Determination of the diffusion coefficient

The diffusion coefficient is constant for ideal systems as discussed here and can be determined by a permeation method, i.e. the time-lag method. If the membrane is free of penetrant at the start of the experiment the amount of penetrant (Qj) passing through the membrane in the time t is given by [19] 

If the linear plot of Q / (C.Cj) versus t is extrapolated to the time axis, the resulting intercept, 0, is called the time lag, i.e. 

Instead of measuring a flow, the increment of the permeate pressure (pj) can be monitored as well. In this way the time-lag can be obtained from a p versus time plot. 

The time-lag method is very suitable for studying ideal systems with a constant diffusion coefficient. The permeability coefficient P can be obtained from the steady-state part of this permeation experiment (eq, V - 106), which means that both the diffusion coefficient and the permeability coefficient can be determined fiom one experiment More 

Once the diffusion coefficient D and the permeabilin coefficient P have been determined the solubility coefficient is known as well from the ratio P over D (see eq. V - 80). 

---

Figure 13. Voltage relaxation method for the determination of the diffusion coefficients (mobilities) of electrons and holes in solid electrolytes. The various possibilities for calculating the diffusion coefficients and from the behavior over short (t L2 /De ) and long (/ L2 /Dc ll ) times are indicated cc h is the concentration of the electrons and holes respectively, q is the elementary charge, k is the Boltzmann constant and T is the absolute temperature.

Farng and Nelson [1] applied the capillary method to the determination of the diffusion coefficient of salicylic acid in the presence of polyelectrolytes. The reported variability in terms of the coefficient of variation ranged from 0.89% to 8.3%. Stout et al. [2] showed the tube method to be useful for determining diffusion coefficients of water-insoluble pharmaceuticals such as sulfonamides and steroids. The coefficient of variation associated with the diffusion coefficient for sulfisoxazole is 5.5%. 

Values for G(unknown) were experimentally determined by using the previously calibrated cells, and these data were used to calculate values for D(unknown) using the cell constants. The overall average value of D(unknown) was 1.11 x 1(T5, which compares well with a reported value of 1.1 X 10 5. The coefficient of variation associated with the diffusion coefficient was 2.7% for one cell and 1.7% for a second cell. This calibration procedure thus provided information about the accuracy and precision of the method as well as the effect of temperature and concentration on the determination of the diffusion coefficient. 

Experimental determination of the diffusion coefficient matrix is time-consuming and labor-intensive. Nonetheless, diffusion studies have advanced significantly in recent years. Hence, with persistence and concerted effort, it is possible that reliable and reproducible diffusivity matrices for major components in some natural melts will become available in the near future. 

The ail viscosity can be found in Table 1.2, Appendix I, where we find that the viscosity at 644 K is about 3.1 X 10 5 kg/m s. The Fuller—Schettler—Giddings equation is proposed for the determination of the diffusion coefficient of nonpolar gases in binary mixtures at... 

Equation (26) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. This is the basis for the experimental determination of the diffusion coefficient. Equation (26) is solved for the boundary conditions that apply to a particular experimental arrangement. Then, the concentration of the diffusing substance is measured as a function of time and location in the apparatus. Fitting the experimental data to the theoretical concentration function permits the evaluation of the diffusion coefficient for the system under consideration. 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

Diffusion-Controlled Reactions. The specific rates of many of the reactions of elq exceed 10 Af-1 sec.-1, and it has been shown that many of these rates are diffusion controlled (92, 113). The parameters used in these calculations, which were carried out according to Debye s theory (41), were a diffusion coefficient of 10-4 sec.-1 (78, 113) and an effective radius of 2.5-3.0 A. (77). The energies of activation observed in e aq reactions are also of the order encountered in diffusion-controlled processes (121). A very recent experimental determination of the diffusion coefficient of e aq by electrical conductivity yielded the value 4.7 0.7 X 10 -5 cm.2 sec.-1 (65). This new value would imply a larger effective cross-section for e aq and would increase the number of diffusion-controlled reactions. A quantitative examination of the rate data for diffusion-controlled processes (47) compared with that of eaq reactions reveals however that most of the latter reactions with specific rates of < 1010 Af-1 sec.-1 are not diffusion controlled. 

In order to establish the most appropriate conditions for the determination of the diffusion coefficients of both electroactive species by using Eqs. (4.38) and (4.39), it has been reported that when the reaction product is absent (i.e., cR = 0) neither planar electrodes nor ultramicroelectrodes can be used in DPC for determining diffusion coefficients (see Eqs. (4.48) and (4.50)) because in these situations the anodic limiting current is either independent of DR or null, respectively. 

Here the PFGE results for two elastomers EPDM and polyisoprene [24] are compared. The EPDM investigated here is, for an elastomer, highly crystalline (30%). While crystalline domains are expected to behave as diffusion barriers for Xe, this was thought to be an interesting case for the determination of the diffusion coefficient as a function of the diffusion time A. As a comparison the completely amorphous polyisoprene was used. 

This equation allows the determination of the diffusion coefficient, D, of the atoms of the dissolving substance across the diffusion boundary layer, knowing the value of the dissolution-rate constant, k, and vice versa. It is essential to remember, however, that equation (5.6) holds for Schmidt s numbers, Sc, exceeding 1000. 

This is Barrer s equation for determining of the diffusion coefficient using permeation measurements (Fig. 9-1). The steady state permeation flux is given by the slope of the straight line (7-69) ... 

