Dual sorption

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,... 

Although all of these predictions are qualitatively correct, the differences between the behavior of an ideal polymer and an actual polymer are important in selecting the optimum material for a particular separation. The usual starting point for this fine-tuning is the dual-sorption model originally proposed by Bar-rer et al. [44], This model has since been extended by Michaels et al. [45], Paul et al. [46], Koros et al. [47] and many others. 

According to the dual-sorption model, gas sorption in a polymer (cm) occurs in two types of sites. The first type is filled by gas molecules dissolved in the equilibrium free volume portion of material (concentration cH). In rubbery polymers this is the only population of dissolved gas molecules, but in glassy polymers a second type of site exists. This population of dissolved molecules (concentration cD) is dissolved in the excess free volume of the glassy polymer. The total sorption in a glassy polymer is then... 

The form of the sorption isotherm predicted from the dual sorption model is shown in Figure 2.28. Because the expressions for sorption contain three adjustable parameters, good agreement between theory and experiment is obtained. 

Permeation of gases in glassy polymers can also be described in terms of the dual sorption model. One diffusion coefficient (Do) is used for the portion of the gas dissolved in the polymer according to the Henry s law expression and a second, somewhat larger, diffusion coefficient (DH) for the portion of the gas contained in the excess free volume. The Fick s law expression for flux through the membrane has the form... 

The sorption of simple gases in glassy polymers follows a more complex relation and is well described by the so-called dual sorption model [11-17]... 

The permeation of simple gases in glassy polymers is more complex than in rubbery polymers. An extension of the dual sorption model of permeation leads to a relation, when the downstream pressure is small, of the following form... 

Figure 19.8 Route for the preparation of the dual sorption and sensing material Mil (first step) and analytical as well as sequestration reaction with Hg2+ (second step).

Because of the assumed dual sorption mechanism present in glassy polymers, the explicit form of the time dependent diffusion equation in these polymers is much more complex than that for rubbery polymers (82-86). As a result exact analytical solutions for this equation can be found only in limiting cases (84,85,87). In all other cases numerical methods must be used to correlate the experimental results with theoretical estimates. Often the numerical procedures require a set of starting values for the parameters of the model. Usually these values are shroud guessed in a range where they are expected to lie for the particular penetrant polymer system. Starting from this set of arbitrary parameters, the numerical procedure adjusts the values until the best fit with the experimental data is obtained. The problem which may arise in such a procedure (88), is that the numerical procedures may lead to excellent fits with the experimental data for quite different starting sets of parameters. Of course the physical interpretation of such a result is difficult. 

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 sorption isotherm for many substances in polymers in the glassy state, as well as water in cellulose, can be described by two processes that are independent of one another (dual sorption 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). 

Early investigations of the dual-sorption model started from the assumption that only Henry s part of the sorbed gas contributed to the gas transport, whereas the Langmuir part would not contribute to it, due to immobilisation. Then the transport flux would be... 

While the permeability of most rubbery polymers to most gases is pressure independent, the permeability of glassy polymers usually is observed to decline with increasing pressure. This is accounted for by the dual sorption theory if one assumes that both the Henry and Langmuir type sites contribute to per-... 

The fit of these expressions to experimental results is very good. At low pressure regimes, the fit was shown to be even better than that of dual sorption expressions. Except for these regimes, the two models seem to do equally well in describing sorption and permeability data. Concentration dependent diffusivity and permeability have been considered before mainly for vapors. The new aspect of the matrix model is that it broadens these effects to fixed gases. The important difference between the matrix and dual sorption models is in the physical picture they convey of gas transport and interaction with the polymer. Additional experimental evidence will be needed to determine the preference of these different physical representations. 

In glassy polymers the interactions of the penetrant molecules with the polymer matrix differ from one sorption site to another. A limiting description of the interaction distribution is known imder the name of the dual-sorption model [59, 60]. In this model, the concentration of the penetrant molecules consists of two parts. One obeys Henry s law and the other a Langmuir isotherm ... 

In the case of chitosan derivatives, noble metal ions are sorbed according to several kinetic models based on pure sorption, pure reduction and dual sorption-reduction mechanisms [149, 150]. Moreover, the optimum acid pH for noble metal ions sorption depends on the metal. For platinum and palladium, it was equal to 2. For CS cross-linked by glutaraldehyde (CS-GA) for... 

Fig.2 (c)(d) Applications of the 8-channel multifunctional valve, c, merging zones application in which injected sample and reagent zones merge at point M. d. application for dual sorption column preconcentration with which a second valve is required to elute the two columns sequentially. V, V/, multifunctional valves V//, 2-way valve ... 

WAN Wang, N.-H., Ishida, S., Takishima, S., and Masuoka, H., Sorption measmement of nitrogen in polystyrene at high pressure and its correlation by a modified dual-sorption model, Kagaku Kogaku Ronbunshu, 18, 226, 1992. 

The use of Equation (11.37) allows for an estimation of the b value for hexane, based on known dual sorption parameters [50]. The resulting value of 240 lOOatm is two orders of magnitude greater than that available for CO2 (0.51 attn [51]), and indicates that hexane has a much stronger affinity for the micro-voids than CO2. This is consistent... 

Both Equations [18.1] and [18.5] describe ideal cases. In many experimental situations, sorption isotherms are non-linear, showing a negative behavior compared to Henry s law. Some cases can be described using a dual sorption theory (Paul and Koros, 1976) in which the global sorption capacity is the result of two different contributions, one that follows Henry s law and a second that follows Langmuir behavior ... 

Toi et al. applied the dual sorption mechanism to analysis of the time-lag diffusion (permeation) under the constraint that the penetrant fraction attributed as the Langmuir component is completely immobilized, but in local equilibrium with the Henry s law dissolution component [1]. They yielded a mathematical description of transient permeation, consisting of a nonlinear partial differential equation. This equation was then solved by a finite-difference technique for the case of permeation... 

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