Equation Vogel-Tammann-Fulcher

For compositions above 60 wt% P2O5, considerable deviations from linearity can be detected, that is, according to the Vogel-Tammann- Fulcher equation Eq. (12.14) To is not negligibly small compared with T. Hence the critical temperature To increases with increasing concentration and approaches the temperature range of the viscosity data. 

Figure 12.15 Critical temperature To determined by fitting the viscosity data in Figure 12.14 from [8, 15] with the Vogel-Tammann-Fulcher equation. Glass transition temperatures taken from [42].

Before proceeding further, it is appropriate to discuss some aspects of molecular mobility of amorphous solids as it affects stability. The temperature dependence of molecular motion in amorphous systems is described by the empirical Vogel-Tammann-Fulcher (VTF) equation ... 

If the relaxation time of the supercooled liquid is described by the Vogel-Tammann-Fulcher (VTF) equation... 

Figure 1.18. Arrhenius plots for the relaxation time of the total energy of M13 at (a) p = 4 and (b) p = 14. Circles are mean relaxation times from the master equation, dashed lines are fits to the Arrhenius form, and the solid line in (a) is a fit to the Vogel-Tammann-Fulcher (VTF) form.

The conductivity of ionic liquids often exhibits classical linear Arrhenius behavior above room-temperature. However, as the temperature of these ionic liquids approaches their glass transition temperatures (Tg) the conductivity displays significant negative deviation from linear behavior. The observed temperature-dependent conductivity behavior is consistent with glass-forming liquids, and is often best described using the empirical Vogel-Tammann-Fulcher (VTF) equation. 

The K values shown in Table 14.3 for sample C can be well fitted by the Vogel-Tammann-Fulcher (VTF) equation or the Williams-Landel-Ferry (WLF) equation.Prom the VTF equation with the parameters obtained from the fitting, the K values at 127.5 and 93.7gC are calculated and listed in Table 14.3, with the former also listed in Table 14.1. The result of K (andrs) at 93.7gC is used in sections 14.8 and 14.10.a where the structural relaxation time and the length scale at Tg are defined or studied. 

One of the important properties of a polymer electrolyte leading to its development activity is the ionic conductivity. Temperature dependence on the conductivity of amorphous polymer electrolytes generally follows the Vogel-Tammann-Fulcher [VTF] equation [14] ... 

The viseosities of the 1,3-dialkylimidazoilium aluminium ehloride and l-mefliyl-3-ethylimidazolium aluminium bromide ionie liquids have also been reported for different eompositions and temperatures. For both the ehloroaluminate and bromoaluminate ionie liquids the temperature dependence was found not to have an Arrhenious type curve, with non-linear plots of Inq vs. 1/T. In these studies the temperature range used was wider than that of the N-alkylpyridinium. This non-Arrhenius behavior is characteristic of glass forming melts. Here the three parameter Vogel-Tammann-Fulcher (VFT) equation ... 

Limitations of the widely used KWW equation have been discussed and addressed by employing the Vogel-Tammann-Fulcher (VTF) and AG equations in order to calculate the relaxation time above and below the glass transition temperature more precisely (Shamblin et al. 1999). 

The temperature dependence of the dynamic viscosity t of a liquid close to its glass temperature Tg can be described by the Vogel-Tammann-Fulcher (VTF) equation [43-45] or by the Theory of free volume introduced by Doolittle [46 8], Cohen and Turnbull [49, 50]. An exponential dependence from the reciprocal temperature 1/T is found (see (8.8)). 

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

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