Solute diffusion in polymers

Pawlisch, C. A. Brie, J. R. Laurence, R. L., "Solute Diffusion in Polymers. 2. Fourier Estimation of Capillary Column Inverse Gas Chromatography Data," Macromolecules, 21, 1685 (1988). 

Solute Diffusion in Polymers by Capillary Column Inverse Gas Chromatography... 

Solute diffusion in polymers by capillary inverse gas chromatography, in Inverse Gas Chromatography (eds D.R. Lloyd, T.C. Ward,... 

Pawlisch, C.A., Maoris, A., and Laurence, R.L. (1988) Solute diffusion in polymers. 2. Fourier estimation of capillary column inverse gas chromatography data. Macromolecules, 21 (6), 1685-1698. 

Maklakov AI, Skirde VD, Fatkullin NF (1987) Self-diffusion in polymer solutions and melts (in Russian). University Publ, Kazan... 

It can be noted that other approaches, based on irreversible continuum mechanics, have also been used to study diffusion in polymers [61,224]. This work involves development of the species momentum and continuity equations for the polymer matrix as well as for the solvent and solute of interest. The major difficulty with this approach lies in the determination of the proper constitutive equations for the mixture. Electric-field-induced transport has not been considered within this context. 

The Mackie-Mears expression has been extensively used in the analysis of diffusion in polymers where it is assumed that the obstacles, i.e., the polymer fibers, are of the same order of magnitude as the radius of the solute. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

Since the prediction of the solute diffusion coefficient in a swollen matrix is complex and no quantitative theory is yet possible, Lustig and Peppas [74] made use of the scaling concept, arriving at a functional dependence of the solute diffusion coefficient on structural characteristics of the network. The resulting scaling law thus avoids a detailed description of the polymer structure and yet provides a dependence on the parameters involved. The final form of the scaling law for description of the solute diffusion in gels is... 

JS Vrentas, JL Duda. Molecular diffusion in polymer solutions, AIChE J 25 1-24, 1979. 

SR Lustig, NA Peppas. Solute diffusion in swollen membranes. IX. Scaling laws for solute diffusion in gels. J Appl Polym Sci 36 735-747, 1988. 

W Brown, K Chitumbo. Solute diffusion in hydrated polymer networks. Chem Soc Faraday Trans I 71 1-11, 1975. 

Deff effective diffusivity of solute in porous membrane Dim diffusivity of solute i in polymer membrane... 

Finally, two articles review recent experimental and theoretical developments with respect to self-diffusion in polymer solutions, either concentrated A5), or dilute and semidilute A6) with emphasis on scaling concepts. Both include references to the recent PGSE literature. 

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. Such a case, for example, is the diffusion in polymer systems characterized by concentration-dependent diffusion coefficients.This chapter gives an overview of the most powerful numerical methods used at present for solutions of the diffusion equation. Indeed the application of these methods in practice needs the use of adequate computer programs (software). 

The characteristics of pore structure in polymers is a key parameter in the study of diffusion in polymers. Pore sizes ranging from 0.1 to 1.0 pm (macroporous) are much larger than the pore sizes of diffusing solute molecules, and thus the diffusant molecules do not face a significant hurdle to diffuse through polymers comprising the solvent-filled pores. Thus, a minor modification of the values determined by the hydrodynamic theory or its empirical equations can be made to take into account the fraction of void volume in polymers (i.e., porosity, e), the crookedness of pores (i.e., tortuosity, x), and the affinity of solutes to polymers (i.e., partition coefficient, K). The effective diffusion coefficient, De, in the solvent-filled polymer pores is expressed by ... 

The diffusion of small molecules in polymeric solids has been a subject in which relatively little interest has been shown by the polymer chemist, in contrast to its counterpart, i.e., the diffusion of macromolecules in dilute solutions. However, during the past ten years there has been a great accumulation of important data on this subject, both experimental and theoretical, and it has become apparent that in many cases diffusion in polymers exhibits features which cannot be expected from classical theories and that such departures are related to the molecular structure characteristic of polymeric solids and gels. Also there have been a number of important contributions to the procedures by which diffusion coefficients of given systems can be determined accurately from experiment. It is impossible, and apparently beyond the author s ability, to treat all these recent investigations in the limited space allowed. So, in this article, the author wishes to discuss some selected topics with which he has a relatively greater acquaintance but which he feels are of fundamental importance for understanding the current situation in this field of polymer research. Thus the present paper is a kind of personal note, rather than a balanced review of diverse aspects of recent diffusion studies. 

If D only depends on temperature (and thus not on concentration or time), the diffusion process is called Fickian. Simple gases show Fickian diffusion and so do many dilute solutions (even in polymers).The diffusivity can be determined directly either from sorption or from permeation experiments. In the first case the reduced sorption, c(t) / (cfX, cG), is plotted versus the square root of the sorption time and D is calculated from the equation ... 

Vrentas JS, and Duda JL, "Molecular Diffusion in Polymer Solutions , AlChE J 25 (1979) 1-24. 

Farinas, K. C. et al. Characterisation of solute diffusion in a polymer using ATR-FTIR spectroscopy and bulk transport techniques. Macromolecules 27(18) 5220-5222, 1994. 

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