Dual mode model dependence

The permeability of a polymer to a penetrant depends on the multiplicative contribution of a solubility and a mobility term. These two factors may be functions of local penetrant concentration in the general case as indicated by the dual mode model. Robeson (31) has presented data for CO2 permeation in... 

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... 

The basic difference between Mconcentration-dependentM and dual-mode, models is in their assumption about penetrant-polymer interactions. Concentration-dependent sorption and transport models are based on the assumption that the concentration-dependence of the solubility and diffusion coefficients arises... 

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... 

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. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

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). 

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). 

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. 

Due to the contribution of the hole-filling mechanism, the dual-mode model predicts that the sorption potential of NOM is concentration-dependent (i.e., nonlinear) and that sorption is subject to competitive effects. 

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 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. 

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)]. 

The experimental evidence presented here and in the literature (15) show that the real diffusion coefficient depends on concentration. These results are incompatible with the notion of concentration-independent diffusion coefficients for the dissolved and Langmuir sorbed molecules [D and Djj in equation (15)] as proposed by the dual-mode sorption and transport model ( 13). 

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... 

FIGURE 9.4 Dependence of constants (a, b, and c present Henry constant, sorption affinity constant, and Langmuir sorption capacity respectively) of the model of dual-mode sorption of hydrocarbons by glassy polyphenylene oxides on boiling temperatures of hydrocarbons Z), is pDMePO, poly-2,6-dimethyl-l,4-phenylene oxide o is pDPhPO, poly-2,6-diphenyl-l,4-phenylene oxide is pDMePO/pDPhPO copolymer (97.5/2.5% mol) v is pDMePO/pDPhPO copolymer (75/25% mol). (From analysis of results presented in Lapkin, A.A., Roschupkina, O.P., and Ilinitch, O.M., J. Membr. Sci., 141, 223, 1998.)... 

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]. 

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. 

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 ... 

Composition-Dependent Permeability. There are some experimental data 8"19 indicatiug the existence of competition between penetrants in accordance with the postulations of the generalized dual-mode transport model [Eq. (20.6-40)]. It is of interest to examine the magnitude or this competition effect on the overall performance of an actual separator. This is illustrated in Tables 20.6-5 and 20.6-6. Equation... 

Care should always be exercised when using solubility data for glassy or crystalline polymers (not included here), because SCF sorptiOTi occurs preferentially in the amorphous phase, which may additirmally experience swelling-related stress. Solubihty data for CO2 in solid polymers is compiled in [5]. Often, the pressure dependence of SCF sorption in glassy polymers follows a dual-mode sorption model, with substantial deviations from Henry s law. 

At elevated pressures, the dependence of the solute concentration on the ideal gas pressure can be represented as a sum of linear terms, bjp (for sites j where bjp 1), and Langmuir-like terms, bjP/(l -i- bjp) (for sites] where the term bjp is at least comparable to unity). If a system consists only of linear sites and a set of identical Langmuir sites, the well-known dual-mode-sorptiorr model [2] is obtained [54],... 

When the temperature is lower than the critical temperature 7 of gas the solubility of the permeant gas in the membrane may become so high that the diffusion constant no longer remains constant. A modification was done in the above dual transport model with partial immobilization of the Langmuir sorption mode to include the concentration dependency of the diffusion coefficient [153]. Instead of Equation 5.204 the following equation expresses the permeability coefficient ... 

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