Free volume theory solute diffusion

The theories that describe diffusion in concentrated polymer solutions are approximate in nature. Among them, only one seems sufficiently developed to offer a good description of mass transfer in polymer-solvent systems the free-volume theory of diffusion. Though it affords good correlative success, it needs further testing. 

At present, all these theories are approximate, since all attempts to derive them using molecular mechanics have been largely unsuccessful, because there is a large number of degrees of freedom in describing concentrated polymer solutions. Among these approximate theories, such as those developed by Barrer (IQ), DiBenedetto (ID, and van Krevelen (12), the free-volume theory of diffusion is the only theory sufficiently developed to describe transport processes in concentrated polymer solutions. 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

The PERVAP simulator (tubular module) was developed by Alvarez (2005), using FORTRAN language (Compaq Visual Fortran Professional Edition 6.6.a). The mathematical model applied is based on the solution-diffusion mechanism. Activity coefficients of the components in the feed phase (jj) were determined using the UNIFAC method (Magnussen et al, 1981). The prediction of diffusion coefficient (Z) ) was carried out using the free-volume theory. 

Solute Diffusion in Biopolymers as a Function of Water Activity Using a Modified Free Volume Theory... 

From the free volume theory (Vrentas and Vrentas, 1994), the following expression for concentration and temperature dependence of a solute diffusion coefficient in a concentrated polymer solution has been proposed ... 

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...

Because the solute molecules are generally large organic molecules, the temperature dependence of D close to T, is described not by an Arrhenius equation but by one based on free volume theories [210]. At zero concentration of diffusing substance ... 

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 structure of homogeneous PHEMA, that is materials polymerized in solutions with less than approximately 45 wt. % water in the reaction mixture 28), has been examined in greater detail. The earliest estimate of pore size in these PHEMA materials was 0.4 nm for a polymer prepared in the presence of water and ethylene glycol was provided by Refojo 29) who used the relationship between water permeability and average pore diameter developed by Ferry 30). The materials investigated by Refojo contained 39 wt. % water. This method was later applied by Haldon and Lee (57) to similar PHEMA samples of 41-42 wt. % hydration to obtain a pore radius between 0.4 and 0.8 nm. It was acknowledged that the assumptions implicit in the use of the Ferry equation resulted in underestimation of the pore size. This was highlighted by the observation that sodium fluorescein, with a radius of 0.55 nm, could readily diffuse into PHEMA. Later Kou et al. 32) demonstrated that solutes of radius 0.6 nm were able to penetrate PHEMA and copolymers of HEMA with methacrylic acid, and that the rate of diffusion was consistent with free volume theory. 

The parameters for the free volume theory of binary solution systems can be found in the literatures,and they have been effectively used in modeling drying process. But there are many difficulties in estimating diffusion coefficient for the ternary systems. Until now, almost all the empirical and theoretical correlations arc restricted to the binary solution systems. 

A new rate model for free radical homopolymerization which accounts for diffusion-controlled termination and propagation, and which gives a limiting conversion, has been developed based on ft ee-volume theory concepts. The model gives excellent agreement with measured rate data for bulk and solution polymerization of MMA over wide ranges of temperature and initiator and solvent concentrations. 

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