Diffusivities in polymers

J. Crank and G. S. Park, eds.. Diffusion in Polymers, Academic Press, London, 1968. 

Crank G, Park GJ (1968) Diffusion in polymers, chapt. 2. Academic Press, London... 

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

Meerwall v., E. D. Self-Diffusion in Polymer Systems. Measured with Field-Gradient Spin Echo NMR Methods, Vol. 54, pp. 1—29. 

Friedrich, K. Crazes and Shear Bands in Semi-Crystalline Thermoplastics. Vol. 52/53, pp. 225-274. Fujita, H. Diffusion in Polymer-Diluent Systems. Vol. 3, pp. 1-47. 

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. 

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. 

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. 

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. 

J Crank, GS Park. Diffusion in Polymers. New York Academic Press, 1968. 

H Fujita. Diffusion in polymer-diluent systems. Fortschr Hochpolym-Forsch 3 1 — 21, 1961. 

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. 

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

JC Wu, NA Peppas. Numerical simulation of anomalous penetrant diffusion in polymers. J Appl Polym Sci 49 1845-1856, 1993. 

JS Vrentas, CM Jarzebski, JL Duda. Deborah number for diffusion in polymer-solvent systems. AIChE J 21 894-902, 1975. 

GF Billovits, CJ Durning. Polymer material coordinates for mutual diffusion in polymer-penetrant systems. Chem Eng Commun 82 21-44, 1989. 

EJ Lightfoot. Kinetic diffusion in polymer gels. Physica A 169 191-206, 1990. 

BK Davis. Diffusion in polymer gel implants. Proc Natl Acad Sci 71 3120-3123, 1974. 

Topological Nature and Chain Sliding Diffusion in Polymer Nucleation... 

Reactions Limited by Rotational Diffusion in Polymer Matrix... 

The theory of diffusion in polymers as heterogeneous media was discussed in Refs. [68,74,81-85], The correlation between the frequency of rotation vT of the nitroxyl radical (TEMPO) and diffusion coefficient of oxygen D (298 K) was found [86]. 

All reactions collected in Table 19.6 are slow. They occur with rate constants that are sufficiently lower than the rate constants of diffusion in polymer, as well as the frequency of reactant orientation in the cage (vor =vrx P). Hence, physical processes are not limited by the rates of these reactions. However, polymer media influences the kinetics of these reactions. 

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