Diffusion transient permeation

A major breakthrough in the study of gas and v or transport in polymer membranes was achieved by Daynes in 1920 He pointed out that steady-state permeability measurements could only lead to the determination of the product EMcd and not their separate values. He showed that, under boundary conditions which were easy to achieve experimentally, D is related to the time retired to achieve steady state permeation throu an initially degassed membrane. The so-called diffusion time lag , 6, is obtained by back-extrapolation to the time axis of the pseudo-steady-state portion of the pressure buildup in a low pressure downstream receiving vdume for a transient permeation experiment. As shown in Eq. (6), the time lag is quantitatively related to the diffusion coefficient and the membrane thickness, , for the simple case where both ko and D are constants. 

Geus et al. [75] reports diffusion data at 21 and 145°C for Hj, N2, CH4, CO2 and CF2CI2 in silicalite membranes on a clay support which are obtained with the similar transient permeation technique as used above by Vroon. The diffusion coefficients for methane are about two orders of magnitude smaller than those obtained by PF-NMR methods. Usually this last technique gives relatively large diffusion coefficient values, which in the case of n-butane are of the same order of magnitude as reported for FR techniques and membrane techniques as reported by Kapteyn. 

The pure gas (99.99% or higher purity) transport properties were tested with an instrument (GKSS, Geesthacht, Germany) with eonstant permeate volume described elsewhere [14], The short response time of the instrument allows one to record transient permeation behaviours of less than 1 second. The pure gas permeability is the amount of gas permeating in the unit time, multiplied by the thickness of the membrane and normalized for the membrane surface and the pressure gradient. The recorded pressure vs. time plots were used to derive diffusion coefficients from the initial transient permeation, and permeability at the steady state. The time-lag 9 is the intercept of the linear part of the pressure vs. time curve with the axis of time. In a homogeneous membrane in which the solubility of a gas obeys Henry s law, its diffusion coefficient D can be calculated from the ratio ... 

It must be pointed out that the information on gas diffusivity and solubility reported in Figure 6.4 are qualitative, since they are derived from a simplified treattnent of the transient permeation behaviour equilibrium gas sorption experiments would be required to obtain detailed information on the different role of polymer and zeolites in these mixed matrix membranes. 

Toi et al. applied the dual sorption mechanism to analysis of the time-lag diffusion (permeation) under the constraint that the penetrant fraction attributed as the Langmuir component is completely immobilized, but in local equilibrium with the Henry s law dissolution component [1]. They yielded a mathematical description of transient permeation, consisting of a nonlinear partial differential equation. This equation was then solved by a finite-difference technique for the case of permeation... 

The diffusion coefficient may be determined from the transient permeation current density at various times tyfor Tyand selected foil thickness. 

Slow relaxation processes appear to be the dominant factors causing the long times required to reach steady state permeation rates. Transient permeation experiments would yield incorrect diffusion coefficients for membrane materials exhibiting this behavior. Relaxation processes, highly concentration dependent diffusion coefficients and solubility coefficients, therefore, require a more detailed approach to studying transport. This paper describes a preferred method of analyzing diffusion process. 

Marais, S., Metayer, M., and Labbe, M. (1999), Water diffusion and permeability in unsaturated polyester resin films characterized by measurements performed with a water-specific perme-ameter Analysis of the transient permeation. Journal of Applied Polymer Science 74(14), 3380-3395. 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods ate used however, the simplest method uses the Permatran-C instmment (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is appHed to the other side of the film. AH the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeabiUty. Figure 18 shows how the diffusion coefficient can be deterrnined in this type of experiment. The time lag is substituted into equation 21. The solubiUty coefficient can be calculated with equation 2. 

Fig. 20.21 J-l transients for the permeation of hydrogen through ferrous alloys. The normal transient enables the diffusion coefficient ) to be evaluated from the relationship /, = L /6D, where /, is the time at which J attains a value of 0-63 of the steady-state permeation J...

The squares and full lines of Fig. 11 summarize their results. The scatter of the experimental points seems mainly due to the analysis of the transient behavior the diffusion coefficient D and hence the solubility s = P/D fluctuate much more than the steady-state permeation coefficient P. Their Arrhenius lines are described by ... 

Indeed, it is worth noting that by itself, a permeation rate proportional to p°50 could be consistent with any value whatever for the ratio of monatomic to diatomic species in the solid, if the diatomic species is very immobile. For in such case, the permeation flux would be carried entirely by the monatomic species, whose concentration always goes as p0 50. However, a sizable diatomic fraction would significantly modify the transient behavior of the permeation after a change in gas pressure. Although neither Van Wieringen and Warmholtz nor Frank and Thomas published details of the fit of their observed transients to the predictions of diffusion theory, it is unlikely that any large discrepancies would have escaped their attention. 

Figure 26. Rising permeation transient based on a solution to Pick s second law of diffusion.

