Permeability, diffusion and solubility coefficients

Where the flux J is the flow of gas. The flux is the amount of material transported per unit time through a unit surface area and according to Eq. (9-1) is proportional to the constant partial pressure difference Ap between both sides of the membrane. The proportionality constant P is called the permeability coefficient and is the product of the diffusion coefficient D and the solubility coefficient S (or sorption constant) of the gas in the membrane  

S is the ratio of the concentration c of the gas in the polymer and the partial pressure p of the gas in the gas phase in equilibrium with the membrane  

The differing permeability of a given gas through various materials is expressed in terms of the gas and material specific parameter P. Its dimension results from Eq. (9-1)  

The amount of permeated substance can be expressed in mass, mole or volume units. For gases, volume is preferred, expressed as the amount permeating under conditions of standard temperature and pressure (STP), which corresponds to the standard temperature of 273.15 K and standard pressure of 1.01325 105 Pa. The corresponding dimensions for D and S are obtained at the same time from Eq. (9-4). The standard ambient temperature and pressure (SATP) are set at p°= 1 bar= 105 Pa = 0.9678 atm and T = 298.15 K. The data in handbooks however is still mostly expressed with p = 1 atm as standard pressure. For practical purposes the difference between these two conventions is insignificant compared to the variability of the materials themselves. 

The value for P is presented in various dimensions in the literature. One can convert from various dimensions to the desired ones with the help of the conversion factors listed in Table 9-1. 

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Comparative Permeability of C2, C3, and C4 olefins. It is interesting to compare permeability, diffusion, and solubility coefficients of various olefins, as well as ideal olefin/paraffin selectivity in a glassy polymer. Such comparison can be made,... 

The permeability, diffusion, and solubility coefficients for the gases as well as the ideal separation factors for gas pairs have been determined. A poly(imide-siloxane) with 20% siloxane content shows the best compromise with very high permeability and a still high permselectivity for the CO2/CH4 gas pair [12]. 

Table 3.7 Permeability, diffusion and solubility coefficients of gases in PTMSN and PTMSP127P...

Relations between Permeability, Diffusion, and Solubility Coefficients... 

When the permeability, diffusion, and solubility coefficients are functions of pressure their experimental values are mean values (P,D, and S) for the pressures applied at the membrane interfaces, cf., eqs. (61.6-61.8). Equations (61.12-61.14) are applic able als o to P, D, and 5 over a limited range of temperatures. The activation energies Ep and Ed commonly decrease with increasing pressure. 

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. 

When the concentration of penetrant gases in glassy polymers becomes sufficiently high to plasticize the polymers, the permeability, diffusion, and solubility coefficients will deviate from dual-mode sorption behavior and increase as the pressure is raised. 

Andrio et al. reported the effect of temperature and pressure on the permeability, diffusion and solubility coefficients of N2, O2 and CO2 in NR reinforced with different amounts of cellulose. Rubber compounds were prepared by adding cellulose II, as 10% cellulose xanthate aqueous solution, to NR latex under stirring. Analysis of the results obtained by these authors... 

Permeability, Diffusion and Solubility Coefficients of Alkanes Through Santoprene (Blend of Ethylene-Propylene Copolymer and Isotactic Poly(propylene)) Permeability, Diffusion and Solubility Coefficients of Esters Through Poly(epichlorohydrin)... 

TABLE 11. PERMEABILITY, DIFFUSION AND SOLUBILITY COEFFICIENTS OF ALKANES THROUGH SANTOPRENE ... 

Permeability, Diffusion and Solubility Coefficients of Esters through Poly(epichlorohydrin) VI/567... 

Table 4 contains some selected permeability data, including diffusion and solubility coefficients foT flavors in polymers used in food packaging. Generally, vtuylidene chloride copolymers and glassy polymers such as polyamides and EVOH are good barriers to flavor and aroma permeation, whereas the polyolefins are poor barriers. Comparison to Table 2 shows lhat the large-molecule diffusion coefficients are 1000 or more times lower tli an the small-molecule coefficients. 

The diffusion and solubility coefficients for oxygen and carbon dioxide in selected polymers have been collected in Table 5. Determination of these coefficients is neither common, nor difficult. Methods are discussed later. The values of S for a permeant gas do not vary much from polymer to polymer. The large differences that are found for permeability are due almost entirely to differences in D. 

