Finite thickness, diffusion layer

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

For a finite thickness diffuse layer, the velocity drops continuously across the layer to zero at the wall and the constant EO velocity represents the velocity at the edge of the diffuse layer. 

Diffusion layer of finite thickness (diffusion + convection). We now use the Nemst hypothesis, which assumes that the concentration of the reacting species that diffuse changes linearly in a layer of thickness 5n and is constant thereafter. 

A finite length diffusion layer thickness cannot only be caused by constant concentrations of species in the bulk of the solution but also by a reflective boundary, that is, a boundary that cannot be penetrated by electroactive species (dc/dr = 0). This can happen when blocking occurs at the far end of the diffusion region and no dc current can flow through the system, for example, a thin film of a conducting polymer sandwiched between a metal and an electrolyte solution [6]. The impedance in this case can be described with the expression... 

For a semi-infinite diffusion process at cathode represented by Warburg impedance, the Nyquist plot appears as a straight line with a slope of 45°. The impedance increases linearly with decreasing frequency. The infinite diffusion model is only valid for infinitely thick diffusion layer. For finite diffusion layer thickness, the finite Warburg impedance converges to infinite Warburg impedance at high frequency. At low frequencies or for small... 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

Here F is the Faraday constant C = concentration of dissolved O2, in air-saturated water C = 2.7 x 10-7 mol cm 3 (C will be appreciably less in relatively concentrated heated solutions) the diffusion coefficient D = 2 x 10-5 cm2/s t is the time (s) r is the radius (cm). Figure 16 shows various plots of zm(02) vs. log t for various values of the microdisk electrode radius r. For large values of r, the transport of O2 to the surface follows a linear type of profile for finite times in the absence of stirring. In the case of small values of r, however, steady-state type diffusion conditions apply at shorter times due to the nonplanar nature of the diffusion process involved. Thus, the partial current density for O2 reduction in electroless deposition will tend to be more governed by kinetic factors at small features, while it will tend to be determined by the diffusion layer thickness in the case of large features. 

The next set of models treats the catalyst layers using the complete simple porous-electrode modeling approach described above. Thus, the catalyst layers have a finite thickness, and all of the variables are determined as per Table 1 with a length scale of the catalyst layer. While some of these models assume that the gas-phase reactant concentration is uniform in the catalyst layers,most allow for diffusion to occur in the gas phase. 

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

At the other boundary, bulk concentration of A must be maintained at some finite distance from the electrode, while the concentration of B will be zero at the same point. This distance may be regarded as the diffusion layer thickness. In terms of the simulation, the establishment of the semi-infinite boundary condition requires the determination of the number of volume elements making up the diffusion layer. This will be a function of the number of time iterations that have taken place up to that point in the simulation. At any time in the physical experiment, the diffusion layer thickness is given by 6(Dt)1/2. This rule of thumb may be combined with Equation 20.7 to calculate Jd, the number of volume elements in the diffusion layer ... 

The hydrated layer has finite thickness, therefore the exchanging ions can diffuse inside this layer, although their mobility is quite low compared to that in water (n 10-11cm2s-1 V-1). As we have seen in the liquid junction, diffusion of ions with different velocities results in charge separation and formation of the potential. In this case, the potential is called the diffusion potential and it is synonymous with the junction potential discussed earlier. It can be described by the equation developed for the linear diffusion gradient, that is, by the Henderson equation (6.24). Because we are dealing with uni-univalent electrolytes, the multiplier cancels out and this diffusion potential can be written as... 

FIGURE 1.21 An example of a complex-plane impedance plot (Nyquist plane) for an electrochemical system under mixed kinetic/diffusion control, with the mass transfer and kinetics (charge transfer) control regions, for a finite thickness 8N of the diffusion layer. Assumption was made that Kf Kh at the bias potential of the measurement, and D0I = Dmd = D, leading to RB = RCT (krb8N/ >). 

In the same manner, one obtains a solution to the diffusion equation starting with a colored layer having a finite thickness 2d and an initial concentration Co in both directions of the unbounded x-axis (Fig. 7-7) ... 

Figure 7-7 Two sided diffusion from a finitely thick layer.

Figure 7-8 Single sided diffusion from a finite thick layer into a finite layer of the same material.

Diffusion layer — The diffusion layer is an imaginary layer of predominant occurrence of diffusion or a heterogeneous concentration by electrolysis, as shown in the Figure [i]. It is used conveniently when we want to conceptually separate a domain of charge transfer, a domain of diffusion, and a bulk. The diffusion layer thickness can be estimated (Dt) for the electrolysis time, t, and ranges from 0.01 mm to 0.1 mm. When current is controlled by diffusion, the thickness can also be estimated from the current density, j, through DcF/j, which is due to finite values of the differentiation in j = FD (r)c/r)x). The concept of the diffusion layer is im-... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

The bounded Warburg element (BW) describes linear diffusion in a homogeneous layer with finite thickness. Its impedance is written as... 

The following model is a bounded Randles cell also accounting for a linear but finite diffusion, with a homogeneous layer of finite thickness. The structure of the model is shown in Figure 4.18a. The corresponding impedance is... 

Diffusion through a stagnant layer of finite thickness can also yield a uniformly accessible electrode. The diffusion impedance response of a coated (or film-covered) electrode, imder the condition that the resistance of the coating to diffusion is much larger than that of the bulk electrol5M e, is approximated by the diffusion impedance of file coating. This problem is also analyzed in Section 15.4.2. 

The appropriate boundary conditions for a diffusion layer of finite thickness are that... 

The set of equations are solved with relevant boundary conditions, such as C(m)s at the outside surface of the diffusion layer, x = d, are equal to the values in the bulk solution in equibrium. Since the equations cannot be solved analytically, numerical solution was made by calculus of finite differences Gattrell et al. treated the problem more rigorously, and recently published the results. Figure 2 shows the variation of pH at the electrode surface with the concentration of the electrolyte HCO f the pH values are given for two different thicknesses of the diffusion layer and two constant current electrolysis at 5 and 15 niAcm ... 

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