These parameters are in many ways analogous to permeation time lags but the relevant expressions for the case of S(X), DT(X) are considerably more complicated (even in the case of the First njoment which is the simplest)171> than the corresponding time lag formulae 4). Accordingly, moments represent a less efficient way of making use of the information contained in transient diffusion data than the methods discussed above. In spite of these limitations, further study of moments should prove worthwhile. 

Reverse osmosis, pervaporation and polymeric gas separation membranes have a dense polymer layer with no visible pores, in which the separation occurs. These membranes show different transport rates for molecules as small as 2-5 A in diameter. The fluxes of permeants through these membranes are also much lower than through the microporous membranes. Transport is best described by the solution-diffusion model. The spaces between the polymer chains in these membranes are less than 5 A in diameter and so are within the normal range of thermal motion of the polymer chains that make up the membrane matrix. Molecules permeate the membrane through free volume elements between the polymer chains that are transient on the timescale of the diffusion processes occurring. 

The permeation technique as discussed in the context of permeation membranes in Section II.2, can also be used to determine diffusion coefficients and minority conductivities (in fact steady-state oxygen flux that is given by (see Eq. (34))... 

Diffusion of allyl chloride in polyvinyl acetate. II. The transient state of permeation. J. Polymer Sci. 27, 405 (1958b). 

The passive permeability of lipid membranes is another fluidity related parameter. In general, two mechanisms of membrane permeability can operate in the membrane (8). For many nonpolar molecules, the predominant permeation pathway is solubility-diffusion, which is a combination of partitioning and diffusion across the bilayer, both of which depend on lipid fluidity. In a few cases, such as permeation of positively charged ions through thin bilayers, an alternative pathway prevails (9, 10). It is permeation through transient pores produced in the bilayer by thermal fluctuations. This mechanism, in general, correlates with membrane fluidity. However, for model membranes undergoing the main phase transition, permeation caused by this mechanism exhibits a clear maximum near the phase transition point (11). 

The rate of transmembrane diffusion of ions and molecules across a membrane is usually described in terms of a permeability constant (P), defined so that the unitary flux of molecules per unit time [J) across the membrane is 7 = P(co - f,), where co and Ci are the concentrations of the permeant species on opposite sides of membrane correspondingly, P has units of cm s. Two theoretical models have been proposed to account for solute permeation of bilayer membranes. The most generally accepted description for polar nonelectrolytes is the solubility-diffusion model [24]. This model treats the membrane as a thin slab of hydrophobic matter embedded in an aqueous environment. To cross the membrane, the permeating particle dissolves in the hydrophobic region of the membrane, diffuses to the opposite interface, and leaves the membrane by redissolving in the second aqueous phase. If the membrane thickness and the diffusion and partition coefficients of the permeating species are known, the permeability coefficient can be calculated. In some cases, the permeabilities of small molecules (water, urea) and ions (proton, potassium ion) calculated from the solubility-diffusion model are much smaller than experimentally observed values. This has led to an alternative model wherein permeation occurs through transient hydrophilic defects, or pores , formed by thermal fluctuations of surfactant monomers in the membrane [25]. 

It is convenient to distinguish between permeation measurements in which the flux is measured under a known (and constant) pressure gradient and those in which the flux of a component i is driven by a concentration difference between the membrane faces under a constant and equal total pressure at both sides (Wicke-Callenbach [3]). Either of these two main methods may be performed imder steady state or under transient conditions. Whether or not component fluxes cmd diffusivities measured with both methods give similar or different values depends on the conditions and on the type of the dominant diffusion mechanism. 

A schematic picture of different t5q)es of pores is given in Fig. 9.1 and of main types of pore shapes in Fig. 9.2. In single crystal zeolites the pore characteristics are an intrinsic property of the crystalline lattice [3] but in zeolite membranes other pore types also occur. As can be seen from Fig. 9.1, isolated pores and dead ends do not contribute to the permeation under steady conditions. With adsorbing gases, dead end pores can contribute however in transient measurements [1,2,3]. Dead ends do also contribute to the porosity as measured by adsorption techniques but do not contribute to the effective porosity in permeation. Pore shapes are channel-like or slit-shaped. Pore constrictions are important for flow resistance, especially when capillary condensation and surface diffusion phenomena occur in systems with a relatively large internal surface area. 

Here Cq = c(z = 0,t) and Dg is the effective diffusion coefficient (porosity and tortuosity effects are incorporated in Dg). If the upstream (high) pressure is constant and much larger than the downstream (low) pressure, the slope of the asymptote will correspond to the steady state and so it is possible to determine the diffusivity under both steady state and transient conditions from a single permeation experiment. With a narrow and unimodal pore size distribution both methods yield reasonable consistent values. Large discrepancies point to strong microstructural effects (bimodal broad distribution, many dead ends, many defects). 

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