Pj= Dj Sj Permeability coefficient in terms of diffusion and solubility coefficients... 

The crystallinity degree does not reveal any change in the solubility coefficient whatever the temperature and the relative humidity are [120]. Consequently the permeability coefficient seems to be controlled by the diffusion coefficients which increase with the crystallinity degree at certain temperature and relative humidity [121]. This is in contradiction with the decrease in permeability coefficients, reported by Tsuji et al. and Shogren [120, 136]. Indeed, in their study the diffusion and solubility coefficient decrease slightly with the crystallinity degree of PLLA [138]. 

The permeability coefficient, P, combines the effects of the diffusion and solubility coefficients. The barrier characteristics of a polymer are commonly associated with its permeability coefficient values. The well-known relationship P = DS holds when D is concentration independent and S follows Henry s law. Standard methods for measuring the permeability of organic compounds are not yet available. ASTM E96 describes a method for measuring the water vapor transmission rate. ASTM D1434 describes a method for the determination of oxygen permeability. 

Membrane separation of gaseous small molecules through dense (non-porous) polymeric membranes occurs because of differences in solubility and diffusivity, while membrane performance is characterized by permeability and selectivity. The permeability of component i. Pi, is defined as the product of the diffusion and solubility coefficients (A and Si, respectively) (Eq. (9-13)). 

From the solubility and diffusivity data collected in the sorption experiments the value for the permeabihty of the two penetrants in the different mixed matrices was calculated. If the solution-diffusion model is considered to hold true, and Fick s law is suitable to represent the diffusive flux, the permeability can be calculated as the product of diffusivity and solubility coefficient. The calculated permeability is shown in Figures 7.8a and b, where it is clear that the addition of filler increases the permeability of the membrane for both penetrants. 

As noted earlier, when both the diffusion and solubility coefficients are constants, the simple form shown in Fig. 20.3-2u results. In all other cases, nonconstamty of either the sorption or diffusion coefficients can cause nonconstancy of the observed permeability and selectivity. As illustrated, separation of the two component contributions is straightforward if both equilibrium solubility and steady-state permeation data are availabie. 

They also carefully examined the contribution of diffusivity and solubility coefficients to permeability. They found that the diffusion coefficient followed a different trend from permeability, as D(02)>D(C02)>D(N2). However, solubility followed the order according to critical temperature S(C02)>S(02)>S(N2>. The solubility value was... 

The positive diffusional activation energy is larger in absolute value than the negative AH, and so the overall permeability increases as temperature increases, but to a lower degree than the diffusion coefficient itself It should be noted that the use of equation 34 is only strictly valid when the diffusion and solubility coefficients are independent of concentration. 

The effect of copolymer composition on free volume and gas permeability of PECT copolymers as well as PET and PCT homopolymers was studied by Hill et al. (97). The free volume was studied by positron annihilation lifetime spectroscopy (PALS) in order to determine the relative size and concentration of free volume cavities in the copolymers. The logarithm of the permeability to oxygen and carbon dioxide increased linearly with the %mol content of 1,4-CHDM units in the copolymer, which was in agreement with the free volume cavity size and relative concentration observed by PALS measurements. Light et al. (98) studied the effect of sub-T relaxations on the gas transport properties of PET, PCT and PECT polyesters. They observed that modification of PET with 1,4-CHDM increased the magnitude of the p-relaxation, as well as the diffusion and solubility coefficients for oxygen and CO. ... 

Slavutsky et al. (2014) prepared starch/cellulose nanocrystals (CNCs) films and their water barrier properties were studied. The measured film solubility, contact angle, and water sorption isotherm indicated that reinforced starch/CNC films have a lower affinity to water molecules than starch films. Permeability, dififusivity, and solubility coefficients indicated that the permeation process was controlled by the water diffusion and was dependent on the tortuous pathway formed by CNC incorporation. The decrease in surface hydrophilicity and the improvement in water vapor barrier properties with the addition of CNC showed that these nanocomposites present excellent potential as a new biomaterial for application in food packaging and conservation. 

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