Models for diffusion in polymers

The literature contains a very large amount of both experimental and theoretical information on the diffusion of small molecules, especially gases, in polymers. Since the pioneering works (1,2) on the diffusion of gases through rubber septa interest in diffusion phenomena in polymers has continously increased and diversified. The considerable interest and the concentrated academic and industrial research efforts in the study of diffusion in polymers arises from the fact that important practical applications for these materials depend to a great extent on diffusion phenomena. In the last five decades a series of classic books and reviews have been devoted to the presentation of the main topics, experimental results, theories and applications for the diffusion of small penetrants in and through polymers (3-19). An interested reader will most certainly find in one of these references information which apply to her/his special area of interest. 

The first section of this chapter is devoted to the presentation of some diffusion models constructed on the basis of phenomenological considerations. These so-called heuristic models are often cited in the literature and used for the interpretation of experimental results. A special emphasis is to discuss how the mathematical formulae of these models can correlate with experimetal data and moreover to predict diffusion coefficients beyond the ranges experimentally investigated. This latter aspect is of great interest not only from a fundamental point of view but also in many practical fields where the possibility to predict a diffusion process might be a more economic alternative to its experimental investigation. 

The last section gives a presentation on the present status of so-called ab initio computer simulations of the diffusion of small penetrants in polymer matrices. This is a domain currently in rapid development and there are many expectations that, computer simulations will one day become a practical and performant tool to predict diffusion phenomena in complex penetrant polymers system. 

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Most free-volume models for diffusion in polymers follow the phenomenological basis set in (55) where the self-diffusion of an ideal liquid of hard spheres ( molecules ) has been analysed. These molecules are confined - for most of the time - in a cage formed by their immediate neighbours. A local fluctuation in density may open a hole within a cage, large enough to permit a considerable displacement of the sphere contained by it. This displacement gives rise to diffusion only if another sphere jumps into the hole before the first sphere returns to its initial position. Diffusion occurs not as a result of an activation process in the ordinary sense but rather as a result of the redistribution of the free-volume within the liquid of hard spheres. 

The model of Meares is only of historical interest because it was the first molecular model for diffusion in polymers. Meares (12) found that the activation energy for diffusion correlates linearly with the square of the penetrant diameter, but not with the diameter cubed. Therefore, he inferred that the elementary... 

This classification should in principle be valid for both rubbery and glassy polymers. However, as will be shown in this section, until now more detailed and true microscopic" treatments have mainly been models for diffusion in rubbery polymers. An explanation for this may be the much more complex nature of the diffusion process in glassy polymers (9,13,32-34). 

MERCEA, p.. Models of Diffusion in Polymers, in Piringer, O.-G. and Baner, A. L., 2000, Plastic Packaging Materials for Food, Wiley-VCH, Weinheim, New York. 

The concept of dual mode sorption was first dearly demonstrated and quantified by Michaels, Vieth and Barrie in 1963 The same authors also discussed its effect on the diffusion process itself. Vieth and his co-workers aibsequently extended these findings to a number of polymer-gas systems and elaborated the theoretical aspects of the problem In particular, a model for diffusion in glassy polymers, which has come to be known as the totd inunobilization model, was developed by Vieth and Sladek ... 

Cohen DS. Theoretical models for diffusion in glassy polymers. J. Polym. Set B Polym. Phys. 1983 21 2057-2065. 

MODELS FOR DIFFUSION IN BOTH RUBBERY AND IN GLASSY POLYMERS Some models are applicable to diffusion of small molecules in glassy as well as in rubbery polymers. These too fall into the general categories of molecular models and free-volume models. Recent molecular dynamics simulations of simple polymer/penetrant systems will also be discussed. 

Since the tortuosity for TMA is 2, it is reasonable to assume that this represents the intrinsic tortuosity for the extracellular space (i.e., Aeff/AA,pore = 2). The additional tortuosity from the extracellular space should depend on molecular size, and can be estimated from models of diffusion in polymer solutions or gels (as shown in Figures 4.10 and 4.12). 

Vieth WR, Sladek KJ (1965) A model for diffusion in a glassy polymer. J Colloid Sci 20... 

Saltzman, W. M., Pasternak, S. H., and Langer, R., Micro-structural models for diffusive transport in porous polymers, in Controlled-Release Technology, ACS Symposium Series 348... 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

Bartle et al. [286] described a simple model for diffusion-limited extractions from spherical particles (the so-called hot-ball model). The model was extended to cover polymer films and a nonuniform distribution of the extractant [287]. Also the effect of solubility on extraction was incorporated [288] and the effects of pressure and flow-rate on extraction have been rationalised [289]. In this idealised scheme the matrix is supposed to contain small quantities of extractable materials, such that the extraction is not solubility limited. The model is that of diffusion out of a homogeneous spherical particle into a medium in which the extracted species is infinitely dilute. The ratio of mass remaining (m ) in the particle of radius r at time t to the initial amount (mo) is given by ... 

Microstructural Models for Diffusive Transport in Porous Polymers... 

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