Vogel-Fulcher temperature

Tk is often close to the Vogel-Fulcher temperature T0 discussed in connection with Figure 2, which is determined by fitting the Vogel-Fulcher relation5-8 to the temperature dependence of the structural relaxation time of the melt9 using Eq. [3] ... 

Applying this prediction to the cooling rate dependence of a break points in the specific volume curves, one obtains a Vogel-Fulcher temperature of To = 0.35 that agrees well with that determined from the temperature dependence of the diffusion constant in this model, which is T = 0.32. 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

The non-Arrhenius temperature-dependence of the relaxation time. It shows a dramatic increase when the glass transition temperature region is approached. This temperature dependence is usually well described in terms of the so called Vogel-Fulcher temperature dependence [114,115] ... 

To is known as the Vogel-Fulcher temperature and is located about 30 K below Tg. r is the asymptotic value of the relaxation time of the correlator 4> for T—>oo. Also the rheological shift factors a (T) mentioned above approximately follow such temperature dependences [34] ... 

Figure 26. Temperature dependence of xmax = 1 / (2nfmax) for a glycerol sample and its mixtures with water. The solid curves show the description by the VFT model, where values of the Vogel-Fulcher temperature (TV) are shown with the legend. The preexponential factor (xv) is almost independent with concentration In (xv) = —35.9 for 100mol% of glycerol, —36.1 for 95 to 65%, —36.2 for 60%, —36.3 for 60% to 45%, and —36.4 for 40mol% of glycerol-water mixtures. D = 22.7 for all concentrations presented here. (Reproduced with permission from Ref. 208. Copyright 2005, American Chemical Society.)...

These are the Vogel-Fulcher equations [44]. In addition to the prefactors, two common parameters appear, namely the activation temperature 7, typically 7 = 1000 -2000 K, and the Vogel-Fulcher temperature 7y, whieh is generally 30- 70 K below the glass temperature. Using the Vogel-Fulcher equations, Williams, Landel and Ferry derived an expression for the shift parameter log a. This expression is known in the literature under the name WLF equation [45, 46] ... 

The onset temperature of the fast process was investigated for many kinds of polymers, revealing that the onset temperatures are not always the Vogel-Fulcher temperature. In Fig. 15, for example, dynamic scattering laws of atactic polystyrene (PS) [94] are shown in a temperature range well below Tg (=373 K). At a very low temperature such as 21 K, the boson peak is clearly observed while... 

U >J is linearly proportional to the logarithm of the viscosity as shown in Fig. 20 for PB. These results support the free-voliune concept [114] and it was suggested that the free-voliune could be replaced by the mean square displacement of the fast motion. The fact that the fast process of PB disappears at the Vogel-Fulcher temperature Tq also supports this concept since the free-vol-ume is assumed to vanish at Tq in the derivation of the Vogel-Fulcher law within the free-volume concept. 

Since >oo was measured by a simulation at infinite temperature (yielding Doo = 4.7-10 " ), eq. (6.27) only possesses two fit parameters, which were determined to be A = 0.652 0.012 and To = 0.097 0.009. Hence a nonzero Vogel-Fulcher temperature of To 0.1 results from this fit. However, again one has to realize that the two-dimensional diffusion data only... 

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