 

The free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

This exemplifies the experimental difficulties inherent in determining the absolute value of Tg, which is considered in more detail when thermosets are discussed. Of particular interest is the value that a relaxation-dependent property may have when a system is in the vicinity of the glass transition. This is given by the empirical Williams, Landel and Ferry (WLF) equation ... 

A master curve can be constructed as indicated in Figure 22.8, where the zero-shear-rate viscosity t]q has to be evaluated for each one of the indicated viscosity curves. Both, the effect of temperature and pressure on the viscosity versus shear rate curve can be addressed by considering a shift factor that may be related, for instance, to the free volume of the system by means of the Williams, Landel, and Ferry (WLF) equation [9, 15, 23, 24]. With the aid of this shift factor, the new viscosity curve can be constructed from known viscosity values and the reference curve at the prescribed values of temperature and a pressure. The use of shift factors to take into account the temperature dependence on the viscosity curve was also used by Shenoy et al. [19-21] in their methodology for producing viscosity curves from MFI measurements. 

In this paper, we analyze the effect of fluorine substitution in the polymers listed above by dielectric analysis (DEA), dynamic mechanical analysis (DMA) and stress relaxation measurements. The effect of fluorination on the a relaxation was characterized by fitting dielectric data and stress data to the Williams, Landel and Ferry (WLF) equation. Secondary relaxations were characterized by Arrhenius analysis of DEA and DMA data. The "quasi-equilibrium" approach to dielectric strength analysis was used to interpret the effect of fluorination on "complete" dipole... 

The temperature dependence of the mean relaxation time thn can be described by the empirical function according to Williams, Landel and Ferry (WLF) ... 

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. 

The polymer SBR is mechanically simple, and it was found that the results for G for a given substrate could be superimposed using the Williams, Landel and Ferry (WLF) technique (see Viscoelasticity - time-temperature superposition). The master curves for each substrate were close to being parallel to each other and to a similar curve for cohesive fracture of the SBR (Fig. 1). 

The viscosity shift factor relates the viscosity p of the solid material to the density q and the temperature T. By applying the assumption that the product of absolute temperature and density is roughly constant, the Williams, Landel and Ferry (WLF) equation (Williams et al, 1955) is obtained ... 

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 ... 

This relationship is analogous to the empirical one established by Williams, Landel, and Ferry (WLF equation) which serves to relate the dielectric and mechanical relaxation times measured at a temperature T with those measured at the reference temperature (here Tg). 

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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