Microdisc electrode

Of course, in order to vary the mass transport of the reactant to the electrode surface, the radius of the electrode must be varied, and this unplies the need for microelectrodes of different sizes. Spherical electrodes are difficult to constnict, and therefore other geometries are ohen employed. Microdiscs are conunonly used in the laboratory, as diey are easily constnicted by sealing very fine wires into glass epoxy resins, cutting... 

D.M. Zhou, H.X. Ju, and H.Y. Chen, Catalytic oxidation of dopamine at a microdisc platinum electrode modified by electrodeposition of nickel hexacyanoferrate and Nafion. J. Electroanal. Chem. 408, 219-223 (1996). 

Galceran, J., Taylor, S. L. and Bartlett, P. N. (1999). Application of Danckwerts expression to first-order EC reactions. Transient currents at inlaid and recessed microdisc electrodes, J. Electroanal. Chem., 466, 15-25. 

Fig. 3 Schematic illustration of a microfabricated multi-element array A comprising 32 interdigitated microsensor electrodes, and B comprising 64 independently addressable microdisc voltametric electrodes. Each device shows the large area counter electrode (middle) and the reference electrode as a band around the counter electrode...

Single-crystal surfaces behave quite differently, almost no peroxide being produced on Pt(lll) in contact with 0.1 M KOH, the authors attributing this to the fact that OH is adsorbed reversibly on this surface, but irreversibly adsorbed on Pt(lOO) and Pt(llO) [61, 70]. Experiments conducted with platinum microdisc electrodes (2.5 to 12.5 jjLva) have unambiguously shown how the apparent number of electron for O2 reduction is dependent on mass transfer [71]. 

The geometry of the microelectrodes is critically important not only from the point of view of the mathematical treatment, but also their performance. Thus, the diffusion equations for spherical microelectrodes can be solved exactly because the radial coordinates for this electrode can be reduced to the point at r = 0. On the other hand, a microelectrode with any other geometry does not have a closed mathematical solution. It would be advantageous if a microdisc electrode, which is easier to fabricate, would behave identically to a microsphere electrode. This is not so, because the center of the disc is less accessible to the diffusing electroactive species than its periphery. As a result, the current density at this electrode is nonuniform. 

For the disk-shaped bead mound and an x scan over the center of the bead spot, the lateral distance of the UME position from the center is r = Ax = x—x0, where x0 is the x coordinate of the spot center. The factor 9 is proportional to the steady-state concentration distribution over a microdisc electrode and assumes the following form, where rs is the radius of the bead spot ... 

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 ... 

In view of the expressions of the stationary current-potential responses of microspherical and microdisc electrodes (Eqs. (3.74) and (3.95), respectively), it is clear that an equivalence relationship between disc and hemispherical microelectrodes, like that shown for fast charge transfer processes (see Eq. (2.170) of Sect. 2.7), cannot be established in this case. 

In order to determine the accuracy of the solution proposed in Eq. (3.101) for the case of a microdisc electrode, in Fig. 3.13 numerical results are compared with this equation and also with the Oldham Eq. (3.95). Fully reversible, c[Jisc ss = 1000 /4, quasi-reversible, cj[lsc ss = nj4, and fully irreversible, cj[lsc ss = 0.001 /4, heterogeneous kinetics were considered under steady-state behavior. It is seen that, for fully reversible kinetics, both equations give almost identical results which are in good agreement with the simulated values. As the kinetics becomes less reversible, however, the results given by the two equations diverge from each other, with the simulated result lying between them. The maximum error in the Oldham equation is 0.5 %, and for Eq. (3.101), the maximum error is 3.6 %. 

Fig. 3.13 Simulated (white dots) and analytical steady-state voltammograms for the reduction of a single electro-active species at a microdisc electrode for reversible, quasi-reversible, and irreversible kinetics calculated from Eqs. (3.101) (solid line) and (3.95) (dashed line).

For shallow recessed microdisc electrode arrays, the hemispherical diffusion is larger than that for coplanar microdisc arrays. The minimum interelectrode distance necessary for hemispherical diffusion becomes smaller as recess depth increases [58],... 

Figure 7.36a-c shows the forward and reverse components of the square wave current. When the chemical kinetics is fast enough to achieve kinetic steady-state conditions (xsw > 1.5 and i + k2 > (D/rf), see [58,59]), the forward and reverse responses at discs are sigmoidal in shape and are separated by 2 sw. This behavior is independent of the electrode geometry and can also be found for spheres and even for planar electrodes. It is likewise observed for a reversible single charge transfer at microdiscs and microspheres, or for the catalytic mechanism when rci -C JDf(k + k2) (microgeometrical steady state) [59, 60]. 

