Uniformly accessible electrode

A uniformly accessible electrode is an electrode where, at the interface, the flux and the concentration of a species produced or consumed on the electrode are independent of the coordinates that define the electrode surface. The mass flux at the interface is obtained by solving the material balance equation. If migration can be neglected, the material balance equation for dilute electrol5 c solutions is reduced to the convective-diffusion equation. For an axisymmetric electrode, the concentration derivatives with respect to the angular coordinate 9 are equal to zero, and the convective-diffusion equation can be expressed in cylindrical coordinates as 

If a flow exists with axial velocity Vy independent of the radial coordinate and if the bovmdary condition at y = 0 is also independent of the radial coordinate, then the concentration is only a function of y and the convective-diffusion equation is reduced to 

Equation (11.3) represents a uniformly accessible electrode because the concentration is a function only of time t and the axial position variable y. 

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Albery W J and Bruckenstein S 1983 Uniformly accessible electrodes J. Electroanal. Chem. 144 105... 

Besides the RDE and DME, other uniformly accessible electrodes under laminar flow have been described. They include the rotating hemispherical and rotating cone electrodes, which were developed to obviate the problem of trapped gas bubbles at the centre of an RDE their use is not widespread. 

Equation (120) is true at any particular point on any electrode since the diffusion layer thickness is constant for a uniformly accessible electrode, in this case we simply replace j by i. Otherwise, the equation must be integrated over the area of the electrode. 

Analysis for uniformly accessible electrodes is by plotting (1/M vs. i/k u which leads to ik and hence k. If n is unknown, then plot log i vs. log i/k D to give /u. Other plots are also possible [143]. In the particular case where n = 1 (no adsorption)... 

An interesting way of evaluating rate constants and charge transfer coefficients is the technique of iso-surface concentration voltammetry (ISCVA) [164] where the surface concentration of reactant is held constant over the electrode surface. A uniformly accessible electrode such as the RDE is therefore a prerequisite. At the RDE, the value of Hto1/2 is kept constant and disc potential plotted against current for different ratios of i/co1/2. This yields the kinetic parameters as well as E and the number of electrons transferred. 

It is for these reasons that most of the theoretical treatments have, until recently, considered uniformly accessible electrodes, i.e. the DME and RDE, where 8K and 5N are independent of the electrode coordinate, and also fast homogeneous reactions (5K 5N ). 

In LSV experiments at stationary electrodes, there can be unwanted effects due to natural convection forced convection and a uniformly accessible electrode obviate this problem. The minimum voltage scan rate at which LSV effects appear (i.e. steady-state assumptions fail) will depend on the electrode kinetics and flow parameters. We can immediately identify two extreme situations. 

Note that at uniformly accessible electrodes, spheres and cylinders, the mass flux is identical at all the points of its surface, whereas at non-uniformly accessible ones, discs and bands, the mass flux varies through the radius and the width, respectively. Therefore, in these cases, the surface gradient should be calculated by integrating the flux over the electrode surface such that the current is given by (see Scheme 2.5) ... 

In reference [18], the authors show the expression for the stationary current obtained at uniformly accessible electrodes in the case in which species R is not initially present in the solution (i.e., < ,) = 0). In the case of spherical electrodes, it can be written ... 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

It is worth recalling that, as was indicated in Sect. 2.6, in the cases of non-uniformly accessible electrodes (discs and bands), the current is an average quantity resulting from an average flux over the electrode surface (see for example [17-19]). 

Equation (6.21) is valid for any uniformly accessible electrode, including dropping electrodes, stationary plane electrodes, various hydro-... 

Thus, at a uniformly accessible electrode, /L is directly proportional to electrode area. 

In Sections 6.3-6.5 expressions for the analysis of the voltammograms corresponding to the simple electron transfer process O + ne-— R, obtained for uniformly accessible electrodes such as the rotating disc electrode, were presented. In this section these expressions will be applied to hydrodynamic electrodes in general. 

For the rotating disc electrode, a uniformly accessible electrode, the concentration only varies in the direction perpendicular to the electrode hence the time-dependent convection-diffusion equation is given by equation (10.18) ... 

The first step in developing an equivalent electrical circuit for an electrochemical system is to analyze the nature of the overall current and potential. For example, in the simple case of the uniformly accessible electrode shovm in Figure 9.1(a), the overall potential is the sum of the interfacial potential V plus the Ohmic drop Rgi. Accordingly, the overall impedance is the sum of the interfacial impedance Zo plus the electrolyte resistance Re- At the interface itself, shown in Figure 9.1(b), the overall current is the sum of the Faradaic current if plus the charging current I c through the double layer capacitor C. Thus, the interfacial impedance results from the double-layer capacity in parallel with the Faradaic impedance Zf. 

Figure 9.1 Electrical circuit corresponding to a single reaction on a uniformly accessible electrode a) series combination of the electrolyte resistance and the interfacial impedance and b) parallel combination of the Faradaic impedance and the double-layer capacitance, which comprise the interfaciai impedance.

Some issues pertaining to mass transfer to electrodes are described in Section 5.6, and the associated issues for cell design are considered further in Section 8.1.2. In many cases, a uniformly accessible electrode cannot be used. The time-constant dispersion that can arise as a result of nonuniform mass transfer is discussed in Section 13.2. 

Equation (11.20) represents a generalized form of the impedance response of a uniformly accessible electrode. Note that, in die limit that Rt Zd, equation (11.20) yields the impedance response of a single electrochemical reaction as discussed in Chapter 10. 

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. 

We now turn to the second criterion, in particular bearing in mind the criticism, alluded to above, about the difficulty associated with the theoretical description of processes at non-uniformly accessible electrodes. Again, we will compare and contrast the channel electrode and the RDE. Now the theoretical description of electrode reactions involves, typically, the solution of perhaps several coupled steady-state convective-diffusion equations of the form... 

Method ofMirkin and Bard (33). If the voltammetry is based on steady-state currents, one can analyze a quasireversible wave very conveniently in terms of two differences, Eu — and IE3/4 - Ey. Mirkin and Bard have published tables correlating these differences with corresponding sets of and a hence one can evaluate the kinetic parameters by a look-up process. Reference 33 contains a table for uniformly accessible electrodes, which applies to a spherical or hemispherical UME. A second table is given for voltammetry at a disk UME, which is not a uniformly accessible electrode. 

In hydrodynamic systems Planar diffusion to a uniformly accessible electrode, e.g. for rotating disk electrodes (hypothetical Nernst model with S = diffusion layer thickness)... 

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