Vogel-Fulcher plot

Fig. 6.4. Vogel-Fulcher plot of the chain diffusion constants D for the three different polycarbonate modifications, as indicated in the figure, for N = 20 model monomers...

Figure 22. Vogel-Fulcher plot of the relaxation time T of 8CB in a silica gel [219].

Fig. 5.10. Plot of the inverse logarithm of the self-diffusion constant of BPA-PC, for a length N = 20 of the coarse-grained chains, vs. temperature. Straight line indicates the Vogel-Fulcher [187] fit. From [28]...

Fig. 2.23 Vogel-Fulcher-Tamman-Hesse plots for P2tBCHM ( ) and P4tBCHM ( ). (From ref. [32])...

The Arrhenius plot of the viscosity of the ILs is not a straight line but a Vogel-Fulcher-Tamman (VFT) type curve. Since ionic conductivity is the inverse of the viscosity (Eq. (3.8)), it also obeys the VFT equation. 

Since most of ILs including polymer systems show upper convex curvature in the Arrhenius plot, and not a straight line, the temperature dependence of the ionic conductivity is expressed by Vogel-Fulcher-Tamman (VFT) equation [7] ... 

Figure 11. Self-diffusion coefficient plotted vs. temperature, for several lattice sizes. The solid and the dotted line are fits to the Vogel-Fulcher and Adam-Gibbs equation, respectively. Prom (Binder, K. Baschnagel, J. Bdhmer, S. Paul, W. Phil Mag. 5, in press.).

Dielectric spectroscopy (DS) provides a means of studying chain dynamics of amorphous, crystalline, and liquid crystalline (LC) polymers (1,3-5,11). As one example Figure 3 shows plots of s" against log / /Hz for the amorphous polymer poly(vinyl acetate) at different temperatures. The a loss peak (Fig. 3a) is due to the large-scale motions of chain dipoles and moves to ultralow frequencies as Tg is approached. VaiT)) obeys the Vogel-Fulcher-Tammann equation... 

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.

It was shown in Figure 12.4 that, in the case of glass-rubber relaxation, the linearity of the plot of peak E" or the peak of tan d versus the reciprocal of the peak temperature, was lost. Consequently, the Vogel-Fulcher-Tamman-Hesse... 

To see more clearly the temperature effect on ion conduction, the logarithmic molal conductivity was plotted against the inverse of temperature, and the resultant plots showed apparent non-Arrhenius behavior, which can be nicely fitted to the Vogel— Tamman-Fulcher (VTF) equation ... 

The Tg may also increase as a result of increased interactions between ion pairs and the formation of ion clusters. The increase in Tg is nearly linear with the ratio Li /EO, even up to a ratio of 0.5 for LiCF3S03 solubilized in PM MS-8 or in the identical poly(methacrylate) comb polymer (Figure 3). The rise in Tg is much more rapid for the poly(methacrylate) comb polymer than for the polysiloxane. The free-volume mechanism of ion conduction is confirmed in our system by the linearity of temperature-dependent conductivity plots when the Vogel-Tammann-Fulcher (VTF) expression a = exp [-K T - rj] (in which is the ideal Tg and and K are constants) (i, 2, 9) is applied. 

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 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 ... 

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. 

Figure 6.5. Schematic of an Arrhenius plot for mechanisms commonly observed in polymers. The lines correspond to Arrhenius [Eq. (6.8) for y, p, and Maxwell-Wagner-Sillars (MWS) relaxations] and Vogel-Tammann-Fulcher-Hesse [VTFH Eq. (6.10)], for a and normal-mode (n-) relaxation] temperature dependences for the relaxation time t(T). Relaxations ascribed to small, highly mobile, dipolar units appear in the upper right side of the plot, while those originating from bulky dipolar segments, slowly moving ions, and MWS mechanisms are located in the lower-left part of the plot.

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