Profile diffusion

When the concentration of the solute is low, the measured diffusion profiles are predictable from Eick s second law of diffusion ... 

For high feed loads, the shape of the diffuse traihng profile and the location of the leading front can be predicted from local equihbrium theory (see Fixed Bed Transitions ). This is illustrated in Fig. 16-35 for Tp = 0.4. For the diffuse profile (a simple wave ), Eq. (16-131) gives ... 

The chemical diffusion coefficient at any concentration C in the experimental diffusion profile is dren given by... 

In addition to the analysis of relatively thick metallic coatings and diffusion profiles, dc GD-OES has also been successfully applied to the analysis of thin protective layers (thickness typically <100 nm), e.g. phosphate and chromate layers on steel [4.185]. 

Electron probe microanalysis (composition diffusion profiles local thickness)... 

The experiments further reveal that the spreading of functional PFPE proceeds in a layering fashion, which is apparently different from that of nonfunctional PFPE. It is reported [49] that in spreading of Z-dol multiple layers develop from the liquid front, but each layer exhibits a different thickness. The thickness of the first layer is close to the diameter of gyration of the polymer in bulk and the second layer is nearly twice as thick as the first layer. No such layering was observed in the case of Z-15. Instead, the liquid front evolves smoothly with time, leading to a gradual and diffusive profile. 

Figure 4 Schematic representation of a small section of a diffusion profile illustrating the application of Fick s law to determine the concentration change in the central volume element as a result of the fluxes (F) across the two planes at L and R (see text for details).

It is clear that one of the major limitations of this analysis is the assumption of constant excited-state coverage. Deviations from the behavior described by Eq. (45) in the low frequency range have been observed at photocurrent densities higher than 10 Acm [50]. These deviations are expected to be connected to excited-state diffusion profiles similar to those considered by Dryfe et al. [see Eq. (38)] [127]. A more general expression for IMPS responses is undoubtedly required for a better understanding of the dynamics involved in back electron transfer as well as separation of the photoproducts. 

In a recent study, Zellmer et al. (2000) found a zero-age U-Th isochron for the 1940 Kameni dacite in Santorini (Aegean volcanic arc). The existence of ( Ra/ °Th) ratios lower than 1 in Kameni dacites suggests that plagioclase fractionation took place less than 1 ka before eruption, in agreement with estimates based on Sr diffusion profiles in plagioclases (Zellmer et al. 1999). 

Where Ci is the groundwater U concentration and / is half the thickness of the bone. Using the following dimensionless parameters Millard and Hedges were able to produce generalized diffusion profiles (Z vs. x ) for the parameter t (e.g.. Fig. 3) ... 

Exposure of bulk GaAs Si wafers to a capacitively coupled rf deuterium plasma at different temperatures generates deuterium diffusion profiles as shown in Fig. 1. These profiles are close to a complementary error function (erfc) profile. At 240°C, the effective diffusion coefficient is 3 x 10 12 cm2/s. The temperature dependence of the hydrogen diffusion coefficient is given by (Jalil et al., 1990) ... 

An alternative procedure which is sometimes used is to place a rod in which the concentration of the isotope is constant throughout c0, against a bar initially containing none of the isotope. The diffusion profile then shows a concentration at the interface which remains at one-half that in the original isotope-containing rod during the whole experiment. This is called the constant source procedure because the concentration of the isotope remains constant at the face of the rod which was originally isotope-free. The solution for the diffusion profile is with the boundary condition c = c0/2, x = 0, t > 0 is... 

To obtain the diffusion coefficient from a diffusion profile, use ... 

Radioactive 180 was diffused into a poly crystalline pellet of ZnO at 900°C for 48 h. The diffusion profile had a marked tail showing that extensive grain boundary diffusion had occurred. The variation of the concentration of the radioactive isotope with depth for the tail of the penetration profile is given in the following table. Calculate the grain boundary diffusion coefficient, Dgb, of 180 if the bulk diffusion coefficient at 900°C is 5.53 x 10-21 m2 s-1 and the grain boundary width is taken as 1 nm. 

Some insight on the effect of the parameters on the mathematical solution can be gained through a graphical procedure. The basic idea is to plot the uptake and diffusive fluxes as functions of a variable concentration on the surface cjy, (i.e. c mO o)) and seek their intersection. It is therefore convenient to introduce the diffusive steady-state (dSS, see Section 2.4 below) flux, / ss, or flux corresponding to the diffusion profile conforming to the steady-state situation for a given surface concentration ... 

Table 5.15. Refinement of the fit parameters used to parameterize the Ce diffusion profile of...

Iqdari and Velde (unpub. data, 1992, see Table 8.2) described experiments of Ce diffusion in apatite soaked in CeCl2 with asymmetric diffusion profiles. For one of their runs carried out at 1100°C for 15 days, and described as an example of a non-linear least-square fit in Section 5.2, it has been found that the relationship between the Ce concentration CCt and the distance X to the mineral surface is described by... 

The expression given for X as a function of C leaves us in trouble at both ends of the diffusion profile. X appropriately tends towards + oo and - oo when CCe tends asymptotically towards —0.05 and 1.93, respectively, which are nearly the extreme concentrations in the profile. This is what we expect from an infinite system. However, the integrals of the rational fractions are simply natural logarithms which cannot be evaluated for a zero argument and therefore do not converge when evaluated between C0 and Cl. We will therefore restrict the calculation to the interval between extreme concentrations, say C0 = 1.865 and = 0.012. The flux at both ends will not be strictly zero, since for these values,... 

Fig. 7. Vertical turbulent diffusivity profile corresponding to Eq. (9.11). From Lamb and Duran (1977).

Under neutral conditions the atmospheric lapse rate is adiabatic. Close to the ground the vertical eddy diffusivity profile can be based on Monin-Obukhov similarity theory, in which case = 1 and = ku,z- With this formulation, increases without limit—clearly a physically unrealistic situation. Myrup and Ranzieri (1976) proposed a set of empirical roll off functions for altitudes above the surface layer ... 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

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