Free volume theories, phenomenological

The Kauzmann temperature plays an important role in the most widely applied phenomenological theories, namely the configurational entropy [100] and the free-volume theories [101,102]. In the entropy theory, the excess entropy ASex obtained from thermodynamic studies is related to the temperature dependence of the structural relaxation time xa. A similar relation is derived in the free-volume theory, connecting xa with the excess free volume AVex. In both cases, the excess quantity becomes zero at a distinguished temperature where, as a consequence, xa(T) diverges. Although consistent data analyses are sometimes possible, the predictive power of these phenomenological theories is limited. In particular, no predictions about the evolution of relaxation spectra are made. Essentially, they are theories for the temperature dependence of x.-jT) and r (T). 

The free-volume theory gives an excellent account of many phenomena near Tg and enables much of the phenomenology of the glass transition to be understood in a qualitative and even semiquantitative way. The free volume plays the role of an order parameter, and by fitting the thermal expansivity jump at Tg, the other thermod3mamic results are approximately described. Absolute estimates of free volume are available from crystal densities and van der Waals radii, so the... 

On the other hand, some phenomenological distributions of relaxation times, such as the well known Williams-Watts distribution (see Table 1, WW) provided a rather good description of dielectric relaxation experiments in polymer melts, but they are not of considerable help in understanding molecular phenomena since they are not associated with a molecular model. In the same way, the glass transition theories account well for macroscopic properties such as viscosity, but they are based on general thermodynamic concepts as the free volume or the configurational entropy and they completely ignore the nature of molecular motions. 

In contrast to Vf, the specific volume V in the glassy state of the polymer exhibits a larger slope than above Tg, which is due to the thermal contraction of Vocc ( occ,g O.Sag). The agreement of the slope dVld vh) fl-om the phenomenological relation Eq. (11.6) with dVf/d vh) from Eq. (11.5) above Tg we consider as evidence that o-Ps detects precisely the free volume calculated fl om S-S hole theory. The larger slope of dVld vh) below Tg supports the conclusion from the S-S equation of state calculation that Vocc shows here, as distinct from above Tg, a certain thermal expansion. 

A phenomenological explanation has been given for equations 6.27-6.29 by this author [12] without reference to Schapery s theory. It is a rational analysis of the relationship between macroscopic deformation and the deformation in the segmental level of polymer chains. An increase of free volume resulting from deformation in the dynamic state is the explanation for the strain-time correspondence. 

A second class of models directly relates flow to blend structure without the assumption of an ellipsoidal droplet shape. This description was initiated by Doi and Ohta for an equiviscous blend with equal compositions of both components [34], Coupling this method with a constraint of constant volume of the inclusions, leads again to equations for microstructural dynamics in blends with a droplet-matrix morphology [35], An alternative way to develop these microstructural theories is the use of nonequilibrium thermodynamics. This way, Grmela et al. showed that the phenomenological Maffettone-Minale model can be retrieved for a specific choice of the free energy [36], An in-depth review of the different available models for droplet dynamics can be found in the work of Minale [20]. 

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