Vrentas-Duda free-volume theory

The diffusion coefficient depends on both the temperature of the system and the concentration of volatiles, and can be estimated through the Vrentas-Duda free-volume theory [10]. 

The Pawlisch model [63] was applied, for example, to calculate the partition coefficient and diffusion coefficients of a series of solvents in PMMA-co-BMA (poly(methyl methacrylate-co-butyl methacrylate)) at infinite dilution [66]. Values of K and Dp that best fitted the experimental data were found, and the model predicted perfectly the system behavior. The Vrentas-Duda free volume theory was found to correctly correlate diffusion data above Tg. 

Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. I. Reexamination of the Free-Volume Theory, Journal of Polymer Science Polymer Physics Edition 15, 403, 1977. Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems. II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecnlar Weight, Journal of Polymer Science Polymer Physics Edition 15, 417, 1977. 

JS Vrentas, JL Duda. Diffusion in polymer-solvent systems. I. Reexamination of the free volume theory. J Polym Sci, Polym Phys Ed 15 403-416, 1977. 

Fig. 8.8 Free volume theory prediction of mutual binary diffusion coefficient for the toluene-PS system based on parameters (19). [Reproduced by permission from J. L. Duda, J. S. Vrentas, S. T. Ju and H. T. Liu, Prediction of Diffusion Coefficients, A.I.Ch.E J., 28, 279 (1982).]...

Vrentas, J. S., Duda, J. L., and Ling, H. C. Free volume theories for self-diffusion in polymer-solvent systems. I. Conceptual differences in theories. J. Polym. Sci. 25 275, 1985. 

Application to Polvmer-Solvent Systems. Fujita (231 was the first to use the free-volume theory of transport to derive a free-volume theory for self-diffusion in polymer-solvent systems. Berry and Fox (241 showed that, for the temperature intervals usually considered (smaller than 200°C), the theories that consider a redistribution energy for the voids gives results similar to those of the theories that assume a zero energy of redistribution for the free volume available for molecular transport. Vrentas and Duda (5.61 re-examined the free-volume theory of diffusion in polymer-solvent systems and proposed a more general version of the theory presented by Fujita. They concluded that the further restrictions needed for the theory of Fujita are responsible for the failures of the Fujita theory in describing the temperature and concentration dependence... 

The assumptions and restrictions of the free-volume theory, as well as the significance of its parameters, are discussed in detail by Vrentas and Duda (5.61. For temperatures close to the glass transition temperature, the diffusion process is free-volume dominated and the energy term can be absorbed in the pre-exponential term. Equation 36 becomes... 

Free volume theories emphasize the amount of "empty space" [43-45] available for diffusion. The theory of Vrentas and Duda [46-53] is an especially elaborate example of such theories. 

Figure 2. Flow chart on the use of (i) the free volume theory of Vrentas and Duda, to obtain a "global perspective and (ii) the statistical mechanical model of Pace and Datyner, to obtain an "intermediate perspective on the scale of parameters describing polymer chain segments.

The free-volume theory of diffusion was developed by Vrentas and Duda. This theory is based on the assumption that movement of a small molecule (e.g., solvent) is accompanied by a movement in the solid matrix to fill the free volume (hole) left by a displaced solvent molecule. Several important conditions must be described to model the process. These include the time scales of solvent movement and the movement of solid matrix (e.g. polymer segments, called jumping units), the size of holes which may fit both solvent molecules and jumping units, and the energy required for the diffusion to occur. 

Figure 7.3.9. Concentration and temperature dependence of the binary diffusion coefBcient of a polystyrene-toluene solution according to the free volume theory of Vrentas and Duda. [After...

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]. 

Even with this modification, the resulting equation was no better than Fujita s original equation, which only correlates data at 0i < 0.2. These limitations seem to be absent in the free-volume theory of Vrentas and Duda, in which they obtained a general expression for the mutual diffusion coefficient, D, as... 

The reader who is interested in free-volume theories for polymer solutions might, for instance, take a closer look at the Vrentas-Duda theory [79-85]. This theory has also been used and discussed by several other workers to model the diffusion behavior of small species in polymer-solvent mixtures [78, 86-89 and references therein]. 

Vrentas, J. S. and J. L. Duda. 1977. Diffusion in polymer—Solvent systems. I. Reexamination of the free-volume theory. Journal of Polymer Science Polymer Physics Edition 15 (3) (March) 403-416. doi 10.1002/pol.l977.180150302. http //doi.wiley.eom/10.1002/ pol.1977.180150302. 

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