Viscosity Vogel-Tammann-Fulcher

It is well known that the viscosity of a liquid decreases upon heating. This would usually be expected to fit to an Arrhenius type of behavior. That is, the natural logarithm of the viscosity should vary in a linear manner with temperature. All of the I Ls for which the temperature dependence of the viscosity has been studied deviate from this behavior [7, 9, 11, 14, 20]. Rather, they fit a Vogel-Tammann-Fulcher interpretation where the viscosity of the IL at any given temperature is better related to a material-specific temperature such as the difference between the temperatare of the study and the glass transition temperature of the IL. 

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].

From the compiled vapor pressure and conductivity data, the evaporation enthalpy and the activation enthalpy for proton conduction were calculated as a function of composition. The critical temperature according the Vogel-Tammann-Fulcher law was determined from the viscosity data and compared with glass transition temperatures from other studies using NMR spectroscopy. A correlation between dynamic viscosity and molar conductivity was found. As expected, a considerable decoupling between ionic conduction and viscous flow can be determined from a Walden plot, which is based on proton-hopping mechanisms in phosphoric acid. 

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 viscosity of an IL is heavily influenced by temperature [14]. Specifically, an increase in temperature leads to an increase in the Brownian motion of the constituent molecules of an IL [39], Okoturo et al. investigated the temperature dependence of IL viscosity and observed that ILs containing un-functionalized asymmetric cations typically obeyed the Arrhenius law [56]. On the other hand, the majority of ILs containing small, symmetric cations with low molar masses obeyed the Vogel-Tammann-Fulcher laws [56]. ILs that obeyed neither the Arrhenius law nor the Vogel-Tammann-Fulcher laws generally contained functionalized asymmetric cations with higher molar masses [56]. 

At the time of development of free volume theory, two important empirical equations of viscosity were known. They are the Doolittle (1951) equation (3.01) and the Vogel, Tamman and Fulcher (VTF) equation (3.02) (Vogel, 1921, Fulcher, 1923, Tammann and Hesse, 1926), which are given below. 

Most viscosity-temperature relationships for glasses take the form of an Arrhenius expression, as was the case for binary metal alloys. The Vogel-Fulcher-Tammann (VFT) equation is one such relationship. 

In the a-process, the viscosity and consequently the relaxation time increase drastically as the temperature decreases. Thus, molecular dynamics is characterized by a wide distribution of relaxation times. A strong temperature dependence presenting departure from linearity or non-Arrhenius thermal activation is present, owing to the abrupt increase in relaxation time with the temperature decrease, thus developing a curvature near T. This dependence can be well described by the Vogel-Fulcher-Tammann-Hesse (VFTH) equation [40, 41], given by Equation 2.1 ... 

It is remarkable that the free-volume-based Doolittle equation devised to explain the temperature dependence of the viscosity of a generic simple atomic liquid can account so well both for the viscous behavior of metallic glass alloys through the Vogel, Fulcher, and Tammann (VFT) equation representation discussed in Chapter 1 and for the behavior of the much more complex sub-cooled... 

Fig. 19. Temperature dependence of the shift factors of the viscosity (T), terminal dispersion ( ), and softening dispersion (0) of app from Ref. 73. The temperature dependence of the local segmental relaxation time determined by dynamic light scattering ( ) (30) and by dynamic mechanical relaxation (o) (74). The two solid lines are separate fits to the terminal shift factor and local segmental relaxation by the Vogel-Fulcher-Tammann-Hesse equation. The uppermost dashed line is the global relaxation time tr, deduced from nmr relaxation data (75). The dashed curve in the middle is tr after a vertical shift indicated by the arrow to line up with the shift factor of viscosity (73). The lowest dashed curve is the local segmental relaxation time tgeg deduced from nmr relaxation data (75).

That the viscosities of the ILs are highly sensitive to the temperature is evident from the data presented in Table 7.1. The temperature dependence of the viscosity of morpholinium ILs, depicted in Figure 7.2, is found to be better represented by the Vogel-Fulcher-Tammann (VFT) equation [63], an equation widely used to describe the temperature dependence of the viscosity of glass-forming substances compared to the Arrhenius equation. 

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