Pressure dependence, permeability polymers

The steady-state equation for gas transmission only applies when the gases are sparingly soluble in the polymer (< 0.2%), and when there is no chemical association. This is true for the air gases at pressures below lO Pa, but often not true for water and organic vapors, which can have pressure-dependent permeabilities. Vapors may also act as plasticisers and so increase diffusion. 

To effectively use ionomer membranes for dehydration applications it is necessary to understand water transport in these polymers. Molecular diffusion in swollen polymers does not follow the classical Fickian behavior. Fickian behavior is observed for diffusion of gases at low pressure through rubbery polymers at temperatures well above Tg. Under these conditions permeability is independent of gas pressure. Glassy polymers show pressure dependent permeabilities. These effects disappear at higher pressures and can be explained by dual mode theory. Similarly, permeabilities of vapors such as water in hydrophobic or mildly hydrophilic membranes are independent of water vapor pressure. 

Nonlinear, pressure-dependent solubility and permeability in polymers have been observed for over 40 years. Meyer, Gee and their co-workers (5) reported pressure-dependent solubility and diffusion coefficients in rubber-vapor systems. Crank, Park, Long, Barrer, and their co-workers (5) observed pressure-dependent sorption and transport in glassy polymer-vapor systems. Sorption and transport measurements of gases in glassy polymers show that these penetrant-polymer systems do not obey the "ideal sorption and transport eqs. (l)-(5). The observable variables,... 

Nonlinear, pressure-dependent sorption and transport of gases and vapors in glassy polymers have been observed frequently. The effect of pressure on the observable variables, solubility coefficient, permeability coefficient and diffusion timelag, is well documented (1, 2). Previous attempts to explain the pressure-dependent sorption and transport properties in glassy polymers can be classified as concentration-dependent and "dual-mode models. While the former deal mainly with vapor-polymer systems (1) the latter are unique for gas-glassy polymer systems (2). 

Pressure Dependence of Hydrocarbon Permeability in Rubbery Polymers Effect... 

Accurate description of barrier films and complex barrier structures, of course, requires information about the composition and partial pressure dependence of penetrant permeabilities in each of the constituent materials in the barrier structure. As illustrated in Fig. 2 (a-d), depending upon the penetrant and polymer considered, the permeability may be a function of the partial pressure of the penetrant in contact with the barrier layer (15). For gases at low and intermediate pressures, behaviors shown in Fig. 2a-c are most common. The constant permeability in Fig.2a is seen for many fixed gases in rubbery polymers, while the response in Fig. 2b is typical of a simple plasticizing response for a more soluble penetrant in a rubbery polymer. Polyethylene and polypropylene containers are expected to show upwardly inflecting permeability responses like that in Fig. 2b as the penetrant activity in a vapor or liquid phase increases for strongly interacting flavor or aroma components such as d-limonene which are present in fruit juices. 

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 ... 

FIGURE 20.3-9 Pressure dependence of the permeability of various polymers to COj. The data wises measured with a vacuum downstream except for the cellulose acetate (CA). which was estimated from a variety of sources.21... 

The ideal separation foctor, equal to the ratio of the permeabilities of the two components, is also inteipretable as a product of two factors a "solubUiQr selectivity and a "mobiUty selectivity. These two selectivity contributions, consisting of the ratios of the respective conqionent solubilities and diffiisivities, indicate the relative importance of thermodynamic and kinetic fecKirs in the permselection process. Unfortunately, optimization of product permeability and membrane selectivity is oftmi difficult, and trade-offs in the two parameters may be necessary on economic grounds. A brief discussion of characterization methods and typical forms of sorption isotherms and local difiiiaon coeffidents for gases and vapors in polymers is presented below. This discussion serves as a background for rationalizing pressure dependencies of permeabilities and selectivities. 

Determination of concentration-dependent diflusion coefficient data usually is done indirectly. Typically, one determines D(.C) from the observed pressure dependence of die permeability in conjunction with the independently (tetermined dC/dp data discussed in die ptecedii section. When significant concentration dependence is observed for the local difliision coefficients, the fi m generally resembles one of die corves shown in Fig. 20.3-fi. Of course, for low-sorbing gases such as Hj and N2 in rubbery polymer, both D and dC/dp are essentially constant resulting in adherence to die simple situation indicated Iqr Eq. (20.1-5). 

The dependence of permeability, diffusion, and solubility coefficients on penetrant gas pressure (or concentration in polymers) is very different at temperatures above and below the glass transition temperature, Tg, of the polymers, i.e., for mbbery and glassy polymers, respectively. Thus, when the polymers are in the rubbery state the pressure dependence of these coefficients depends, in turn, on the gas solubility in polymers. For example, as mentioned in Section 61.2.4, if the penetrant gases are very sparsely soluble and do not significantly plasticize the polymers, the permeability coefficients as well as the diffusion and solubility coefficients are independent of penetrant pressure. This is the case for supercritical gases with very low critical temperatures (compared to ambient temperature), such as the helium-group gases, Ha, Oa, Na, CH4, etc., whose concentration in rubbery polymers is within the Heruy s law limit even at elevated pressures. 

