Modified free-volume model

Cohen and Grest (1979) have assigned on the basis of their modified free volume model that the tunneling states arise at very low temperatures (T Tg) because the volume dependent free energies of the cell in their model (see Chapter 3) possess curvature, which lead to local free energy minima. These minima are separated by solid-like barriers of... 

Local density fluctuations occur in penetrant polymer systems both above and below Tg. It is then reasonable to expect that a free-volume diffusion model should also provide an adequate description of the diffusion of small penetrants in glassy polymers. To reach this goal the free-volume model for diffusion of small penetrants in rubbery polymers, second part of Section 5.1.1, was modified to include transport below Tg (64,65,72,91-93). 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

To account for the variation of the dynamics with pressure, the free volume is allowed to compress with P, but differently than the total compressibility of the material [22]. One consequent problem is that fitting data can lead to the unphysical result that the free volume is less compressible than the occupied volume [42]. The CG model has been modified with an additional parameter to describe t(P) [34,35] however, the resulting expression does not accurately fit data obtained at high pressure [41,43,44]. Beyond describing experimental results, the CG fit parameters yield free volumes that are inconsistent with the unoccupied volume deduced from cell models [41]. More generally, a free-volume approach to dynamics is at odds with the experimental result that relaxation in polymers is to a significant degree a thermally activated process [14,15,45]. 

A modified version of the free-volume theory is used to calculate the viscoelastic scaling factor or the Newtonian viscosity reduction where the fractional free volumes of pure polymer and polymer-SCF mixtures are determined from thermodynamic data and equation-of-state models. The significance of the combined EOS and free-volume theory is that the viscoelastic scaling factor can be predicted accurately without requiring any mixture rheological data. 

The FH model provides a first approximation for polymer solutions. As shown, both the combinatorial and the energetic terms need substantial improvement. Many authors have replaced the random van-Laar energetic term by a nonrandom local-composition term such as those of UNIQUAC, NRTL, and UNIFAC models. The combinatorial term should be extended/modified to account for the free-volume differences between solvents and polymers. 

Figure 12,4 Density and fractional free volume of PebaxVPEC as function of volume fraction of PEC determined by additive model and buoyancy method [modified plot from ref 24]...

In dense polymers, the self-diffusion of small plasticising solvent molecules has been measured PGSE methods [105]. For polymers above the glass-transition temperature, it is common to model the solvent diffusion using the free volume theory as modified for polymer systems by Fujita [122] and by Vrentas and Duda [123]. For solvent diffusion in dense polymers below Tg, an alternative model has been given by Frisch and Stern [124]. 

Independently of what one thinks of their formation mechanism, Ps atoms need space to be formed. In gasses, there is an enormous empty space between gas molecules, so space (or the shortage of it) is not a limiting factor of Ps formation. In fluids, however, Ps creates a small empty space, a bubble around itself. The bubble model was developed a long time ago (Ferrel 1957) and has been modified continually. Its present form tries to synthesize the results of the spur model and modern physical chemistry (Stepanov et al. 2000). In solids, structural free volumes might serve the empty space needed for positronium formation. 

As already mentioned above, the calculation methods are strongly dependent on the data model. The geometries can be illustrated differently from point, to lines, to surface, to volume models and provide different advantages in the CAD program depending on the objectives (Figure 2.20). Thus, for example, free-form surfaces can be produced and modified on surface models. 

Two distinct models have been used for interpreting the influence of features such as chemical stmcture, molecular mass, cross-linking and plasticisers on the glass transition in amorphous polymers. The first approach considers changes in moleeular flexibility, which modify the ease with which conformational changes can take place. The alternative approach relates all these effects to the amount of free volume, which is assumed to attain a critical value at the glass transition. 

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