Concentration dependence of the diffusion coefficient

At dilute concentrations DP is usually constant. When swelling is caused by either fat, water, essential oils or other organic components found in the product then Dp can become concentration-dependent in the region of a boundary layer in P. In such cases the diffusion equation (7-12) is no longer valid and the general form of the diffusion equation (7-11) must be used. 

In this case D = D(c) is a function of the concentration c of the substance causing the swelling. The literature holds numerous recommended solutions for treating such cases, none of which are universally applicable. 

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When (DEB), is much smaller than unity, the polymer relaxation is relatively rapid compared to diffusion. In this case, conformational changes take place instantaneously and equilibrium is attained after each diffusional jump. This is the type of diffusion encountered ordinarily and is called viscous diffusion. Therefore, the transport will obey classical theories of diffusion. When (DEB), is much larger than unity, the molecular relaxation is very slow compared to diffusion and there are no conformational changes of the medium within the diffusion time scale. In this case, Fick s law is generally valid, but no concentration dependence of the diffusion coefficient is expected. This is termed elastic diffusion. When (DEB), is in the neighborhood of unity, molecular rearrangment... 

The composition at the permeate-phase interface depends on the partial pressure and saturation vapour pressure of the component. Solvent composition within the membrane may vary considerably between the feed and permeate sides interface in pervaporation. By lowering the pressure at the permeate side, very low concentrations can be achieved while the solvent concentration on the feed-side can be up to 90 per cent by mass. Thus, in contrast to reverse osmosis, where such differences are not observed in practice, the modelling of material transport in pervaporation must take into account the concentration dependence of the diffusion coefficients. 

Figure 2. Concentration dependence of the diffusion coefficient for polystyrene in two solvents at the theta point for various molecular weights. The line is the theoretical curve. (Reproduced from Ref. 12. Copyright 1983 American Chemical Society.)...

Tomozawa M. (1985) Concentration dependence of the diffusion coefficient of water in Si02 glass. Am. Ceram. Soc. C, 251-252. 

Self-difFusion coefficients were measured with the NMR spin-echo method and mutual diffusion coefficients by digital image holography. As can be seen from Figure 4.4-3, the diffusion coefficients show the whole bandwidth of diffusion coefficient values, from 10 m s on the methanol-rich side, down to 10 on the [BMIM][PFg]-rich side. The concentration dependence of the diffusion coefficients on the methanol-rich side is extreme, and shows that special care and attention should be paid in the dimensioning of chemical processes with ionic Hquids. 

Several other empirical relations for diffusion coefficients have been suggested Olson and Walton (01) have devised a means for estimating diffusion coefficients of organic liquids in water solution from surface-tension measurements. Hill (H5) has proposed a method based on Andrade s theory of liquids which allows for the concentration dependence of the diffusion coefficient in a binary liquid mixture. The formula of Arnold (A2, T6, p. 102) does not seem generally useful inasmuch as it contains two constants ( abnormality factors ) characteristic of the solute and of the solvent. 

Fig. 10. Concentration dependence of the diffusion coefficient of Dye I in Nylon 6 determined from diffusion profiles (film-roll method), in comparison with theoretical calculations from Eq. (25) with DT2 = 1.5 x lO-10 cm2/s and DT2/DX] =0.7 (full line) 82-84>. The individual contributions of the first and second terms of Eq. (25) are shown by the broken lines A and B respectively...

The sorption of a weak electrolyte by a charged polymer membrane is another case where Nernst + Langmuir-like dual mode sorption, involving the undissociated and dissociated species respectively, may be expected. The concentration of each species in solution follows, of course, from the dissociation constant of the electrolyte. The sorption isotherms of acetic acid and its fluoroderivatives have been analysed in this manner, and the concentration dependence of the diffusion coefficient of acetic acid interpreted resonably successfully, using Nylon 6 as the polymer substrate 87). In this case the major contribution to the overall diffusion coefficient is that of the Nernst species consequently DT2 could not be determined with any precision. By contrast, in the case of HC1, which was also investigated 87 no Nernst sorption or diffusion component could be discerned down to pH = 2 and the overall diffusion coefficient obeyed the relation D = DT2/( 1 — >1D), which is the limiting form of Eq. (25) when p — 00. 

Amandykov and Gurov [155,184] have extended Kikuchi s approach to the case when the TSM is used (QCA, R — 1). According to these authors, the expression for the correlation cofactor depends on the parameter of the AC interaction with the neighboring species. Therefore, this behavior exerts a significant effect on the concentration dependence of the diffusion coefficients. The potential of atoms interaction at any distances has been applied to the construction of the kinetic equation describing the diffusion decomposition of solid solutions [185]. 

