Vrentas-Duda theory

Solvent power parameter entering Flory s theory of dilute solutions, degree of neutralization in polyelectrolyte solutions, free-volume parameter entering Vrentas-Duda theory subscript (1,24) denotes molecular species in solution. 

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]. 

The diffusivity data were analyzed using the Vrentas-Duda version of the foee-volume theory. Hie basic equation describing the solvent and temperature dependence of the diffusion coefficient in the limit of zero mass fraction above the glass transition temperature is given by the expression... 

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. 

The development of a scientific understanding of diffusion in liquid-phase polymeric systems has been largely due to Duda et al. (1982), Ju et al. (1981), and Vrentas and Duda (1977a,b, 1979) whose work in this area has been signal. In their most recent work, Duda et al. (1982) have developed a theory which successfiilly predicts the strong dependence of the diffusion coefficient on temperature and concentration in polymeric solutions. The parameters in this theory are relatively easy to obtain, and in view of its predictive capability this theory would seem to be most appropriate for incorporating concentration-dependent diffusion coefficients in the diffusion equation. 

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... 

There are some important considerations in the Duda-Vrentas theory that bear some examination for example, the effect of the solvent size on diffusional behavior, and the behavior of the diffusion process near the glass transition. 

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... 

Diffusion Below the Glass Transition. The fiee-volume theory can also be used to analyze the influence of the glass transition temperature on the diffusivity of the PMMA/methanol system. Tile use of the fiee-volume theory both above and below the glass transition is discussed by Vrentas and Duda Q). According to these authors, below the glass transition temperature, Equation 39 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. 

The free-volume model proposed by Vrentas and Duda (67-69) is based on the models of Cohen and Turnbull and of Fujita, while utilizing Bearman s (7j0) relation between the mutual diffusion coefficient and the friction coefficient as well as the entanglement theory of Bueche (71) and Flory s (72) thermodynamic theory. The formulation of Vrentas and Duda relaxes the assumptions deemed responsible for the deficiencies of Fujita s model. Among the latter is the assumption that the molecular weight of that part of the polymer chain involved in a unit "jump" of a penetrant molecule is equal to the... 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

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.

Figure 5. Correlation between penetrant diameter d used in the model of Pace and Datyner and size parameter used in the theory of Vrentas and Duda.

Figure 6. Best fit of the logarithm of the diffusion coefficient D calculated by the theory of Vrentas and Duda to the logarithm of D calculated by the model of Pace and Datyner, at 298.15K, for an idealized completely amorphous sample of PVDC, as a function of the penetrant diameter d.

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...

---