All symbols have their usual meaning and only more important ones are defined here. Cj is the concentration of component j in the aqueous phase (e.g. polymer, tracer, etc.). The viscosity of the aqueous phase, rj, may depend on polymer or ionic concentrations, temperature, etc. Dj is the dispersion of component j in the aqueous phase Rj and qj are the source/sink terms for component j through chemical reaction and injection/production respectively. Polymer adsorption, as described by the Vj term in Equation 8.34, may feed back onto the mobility term in Equation 8.37 through permeability reduction as discussed above. In addition to the polymer/tracer transport equation above, a pressure equation must be solved (Bondor etai, 1972 Vela etai, 1974 Naiki, 1979 Scott etal, 1987), in order to find the velocity fields for each of the phases present, i.e. aqueous, oleic and micellar (if there is a surfactant present). This pressure equation will be rather more complex than that described earlier in this chapter (Equation 8.12). However, the overall idea is very similar except that when compressibility is included the pressure equation becomes parabolic rather than elliptic (as it is in Equation 8.12). This is discussed in detail elsewhere (Aziz and Settari, 1979 Peaceman, 1977). Various forms of the pressure equation for polymer and more general chemical flood simulators are presented in a number of references (Zeito, 1968 Bondor etal, 1972 Vela etal, 1974 Todd and Chase, 1979 Scott etal, 1987). 

This plasticization phenomena can be observed in some glassy polymers exposed to high pressures of condensable gases, such as CO2. Moreover, a pressure dependence of permeability occurs for glassy polymers prior to plasticization, whereas rubbery materials tend to have essentially pressure-independent permeabilities. The decrease in permeability of component A with increasing partial pressure of either component A or B prior to the onset of plasticization (70) shown in Figure 21 (71) is characteristic of glassy polymers. 

Permeability (P) is defined as the product of solubility (S) and diffusivity (D) of the gas into the polymer membrane and is temperature and pressure dependent. The permeability, diffusivity and solubility values of different gases in polydimethylsiloxane (PDMS) are collected in Table 22.1. 

In many cases, Q is a constant for a given gas and polymer combination but for other combinations, particularly with vapours, Q varies with, for example, test piece thickness or pressure difference. Hence, it is necessary to know the dependence of Q on all possible variables in order to characterise the permeability of the material completely. 

In addition to its major use in determining the number-average molecular weight (Ma) of polymers, osmometry has also been used to determine M of block copolymer micelles. The method involves determining the osmotic pressure (77) across a membrane that is permeable to solvent only. Because osmotic pressure is a colligative property, it depends on the number of particles, and hence yields Ma. It also depends on the interactions between particles, and thus... 

Equation (2.79) expresses the driving force in pervaporation in terms of the vapor pressure. The driving force could equally well have been expressed in terms of concentration differences, as in Equation (2.83). However, in practice, the vapor pressure expression provides much more useful results and clearly shows the connection between pervaporation and gas separation, Equation (2.60). Also, the gas phase coefficient, is much less dependent on temperature than P L. The reliability of Equation (2.79) has been amply demonstrated experimentally [17,18], Figure 2.13, for example, shows data for the pervaporation of water as a function of permeate pressure. As the permeate pressure (p,e) increases, the water flux falls, reaching zero flux when the permeate pressure is equal to the feed-liquid vapor pressure (pIsal) at the temperature of the experiment. The straight lines in Figure 2.13 indicate that the permeability coefficient d f ) of water in silicone rubber is constant, as expected in this and similar systems in which the membrane material is a rubbery polymer and the permeant swells the polymer only moderately. 

Pressure applied to the external solution would also increase the pressure inside the capsule, and in the absence of fluid compressibility there would be no change in the capsule volume. Without access to the inside of the capsule we cannot apply a pressure difference to investigate Darcy flow through the membrane. One possiblity, yet to be tested experimentally, is to add to the external solution an uncharged polymer which cannot pass through the membrane. The external chemical potential of water is thereby reduced [9], and the resulting flow out of the capsule can be shown to depend upon the permeability k of the membrane. 

When the gas or vapor feed stream contains a component that is highly soluble in the polymer membrane and causes plasticization, then the selectivity as defined by Equation 4.6 will depend on the partial pressure or the amount of the plasticizing component sorbed into the membrane. Furthermore, pure-gas permeation measurements are generally not a good indicator of the separation performance, and mixed-gas permeation measurements will be needed [21-23]. Often, the mixed-gas selectivity is less than predicted from pure-gas measurements [8] however, the opposite has been observed [24], Competitive sorption effects can also compromise the prediction of mixed-gas behavior from pure-gas measurements [25], For gas pairs where each component is less condensable than C02, like 02/N2, it is generally safe to conclude that the selectivity characteristics can be accurately judged from pure-gas permeabilities at all reasonable pressures. When the gas pair involves a component more condensable than C02, plasticization is likely to be a factor and pure-gas data may not adequately reflect mixed-gas selectivity. When C02 is a component, the situation depends on the partial pressures and the nature of the polymer. 

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