Diffusion in glassy polymers

Because of the assumed dual sorption mechanism present in glassy polymers, the explicit form of the time dependent diffusion equation in these polymers is much more complex than that for rubbery polymers (82-86). As a result exact analytical solutions for this equation can be found only in limiting cases (84,85,87). In all other cases numerical methods must be used to correlate the experimental results with theoretical estimates. Often the numerical procedures require a set of starting values for the parameters of the model. Usually these values are shroud guessed in a range where they are expected to lie for the particular penetrant polymer system. Starting from this set of arbitrary parameters, the numerical procedure adjusts the values until the best fit with the experimental data is obtained. The problem which may arise in such a procedure (88), is that the numerical procedures may lead to excellent fits with the experimental data for quite different starting sets of parameters. Of course the physical interpretation of such a result is difficult. 

However, the mathematical formulae of DST satisfactorily present the dependence of the solubility and diffusion coefficients for simple gases and organic vapors on the concentration of the penetrant in the glassy polymer (9,11,13,15,17,33,34,89). 

From the point of view of earlier discussions, namely the true prediction of diffusion coefficients for volatile and nonvolatile organic penetrants in glassy polymers, the diffusion equations derived in the framework of the DST have only a limited usefulness. That means that, because the parameters of the DST models are not directly related to first principles , the equations can be used with success to correlate experimental results, but not to truly predict diffusion coefficients. 

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 

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). 

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T Alfrey, EF Gurnee, WG Lloyd. Diffusion in glassy polymers. J Polym Sci Part C 12 249-261, 1966. 

JC Wu, NA Peppas. Modeling of penetrant diffusion in glassy polymers with an integral sorption Deborah number. J Polym Sci Polym Phys Ed 31 1503-1518,... 

Dual-Mode Gas Sorption and Diffusion in Glassy Polymer Membranes. 97... 

Application of the dual mode concept to gas diffusion in glassy polymers was originally subject to the limitation that DT2 = OinEq. (6) ( total immobilization model )6-Later this simplifying assumption was shown to be unnecessary, provided that suitable methods of data analysis were used 52). Physically, the assumption DX2 = 0 is unrealistic, although it is expected that DT2 < DX1 52). Hence, this more general approach is often referred to as the partial immobilization model . 

The above results indicate that the general characteristics of gaseous diffusion in glassy polymer membranes can be represented reasonably well in terms of the dualmode concept. The basic reason for the observed increasing D(C) function is seen to be the concurrent increasing proportion of less strongly sorbed (and hence more easily activated) penetrant molecules. The model is, no doubt, highly idealized, but is nevertheless shown to be physically reasonable and consistent with the correspond-... 

Sefcik M. D., Raucher D. The Matrix Model of Gas Sorption and Diffusion in Glassy Polymers, to be published... 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

In the following chapter we present the matrix model of gas sorption and diffusion in glassy polymers which is based on the observation that gas molecules interact with the polymer, thereby altering the solubility and diffusion coefficients of the polymer matrix. 

Most of the existing theories on diffusion in glassy polymers consider the transport of a single penetrant, namely the solvent. 

Odani, H., J. Hayashi and M. Tamura Diffusion in glassy polymers. II. Effects of polymer-penetrant interaction diffusion of methyl ethyl ketone in atactic polystyrene. Bull. Chem. Soc. Japan 34, 817 (1961). 

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. 

By comparison, glassy polymers have (on the average) very long relaxation times. Hence, "in the presence of a penetrant, the motions of whole polymer chains or of portions thereof are not sufficiently rapid to completely homogenize the penetrant s environment. Penetrant (molecules) can thus potentially sit in holes or irregular cavities with very different intrinsic diffusional mobilities" ( ). In other words, there could exist more than one mode of penetrant absorption and diffusion in glassy polymers. 

R. M. Barrer, Diffusivities in glassy polymers for the dual model sorption model, J. Membr. ScL, 18, 25-35 (1984). 

Z. J. Grzywna, J. Stolarczyk, Diffusion in glassy polymers - from random walks to partial differential equations, Acta Phys. Pol. B, 36, 1001 (2005). 

J.S. Vrentas, C.M. Vrentas, Solvent self-diffusion in glassy polymer-solvent systems, Macromolecules 27 (1994) 5570-5576. 

Four methods based on transition-state theory were then discussed. The frozen polymer method is inappropriate for penetrant diffusion in glassy polymers, because chain fluctuations are intricately coupled to the diffusion mechanism. Path-based methods are very sound, but the resulting high dimensionality could pose computational problems. 

This work offers a contribution to the understanding of some fundamental aspects of sorption and diffusion in glassy polymers. The research focuses on an extensive experimental study of sorption and mass transport in a specific polymeric matrix. A high free volume polymer, (poly l-trimethylsilyl-l-propyne) [PTMSP], has been used here in order to emphasise aspects of sorption and transport which are peculiar to polymer/penetrant mixtures below the glass transition temperature. The discussion of the experimental data presented in this work permits a clarification of concepts which are of general validity for the interpretation of thermodynamic and mass transport properties in glassy systems. 

Prediction of gas diffusivity in glassy polymers. CEDis useful as it can be estimated conveniently without experimental measurements. In contrast, Vf values require measurements of polymer density. Accordingly, we attempted to correlate CED and D for several families of glassy polymers. Figure 7 presents this correlation for polyimides(7-5), polycarbonates(77,33), polysulfones(3 ), polystyrenes(i9), and the polyimides which we have studied. Overall, a good correlation between log(D) and CED was observed. 

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