VFTH equation

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],... 

The inset to Fig. 6 exhibits as depending sensitively on the polymer class, but relatively weakly on molar mass. The temperature variation of Sc T)/sl is roughly linear for small 6T near Tq, consistent with the empirical VFTH equation, as noted earlier. On the other hand, this dependence becomes roughly quadratic in 6T at higher temperatures, where Sc achieves a maximum s at 7a-Attention in this chapter is primarily restricted to the broad temperature range (To 7 7a), vdiere a decrease of Sq vith T is expected to correspond to an... 

Because the AG equation (33) for r reduces exactly to the VFTH equation (34) over the temperature range in which ScT is proportional to 5T, the correspondence between these expressions for r uniquely establishes a relation between the kinetic fragihty parameter D = I/Kg and thermodynamic fragihty St T)T/hT. Specifically, in the temperature regime near the glass transition... 

Interest in the pressure dependence of structural relaxation in fluids has been stimulated by recent applications [175, 176] of a simple pressure analogue of the VFTH equation for the relaxation time x at a constant pressure P to the analysis of experimental data at variable pressures. Specifically, x(P) for both polymer and small molecule fluids has been found to extrapolate to infinity at a critical pressure Pg, and this divergence takes the form of an essential singularity,... 

The temperature dependence of T in the high temperature regime (Z > Zt) is often modeled by an alternative VFTH equation. 

The parameters for P4tBCHM are summarized in Table 2.6 The temperature dependence of the a relaxation in the frequency domain can be conveniently analyzed by means of the Vogel-Fulcher-Tamman-Hesse (VFTH) equation [88-90] which was empirically formulated as ... 

Using these ideas. Miller (1978) has given a simple explanation for the validity of the VFTH equation using a rotational isomeric state (RIS) model for polymers, which also provides a molecular interpretation of its parameters, A and Tp. Miller assumes that... 

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... 

The parameters of the VFTH equation can be calculated from the values of Ci and C2 by using the expressions of Eq. (8.39). An alternative way of obtaining m and is to plot Inaor versus / T — Too) and determine by trial and error the value of Too that best fits the plot to a straight line. It should be pointed out that in most cases Tg — 50 K. By comparing Eqs. (8.33) and (8.35), the volume fraction and the coefficient of expansion at 7k are given by (16)... 

The experimental evidence indicates than when a non crystallizable liquid is cooled, a temperature is reached at which the a 3 absorption splits into two relaxations the slow a relaxation, which obeys the VFTH equation and remains kinetically frozen at temperatures below Tg, and the faster 3 relaxation, which follows Arrhenius behavior and remains operative below Tg. This behavior, illustrated (9) in Figure 12.5, is exhibited by low and high molecular weight liquids. To interpret this bifurcation it is convenient to consider that condensed phases owe their existence to interactions between the constituent particles atoms, ions, or molecules. These interactions are embodied in a potential energy function (ri, F2,. .., r ) that depends on the local position of those particles, a schematic representation of which is given in Figure 12.6(2). 

Figure 6. Temperature dependences of the structural (a) and the secondary y-relaxation times for octa-O-acetyl-lactose, as well as the temperature dependence of secondary f- and y-relaxation times for lactose. The solid lines were obtained from fitting experimental data for the secondary relaxations to the Arrhenius law whereas the dashed line represents a fit the temperature dependence of a-relaxation times to the VFTH equation.

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 ... 

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