Sorption dual model

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... 

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... 

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 ... 

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 dual sorption model invokes the existence of two thermodynamically distinct populations of the penetrant gas, namely, molecules dissolved in the polymer by an ordinary dissolution mechanism (obeying Henry s law), and... 

Thus, the dual sorption model predicts that the isothermal plot of vs. p will consist of a low-pressure linear region and a high-pressure linear region that are connected by a nonlinear region. Some experimental results arc illustrated in Figure 5.3, for typical examples of dual sorption. 

The membrane transport of the gas molecule was formulated by the dual sorption model with the following basic assumptions ... 

In the analysis of the sorption isotherms. Equation 1, derived by a dual sorption model, was used. The model consists of two modes of the sorption, i.e., electrostatic (Langmuir) and nonelectrostatic (partition)... 

Dual-sorption models are based on the premise that whilst some penetrant molecules diffuse normally in the polymer matrix, others will be affected by polymer-penetrant interactions or micro-void filling. Carter and Kibler (1978) addressed this problem in terms of the probability that a water molecule may react with a polymer molecule. Their model is based on the theory that moisture in a polymer network can be either bound or free. The probability that a free water molecule becomes bound is y and the probability that a bound water molecule is... 

The sorption isotherms of C2-C3 hydrocarbons in polyphenylene oxides were analyzed in terms of the dual sorption model [44] using the least squares best fit approach. The calculated parameters of the model equation... 

The above case of dual-sorption model can be considered a special case of what has been termed anomalous diffusion in polymers. Because organic vapors or liquids can interact strongly with a polymer and cause it to swell, an extreme case of anomalous diffusion occurs when the mass uptake (time-integrated flux) into the polymer is totally controlled by the stress gradient between the swollen and unswollen regions rather than by the concentration gradient. This was first characterized by Alfrey and coworkers [31] and referred to as case II diffusion. Fickian diffusion leads to an initial mass uptake of a polymer film or sheet exposed to a swelling solvent that follows the expression [32] ... 

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