Free-volume theory of diffusion

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 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. 

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... 

Foam fractionation Fractional extraction Fractionation, seeDistillation Free-volume theory of diffusion Freezing-point determination Fugacity of nitrogen standard state Fugacity coefficient composition dependence of acetic acid vapor... 

The other approach derives from the free volume theory of diffusion. In this point of view, it is assumed that the fluctuations of local density in the polymer result in free volume or hole formation. When the hole is of sufficiently large size, and if it is formed near the penetrant molecule, the molecule can move or jump into it. The diffusion is assumed to be proportional to the probability of forming such holes of right size. The effect of the diffusing molecules on the free volume formation can be taken into consideration, too. 

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. 

Application of this theory to both polymeric and small molecule liquids has given jump distances 6 which correspond roughly to the dimensions of small molecules (i.e., from 2 to 20 A). Typical jump frequencies range from 10 to 10 per second. These and other free volume theories of diffusion are described in more detail in Crank and Park [4]. The important concept, common to all free volume theories, is that diffusion occurs in polymers through free volume obtained by minor displacements of side groups or segments of the chain but without net translational movements of the centre of mass of the polymer. 

FREE-VOLUME THEORY OF DIFFUSION IN RUBBERY POLYMERS... 

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 ... 

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]...

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). 

General comparisons have been presented between diffusion coefficients and apparent activation energies derivable from the "global" free volume theory of Vrentas and Duda, and the statistical mechanical model of Pace and Datyner, as an initial step in the utilization of these theories. 

A well-known and simple theory for describing molecular transport in a liquid is the free-volume theory of Cohen and Turnbull [1959, 1970]. Employing statistical mechanics, these authors showed that the most probable size distribution of the free volume per molecule in a hard sphere liquid may be described by an exponential decreasing function. It was assumed that diffusion of the hard-spheres can only take place when, due to thermal fluctuations, holes are formed whose size is greater than a critical volume. When applying this theory to a structural relaxation process in a liquid, its (circular) frequency o) = r = 2jtv is expressed by... 

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...

The surface concentration dependence of the lateral mobility of Fig. 7 was analyzed in terms of the free-volume theory of hard sphere liquids of Cohen and Turnbull [55, 56], as well as in view of the Enskog theory of dense gases [57] extended by Alder s molecular dynamics calculations to liquid densities [58]. The latter approach was particularly successful. It revealed that the lateral diffusion constant of the Fc amphiphiles does follow the expected linear dependence on the relative free area, Af/Ao, where Af = A — Ao, A = MMA, and Aq is the molecular area of a surfactant molecule. It also revealed that the slope of this dependence which is expected to inversely depend on the molecular mass of a diffusing particle, was more than 3 orders of magnitude smaller [54]. Clearly, this discrepancy is due to the effect of the viscous drag of the polar head groups in water, a factor not included in the Enskog theory. 

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... 

H. L. Frisch, D. Klempner, and T. Kwei, Modified Free Volume Theory of Penetrant Diffusion... 

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