Diffusion free-volume theory

It is also useful to note that other approaches to describe diffusion in solvent-polymer systems have been developed using free-volume theory [408-410]. 

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. 

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

Yasuda et al. [64] developed a free volume theory describing the diffusion... 

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. 

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 ... 

Figure 14 The free volume theory of Yasuda and coworkers holds for the diffusion of acetaminophen in swollen 10 X 4 poly(lV-isopropyl acrylamide) gel. (Adapted from Ref. 176.)...

In the free volume theory, translational diffusion of a lipid molecule in the bilayer occurs only when a free volume larger than a certain critical size appears in the vicinity of the lipid molecule. The free volume theory implies that the smaller the overall volume, the lower the probability for a molecule to associate with a free volume of a critical size. The molecules diffuse slower if the probability for a molecule to associate with a free volume of critical size is small. With increasing pressure, the overall volume decreases and the lateral diffusion is thus reduced. The activation volume for diffusion in the LC phase was calculated using the expression ... 

Yapel RA, Duda JL, Lin X and Von Meerwall ED (1994) Mutual and self-diffusion of water in gelatin experimental measurement and predictive test of free-volume theory. Polymer 35, 2411-2416. 

Polyvinylchloride was the host polymer in a study of the diffusion of dimethyl-phthalate, dibutylphthalate, and dioctylphthalate, performed by Maklakov, Smechko, and Maklakov 60) between room temperature and 110 °C. Azancheev and Maklakov 61) extended this work to include polystyrene as host, and to dependences of diffusion on concentration. They concluded that the macromolecules did constrain and trap the phthalate molecules at high polymer concentration, but without inhibiting the mobility of these diluents at lower polymer concentrations, e.g., in the gel. They used a version of the free volume theory to give a semi-quantitative explanation of the temperature and molecular size dependence of phthalate diffusion. 

Fig. 16. The diffusion coefficient of acetaminophen in 10 x 4 PNIPAAm gels falls as the swelling degree (Q) of the gel decreases due to increasing temperature. Below the transition temperature of the gel, the linear relationship between log D and (Q — 1) 1 predicted by the free volume theory of Yasuda et al. [10] is observed. Above the transition temperature, the theory underestimates D by 35 times. Reprinted from the Journal of Controlled Release (1992) 18 1, by permission of the publishers, Elsevier Science Publishers BV [70]...

Fig. 17. Vitamin B12 diffusion coefficients vs polyfacrylamide-co-sodium methacrylate) gel swelling. The variation is that expected from free volume theories. Reprinted with permission from Chemical Engineering Science, 44,...

A comprehensive model which is based on the free-volume theory and which accounts for the effect of molecular weight and solvent on chain entanglements and glassy-state transition has been recently developed by Marten and Hamielec (7 ). This model accounts for diffusion-controlled termination and propagation... 

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).]...

Summing up, the two-phase model is physically consistent and may be applied for designing industrial systems, as demonstrated in recent studies [10, 11], Modeling the diffusion-controlled reactions in the polymer-rich phase becomes the most critical issue. The use of free-volume theory proposed by Xie et al. [6] has found a large consensus. We recall that the free volume designates the fraction of the free space between the molecules available for diffusion. Expressions of the rate constants for the initiation efficiency, dissociation and propagation are presented in Table 13.3, together with the equations of the free-volume model. 

Similarly to Fig. 5-4 for other glassy polymer-solvent systems also the predictions of this free-volume theory are in general agreement with experimental data on the temperature dependence of D in the vicinity of Tg2. In particular, the theory predicts a step change in Ed at Tg2, and this is consistent with most experimental investigations of polymer-solvent diffusion at temperatures just above and below the glass transition temperature (6,11,15). 

Thus measurements of the viscosity 9 (0,7) over a range of temperature allow determination of f(0,T) as a function of T, provided the value of /(0,T) at a certain temperature T is known from other source. For this purpose we may utilize the measurement of viscosity as a function of diluent concentration at the given T ] the substitution of such data into Eq. (40) may lead to the determination of the required f(0,T ). It is to be expected that, if the free volume theories of viscosity and diffusion developed above are at all correct, the values of /(0,7) thus derived from y data should agree with those obtained from ae data by application of Eq. (40) and also with those from DT data analyzed in terms of Eq. (36). 

Although the WLF equation has been tested extensively, as yet only a few examples can be quoted for the comparison of the free volume theories of diffusion and viscosity with experimental data on polymeric systems. As for previous comparisons the reader should consult recent articles of Fujita and his coworkers [Fujita, Kishimoto and Matsu-moto (1960) Fujita and Kishimoto (1960 1961)]. Here we shall show some new data to illustrate the applicability and limitations of these theories. 

Fig. 16gives a value of 0.046 for /(0.T) at this temperature. Again, this value may be compared favorably with f(0,Ts) = 0.049 predicted from the WLF equation. Summarizing, we may conclude that for this polymer-solvent system the validity of the free volume theory of diffusion has been reasonably checked with experiment. Results of similar nature can be found in a paper by Fujita, Kishimoto and Matsumoto... 

The real reason for this serious limitation of the free volume theory is as yet not clarified. Kishimoto, Maekawa and Fujita (1960) have considered that diffusion of small molecules such as water requires for their jumping only a very local cooperation of the solid-like vibrations of two or three monomers and, therefore, their rate of diffusion would not depend on physical factors through the average free volume of the system. It would simply increase with temperature as a result of enhanced thermal agitations of individual monomer units as well as the penetrant molecules themselves. However, an explanation like this is as yet no... 

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