Diffusion between finite layers metals

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... 

Fig. 13 A cartoon of a profile of a smooth electrochemical interface. The half-space z < 0 is occupied by the metal ionic skeleton that, within the jellium model, is described as a continuum of positive charge density (n+) and the dielectric constant due to bound electrons (ei,), the value of which lies typically between 1 and 2. The gap accounts for a finite distance of closest approach of solvent molecules to the skeleton the gap is determined by the balance offerees that attract the molecules to the metal and the Pauli repulsion of the closed shells of the molecules from the free electron cloud of the metal of density n(z). The regions a < z < a + d and z> a + d correspond, respectively, to the first layer of solvent molecules (which can be roughly characterized by charge-dependent effective dielectric constant) and the diffuse-layer part.

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... 

Several additional physical processes potentially exist that can create somewhat more complicated equivalent circuit diagrams, as is evident for example from discussions in Chapter 10 on conductive polymer films. For instance, additional macrodefect corrosion and diffusion effects may develop between the coating and the surface, with an additional low frequency relaxation becoming visible in the Nyquist plot. The interface between a pocket of solution and the bare metal is modeled as a double-layer capacitance in parallel with a kinetically controlled charge-transfer resistance R, which can also often include the diffusion element associated with corrosion products in series with R. If the diffusion element represents a finite diffusion, an additional Rpiipp I element appears and a third relaxation at low frequencies,... 

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