Fig. 24. Determination of the diffusion coefficients from the observed diffusion fronts shown in Fig. 23 (a) Diffusion along moisture gradient (b) self-diffusion scenario for 30% w/b moisture content. (Reproduced from Ref. 90 with permission from Elsevier.)...

Yu, G.T. and S.K. Yen, Determination of the diffusion coefficient of proton in CVD gamma aluminum oxide thin films. Surface and Coatings Technology, 166, 195 (2003). 

B. Tribollet, J. Newman, and W. H. Smyrl, "Determination of the Diffusion Coefficient from Impedance Data in the Low Frequency Range," Journal of The Electrochemical Society, 135 (1988) 134-138. 

Thus, for example, the deviations from the expected behavior for the higher homologs in the determination of the diffusion coefficients of the m-alkoxyphenol and alkyl p-hydroxybenzoate homologous series, in alkaline aqueous ethanol solution, was attributed in part to solute adsorption on the walls of the Teflon dispersion tube. ... 

One area of material science where ET-IR imaging has proved to be of extraordinary importance, in terms of scientific and practical aspects, is that of polymer analysis and polymer physics. In order to illustrate the broad range of appUcability in these disciplines, we will now discuss some selected examples in detail, ranging from phase separation in biopolymer blends, the use of polarized radiation to produce anisotropy images of inhomogeneously deformed polymer films, and determination of the diffusion coefficient of D2O in an aliphatic polyamide. 

Determination of the diffusion coefficient by permeability experiments [42], when a liquid membrane is clammed between a feed and receiving phase, with a membrane solvent. At time t = 0, a carrier which is substituted with a chromophoric group is added to the feed phase ([cf]o). The carrier diffuses through the membrane and the increase of concentration in the receiving phase ([cj() is monitored by UV/Vis spectroscopy (dj as a function of time. The transport through the pores of the membrane is assumed to be rate limiting and Eq. (25) is derived ... 

Interestingly, this approach allows the determination of the diffusion coefficient and concentration of species in the bottom phase with the tip positioned in the top phase. The microprobe does not have to enter or contact the second phase, which may allow measurements in the media unsuitable for electrochemical experiments (e.g., very nonpolar solvents) (56c). 

L.D. Coo, M. Sadek, R.T.C. Brownlee, T.J. Cardwell, R.W. Cattrall, S.D. Kolev, Determination of the diffusion coefficient of l-(2 -pyridylazo)-2-naphthol in ethanol-water solutions using flow injection and nuclear magnetic resonance techniques, Anal. Chim. Acta 386 (1999) 137. 

Analysis of deposition transients shows that deposition of copper on TiN from 50 mM copper (II) pyrophosphate solution proceeds through instantaneous nucleation of three dimensional hemispherical clusters and diffusion limited growth. Determination of the diffusion coefficient from the current maximum and analysis of the current decay using the Cotrell equation yielded values of 1 x 10 6 to 2 x 10 6 cm2 s, slightly lower than the value for Cu2t ions due to the presence of the pyrophosphate ligand. The potential dependence of inux and t,n suggest that the nucleus density is the only potential dependent parameter. 

Fig. 27 Determination of the diffusion coefficient of partially per-deuterated methanol into a water (D2O) swollen PNIPAAm gel at 21 °C. The increase of magnetisation of a thin sample layer (thickness about 100 pm) is measured. The layer is located at a distance of 4.8 mm from the sample surface...

Krom, M., and Berner, R. A. (1980). The experimental determination of the diffusion coefficients of sulfate, ammonium, and phosphate in anoxic marine sediments. Limnol. Oceanogr. 25, 321-337. 

The diffusion coefficient for catalyst pores is usually calculated approximately. Roiter ef al. (140, 141) worked out a method for experimental determination of the diffusion coefficient and calculation of reaction rates without errors induced by diffusion. 

Conrath, G., et al. In situ determination of the diffusion coefficient of a solute in a gel system using a radiotracer. Journal of Controlled Release, 1989, 9,159-168. 

The goal of these experimental investigations is the determination of the diffusion coefficient D as a function of the temperature. In the typical molecular crystals naphthalene or anthracene, the diffusion is thermally activated ... 

Saltzman ES, King DB, Holmen K, Leek C (1993) Experimental determination of the diffusion coefficient of dimethylsulfide in water. J Geophys Res 98(C9) 16481-16486... 

Experimental determination of the diffusion coefficients D is enough complicated and labor intensive process, whereas the experimental determination of the coefficients of viscosity doesn t strike the great eomplieations. Established long ago the empirical Walden s rule... 

Note the lack of dependence on electrode area or the number of electrons passed, both unknown quantities that have to be assumed or determined in the Cottrell determination of the diffusion coefficient. As Kuwana points out, the solution is identical to Beer s law if we allow the path-length I to be given by... 

The channel-flow cell technique may also be used for transient absorption measurements (chronoabsorptometry) with numerical simulation of the data. The kinetics of the dimerization of TMPD+ and MV+ were both on the order of 10" s k The technique differs from the usual OTE method in that it allows the determination of the diffusion coefficient of the product. Dr [209]. 

This chapter has addressed the method of time lag, and we have shown its application to a large number of diffusion and adsorption problems to show its utility in the determination of the diffusion coefficient as well as adsorption parameters. The central tool in the time lag analysis is the Frisch s method, and such a method has allowed us to obtain the expression of the time lag without any recourse to the solution of the concentration distribution within the medium. We shall present in the next few chapters other methods and they all complement each other in the determination of parameter. 

---