Reduced sorption curves

For a classical diffusion process, Fickian is often the term used to describe the kinetics of transport. In polymer-penetrant systems where the diffusion is concentration-dependent, the term Fickian warrants clarification. The result of a sorption experiment is usually presented on a normalized time scale, i.e., by plotting M,/M versus tll2/L. This is called the reduced sorption curve. The features of the Fickian sorption process, based on Crank s extensive mathematical analysis of Eq. (3) with various functional dependencies of D(c0, are discussed in detail by Crank [5], The major characteristics are... 

Various anomalous diffusional behaviors have been observed and documented for both sorption and permeation experiments. Detailed discussions of these anomalies can be found elsewhere [12,35,36], Here only a brief summary of major findings is given. First, for the sorption anomaly, it has been observed that the reduced sorption curve has a distinctive thickness dependence. In this case, the reduced absorption and desorption curves obtained at various thick-... 

Various methods have been proposed for the evaluation of D as a function of penetrant concentration from sorption measurements. They all are applicable only for sorption data of the Fickian type, and may be classified into two groups. Methods belonging to one group utilize data for the initial slope of the reduced sorption curve, while the ones belonging to the other group resort to rates at which M (<) approaches the equilibrium value Since little work has yet been done to adapt the approach-to-equihbrium data for concentration-dependent D, the subsequent discussion will be confined to methods of the former group only. 

Time Dependence. Time dependence may be observed in the diffusion of organic vapors in polymers below their glass transition temperature (Tg) (5). At these temperatures, the rate of diffusion is comparable with the rate of motion of the polymer segments. As a result, the value of the diffusion coefficient attained at a given concentration in an element of the polymer will depend on the time for which this concentration has existed at the element. D has more time in which to approach its equilibrium value in thicker films. Therefore, sorption proceeds more rapidly the thicker the film, and the reduced sorption curves do not coincide as required by Equation 1 describing Fickian diffusion. 

Sorption Data. The results were plotted as reduced sorption curves, Mt/Moo vs. The apparent diffusion coefficient was calculated to be... 

Unlike conventional rubbers, however, the results are time dependent since reduced sorption curves for films of varying thicknesses do not coincide with each other (Figure 2). This anomaly, which is characteristic of glassy materials, is a sign of the inability of the polymer molecules to respond quickly to the changing concentration. The diffusion coefficient (D) depends on the time for which the polymer and penetrant have been in contact. D has more time in which to approach its equilibrium value in thicker films, and therefore sorption proceeds relatively more quickly the thicker the film. 

Figure 6. Reduced sorption curve of oxygen in polyaniline doped with 1 M HCl at 137.9 kPa oxygen. The solid line is from theoretical predictions by Pick s law with D=1.88 X 10 cmVsec.

Figure 17. A typical reduced sorption curve for the uptake of water by an epoxy resin.

A procedure for characterizing the rates of the volume change of gels has not been uniformly adopted. Often, the kinetics are simply presented as empirical sorption/desorption curves without quantitative analysis. In other cases, only the time required for a sample of given dimensions to reach a certain percentage of equilibrium is cited. One means of reducing sorption/desorption curves to empirical parameters is to fit the first 60% of the sorption curve to the empirical expression [119,141]... 

The behavior predicted by this equation is illustrated in Fig. 16-33 with N = 80. xF = (evtp/L)/[( 1 - e)(p K, + ,)] is the dimensionless duration of the feed step and is equal to the amount of solute fed to the column divided by tne sorption capacity. Thus, at xF = 1, the column has been supplied with an amount of solute equal to the stationary phase capacity. The graph shows the transition from a case where complete saturation of the bed occurs before elution (xF= 1) to incomplete saturation as xF is progressively reduced. The lower curves with xF < 0.4 are seen to be nearly Gaussian and centered at a dimensionless time xm — (1 — tf/2). Thus, as xF -> 0, the response curve approaches a Gaussian centered at t = 1. 

Figure 2.40 Blocking of hydrogen in hydrogen/sulfur dioxide gas mixture permeation experiments with finely microporous membranes [63] as a function of the amount of sulfur dioxide adsorbed by the membrane. As sulfur dioxide sorption increases the hydrogen permeability is reduced until at about 140 cm3 (SO2) (STP) /g, the membrane is completely blocked and only sulfur dioxide permeates. Data obtained at several temperatures fall on the same master curve ( , 0°C A. —10 °C , — 20.7 °C A, —33.6°C). Reprinted from R. Ash, R.M. Barrer and C.G. Pope, Flow of Adsorbable Gases and Vapours in Microporous Medium, Proc. R. Soc. London, Ser. A, 271, 19 (1963) with permission from The Royal Society...

Normally, the moisture sorption-desorption profile of the compound is investigated. This can reveal a range of phenomena associated with the solid. For example, on reducing the RH from a high level, hysteresis (separation of the sorption-desorption curves) may be observed. There are two types of hysteresis loops an open hyteresis loop, where the final moisture content is higher than the starting moisture content due to so-called ink-bottle pores, where condensed moisture is trapped in pores with a narrow neck, and the closed hysteresis loop may be closed due to compounds having capillary pore sizes. 

This result indicates that the velocity of the solute front is inversely proportional to the slope of the isotherm. We can illustrate this result using a Type I isotherm (Figure 7.6). During the adsorption step, the direction is from the lower left to upper right portion of the curve. So, dg/dT is largest V is slowest) during the initial portion of sorption. This is the rate-limiting step so the entire front moves as a discontinuous wave (stoichiometric front). A balance across this wave shows that Aq/AY reduces to Aq/AY, the chord from the initial state to the saturated state in the column. 

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