In the case of chemical compounds of constant composition, application of these equations is, on the one hand, facilitated by the obvious fact that it is not necessary to take into account the concentration dependence of the diffusion coefficient. On the other hand, however, there arise serious, if not insurmountable, difficulties with the direct use of those equations because no concentration gradient can evidently exist in any growing layer, if a compound has no homogeneity range. It is therefore not surprising that... 

Firsly, the concentration dependence of the diffusion coefficient can be neglected. Secondly, the concentration of components at the interfaces of any growing layer can be assumed to be equal to the limits of the homogeneity range of a compound according to the equilibrium phase diagram of the A-B binary system. Thirdly, the concentration distribution of the components across a compound layer at any moment of time can reasonably be assumed to be close to linear (Fig. 1.22a), so that... 

Table 5.9 characterise the diffusion in saturated solutions rather than in the pure solvent. It must therefore be quite clear that the rotating disc method cannot be employed to find out the concentration dependence of the diffusion coefficient of A in B. 

One of the simplest early free-volume diffusion models was formulated in (51,52,60). The concept of this model was considered an advance, because some of the parameters required to describe the concentration dependence of the diffusion coefficient could be obtained from the physico-chemical properties of the polymer and penetrant. The relation proposed for the calculation of the thermodynamic diffusion coefficient, DT, was (51,60) ... 

The above argument brings out an important point about the limitations of the Nernst-Einstein equation. It does not matter whether the diffusion coefficient and the equivalent conductivity vary with concentration to introduce deviations into the Nernst-Einstein equation, D and A must have different concentration dependencies. The concentration dependence of the diffusion coefficient has been shown to be due to nonideality (f 1), i.e., due to ion-ion interactions, and it will be shown later that the concentration dependence of the equivalent conductivity is also due to ion-ion interactions. It is not the existence of interactions perse that underlies deviations from the Nernst-Einstein equation otherwise, molten salts and ionic crystals, in which there are strong interionic forces, would show far more than the observed few percent deviation of experimental data from values calculated by the Nernst-Einstein equation. The essential point is that the interactions must affect the diffusion coefficient and the equivalent conductivity by different mechanism and thus to different extents. How this comes about for diffusion and conduction in solution will be seen later. 

Let us now analyze the concentration dependence of the diffusion coefficient of polymer solutions. By using the methods outlined above, we find that... 

Adsorption plays an important role in permeation through microporous membranes. First of all, steps 1 and 5 involve adsorption and desorption processes. Second, the concentration dependence of the diffusion coefficient is often described by the adsorption isotherm. Some data on adsorption in zeolites will be presented in Section III.D. 

The D values determined in this way may be substantially lower than the actual ones. The difference is probably due to concentration dependence of the diffusion coefficient. Experimentally established concentration profiles may likewise differ from the theoretical ones (cf. Sanders and Schaeffer, 1976). Nevertheless, the volatilization rate may be described well by means of D [Pg.70]

The concentration dependence of the diffusion coefficient is plotted in Fig. 8.9 in the scaling form suggested by Eq. (8.85) for polymer solutions in good solvents. The expected exponent is observed over a limited range of approximately one decade above the overlap concentration 0 and a stronger concentration dependence is seen at higher concentrations, where entanglements become important. 

Here, is a measure of the average mesh size of the entangled network. Understanding of these molecular parameters, for example, the concentration dependence of the diffusion coefficient is well developed for neutral polymer solutions (25) and polyelectrolytes with a large amount of simple salts. However, for salt-free polyelectrolytes, reliable data are lacking and understanding is rather poor. 

Gallagher, W. H., and Woodward, C. K. (1989) The concentration dependence of the diffusion coefficient for bovine pancreatic trypsin inhibitor a dynamic light scattering study of a small protein. Biopolymers. 28, 2001-2024... 

SOLVENT MOBILITIES. One check on the physical significance and the reliability of the data representing the concentration dependence of the diffusion coefficient is to convert these results to solvent mobilities. The values should increase rapidly with increasing concentration and extrapolate to the self-diffusion coefficient for toluene. The procedure for carrying out the calculations was outlined in previous publication (11) and is repeated here in a brief form for convenient reference. The diffusion coefficient obtained directly in the vapor sorption experiment is a polymer, mass-fixed, mean diffusion coefficient, D, in the sorption interval. Duda et. al. (12) have shown that, if the concentration interval is small, the true diffusion coefficient, D, is simply related to the mean diffusion coefficient at a prescribed intermediate concentration in the interval ... 

The latter was obtained by transferring samples which had reached equilibrium in a particular salt solution to a fresh but more dilute salt solution. The concentration dependence of the diffusion coefficient is shown in Figures 2 and 3 where it can be seen that the data is in agreement with the theory. The diffusion coefficients obtained by the above methods were, of course, average values over the concentration range existing in each measurement. This range was of the order of 0.27., sufficiently small for the value of D obtained to be considered representative of the value at the mean concentration. 

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