Vogel-Tammann-Fulcher equation viscosity

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].

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)). 

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. 

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. 

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. 

The viscosities measiued for NBS 711 glass were compared to those obtained using the Vogel-Fulcher-Tammann [9] equation issued by the NBS (Figure 10.17) ... 

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 ... 

The average a-relaxation time of polymers exhibit a dramatic sensitivity to temperature as Tg is approached. Figure 3.1b shows the temperature dependence of the average a-relaxation time as a function of inverse temperature normalized to Tg for an amorphous polymer. The temperature dependence of the average a-relaxation (or viscosity) can be well described by the Vogel-Fulcher-Tammann-Hesse (VFTH) equation [64-66]... 

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