Sorption curve

The anomalous increase of the water uptake observed in Fig. 10 when approaching equilibrium at 60 °C has been associated to the damage. The abrupt upturn of the sorption curve may be explained considering a possible crazing of the low crosslinked internodular matrix induced by the differential swelling stresses that can arise, at high water contents, between areas of different crosslinking density. 

Sorption curves obtained at activity and temperature conditions which have been experienced to be not able to alter the polymer morphology during the test, i.e. a = 0.60 and T = 75 °C, for as cast (A) and for samples previously equilibrated in more severe conditions, a = 0.99 and T = 75 °C (B), are shown in Fig. 13. According to the previous discussion, the diffusion coefficient, calculated by using the time at the intersection points between the initial linear behaviour and the equilibrium asymptote (a and b), for the damaged sample is lower than that of the undamaged one, since b > a. The morphological modification which increases the apparent solubility lowers, in fact, the effective diffusion coefficient. 

FIGURE 2.12 Sorption curves for different nanocomposites at 45°C (solvent methyl ethyl ketone [MEK]). (From Maiti, M. and Bhowmick, A.K., J. Appl. Polym. Sci., 105, 435, 2007. Courtesy of Wiley InterScience.)... 

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... 

The sorption curves eventually become concave against the time axis. 

Criteria 1-3 are the cardinal characteristics of Fickian diffusion and disregard the functional form of D(ci). Violation of any of these is indicative of non-Fickian mechanisms. Criterion 4 can serve as a check if the D(ci) dependence is known. As mentioned, it is crucial that the sorption curve fully adhere to Fickian characteristics for a valid determination of D from the experimental data. At temperatures well above the glass transition temperature, 7 , Fickian behavior is normally observed. However, caution should be exercised when the experimental temperature is either below or slightly above 7 , where anomalous diffusion behavior often occurs. 

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-... 

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]... 

Fig. 5.4.3. Sorption curve of Cn-LAS and Cn-SPC in sediment, and Cn-LAS in microalgae. In the figure the actual partition coefficient of microalgae was multiplied by...

The data are "normalized" with regard to the ion exchange capacity C of the sorbents. The sorption curves of the illite and of the < 40-pm chlorite are strongly non-linear, whereas that of the montmorillonite approaches linearity. 

Figures 2-5 show the sorption curves of mass uptake versus time for the extract, O-methylated, O-butylated, and O-octylated extract. Three different experiments are shown in each figure a single integral sorption and two subsequent incremental sorptions. Both the extract and the O-methylated extract required 30 to 150 hours to reach equilibrium, depending on the particular experiment. The O-butylated extract sorbs benzene considerably faster, requiring less than 20 hours to reach equilibrium at all pressures. Finally, the O-octylated extract required less than one hour to reach equilibrium at each pressure.

The sorption-desorption isotherm for the O-octylated extract is noticably different. The shape of the sorption curve is concave upward and the hysteresis effect is much less pronounced. Nearly all of the benzene (97% wt) could be desorbed under vacuum at 30" C. Clearly, this material is responding much differently than the untreated and O-methylated extracts. 

EXTRACT and O-METHYLATED EXTRACT. The sorption of benzene by the extract and the O-methylated extract is characterized by a rapid, initial uptake followed by a very slow approach to equilibrium. Such sorption behavior is very similar to that of glassy polymers. Thus we have chosen to interpret the sorption curves shown in Figures 2 and 3 in terms of the Berens-Hopfenberg model developed for the sorption of organic vapors into glassy polymers.(lS) By doing so, we attempt to correct the total sorption values for surface adsorption in order to calculate x parameters. 

Scheinost, A. Schwertmann, U. (1995) Prediction of phosphate sorption curves in a soils-cape on molasse sediments. Soil Sci. Soc. 

