Glassy polymers dual-mode models

Nonlinear, pressure-dependent sorption and transport of gases and vapors in glassy polymers have been observed frequently. The effect of pressure on the observable variables, solubility coefficient, permeability coefficient and diffusion timelag, is well documented (1, 2). Previous attempts to explain the pressure-dependent sorption and transport properties in glassy polymers can be classified as concentration-dependent and "dual-mode models. While the former deal mainly with vapor-polymer systems (1) the latter are unique for gas-glassy polymer systems (2). 

The ex erimental results resented in the preceding chapter and in the literature are inconsistent with the assumptions and the physical interpretation implicit in the dual-mode model and strongly suggest that the sorption and transport in gas-glassy polymer systems should be represented by a concentration-dependent type model. 

This equation has become known as the "dual sorption model", because obviously two separate sorption mechanisms are operative for gases in glassy polymers. One mode (first term on the right in Eq. (18.36)) follows the Henry s law the other mode (second term) follows a Langmuir form. This additional mode is attributed to sorption into micro-voids that apparently pre-exist in the glassy state of the polymer (and only there ) it disappears above Tg (see Fig. 18.9b). 

The solubility in glassy polymers is usually described by the so-called dual-mode model, which implies that there is a need for a more detailed definition of the sorption, c, in the flux Equation 4.1. Equations 4.20 and 4.21 illustrate this and can relate to... 

Solubility data for methanol at low activities in glassy PVC, illustrated in Figure 2, again show the downward curvature of the dual-mode model. These data also exhibit a pronounced dependence on the previous thermal history of the polymer it appears that the Langmuirian sorption capacity of the polymer parallels the history dependence of free volume in the glassy state (6.9). 

These methods are interested in studying the distinction between the pure dualmode sorption/transport model curve and the actual sorption and permeation experiment curve that seems to contain the various unsolved appearances for diffusion and sorption of gases in glassy polymer. It becomes obvious that the deviations from the Fickian model in an experimental transport or sorption/desorp-tion curve for a gas in a glassy polymer are not necessarily consistent with the onset of concomitant diffusion and relaxation [11], but are just owing to the dual-mode model. That is to say, only the combinations of parameters of the dual mode model make the curve either fit with or deviation from the Fickian model curve. 

S. Kanehashi and K. Nagai, Analysis of dual-mode model parameters for gas sorption in glassy polymers, J. Membr. Sci., 253 (2005) 117-138. 

The following three multicomponent transport models have been used to explain the depression of the permeability of a component in a mixture relative to its pure component value (Fig. 21) the Petropoulos model and the competitive sorption model, both of which assume that direct competition for diflfiisive pathways within the glass is negligible, and a more general permeability model in which direct competition can occur between penetrant molecules for both sorption sites and diffusion pathways. All three of the models presented here are based upon the framework of the dual-mode model. It is worth mentioning that the site-distribution model has recently been extended to accoimt for diffusion (98) and that free volume models exist for transport in glassy polymers (99). 

On the other hand, estimation of solubility and diffiisivity data in glassy polymers requires direct experimental measurements. The results of these measurements can be correlated through the use of empirical models. The solubility of low molecular weight species in glassy polymers is usually described in terms of the dual mode model (7,2). The dual mode model is a powerful tool for the representation of most gas solubility isotherms. Its utility has been demonstrated in many studies (i-5). However, the parameters entering the model are endowed with a physical meaning not always consistent with the experimental observations (6). Thus, despite some interesting attempts to interpret the dependence of these parameters on polymer... 

The sorption isotherms for ethanol and methanol reported in Figure 4 and 5 cannot be interpreted on the basis of the well known dual mode model (/,2). This model assumes that the penetrant content in the glassy polymer matrix may be expressed as fimction of pressure through the sum of two contributions the first refers to the penetrant molecules which are considered to be adsorbed onto the surface of microvoids in the interior of the solid polymer, and the second represents the contribution due to penetrant molecules which are strictly dissolved into the solid phase. In the original formulation of the dual mode model the first contribution is expressed as fimction of pressure in terms of the Langmuir equation and the second through Henry s law. 

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 . 

It is particularly interesting and instructive to note that application of Henry + Langmuir dual-mode sorption and diffusion models is not confined to glassy polymer-gas systems. Sorption and transport of high affinity ionic species, exemplified by anionic dyes, in charged polymers, exemplified by polyamides at low pH, has been treated in the same way. These systems are of considerable importance both from the bio-mimetic and from the textile processing point of view, but have received limited atten-... 

One can easily show that the appropriate equation derived from the dual mode sorption and transport models for the steady state permeability of a pure component in a glassy polymer is given by Eq (7) (18) when the downstream receiving pressure is effectively zero and the upstream driving pressure is p ... 

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 dual-mode sorption and transport model the pressure-dependence of a (= C/p), P and 0 in gas-glassy polymer systems arises from the pressure-dependent distribution of the sorbed gas molecules between Langmuir sites and Henry s law dissolution. Although k, Dg and are assumed to be constant, the average or effective solubility and diffusion coefficients of the entire ensemble of gas molecules change with pressure as the ratio of Henry s to Langmuir s population, C /C, changes continuously with pressure [eq. (14)]. 

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 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 ... 

To analyze sorption of penetrants, including hydrocarbons, in glassy polymers, the dual-mode sorption model is most frequently used. For a number of glassy polymers, correlations between the constants of the dual-mode sorption model and the condensabUity of hydrocarbons have been established (see, e.g.. Figure 9.4a through 9.4c and data presented in Refs. [18-20]). Temperature dependence of model constants is described by Vant-Hoff equation, where the exponent contains heat of penetrant sorption A//s. This quantity is essentially dependent on the heat of penetrant condensation, AHcond- A//s = AH ond + A//i, where AHi is partial molar enthalpy of penetrant dissolution in the polymer, AHi = [d( AGi/T)/d( l/T)]c, AGi is the partial molar free... 

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]. 

A phenomenological theory known as the "dual-mode sorption" model offers a satisfactory description of the dependence of diffusion coefficients, as well as of solubility and permeability coefficients, on penetrant concentration (or pressure) in glassy polymers (4-6,40-44). This model postulates that a gas dissolved in a glassy polymer consists of two distinct molecular populations ... 

Pace and Datyner ( 5, 5) have also proposed a model for the absorption (solution) of small molecules in polymers applicable at temperatures above and below Tg, which incorporates the dual-mode sorption model for the glassy region. The presence of microvoids is assumed for rubbery polymers as well as for polymer glasses. "Hole filling" is suggested as an important sorption mode above as well as below Tg, with one crucial difference between the sorption mechanism in the rubbery and glassy regions hole saturation does not occur in the rubbery state because new microvoids are formed to replace those filled with penetrant molecules. 

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