Langmuir dual-mode sorption model

The sorption and diffusion behaviour of gas mixtures is of particular interest from the point of view of membrane gas separation, which is steadily gaining in importance by virtue of its low energy requirements. On the basis of the dual mode sorption model, one may reasonably expect that sorption of a binary gas mixture A, B in the polymer matrix will exhibit little gas-gas interaction and hence will tend to occur essentially additively. In the Langmuir-like mode of sorption, on the other hand, there will be competition between A and B for the limited number of available sites. These considerations led 67) to the following reformulation of Eqs. (8) and (9)... 

It is particularly interesting and instructive to note that application of Henry + Langmuir dual-mode sorption and diffusion models is not confined to glassy polymer-gas systems. Sorption and transport of high affinity ionic species, exemplified by anionic dyes, in charged polymers, exemplified by polyamides at low pH, has been treated in the same way. These systems are of considerable importance both from the bio-mimetic and from the textile processing point of view, but have received limited atten-... 

NR and — CO groups, measured after prior washing of the resin with water 85), was well represented by a Langmuir + Nernst dual mode sorption model at salt concentrations not exceeding 0.2 mol dm 3. A detailed physical interpretation of the relevant parameters was not given, however, neither was the dual mode concept utilized in a corresponding diffusion study 86). 

Fredrickson and Helfand incorporated coupling terms in the transport equation and showed that "these terms provide mobility to molecules that are absorbed into microvoids, even if the molecules have an intrinsic diffusion coefficient in the hole phase that vanishes." These investigators proposed experiments that could provide information on the size and topological connectivity of microcavities (51). Related studies have also been made by Chern, Koros, et al. (52) and by Barrer (53). However, Petropoulos (54) has expressed the opinion that the treatment of Fredrickson and Helfand (51 ) as well as that of Barrer (53) "introduces more diffusion parameters than can reasonably be expected to be measurable on the basis of past experience". Petropoulos also showed that these treatments impose certain limitations on the physical meaning of the diffusion parameters. In the same study, Petropoulos has examined the modification of his dual-mode sorption model ( 5) which are necessary if the Langmuir domains are sufficiently extensive to constitute a macroscopically recognizable phase rather than scattered individual sites (or microcavities) (54). ... 

Free Volume. The Langmuir sorption capacity, C , of the dual-mode sorption model 12 J 3) characterizes gas sorption in the non-equilibrium excess fiee volume of a glassy polymer. The total sorbed gas concentration in a glassy polymer, which is a summation of Henry s law dissolution (C ) and Langmuir type hole filling (C, ), is given by ... 

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). 

The basic assumptions of the dual mode sorption theory as it applies to the transport model of Vieth and Sladek, have been stated by Vieth et al. in their excellent review of the subject The sorption isotherm was described by the combination of a Henry s law dissolved component, Cp, and a Langmuir hole filling term, Ch, i.e.. 

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 total solubility is the sum of the contributions of both the dissolution mode based on Henry s law and the binding mode using the Langmuir model the dual mode sorption [11], At equilibrium, the total solubility of olefin is shown by Eq. (9-4), where C is the total concentration of the olefin gas absorbed in the sample, p is the applied olefin pressure, ku is the solubility coefficient of the olefin gas for Henry s law mode, K is the olefin binding equilibrium constant of Eq. (9-2), as defined in Eq. (9-3), and Q is the saturated amount of the olefin gas bound to the silver complex. 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. Satisfactory agreement is obtained between integral diffusion coefficient from sorption and desorption rate curves and apparent diffusion coefficient from permeation rate curves (time-lag method). These rate curves were also compared to the curves predicted by Fickian-type diffusion equations. 

The extension of this model to mixed-gases is considerably more difficult than the extension of the simple dual-mode transport models. To include gas mixtures, the Barrer model must be modified to accurately describe the fraction of filled Langmuir sorption sites when more than one gas species is present in the pol5mier. To account for the presence of a second gas, the flux expressions can be rewritten using the following definition for the fraction of filled holes ... 

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). 

The solubility in glassy polymers is usually described by the so-called dual-mode model, which implies that there is a need for a more detailed definition of the sorption, c, in the flux Equation 7.1. The following Equations 7.20 and 7.21 illustrate this and can be referred to Figure 7.6. The term Cg accounts for Henry s law, while Cu is the Langmuir term with b being the hole affinity constant (bar ) and is the saturation constant (cm [STP]/cm ) ... 

The sorption isotherms for ethanol and methanol reported in Figure 4 and 5 cannot be interpreted on the basis of the well known dual mode model (/,2). This model assumes that the penetrant content in the glassy polymer matrix may be expressed as fimction of pressure through the sum of two contributions the first refers to the penetrant molecules which are considered to be adsorbed onto the surface of microvoids in the interior of the solid polymer, and the second represents the contribution due to penetrant molecules which are strictly dissolved into the solid phase. In the original formulation of the dual mode model the first contribution is expressed as fimction of pressure in terms of the Langmuir equation and the second through Henry s law. 

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)... 

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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