Thickness of the Nernst diffusion layer

Horvath and Lin assumed that 6 is equal to the thickness of the Nernst diffusion layer, Dm/kf, with... 

The electrode reaction proper (1) may be considered as very rapid, especially when the electrode potential is considerably more negative than the equilibrium one. Therefore, the ratecontrolling step is the transport of Me ions from the bulk of the solution to the surface of the cathode. The rate of this transport depends on the thickness of the Nernst diffusion layer, which is involved in the coefficient of mass transport the two quantities depend, in turn, on the rate of stirring of the solution near the electrode. Thus, the rate of stirring controls the limiting current density, i.e. the current density at which the concentration of the metal ions at the cathode sur-... 

In reality, as one moves away from the interface towards the bulk solution, the contribution of convection to transport increases while that of diffusion decreases. Rather than treating simultaneously transport by diffusion and convection, the Nernst model makes a clear separation between the two transport mechanisms a total absence of convection inside the Nernst diffusion layer (y < S), and an absence of diffusion outside the Nernst diffusion layer (y > S). The intensity of convection affects the flux at the electrode by fixing the thickness of the Nernst diffusion layer. For the remainder of this book, the Nernst diffusion layer will simply be called the diffusion layer. 

A derivation analogous to the previous yields the Nernst impedance, once the boundary conditions are modified to include the effects of convection y= S, Ac = 0, where represents the thickness of the Nernst diffusion layer. The solution of (5.153) gives, in this case ... 

Note that if r > 5, which occurs at short times, the 1/6 term predominates, and Equation 2S-22 reduces to an equation analogous to Equation 25-5. If r 6, which occurs at long times, the 1/r term predominates, the electron-transfer process reaches a steady state, and the steady-state current then depends only on the size of the electrode. This means that if the size of the electrode is small compared to the thickness of the Nernst diffusion layer, steady state is achieved very rapidly, and a constant current is produced. Because the current is proportional to the area of the electrode, it also means that microelectrodes produce tiny currents. Expressions similar in form to Equation 25-22 may be formulated for other geometries, and they all have in common the characteristic that the smaller the electrode, the more rapidly steady-state current is achieved. 

It can be seen that the mass transport coefficient is related to the thickness of the Nernst diffusion layer by ki — D/Sf and hence, k[, = 0.62D v co . Engineers often prefer the use of mass transport coefficient because it avoids the discussion of the Nernst diffusion layer, a concept useful to an understanding of experiments and widely met in the electrochemical literature but, in fact, ficticious since concentration profiles are never linear. 

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. 

Although it is possible to control the dissolution rate of a drug by controlling its particle size and solubility, the pharmaceutical manufacturer has very little, if any, control over the D/h term in the Nernst-Brunner equation, Eq. (1). In deriving the equation it was assumed that h, the thickness of the stationary diffusion layer, was independent of particle size. In fact, this is not necessarily true. The diffusion layer probably increases as particle size increases. Furthermore, h decreases as the stirring rate increases. In vivo, as GI motility increases or decreases, h would be expected to decrease or increase. In deriving the Nernst-Brunner equation, it was also assumed that all the particles were... 

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. 

Figure 15.7 Schematic representation of an electrode covered by a porous layer Sf is the porous layer thickness, is the Nernst diffusion layer, c is the concentration of the electroactive species which reacts at the interface and diffuses through the porous layer and through the electrolyte.

FIGURE 2.26. Schematic representation of concentration profiles for the substrate and mediator expected for the Case C situation of the Andrieux-Sav ant scheme. These profiles are drawn for the mediated oxidation situation (the second wave) where the substrate is oxidized at the mass-transfer-limited rate. Arrowheads indicate current ratios approaching infinity tails denote current ratios approaching zero. The L and denote polymer layer thickness and the Nernst diffusion layer thickness, respectively. The latter dimensions are not drawn to scale. 

Gonzalez Velasco, J. (2006) On the dependence of the Nernst diffusion layer thickness on potential and sweep... 

Note that an illustration of the Nernst diffusion layer thickness, 8n, is shown in Figure 6.9 for a case of nonstationary conditions when current density is constant. 

Thus, in a spherical field of diffusion (which is achieved for a microelectrode after a time determined by its radius), one obtains an equation similar to that given for semiinfinite linear diffusion, except that the radius of the electrode plays the role of the Nernst diffusion-layer thickness and the limiting current density is independent of time. The validity of Eq. (14.47) is one of the incentives for fabricating... 

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

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. 

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... 

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. 

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

If the characteristic time is defined independently of the disk radius, and diffusion (12.26) results, the Nernst diffusion layer thickness is dependent only on the number of these time units. So if the characteristic time is r and the maximum duration of the experiment is Tmax (giving Tmax = Tmax/r), then the final diffusion layer thickness is 1JDrmax. Then, in dimensionless distance units (normalisation being division by the disk radius a), this becomes, after multiplying by 6 and noting (12.27),... 

The thickness of the Nernst layer increases with the square root of time until natural - convection sets in, after which it remains constant. In the presence of forced convection (stirring, electrode rotation) (see also Prandtl boundary layer), the Nernst-layer thickness depends on the degree of convection that can be controlled e.g., by controlling the rotation speed of a -> rotating disk electrode. See also - diffusion layer. See also Fick s law. 

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