Diffusion layer, Nernst

It was shown by Nernst [1] and coworkers [2] that the rate of electrochemical reactions is given by Eq. 1 if transport processes are rate-determining at steady state. 

Here and denote the concentration of the reacting species in the bulk of the electrolyte and adjacent to the electrode surface respectively. Nernst [1] assumed that the species diffuse through a region of constant width 6 and that the concentration profile can be determined by linear interpolation in this thin layer of liquid. The quantity 6 which is called the thickness of the Nernst diffusion layer has values between 10 and IQ- cm under regular conditions of stirring. 

Nernst s theory of the diffusion layer has played an important role in the development of the kinetics of electrochemical processes. However, Eq. 1 has to be considered an empirical one when applied to an electrolytic solution in motion. Experimental studies [3] demonstrated that liquid motion can be observed at distances of about 10 cm from solid surfaces. Nernst s assumption that the liquid is static within a layer 10 to 1000 times thicker is not in agreement with the empirical evidence. In addition, neither the thickness of the diffusion layer nor its dependence upon the stirring rate is calculable from Nernst s theory. 

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Volt mmetiy. Diffusional effects, as embodied in equation 1, can be avoided by simply stirring the solution or rotating the electrode, eg, using the rotating disk electrode (RDE) at high rpm (3,7). The resultant concentration profiles then appear as shown in Figure 5. A time-independent Nernst diffusion layer having a thickness dictated by the laws of hydrodynamics is estabUshed. For the RDE,... 

Fig. 2. Concentration profile of the reacting ion at an electrode. The so-called Nernst diffusion layer thickness is indicated by <5n. ...

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. 

The dependence of the limiting current density on the rate of stirring was first established in 1904 by Nernst (N2) and Brunner (Blla). They interpreted this dependence using the stagnant layer concept first proposed by Noyes and Whitney. The thickness of this layer ( Nernst diffusion layer thickness ) was correlated simply with the speed of the stirring impeller or rotated electrode tip. 

Barkey, Tobias and Muller formulated the stability analysis for deposition from well-supported solution in the Tafel regime at constant current [48], They used dilute-solution theory to solve the transport equations in a Nernst diffusion layer of thickness S. The concentration and electrostatic potential are given in this approximation... 

In passing, it is good to emphasise that the above analysis illustrates the limitations of the widely used Nernst diffusion layer concept. This concept assumes that there is a certain thin layer of static liquid adjacent to the solid plane under consideration at x = 0. Inside this layer, diffusion is supposed to be the sole mechanism of transport, and, outside the layer, the concentration of the diffusing component is constant, as a result of the convection in the liquid. We have seen that, in contradiction with this oversimplified picture, molecular diffusion and liquid motion are not spatially separated, and that the thickness... 

Nernst Diffusion-Layer Model This model assumes that the concentration of Ox has a bulk concentration up to a distance 8 from the electrode surface and then falls off linearly to Ox x = 0) at the electrode (neglecting the double-layer effect). The Nernst diffusion-layer model is illustrated in Figure 6.11. 

In this approximation, therefore, one can consider that the diffusion occurs across a region parallel to the interface, i.e., across a Nernst diffusion layer of effective thickness 8. 

Fig. 7.95. The Nernst diffusion-layer thickness is obtained by extrapolating the linear portion of the concentration change to the bulk concentration value.

Fig. 5.5 Nernst diffusion layer. C° depends on the electrode potential (see text).

The mathematical solution was first studied by Girina et al. [274] based on the same approximation as that for potential step studies, by dropping the highest-order convective term [237], and by Fried and Elving by using the Nernst diffusion-layer concept [275]. 

Turbulent flow comprises the solution bulk. (2) As the electrode surface is approached, a transition to laminar flow occurs. This is a nonturbulent flow in which adjacent layers slide by each other parallel to the electrode surface. (3) The rate of this laminar flow decreases near the electrode due to frictional forces until a thin layer of stagnant solution is present immediately adjacent to the electrode surface. It is convenient, although not entirely correct, to consider this thin layer of stagnant solution as having a discrete thickness 5, called the Nernst diffusion layer. 

Figure 3.37 illustrates the Nernst diffusion layer in terms of concentration-distance profiles for a solution containing species O. As pointed out previously, the concentration of redox species in equilibrium at the electrode-solution interface is determined by the Nernst equation. Figure 3.37A illustrates the concentration-distance profile for O under the condition that its surface concentration has not been perturbed. Either the cell is at open circuit, or a potential has been applied that is sufficiently positive of Eq R not to alter measurably the surface concentrations of the 0,R couple. 

Although the transition between stagnant and flowing solution is considered to be abrupt in this example, the transition is in reality gradual. Consequently, the profiles will be rounded as shown by the dotted line in Figure 3.37C. However, the hypothetical situation of an abrupt transition is a useful approximation in mathematical treatments as shown below. (See Ref. 6, Chap. 4, for a critique of the Nernst diffusion layer.)... 

On the other hand, as the Nernst diffusion layer model is applied to an unstirred solution, it is expected that the passage of current will cause formation of the depletion layer (Fig. 7.1), whose thickness 5o will increase with time. In time, this layer will extend from the electrode surface to the bulk of the solution over tens of pm. In order to estimate the time-dependence of So, we can use the approximate Einstein... 

The equivalent Nernst diffusion layer thickness is T (4/3) times d. Eq. (3-1) becomes ... 

Mass-transfer boundary layer (Nernst diffusion layer)... 

Consider the process of plating copper on a plane electrode. Near the electrode, copper ions are being discharged on the surface and their concentration decreases near the surface. At some point away from the electrode, the copper ion concentration reaches its bulk level, and we obtain a picture of the copper ion concentration distribution, shown in Fig. 6. The actual concentration profile resembles the curved line, but to simplify computations, we assume that the concentration profile is linear, as indicated by the dashed line. The distance from the electrode where the extrapolated initial slope meets the bulk concentration line is called the Nernst diffusion-layer thickness S. For order of magnitude estimates, S is approximately 0.05 cm in unstirred aqueous solution and 0.01 cm in lightly stirred solution. 

FIGURE 6 Nernst diffusion-layer model. The solid line represents the actual concentration profile, and the dashed line for c0 the Nernst model concentration profile. 

If concentration relaxation occurs in a diffusion layer with a thickness 5N (Nernst diffusion layer), the impedance ZF is then [5,10]... 

Here, c is the surface concentration of protons, S the Nernst diffusion layer thickness,... 

With large K values, that is low solubility of component i in a liquid food, the material transport through A can also be determined from the contribution of diffusion in L under conditions of thorough mixing. Van der Waals attractive forces between the package surface and the molecules of L in intimate contact with P lead to the formation of a thin but immobile layer in which the diffusion coefficient of i in L, DL, controls mass transport (the Nernst diffusion layer). 

Along with a group of models that have shown themselves useful, their particular normalisations will be presented. The first model, the Cottrell system, will also serve to introduce the concept of the Nernst diffusion layer. 

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