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)... 

Application of dual mode sorption models to low affinity ionic species has been reported. Pertinent examples are mentioned here to illustrate the variety of models... 

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). 

To analyze sorption of penetrants, including hydrocarbons, in glassy polymers, the dual-mode sorption model is most frequently used. For a number of glassy polymers, correlations between the constants of the dual-mode sorption model and the condensabUity of hydrocarbons have been established (see, e.g.. Figure 9.4a through 9.4c and data presented in Refs. [18-20]). Temperature dependence of model constants is described by Vant-Hoff equation, where the exponent contains heat of penetrant sorption A//s. This quantity is essentially dependent on the heat of penetrant condensation, AHcond- A//s = AH ond + A//i, where AHi is partial molar enthalpy of penetrant dissolution in the polymer, AHi = [d( AGi/T)/d( l/T)]c, AGi is the partial molar free... 

The solubility of hydrocarbons in rubbery polymers can be described in more detail by several theories of solutions using various criteria of thermodynamic affinity [7,25-28], of which the Flory-Huggins theory is the most popular one. It takes into account the volume content of the penetrant dissolved in the polymer and the change in the length of the polymer s thermodynamic segment as a result of dissolution [7]. However, it should be pointed out that to describe dissolution, a rehned dual-mode sorption model can be used, e.g., the model by Pace and Datyner [7,29,30]. 

Permeability and diffusion coefficients of hydrocarbons in polyphenylene oxides are also essentially dependent on pressure (see Figure 9.23). It can be seen that in the case of ethylene, with the increase in pressure, the permeability coefficients first decrease, and then begin to rise. Ref. [18] quotes constants of the dual-mode sorption model for a number of hydrocarbons permeation through polyphenylene oxide. 

A phenomenological theory known as the "dual-mode sorption" model offers a satisfactory description of the dependence of diffusion coefficients, as well as of solubility and permeability coefficients, on penetrant concentration (or pressure) in glassy polymers (4-6,40-44). This model postulates that a gas dissolved in a glassy polymer consists of two distinct molecular populations ... 

The dual-mode sorption model does not provide a physical description of the polymer environments in which the two penetrant... 

An important modification of the dual-mode sorption model has been made by Koros and coworkers ( ), who have extended it to binary mixtures. These investigators have shown that the predictions of their extended model were in satisfactory agreement with sorption (solubility) and transport measurements. 

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). ... 

Pace and Datyner ( 5, 5) have also proposed a model for the absorption (solution) of small molecules in polymers applicable at temperatures above and below Tg, which incorporates the dual-mode sorption model for the glassy region. The presence of microvoids is assumed for rubbery polymers as well as for polymer glasses. "Hole filling" is suggested as an important sorption mode above as well as below Tg, with one crucial difference between the sorption mechanism in the rubbery and glassy regions hole saturation does not occur in the rubbery state because new microvoids are formed to replace those filled with penetrant molecules. 

A substantial number and variety of models of gas transport in polymers have been proposed during the last 20-30 years, in view of the great practical and scientific importance of this process. Molecular-type models are potentially most useful, since they relate diffusion coefficients to fundamental physicochemical properties of the polymers and penetrant molecules, in conjunction with the pertinent molecular interactions. However, the molecular models proposed up to now are overly simplified and contain one or more adjustable parameters. Phenomenological models, such as the dual-mode sorption model and some free-volume models, are very useful for the correlation and comparison of experimental data. 

Sorption isotherm for the aged PMSP Sorption isotherms for N, O2, CH4, and CO2 in PMSP membrane at 35 C have been reported by one of the authors (9). Sorption isotherms for CjHg in the initial and aged PMSP at 35 C are shown graphically in Figure 2 in the form of plots of the concentration of c of the penetrant ssolved in the polymer versus the penetrant pressure p. Such isotherms are accurately described by the dual-mode sorption model (10) ... 

Care should always be exercised when using solubility data for glassy or crystalline polymers (not included here), because SCF sorptiOTi occurs preferentially in the amorphous phase, which may additirmally experience swelling-related stress. Solubihty data for CO2 in solid polymers is compiled in [5]. Often, the pressure dependence of SCF sorption in glassy polymers follows a dual-mode sorption model, with substantial deviations from Henry s law. 

The permeability coefficients of gases in rubbery polymers can also be used in calculations involving gas mixtures if the gas solubility in the polymers is sufficiently low, e.g., in the Hemy s law fimit. This is due to the fact that imder such conditions the components of gas mixtures commOTily permeate through a polymer membrane independently of each other. By COTitrast, the permeation of the components of gas mixtures in and through glassy polymers is coupled, i.e., each component affects the permeation behavior of the other com-ponent(s). Commonly, the permeation rate of the faster component(s) of a gas mixture is decreased while that of the slower component(s) is increased, thus decreasing the polymer selectivity. This behavior is quantitatively described by the dual-mode sorption model and its extensimis [5— 9,12,17]. 

Solubility The Spatial Fermi Gas and the Dual-Mode Sorption Model)... 

It is often implied that the dual-mode-sorption model has a [ ysical basis in two distinct mechanisms of gas solubility in dense polymers. The first one is assumed to be associated with a liquid-like solubility, while the second one is due to gas solubility in some preexisting holes in a polymer structiue. To check the microscopic basis of this aiqu-oach, one can analyze the distribution d bj values, obtained through numerical evaluation of the solute s distribution function in atomistic micro-structures dense polymers if the dual-mode-sorption model is meaningful for this case, tlten tie value bn should stand out among all others in the spectrum of bj valtes. 

The dual-mode sorption model considers the glassy solid to consist primarily of an equilibrium-densified matrix with a small volume fraction of uniformly... 

For a penetrant/polymer system that follows the dual-mode sorption model of Eqs. 11.5 and 11.13, the lag time is given by the following expression ... 

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 ... 

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