Vogel-Fulcher

Figure C2.1.15. Schematic representation of tire typicai compiiance of a poiymer as a function of temperature. (C) VOGEL-FULCHER AND WILLIAMS-LANDEL-FERRY EQUATIONS...

On the other hand, polymeric materials show universal aspects of glass transition behavior, just like other materials. For instance, the classical Vogel-Fulcher behavior... 

The dramatic slowing down of molecular motions is seen explicitly in a vast area of different probes of liquid local structures. Slow motion is evident in viscosity, dielectric relaxation, frequency-dependent ionic conductance, and in the speed of crystallization itself. In all cases, the temperature dependence of the generic relaxation time obeys to a reasonable, but not perfect, approximation the empirical Vogel-Fulcher law ... 

In spite of the problems associated with the static structure, the coarsegrained model for BPA-PC did reproduce the glass transition of this material rather well the self-diffusion constant of the chains follows the Vogel-Fulcher law [187] rather nicely (Fig. 5.10),... 

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. 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 2 Sketch of typical temperature dependencies of the viscosity r of glass-forming systems. The viscosimetric Tg of a material is defined by the viscosity reaching 1013 Poise. Strong glass formers show an Arrhenius temperature dependence, whereas fragile glass formers follow reasonably well a Vogel-Fulcher (VF) law predicting a diverging viscosity at some temperature T0.

Tk is often close to the Vogel-Fulcher temperature T0 discussed in connection with Figure 2, which is determined by fitting the Vogel-Fulcher relation5-8 to the temperature dependence of the structural relaxation time of the melt9 using Eq. [3] ... 

To interpret the cooling rate dependence of the glass transition temperature, one can use the Vogel-Fulcher law discussed in the section on the... 

Applying this prediction to the cooling rate dependence of a break points in the specific volume curves, one obtains a Vogel-Fulcher temperature of To = 0.35 that agrees well with that determined from the temperature dependence of the diffusion constant in this model, which is T = 0.32. 

Figure 9 Chain center of mass self-diffusion coefficient for the bead-spring model as a function of temperature (open circles). The full line is a fit with the Vogel-Fulcher law in Eq. [3]. The dashed and dotted lines are two fits with a power-law divergence at the mode-coupling critical temperature. 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

The non-Arrhenius temperature-dependence of the relaxation time. It shows a dramatic increase when the glass transition temperature region is approached. This temperature dependence is usually well described in terms of the so called Vogel-Fulcher temperature dependence [114,115] ... 

To is known as the Vogel-Fulcher temperature and is located about 30 K below Tg. r is the asymptotic value of the relaxation time of the correlator 4> for T—>oo. Also the rheological shift factors a (T) mentioned above approximately follow such temperature dependences [34] ... 

Fig. 4.10 a Characteristic relaxation times determined from dielectric measurements [137] (diamonds), and from NSE spectra at (triangles) for triol (open symbols) and PU (solid symbols). The full lines correspond to Vogel-Fulcher and the dotted lines to Arrhenius descriptions, b Relaxation times from NSE spectra have been arbitrarily multiplied by a factor 6 for triol and 40 for PU to build a normalized relaxation map. (Reprinted with permission from [127]. Copyright 2002 Elsevier)... 

Fig. 4.24 Temperature dependence of the characteristic times obtained from the fits of Spair(Q,t) to stretched exponentials with =0.41 at Qmax=l-48 A (filled circle) and 2.71 A (empty circle). Dashed-dotted line corresponds to the Vogel-Fulcher-like temperature dependence of the viscosity and the solid line to the Arrhenius-like temperature dependence of the dielectric -relaxation. (Reprinted with permission from [189]. Copyright 1996 The American Physical Society)...

Fig. 4.35 Right-hand side Monomeric friction coefficients derived from the viscosity measurements on PB [205]. The open and solid symbols denote results obtained from different molecular weights. Solid line is the result of a power-law fit. Dashed line is the Vogel-Fulcher parametrization following [205]. Left hand side Temperature dependence of the non-ergodicity parameter. The three symbols display results from three different independent experimental runs. Solid line is the result of a fit with (Eq. 4.37) (Reprinted with permission from [204]. Copyright 1990 The American Physical Society)...

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