Vogel-Tammann-Hesse-Fulcher

Unfortunately, reliable experimental estimates of the configurational entropy are unavailable to enable explicit application of the AG model for polymer fluids. Instead, the temperature dependence of t in polymer melts is often analyzed in terms of the empirical Vogel-Fulcher-Tammann-Hesse (VFTH) equation [103],... 

A Vogel-Fulcher-Tammann-Hesse equation can be used to characterize the temperature dependence of the relaxation times for these six different degrees of cure, 0.70, 0.75, 0.80, 0.825, 0.90, and 0.95 ... 

Inserting Eq. (4-16) into Eq. (4-10) gives the Vogel-Fulcher-Tammann-Hesse equation, where now the Vogel temperature To = U/A.2R can be computed from the energy U between trans and gauche states. The values of U obtained indirectly in this way, using... 

It appears, however, that the mode-coupling theory is not able to explain some of the most significant slow-relaxation processes of these more complex glass formers. In particular, it cannot explain the success of the Vogel-Fulcher-Tammann-Hesse (VFTH) equation for the temperature-dependence of the relaxation time near the glass transition. The mode-coupling theory predicts instead a power-law dependence of the longest relaxation... 

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 relaxation-time dependence could be described by the Vogel-Fulcher-Tammann-Hesse equation [88] ... 

The polymer dynamics involves several relaxation processes, some of which start within the vitreous region and cross into high-temperature melts. For example, the fast relaxations of the backbone chain or side groups start near the Vogel-Fulcher-Tammann-Hesse (VFTH) temperature and extend io T>Tc. The linearized VFTH expression is... 

If the free volume shows a linear expansion with the temperature, Vf = Ef(T- T ), where Ef=dVf IdT is its specific thermal expansivity and Tq < Tg is the temperature where the (linearly extrapolated) free volume disappears, one obtains from Eq. (11.7) the well-known Vogel-Fulcher-Tammann-Hesse (VFTH) equation [Vogel, 1921 Fulcher, 1925 Tammann and Hesse, 1926] ... 

In the high-temperature range, both relaxation modes exhibit Vogel-Fulcher-Tammann-Hesse (VFTH) behavior ... 

Vogel-Fulcher-Tammann-Hesse equation or temperature vinyl alcohol... 

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).

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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