The peak height of the SWV net current increases in all the cases with the square wave amplitude until it reaches a constant value (plateau) for sw > lOOmV. This value depends on the electrode shape and size and also on the catalytic rate constants. Under steady-state conditions, the plateau current at microspheres and microdiscs is given by... 

The evolution of the peak current (/ dlsc,peak) with frequency (/) is plotted in Fig. 7.37 for the first-order catalytic mechanism with different homogeneous rate constants at microdisc electrodes. For a simple reversible charge transfer process, it is well known that the peak current in SWV scales linearly with the square root of the frequency at a planar electrode [6, 17]. For disc microelectrodes, analogous linear relationships between the peak current and the square root of frequency are found for a reversible electrode reaction (see Fig. 7.37 for the smallest kx value). 

Appendix C. Solutions for Reversible Electrode Reactions at Microspheres and Microdiscs under Steady State Conditions... 

Scheme C.l Geometries and coordinate systems of (a) a microspherical electrode, (b) a microdisc embedded in an insulating surface...

Cyclic voltammograms can be presented in an alternative format to that shown in Fig. 5 by using a time rather than potential axis, as shown in Fig. 8. The equivalent parameters in steady-state voltammetric techniques are related to a hydrodynamic parameter (e.g. flow-rate, rotation speed, ultrasonic power) or a geometric parameter (e.g. electrode radius in microdisc voltammetry). 

For the case of a microdisc electrode convergent diffusion leads to a steady-state limiting current given by (91). 

Fig. 32 Diagrams showing current-voltage curves measured at a microdisc electrode at scan rates corresponding to the limits of (a) convergent diffusion and (b) planar...

Consideration of Fig. 32 implies that chemical information may be extracted from microelectrode experiments either via steady-state measurements or via transient, often cyclic voltammetric, approaches. In the former approach, measurements are made of the mass transport limited current as a function of the electrode size - most usually the electrode radius for the case of a microdisc electrode. This may be illustrated by reference to a general ECE mechanism depicted by (23a)-(23c) where k is the rate constant for the C step. 

For a simple electron transfer [see (1), (2)], it is possible to solve the diffusion equation analytically at steady state, as described for a microdisc by (91) and for a spherical electrode by (99). 

For many mechanisms, the steady-state Eia or N tt value is a function of just one or two dimensionless parameters. If simulations are used to generate the working curve (or surface) to a sufficiently high resolution, the experimental response may be interpolated for intermediate values without the need for further simulation. A free data analysis service has been set up (Alden and Compton, 1998) via the World-Wide-Web (htttp //physchem.ox.ac.uk 8000/wwwda/) based on this method. As new simulations are developed (e.g. for wall jet electrodes), the appropriate working surfaces are simulated and added to the system. It currently supports spherical, microdisc, rotating disc, channel and channel microband electrodes at which E, EC, EC2, ECE, EC2E, DISP 1, DISP 2 and EC processes may be analysed. 

Consider a typical time-dependent mass transport equation, such as (104) for a microdisc electrode. 

Most of the practical electrode geometries (microdisc, microband, channel, wall jet) require simulation of two spatial dimensions. Although a few early simulations used a simple explicit method (Britz, 1988), its relative inefficiency is compounded in multiple dimensions. Two ways of adding some implicit character to multidimensional simulations have been adopted ... 

Fig, 45 Coordinates for the simulation of a microdisc electrode (a) real space (r, z) (b) Michael el al. s (1989) conformal mapping (c) Amatore and Fosset s (1992) closed-space conformal mapping. 

Commercially available microdisc electrodes of radii 0.6-70/xm may be used for steady-state measurements without problems associated with natural convection. Dimensionless rate constants for spherical and microdisc electrode were interpolated from the working curves of Alden and Compton (1997a). 

Hemispherical electrodes are experimentally realized using hanging mercury drops for macroelectrodes and mercury-coated microdisc electrodes for microhemispheres. The lower radius limit is thus governed by the microdisc radius the upper limit has been chosen as 70 fim above which natural convection becomes significant. 

As discussed in Section 4, instrumentation is being developed that allows access to faster kinetic studies through high rates of convective mass transport. Examples include a fast flow cell (Rees et al., 1995a,b), a microjet electrode (Martin and Unwin, 1995) and a rotating microdisc electrode (White and Gao, 1995). 

X spatial coordinate perpendicular to a rotating electrode or microdisc... 

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