NELF model

The characteristic parameters of the polymer can be calculated by best-fitting the LF equation of state to PVT data above Tq, while for the penetrant either PVT or vapour-liquid equilibrium data can be profitably used. 

The number of lattice sites occupied by a molecule in its pure phase is given by  

A first-order approximation is given by F= 1, to use when no specific data for the mixture are experimentally available. 

The pseudo-equilibrium state of the glassy mixture is desaibed by the usual state variables (temperature, pressure and composition) plus the polymer density p2 that accounts for the departure from equilibrium frozen into the glass. 

The phase equilibrium between the external gas phase and the glassy polymer mixture is given by the usual relationship  

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The thermodynamic derivation of the NELF model has been reported in several publicationsP° " l From a more general point of view, such a model represents a special application of the non-equilibrium thermodynamics of glassy polymers (NET-GP) which indicates the relationships existing in general between the thermodynamic properties above and below the glass transition temperature the NET-GP results hold for any thermodynamic model and are not limited to any particular EoS. 

The solubility isotherms obtained from the non-equilibrium models for all these systems are always satisfactory and all the different models used give very similar results. One may notice that the worst case is represented by the PSf-CH4 systems in which the NELF model slightly underestimates the experimental sorption data, especially at the higher pressure range, with an error, however, not exceeding 15%. 

Figure 2.7 (a) CH4 solubility in PS-TMPC blends and (b) CO2 solubility in PMMA-BCPC blends at 35° C, reported as a function of pressure. Experimental data are from respectively. The NELF model results are also reported as predicted on the basis of the binary systems data alone... 

Experimental solubility of (a) n-C4 and (b) n-Cs in the polymeric phase of 2400-based MMM at 25 °C and comparisons with NELF model... 

Table 7.1 NELF model parameters for n-C4, n-C, mixed matrices and their mixtures ...

The FFV in the unpenetrated polymer phase of the MMM calculated from the sorption isotherm of the test penetrant (n-C4) throngh the NELF model can also be used to correlate the diffusivity data collected during sorption tests. The infinite dilution diffusion coefficient in the polymeric phase, T>p(0) determined from the experimental diffusivity data can be related to the FFV initially present in the polymer matrix (FFVm). As shown in Figure 7.7 for diffusion of n-bntane in AF 2400, there is a correlation between the infinite dilution diffusion coefficient and l/FFVS, that agrees with Eqnation (7.16). Similar behaviour is observed for n-Cs, with different valnes of the parameters A and B, as it can be seen from the results reported in Table 12. This is not surprising, because the values of these empirical parameters are expected to be a function of the penetrant nature and size. 

M. G. De AngeUs, G. C. Sarti, Solubility and diffusivity of gases in mixed matrix membranes containing hydrophobic fumed silica correlations and predictions based on the NELF model, Ind. Eng. Chem. Res., 47, 5214-5226 (2008). 

Equation 27 represents the basic equation for the NELF model based on the Sanchez and Lacombe lattice fluid theory it provides the explicit dependence of the chemical potential of each penetrant species of a multicomponent mixture on temperature, volume and composition. In view of equation 12 and equation 14 at given temperature, volume and composition this equation is valid for any pressure... 

The requirement to specify the polymer density may represent a serious limitation for the practical application of the NELF model. Indeed, the dilation of the polymer matrix at high penetrant pressure could be significant and difficult to estimate without specific experimental data. On the other hand, the use of the non-equilibrium polymer (tensity is actually a powerful tool to represent complex non-equilibrium phenomena. It has been shown, for example, that it allows a description of sorpdon-desorption hysteresis (7) as well as the influence of pretreatments on the solubility isotherms of gases in glassy polymers. For such cases, the different pseudo-equilibrium solubility values at the same prevailing temperature and penetrant fugacity are satisfactorily accounted for by considering die different pseudo-equilibrium polymer densities. 

It is also interesting to observe that the solubility coefficient of a penetrant in a glassy polymer, in the limit of low pressure, is independent of the partial molar volume of the penetrant component in the mixture and can be reliably calculated from the NELF model using only the pure polymer density (14). Thus, also in the absence of volume dilation data the NELF mcidel can be reliably used, albeit in a limited pressure range often in the order of few bars at least (14). 

Solubility of Gaseous Mixtures Comparison with Experimental Data. The experimental determination of the solubility of gas mixture components in a solid polymer is a very complicated task relative to the measurement of solubility of a pure gas or vapour. Indeed, very few mixed gas sorption data are available in the technical literature. To compare with predictions of the NELF model, only the experimental data for the solubility of GO2/C2H4 mixtures in PMMA at 35°C (4, 75, 76) will be considered in this article. Since the volume dilation produced by the gas mixture is not available, the analysis of the data is confined to the low pressure range, where the swelling of the glassy polymer may be neglected. Indeed, in the calculation which follows the non-equilibrium solubility is estimated from the NELF model assuming... 

In Figures 2 and 3, two exanqiles of the solubility isotherms predicted by the NELF model are presented and compared with the experimental data for CXh/CzlU mixtures in PMMA. Such exanqiles show the main features observed in the comparison of the model prediction with all of the experimental data reported in refs (4,15,16). 

In Figure 2 the solubility of in PMMA is considered as a function of CX>2 partial pressure, and the partial pressure of C2H4 is held constant at 2.0610.08 atm. The NELF model correctly predicts the solubility coefficient in the low pressure Kmit... 

It is interesting to note that a polymeric material may be prepared in the glassy state at the same temperature with different density values, as a consequence of different piehistories. According to the NELF model those states result in different pseudo-equilibrium solute contents. 

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