Diffuse layer model 380 example calculation

This set of equations can be approximated with hand calculations or solved using a computer program such as the one described by Dzombak and Morel (1990). An example fit of the diffuse layer model is indicated in Figure 6.6 for copper adsorption to hydrous ferric oxide. 

Figure 7. The effect of ligands and metal ions on surface protonation of a hydrous oxide is illustrated by two examples (1). Part a Binding of a ligand (pH 7) to hematite, which increases surface protonation. Part h Adsorption of Pb2+ to hematite (pH 4.4), which reduces surface protonation. Part c Surface protonation of hematite alone as a function of pH (for comparison). All data were calculated with the following surface complex formation equilibria (1 = 5 X 10"3 M >. Electrostatic correction was made by diffuse double layer model.

Other publications postulate specific adsorption between the surface and the /3-plane. For example Barrow and Bowden [69] interpreted adsorption of anions on goethite in terms of the mentioned above four layer model. The four layers are (in order of increasing distance from the surface) surface layer (H" and OH ions), the layer of specifically adsorbed anions, the first layer of inert electrolyte counterions (analogous to the /3-layer in TLM). and diffuse layer. This model requires an additional adjustable parameter, namely, the capacitance between the surface and the layer of specifically adsorbed anions. Barrow and Bowden report 2.99 F m for phosphate and 60,000 F m ( ) for silicate. The fit in the four layer model was substantially better than with simpler models for fluoride adsorption, but for other anions equally good fit could be obtained without introducing the additional electrostatic plane. In another paper of this series the capacitance of 3-5 F was used in model calculations of phosphate adsorption on aluminum and iron oxides [92]. Similar approach was used by Venema et al. [93] who applied the 1-pK model to interpret the Cd binding by goethite. The ions were assumed to... 

In an attempt to choose an adequate treatment, the predictions for both models were compared with experimental data for PC samples of various metals. Qualitatively the conclusions of the model of independent electrodes turned out to be in agreement with experimental data for such metals as Ag, Au, Cu and so on, see reviews in Ref [20, 34], contrary to the failure of the common diffuse layer (CDL) model, Eq. (36a). However, in quantitative terms the effects of crystallographic inhomogeneity were smaller than predicted by the model based on Eq. (34). Eor example, the capacitance at the minimum, for PC Ag diminishes more rapidly than in its theoretical estimates. The experimental slope of the PZ plots calculated with no account for the surface heterogeneity effects is close to 1 for Bi, within a very wide range of the concentration, in contrast to the model analysis. The value of R found from the same plot is close to 1 for some PC Ag electrodes in rather concentrated solutions. 

We shall present several examples of calculations obtained for the aforementioned model system, assuming that Cj =0.01 M, and the values of stability constants are as follows log = 8, log P2 = 14. An example of distribution of these particles in the diffusion layer, when there is an excess of ligand in the solution, is presented in Figure 3.4. Concentration of each particle therein Cj x) is normalized with respect to a corresponding bulk Cy. 

The impurity profile in the solidified layer cannot be calculated analytically and a numerical [9] procedure must be used. In the model the impurity is allowed to diffuse in the liquid layer according to the concentration gradient and the known diffusivities. The advancing solid front rejects impurities into the liquid according to the fitting segregation coefficient parameter. An example is shown in Fig.8. The concentration profile in the so-... 

Several advantages of the inlaid disk-shaped tips (e.g., well-defined thin-layer geometry and high feedback at short tip/substrate distances) make them most useful for SECM measurements. However, the preparation of submicrometer-sized disk-shaped tips is difficult, and some applications may require nondisk microprobes [e.g., conical tips are useful for penetrating thin polymer films (18)]. Two aspects of the related theory are the calculation of the current-distance curves for a specific tip geometry and the evaluation of the UME shape. Approximate expressions were obtained for the steady-state current in a thin-layer cell formed by two electrodes, for example, one a plane and the second a cone or hemisphere (19). It was shown that the normalized steady-state, diffusion-limited current, as a function of the normalized separation for thin-layer electrochemical cells, is fairly sensitive to the geometry of the electrodes. However, the thin-layer theory does not describe accurately the steady-state current between a small disk tip and a planar substrate because the tip steady-state current iT,co was not included in the approximate model (19). 

The last equation demonstrates that the starting point for the solution of the problem is the calculation of ci(double layer (this makes low-frequency dielectric dispersion [LFDD] measurements a most valuable electrokinetic technique). Probably, the first theoretical treatment is the one due to Schwarz [61], who considered only surface diffusion of counterions (it is the so-called surface diffusion model). In fact, the model is inconsistent with any explanation of dielectric dispersion based on double-layer polarization. The generalization of the theory of diffuse atmosphere polarization to the case of alternating external fields and its application to the explanation of LFDD were first achieved by Dukhin and Shilov [20]. A full numerical approach to the LFDD in suspensions is due to DeLacey and White [60], and comparison with this numerical model allowed to show that the thin double-layer approximations [20,62,63] worked reasonably well in a wider than expected range of values of both and ku [64]. Figure 3.12 is an example of the calculation of As. From this it will be clear that (i) at low frequencies As can be very high and (ii) the relaxation of the dielectric constant takes place in the few-kHz frequency range, in accordance with Equations (3.56) and (3.57). 

There should be some caution in broadly applying (9.1) to all types of carrier transport at interfaces. For example, the relationship does not accurately model the transit time of ballistic transport because the calculation of Xt depends on the mobility, which is only accurate in so far as it measures a diffusive process, i.e., one that involves multiple scattering events [9]. Because the small polaron conductors have transport mediated by lattice vibrations, numerous scattering events will occur as the carriers cross the space charge layer. Therefore, the transit times as calculated by (9.1) should be representative of the behavior for this class of materials [10]. 

To use the above model to predict the durability of the epoxy/mild steel joints (see Fig. 8.6) first requires a knowledge of the rate of diffusion of water into the adhesive layer in the joint. The diffusion and solubility coefficient of water into the bulk adhesive was measured and the diffusion process was also found to be of the concentration independent Fickian type [147]. Further, work [147,149,150] has also shown that the diffusion of water into the adhesive layer may often be modelled by assuming Fickian diffusion. Support for this comes, for example, from studies which have employed tritiated water and so enabled the water concentration in the joint to be directly measured good agreement with the calculated water concentration profile was obtained. 

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