Diffusion in rubbery polymers

By analyzing the interaction between a chain center and the centers from adjacent chains, a relation for the potential interaction energy in both the normal and activated state was obtained (43). This formula was eventually used to develop a relationship for the calculation of the diffusion activation energy Ed. To use this relation 

One can state that a diffusion model, which allows only such calculations and extrapolations of D, exhibits only correlative and semi-predictive capabilities. All classical diffusion models presented in this section fall into this category. 

One of the most popular and detailed statistical molecular models for the diffusion of simple penetrants in amorphous rubbery polymers was proposed in refs. (45,46). This model is based on features taken from those presented briefly above. Because it is frequently cited in the literature, it will be presented here in some detail. 

To construct the model, it has been assumed that the amorphous polymer regions posses an approximately paracrystalline order with chain bundles locally parallel. A penetrant molecule may diffuse through the matrix of the polymer by two modes of motion, Fig. 5-2. 

A detailed discussion on how the equations of this model have been used to analyse the diffusion of simple gaseous penetrants in a series of polymers was given (46,47). The results reported show that the model allows a reasonable correlation of the experimental Ed with the diameters of the diffusant species. The main advantage of this classical molecular model is that the Ed can be estimated without using any adjustable parameters derived from correlation with experimetal diffusion data. However to use the equation proposed in the framework of this model to evaluate an Ed the knowledge of a series material, structural and thermodynamic parameters of the host polymer as well as the dimension and shape of the penetrant molecules must be known (45,46). The determination of these data, as far as they have not already been tabulated in some publication, depends on the availability of certain experimental data and is not always a simple task. 

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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). 

One possible solution to this problem is to develop microscopic diffusion models for glassy polymers, similar to those already presented for rubbery polymers. Ref. (90) combines some of the results obtained with the statistical model of penetrant diffusion in rubbery polymers, presented in the first part of Section 5.1.1, with simple statistical mechanical arguments to devise a model for sorption of simple penetrants into glassy polymers. This new statistical model is claimed to be applicable at temperatures both above and below Tg. The model encompasses dual sorption modes for the glassy polymer and it has been assumed that hole"-filling is an important sorption mode above as well as below Tg. The sites of the holes are assumed to be fixed within the matrix... 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

Vrentas, J.S. and Vrentas, C.M. Solvent self diffusion in rubbery polymer-solvent systems, Macromolecules, 27, 4684, 1994. 

The differences in the transport and solution behavior of gases in rubbery and glassy polymers are due to the fact that, as mentioned previously, the latter are not in a state of true thermodynamic equilibrium (1,3-8). Rubbery polymers have very short relaxation times and respond very rapidly to stresses that tend to change their physical conditions. Thus, a change in a temperature causes an immediate adjustment to a new equilibrium state (e.g., a new volume). A similar adjustment occurs when small penetrant molecules are absorbed by a rubbery polymer at constant temperature and pressure absorption (solution) equilibrium is very rapidly established. Furthermore, there appears to exist a unique mode of penetrant absorption and diffusion in rubbery polymers (2) ... 

At such extraordinarily low penetrant concentrations, plasticization of the overall matrix is not anticipated. As noted by Michaels in the previous discussion concerning crystallinity, motions involving relatively few repeat units are believed to give rise to most short-term glassy-state properties such as diffusion. In rubbery polymers, on the other hand, longer-chain concerted motions occur over relatively short time scales, and one expects plasticization to be easier to induce in these materials. Interestingly, no known... 

Pick s first and second laws were developed to describe the diffusion process in polymers. Fickian or case I transport is obtained when the local rate of change in the concentration of a diffusing species is controlled by the rate of diffusion of the penetrant. For most purposes, diffusion in rubbery polymers typically follows Fickian law. This is because these rubbery polymers adjust very rapidly to the presence of a penetrant. Polymer segments in their glassy states are relatively immobile, and do not respond rapidly to changes in their conditions. These glassy polymers often exhibit anomalous or non-Fickian transport. When the anomalies are due to an extremely slow diffusion rate as compared to the rate of polymer relaxation, the non-Fickian behaviour is called case II transport. Case II sorption is characterized by a discontinuous boundary between the outer layers of the polymer that are at sorption equilibrium with the penetrant, and the inner layers which are unrelaxed and unswollen. 

During the last decades interest in membrane separation processes and in solutions for packaging problems has given a substantial boost to research on sorption and mass transport in polymeric materials. Several interpretative models for the solubility and diffusivity in rubbery polymers are now available which, at least in principle, allow for the prediction of the permeability of low molecular weight species in polymeric films above their glass transition temperature. 

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

Local density fluctuations occur in penetrant polymer systems both above and below Tg. It is then reasonable to expect that a free-volume diffusion model should also provide an adequate description of the diffusion of small penetrants in glassy polymers. To reach this goal the free-volume model for diffusion of small penetrants in rubbery polymers, second part of Section 5.1.1, was modified to include transport below Tg (64,65,72,91-93). 

This paper reviews some of the more important models and mechanisms of gas diffusion in rubbery and glassy polymers in light of recent experimental data. 

STERN TROHALAKI Gas Diffusion in Rubbery and Glassy Polymers... 

Considerable effort has been made during the last two decades to develop a "microscopic" description of gas diffusion in polymers, which is more detailed than the simplified continuum viewpoint of Fick s laws. It has been known for a long time that the mechanism of diffusion is very different in "rubbery" and "glassy" polymers, i.e., at temperatures above and below the glass-transition temperature, Tg, of the polymers, respectively. This is due to the fact that glassy polymers are not in a true state of thermodynamic equilibrium, cf. refs. (1,3,5,7-11). Some of the models and theories that have been proposed to describe gas diffusion in rubbery and glassy polymers are discussed below. The models selected for presentation in this review reflect only the authors present interests. 

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