Vogel Fulcher equation

Several well-known equations are available for interpreting the temperature dependence of viscosity, diffusion coefficient, and other relaxation rates for T > Tg. The Doolittle equation [18], the WLF equation [19], the Vogel-Fulcher equation [20], and the Adam-Gibbs equation [21] can be expressed in the same form. They are known to fit well with the relaxation data of liquids in equilibrium. The universal functional form is [20]... 

The temperature dependence of x of a DGEBA oligomer has been investigated. Two types of descriptions for the temperature dependence of X are proposed (Table 6) one is the WLF equation [66], the other is the Vogel-Fulcher equation [64, 65]. The WLF equation for the temperature dependence of X is as follows [66] ... 

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] ... 

Figure 25 shows the temperature dependence of relaxation time for the relaxation processes in the internal and interfacial regions of the ultrathin PS1.46M film sandwiched between the SiO layers. Since it was hard to distinguish the temperature-Ta relations between the vacuum deposited and laminated films, each data point was averaged over six independent measurements including both vacuum deposited and laminated films. The average thickness was about 40 nm. For comparison, the dashed curve in Fig. 25 denotes the bulk data obtained by the Vogel-Fulcher equation [72, 73] ... 

Fig. 25 Temperature dependence of for aa relaxation processes in internal and interfacial regions of the PS1.46M ultrathin films sandwiched between SiO layers. The average thickness is about 40 nm. The dashed curve denotes the prediction of the Vogel-Fulcher equation using bulk parameters, whereas the solid curve is the best fit by Vogel—Fulcher equation for the interfacial process...

Figure 10. Comparison of the temperature dependence of the chain s diffusion coefficient, D x), and the end-to-end vector scaling time, Tete ( )- Both quantities can be fitted by a Vogel-Fulcher equation (solid lines). Within the error bars the Vogel-Fulcher activation energy and temperature agree with each other for D and Tete- From (53).

Finally, Figure 15 shows the temperature dependence of the inverse relaxation time for the first three Rouse modes. As for the diffusion coefficient and the relaxation time of the end-to-end vector, l/r decreases by about 2 — 3 orders of magnitude in the studied temperature interval and may be fitted by the Vogel-Fulcher equation with a common (but, compared to D and 1/tr, slightly higher)... 

One equation used to describe the rapid increase of the viscosity with decreasing temperature is the Vogel-Fulcher equation (27,28) ... 

Figure 3. Temporal evolution of the average boundary position at eight different T. The inset shows the logarithm of the boundary mobility as a function of T [open circles versus l/T (top axis) compare the data to an Arrhenius relationship, whereas the filled circles compare to the Vogel-Fulcher equation (bottom axis)]. The nonlinearity of the Arrhenius plot indicates that this relationship does not apply to GB mobility date. Figure 3 was originally published in [16], National Academy of Sciences.

The temperature dependence of the glass—rubber transition follows the Vogel—Fulcher equation, which is essentially a generalization of the WLF equation ... 

It was also shown that many transport properties, Z, such as diffusion coefficient, viscosity or electric conductance of undercooled liquids obey a general Vogel-Fulcher equation [393] Z = Zq exp [az/(T - To)]. 

One of the most convenient tools for practical determination of fictitious temperatures is thermomechanometry [396] see Fig. 54, where the time dependence of fictitious temperature can be obtained on the basis of the Tool-Narayanaswami relation [391,396,400] by the optimization of viscosity measurements (logr] T,Tf versus temperatures) using the Vogel-Fulcher equation again. 

For temperatures not too much greater than the glass transition temperature, the temperature dependence of D /T has been shown to be described quite well by a form of the Vogel-Fulcher equation, ... 

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