Diffusion in Polymers

Example 5.3-1 Diffusion of carbon in iron Experiments show that the diffusion of carbon in body-centered cubic (BCC) iron is 2.4 10 cm /sec at 500 °C, but 1.7 10 cm /sec at 900 °C. Eind an equation which allows estimating carbon diffusion at other temperatures. 

Solution The form of this relation is that of Equation 5.3-2 

D = [6.2. 10- cm /secj exp- kJ/mol)/RT The values for Dq and for AH are slightly smaller than those commonly observed. 

Diffusion coefficients in high polymers are closer to those for liquids than to those for solids. This is true even for crystalline polymers, where the coefficients reflect transport around, not through, the small crystals. Typical values for synthetic high polymers are shown in Eig. 5.4-1. The values of these coefficients vary strongly with concentration. Naturally occurring polymers like proteins are not included in Eig. 5.4-1 because these species are best handled with the dilute-solution arguments in Section 5.2. 

The results in Eig. 5.4-1 show that very different limits exist. The first of these limits occurs in dilute solution, where a polymer molecule is imagined as a solute sphere moving through a continuum of solvent. The second limit is in highly concentrated solution, where small solvent molecules squeeze through gaps in the polymer matrix. 

Diffusion of small molecules in or out of polymers plays important roles in many processes and applications of polymers. A few examples encountered in various chapters include membrane and fiber formation processes, paints, coatings, diffusion of plasticizers and additives, and environmental crazing. 

The simplest law of diffusion in one dimension of a gas or liquid (denoted as species 1) in a polymer (species 2) is the empirical Pick s law of diffusion given by 

2 is called the binary diffusion coefficient of the 1-2 binary mixture (cmVs) 

In the simple form of Equation 11.8, Cj varies only in the x-direction. Unless one is in very dilute conditions, D 2 is not a constant because Equation 1.8 is a simplification of the more accurate representation of the flux that should be written as [26] 

The above equation correctly implies that the driving force of a flux is the gradient of chemical potential that represent the gradient of the Gibbs free energy of 1 and not just the gradient in concentration. Both Li and Pi depend on Ci, temperature, and pressure (or stress). At constant T and p, one can express Equation 11.9 as 

While the main subject of this book is the movement of polymer molecules themselves, in the chapters dealing with electrical and photo phenomena, we have already met the movement of other entities inside polymers. Very often the movement of these species is coordinated with, or controlled by, chain movements in the polymer matrix. The same is true of the diffusive motion of small molecules into, inside and through polymers. Such phenomena are important in a wide variety of uses ranging from packaging, through membrane separation processes, to controlled drug release. 

It is important to distinguish between permeability, P and diffusivity, D. The former is a measure of the amount of material that can be absorbed into one side of a polymer sample and then extracted from the other side. This penetration of a small molecule species into a bulk polymer is called permeation. On the other hand, the statistical random walk of sorbate molecules inside the polymer is called diffusion. Another important term is solubility, S. Whilst the ability of a molecule to move through the matrix is clearly important, its solubility can also be a significant factor in defining the permeability. In fact, the permeability is directly related to the product of the diffusivity and the solubihty  

Many of the technological uses of these molecular processes depend on the permeabihty of the sorbate—polymer system. Permeability measures both the amount of solute molecules in the polymer and the speed with which they can move, as was presented above. Looking at the effect of temperature, as was done 

The effect of similarity and dissimilarity is evident in the comparison of noncondensable gases with liquid or even solvent molecules. The former have little effect on the polymer matrix and so the diffusivity is little affected by the amount of gas absorbed. On the other hand, condensable or solvent species can open up the matrix, when the diffusivity becomes very much a function of the amount of absorbed material present. In such cases, the permeating molecules function rather like the plasticisers discussed in Chapter 4. 

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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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