Figure 3. Surfactant sorption on kaolinite for 0.1 M NaCl. Error bars for some data points are smaller than the symbols. Kaolinite concentration was 100 g/L. (a) SDS sorption curves for three pH values. The solid lines identify pH-dependent doses necessary to obtain equivalent SDS loadings (see text), (b) Sorption envelopes. Surfactant doses were 2 and 1.67 mM for SDS and Tween 80, respectively. Adapted from Ko et al. (1998a).

The diffusion coefficient D is determined using sorption curves as the one shown in Fig. 2.63. Using the slope of the curve, a, we can compute the diffusion coefficient as... 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

The resulting curve is termed the absorption (or desorption) curve, or more generally the sorption curve. However, in order to interpret correctly experimental results in terms of Eq. (1) it is necessary to plot M lf), the weight of sorbed penetrant per unit volume of the swelling or deswelling film, against This requirement arises from the fact that... 

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. 

Sorption curves which exhibit any of these and other types of deviations from Fickian features are usually called "non-Fickian processes or sometimes "anomalous processes. In recent years, a number of interpretations have been presented for the origins of non-Fickian features, but none of them are yet wholly satisfactory. In what follows, we describe basic ideas of some representative theories and typical sorption features predicted from them. In so doing, we shall not necessarily follow the historical sequence in which they appeared but rather attempt to group similar ideas in one scheme. 

Recently, Long and Richman (1960) have shown that both sigmoid and two-stage sorption curves can be derived from Eq. (1) (with D dependent on Cj only) if, in place of Eq. (15), one assumes for Cj a somewhat more general equation of the form ... 

From what we have discussed in (b"), it follows that when the two sets of D(Cj) data derived from sorption experiments and permeation experiments do not agree with one another within the expected limits of accuracy, the sorption values of D are incorrect and the measured sorption processes are non-Fickian. This method of examining whether sorption is Fickian or not is particularly useful when the experimental sorption curve has a shape equivalent to the true Fickian curve. Meares (1958a) demonstrated that the D values for the system polyvinyl acetate-allyl chloride at 40° C. determined from steady-state permeation data in-... 

Fig. 15.5. Calculated metal sorption curves for Pb, Cu and Cd onto the bacterium Bacillus subtilis, shown as a function of pH versus the concentration of sorbed metal. Curves are calculated based on experimental metal sorption data of Fein et al. (1997), and were computed using the geochemical speciation programme JCHESS. The solution depicted contains 1 g 1 bacteria dry wt (155 m g surface area, 8.0 Cm electrical double layer capacitance), 1 mM dissolved CaC03 and 1 iM dissolved lead, copper and cadmium. Adsorption was calculated using a CCM treatment.

The mechanical behavior of the dry-to-wet and wet-to-dry cycles complement each other. In the first sorption curve, blocking of the network structure was explained by sorption of a small quantity of moisture Into free volume of the near saturated matrix. If this small quantity of moisture Is allowed to escape, the network takes on mechanical behavior of the unblocked plasticized state. This transition occurs rapidly during the Initial stages of drying. The peak tan 6 value Is representative of plasticized state properties. As further desorption takes place, mechanical properties approach the Initial dry state values. A qualitative description of the diffusion-mechanical mechanisms associated with each segment of the sorption-desorption cycle is outlined in Figure 7. 

D. M. Ruthven and K. F. Loughlin (University of New Brunswick, Fredericton, N. B., Canada) The data of Kondis and Dranoff illustrate a number of important points relating to the problem of calculating diffusion coefficients from sorption curves. For the analysis of sorption curves, the crystal size distribution should be expressed on a weight (or... 

Figure 1 shows the typical sorption curve presented by Kondis and Dranoff (run 11) together with the theoretical curve for uniform spherical particles, calculated from Equation 2 of the preceding paper, using the value D/a = 7.84 X 10" sec The deviation of the experimental points from the theoretical curve is clearly apparent. By summing the contributions of the individual size fractions of particles it may be shown that, for a system of cubic particles with an approximately normal distribution of size (on a weight or volume fraction basis), the proper expression for the sorption curve is given by (2) ... 

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