Thin layer concentration profile

Thin concentration polarization layer. Short tubes, concentration profile not fully developed. Use arithmetic concentration difference. 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

Fig. 4.11. Characteristic intensity profiles for three different kinds of concentration (a) bulk type (b) particulate type and (c) thin-layer type. The critical angle d>c is determined by total reflection at the substrate ([4.21], after Ref [4.44]).

X-ray scattering studies at a renewed pc-Ag/electrolyte interface366,823 provide evidence for assuming that fast relaxation and diffu-sional processes are probable at a renewed Sn + Pb alloy surface. Investigations by secondary-ion mass spectroscopy (SIMS) of the Pb concentration profile in a thin Sn + Pb alloy surface layer show that the concentration penetration depth in the solid phase is on the order of 0.2 pm, which leads to an estimate of a surface diffusion coefficient for Pb atoms in the Sn + Pb alloy surface layer on the order of 10"13 to lCT12 cm2 s i 820 ( p,emicai analysis by electron spectroscopy for chemical analysis (ESCA) and Auger ofjust-renewed Sn + Pb alloy surfaces in a vacuum confirms that enrichment with Pb of the surface layer is probable.810... 

FIGURE 2.4 Three types of concentration profiles encountered among the thin-layer chromatographic bands (a) symmetrical (Gaussian) without tailing, (b) skewed with front tailing, and (c) skewed with back tailing [14],... 

For the densitograms recorded from the nonlinear thin-layer chromatograms with non-Gaussian concentration profiles, it can be stated that ... 

Applying the concentration profile of Eq. (32) obtained from a thin film to both aqueous diffusion layers at steady state, we have... 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

Experimental measurements of DH in a-Si H using SIMS were first performed by Carlson and Magee (1978). A sample is grown that contains a thin layer in which a small amount (=1-3 at. %) of the bonded hydrogen is replaced with deuterium. When annealed at elevated temperatures, the deuterium diffuses into the top and bottom layers and the deuterium profile is measured using SIMS. The diffusion coefficient is obtained by subtracting the control profile from the annealed profile and fitting the concentration values to the expression, valid for diffusion from a semiinfinite source into a semi-infinite half-plane (Crank, 1956),... 

If electron transport is fast, the system passes from zone R to zone S+R and then to zone SR. In the latter case there is a mutual compensation of diffusion and chemical reaction, making the substrate concentration profile decrease within a thin reaction layer adjacent to the film-solution interface. This situation is similar to what we have termed pure kinetic conditions in the analysis of an EC reaction scheme adjacent to the electrode solution interface developed in Section 2.2.1. From there, if electron transport starts to interfere, one passes from zone SR to zone SR+E and ultimately to zone E, where the response is controlled entirely by electron transport. 

Focusing attention on equation (6.51), another implication of pure kinetic conditions is that Qb/dx 0. Also, the fact that (fia/6y)> 0+ (db/dy)y=( = 0 and (fc) 0 -C (a)y=0 implies that in the thin reaction layer containing the entire concentration profile of B,... 

When the reaction scheme involves first- or pseudo- first-order reactions, fast enough for pure kinetic conditions to be achieved D/k concentration profile of B is squeezed within a thin reaction layer adjacent to the electrode surface as represented in Figure 2.31 (bottom diagram). Starting from the electrode surface, the following relationships apply. 

To calculate more precisely the average uptake or the local variation in uptake in each airway, the local variations in velocity and concentration profiles must be taken into account. For example, thin momentum and concentration boundary layers occur at bifurcations and gradually increase in thickness with distance downstream. Bell and Friedlander showed that particle and gas transfer to the airway wall is greatest where the boundary layers are thinnest, e.g., at the carina or apex of bifurcations. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

The thin-source method is also referred to as the thin-film method. One surface is cut into a plane surface and polished. A very thin layer is then sprayed or spread onto the surface. The thin layer contains the component of interest, which at high temperature diffuses into the interior of the sample from the polished surface. After the experiment, a section is cut perpendicular to the polished surface. Concentration profile is measured as a function of distance away from this surface. If the length of the concentration profile is much greater than (> 100 times) the thickness of the thin layer on the surface, the problem may be treated as a... 

Sometimes the concentration cannot be measured directly. One way to treat such a profile is to measure the total concentration by removing successively thin layers of the surface layers. Hence, the first measurement is the integrated total concentration of the species (such as p-counting) from 0 to oo. Then a thin layer dx is removed and the second measurement is the integration from dx to oo. The procedure is repeated until the whole profile is measured. Every measurement is hence the integral of the concentration profile /C dx from x to oo, i.e.. 

As in the case of diffusion from an initially thin layer, experimental concentration data and theoretical concentration profiles may be compared. From this comparison, the value of... 

In the epilimnion/hypolimnion two-box model the vertical concentration profile of a chemical adopts the shape of two zones with constant values separated by a thin zone with an abrupt concentration gradient. Often vertical profiles in lakes and oceans exhibit a smoother and more complex structure (see, e.g., Figs. 19.1a and 19.2). Obviously, the two-box model can be refined by separating the water body into three or more horizontal layers which are connected by vertical exchange rates. 

Equation 5.18 offers a convenient technique for measuring self-diffusion coefficients. A thin layer of radioactive isotope deposited on the surface of a flat specimen serves as an instantaneous planar source. After the specimen is diffusion annealed, the isotope concentration profile is determined. With these data, Eq. 5.18 can be written... 

Occasionally (e.g., thin-layer electrochemistry, porous-bed electrodes, metal atoms dissolved in a mercury film), diffusion may be further confined by a second barrier. Figure 2.7 illustrates the case of restricted diffusion when the solution is confined between two parallel barrier plates. Once again, the folding technique quickly enables a prediction of the actual result. In this case, complete relaxation of the profile results in a uniform finite concentration across the slab of solution, in distinct contrast to the semi-infinite case. When the slab thickness t is given, the time for the average molecule to diffuse across the slab is calculable from the Einstein equation such that... 

Thus the relaxation time for a concentration profile in the thin layer will vary with the thickness squared. 

Figure 3.14 Concentration profile relaxation in thin-layer cell. (A) Initial condition (solid line, product dashed line, reactant). (B) Relaxation in progress. (C) Final condition.

---