Linear diffusion layer

In terms of diffusion layers, such small values of ps correspond to linear diffusion layer thicknesses that are large compare to those of the constrained diffusion layer (right-hand side of Figure 4.16). 

Under these conditions, as sketched on the left-hand side of Figure 4.16, the linear diffusion layer has become very thin, on the same order as the constrained diffusion layer. The response amounts therefore to the steady-state response of an assembly of nas independent disk microelectrodes. The shape of the S-wave and the location of the half-wave potential is a function of the last term in the denominator on the right-hand side of equation (4.18). The parameter that governs the kinetic competition between electron transfer and constrained diffusion is therefore... 

An approximate resolution of the system above leads to an equation relating the substrate concentration at the active sites, (CA) [=0, to its value at the boundary between the constrained diffusion layer and the linear diffusion layer, (Ca)x=2Rq ... 

The mass transport coefficient is, in general, a complex time and potential-dependent function through the linear diffusion layer thickness, <% ,. Only under certain conditions does this dependence disappear (as, for example, for nemstian... 

Figure 2.1b shows the time dependence of the concentration profiles. It can also be observed that the perturbed region of the solution adjacent to the electrode surface grows with time and the relative difference between the linear diffusion layer and the accurate diffusion layer (determined as the value x for which cQ reaches the 99 % of its bulk value) is greater for shorter times [12]. 

The concentration profiles are very sensitive to the kinetics of the electrode reaction. In this context, the determination of the diffusion layer thickness is of great importance in the study of non-reversible charge transfer processes. This magnitude can be defined as the thickness of the region adjacent to the electrode surface where the concentration of electro-active species differs from its bulk value, and it can be accurately calculated from the concentration profiles. In the previous chapter, the extensively used concept of Nemst diffusion layer (8), defined as the distance at which the linear concentration profile (obtained from the straight line tangent to the concentration profile curve at the electrode surface) takes its bulk value, has been explained. In this chapter, we will refer to it as linear diffusion layer since the term Nemst can be misunderstood when non-reversible processes... 

The faradaic current corresponding to any charge transfer process depends on the surface gradient (dco/dx)x=0, which can be expressed as the ratio of the difference between the bulk and surface concentrations of the oxidized species and the linear diffusion layer, <5 0,... 

In order to gain a deeper understanding of the particularities of non-reversible processes at spherical electrodes, it is useful to define the linear diffusion layer... 

Fig. 3.12 Variation of the ratio between the linear diffusion layer thickness of slow and fast electrode reactions for spherical electrodes, <5 /5, with the electrode kinetics (through the dimensionless parameter k° /t/D) and the electrode size (through rsj aJnDt).

A general mathematical formulation and a detailed analysis of the dynamic behavior of this mass-transport induced N-NDR oscillations were given by Koper and Sluyters [8, 65]. The concentration of the electroactive species at the electrode decreases owing to the electron-transfer reaction and increases due to diffusion. For the mathematical description of diffusion, Koper and Sluyters [65] invoke a linear diffusion layer approximation, that is, it is assumed that there is a diffusion layer of constant thickness, and the concentration profile across the diffusion layer adjusts instantaneously to a linear profile. Thus, they arrive at the following dimensionless set of equations for the double layer potential, [Pg.117]

Consider what happens when a potential step of magnitude E is applied to an electrode immersed in a solution containing a species O. If the reaction is nernstian, the concentrations of O and R at x = 0 instantaneously adjust to the values governed by the Nernst equation, (1.4.12). The thickness of the approximately linear diffusion layer, grows with time (Figure 1.4.5). At any time, the volume of the diffusion layer is... 

Fig. 4.1. Concentration profile of species A in the Cottrell experiment at T = Tmax-The solid line corresponds to the profile obtained with a uniform grid with h = 0.0001 the circle points indicate the position of the spatial nodes when an expanding grid is used with h = 0.0001 and ujx = 1-1. AT = 0.01. The value AT/VTinax is chosen as x-axis since this gives an estimation of the distance relative to the linear diffusion layer thickness X/ /Tmax = xly D t max ...

The value X/VTnax is chosen as 2 -axis since this gives an estimation of the distance relative to the linear diffusion layer thickness of the species X/ /Tmax = / ZDAtmajc. 

If deposition to the macroelectrode is under full diffusion control, the distribution of the concentration C inside the linear diffusion layer is given by the Eq. (2.10) [3] ... 

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