Pressure dual mode model

The pressure dependence of the concentration of sorbed gas was consistent with the dual mode model while the relaxation data addressed itself to the validity of the assumptions made by the model. The assumption of rapid interchange was found to be valid for this system while the assumption of an immobile adsorbed phase could introduce a small error in the analysis It should be possible to reduce this error by more exact measurements of the concentration of sorbed gas as classical pressure experiments could... 

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 concentration-dependent models attribute the observed pressure dependence of the solubility and diffusion coefficients to the fact that the presence of sorbed gas in a polymer affects the structural and dynamic properties of the polymer, thus affecting the sorption and transport characteristics of the system (3). On the other hand, in the dual-mode model, the pressure-dependent sorption and transport properties arise from a... 

Comparing the curves in Fig. 2 shows that representing the permeability versus pressure data by either model provides a satisfactory fit to the data over the pressure range of 1 to 20 atm. However, at pressures less than 1 atm. the two models differ in their prediction regarding the behavior of the permeability-pressure curve [Fig. 2]. While the matrix model predicts a strong apparent pressure dependence of the permeability in this range (solid line), the dual-mode model predicts only a weak dependence (broken line). 

Fig. 3 shows the calculated time lags (solid line) and the experimental time lags reported by Wonders and Paul (15) for CO2 in polycarbonate. The time lags predicted by eq. (15) are in excellent agreement with the experimental results at both low and high pressures. The broken line in Fig. 3 results from calculations using the dual-mode model [eq. (17) in the previous chapter]. 

Using the dual-mode parameter values determined from sorption and permeation experiments, calculated time lags agree with the experimental data only at gas pressures above 5 atm. At lower pressures, dual-mode time lags are appreciably shorter than the observed ones, whereas time lags calculated from the matrix model by eq. (15) agree with the experimental data over the entire pressure range. 

The dual-mode model proved rather successful to describe the isotherms and permeabilities at higher pressures. 

The points in Fig. 20.4-9 were evaluated from the parmeability and sorption concentration data using Eq. (20.3-6), which do not depend ou the dual-mode model in any way. The line through the data poists corresponds to the predictions of err(p) using Eq. (20.4-15) along with ihe independantly determined dual-mode parameters for this system,13 It is also important to note that the form of data in this plot which exhibit a tendency to form an asymptote at high pressures is not typical of plasticization. 

At the lowest temperature (-900C) and at the lowest pressure (3 atm), the xenon spectrum for the sorbed line is rather broad, possibly asymmetric or bimodal. This would indicate an asymmetric or bimodal distribution of sorption environments on a fairly short length scale. More complicated descriptions of amorphous environments might consider the effects of the presence of an amorphous/crystalline interfacial region or of two sorption environments as proposed in the dual mode model. The low pressure, low temperature spectra being discussed have the poorest signal to noise and better spectra are needed to characterize this situation properly. 

The dual-mode model is easily extended to include gas mixtures (81). In terms of partial pressures, the expressions for mixed-gas sorption are nsnally written as... 

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. 

The discussion directly following Eq (6) provides a simple, physically reasonable explanation for the preceding observations of marked concentration dependence of Deff(C) at relatively low concentrations. Clearly, at some point, the assumption of concentration independence of Dp and in Eq (6) will fail however, for our work with "conditioned" polymers at CO2 pressures below 300 psi, such effects appear to be negligible. Due to the concave shape of the sorption isotherm, even at a CO2 pressure of 10 atm, there will still be less than one CO2 molecule per twenty PET repeat units at 35°C. Stern (26) has described a generalized form of the dual mode transport model that permits handling situations in which non-constancy of Dp and Dh manifest themselves. It is reasonable to assume that the next generation of gas separation membrane polymers will be even more resistant to plasticization than polysulfone, and cellulose acetate, so the assumption of constancy of these transport parameters will be even more firmly justified. 

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. 

A number of attempts have been made to explain the nonlinear, pressure-dependent sorption and transport in polymers. These explanations may be classified as "concentration-dependent (5) and "dual-mode (13) sorption and transport models. These models differ in their physical assumptions and in their mathematical descriptions of the sorption and transport in penetrant-polymer systems. 

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)]. 

Permeability and diffusion coefficients of hydrocarbons in polyphenylene oxides are also essentially dependent on pressure (see Figure 9.23). It can be seen that in the case of ethylene, with the increase in pressure, the permeability coefficients first decrease, and then begin to rise. Ref. [18] quotes constants of the dual-mode sorption model for a number of hydrocarbons permeation through polyphenylene oxide. 

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