Model dual mode

Let us evaluate some experimental data. To this end, we use a dual-mode model (Eq. 9-6). This model is a combination of a linear absorption (to represent the sorbate s mixing into natural organic matter) and a Freundlich equation (as seen for adsorption to hydrophobic surfaces or pores of solids like activated carbons) ... 

We have been able to demonstrate the basic validity of the assumptions on which the dual mode model is based and we have shown the usefulness of NMR relaxation techniques in the study of this model,"... 

Whereas the dual sorption and transport model described above unifies independent dilatometric, sorption and transport experiments characterizing the glassy state, an alternate model offered recently by Raucher and Sefcik provides an empirical and fundamentally contradictory fit of sorption, diffusion and single component permeation data in terms of parameters with ambiguous physical meanings (28), The detailed exposition of the dual mode model and the demonstration of the physical significance and consistency of the various equilibrium and transport parameters in the model in the present paper provide a back drop for several brief comments presented in the Appendix regarding the model of Raucher and Sefcik,... 

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... 

We have been able to critically examine the dual mode model by pulsed NMR relaxation techniques ... 

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 only "small error", suggested in Assink s statement concerning the dual mode model s assumptions, deals with the earlier approximation by Vieth and Sladek (34) that was equal to zero. The work of Assink was performed prior to the formulation of the dual mobility model which eliminates this approximation and accomodates values of > 0 (17,18). 

Eq (A-l)], but at the same time makes it easier for the penetrant to move through the matrix [Eq (A-2)]. The dual mode model offers the physical interpretation that concavity in the isotherm arises from a site saturation mechanism related to filling of unrelaxed gaps—a phenomenon which is not only easily visualized and understood, but also has been tested explicitly with independently obtained dilatometric data (See Figure 5). 

In one of their comparisons between the matrix model and the dual mode model, a somewhat misleading presentation of data is unintentionally offered in Figure 4 of the Raucher and Sefcik paper "Matrix Model of Gas Sorption and Diffusion in Glassy... 

On a more basic level, since the matrix model implicitly requires a somewhat inconsistent interpretation for the various model parameters in Eq (A-l) and Eq (A-2), it becomes primarily an empirical means of reproducing the observed pure component data with no fundamental basis for generalization to mixtures. One could, of course envision several extensions based on additional a terms in the denominator of Eq (A-l) and additional 8 terms in Eq (A-2). Such an approach to mixture permeation analyses would be completely empirical and mimic the generalization of Eq (2) and Eq (7) however, without any physical justification. The generalizations of Eq (2) and Eq (7) were natural outgrowths of the fundamental physical basis of the Langmuir isotherm. The fact that the mixture data are so consistent with Eq (7) and Eq (9) provides strong support for the physical basis of the dual mode model. 

An important value of a permeation model is not simply its ability to correlate experimental data, but rather to provide a framework for understanding the principal factors controlling membrane performance. The dual mode model is derived from... 

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. 

Figure 1. Sorption isotherm at 35 °C for CO2 in polycarbonate conditioned by exposure to 20 atm CO2. The experimental data are from Ref. 15. The curves, based on the matrix model (solid line) and the dual-mode model (broken line), are calculated using the parameters given in the text.

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]. 

III. THE MATRIX AND DUAL-MODE MODELS AS PHYSICAL MODELS... 

Predicting fast and slow rates of sorption and desorption in natural solids is a subject of much research and debate. Often times fast sorption and desorption are approximated by assuming equilibrium portioning between the solid and the pore water, and slow sorption and desorption are approximated with a diffusion equation. Such models are often referred to as dual-mode models and several different variants are possible [35-39]. Other times two diffusion equations were used to approximate fast and slow rates of sorption and desorption [31,36]. For example, foraVOCWerth and Reinhard [31] used the pore diffusion model to predict fast desorption, and a separate diffusion equation to fit slow desorption. Fast and slow rates of sorption and desorption have also been modeled using one or more distributions of diffusion rates (i.e., a superposition of solutions from many diffusion equations, each with a different diffusion coefficient) [40-42]. 

Partially immobilising sorption ( dual-mode model)... 

TABLE 18.12 Main formulae of the dual-mode model of permeation ... 

Experiment To measure To find Formula of dual mode model to be used ... 

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