Microdisc

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]. 

Kitzerow et al. recently demonstrated that temperature-induced phase transitions (Iso-N) and electric field-induced reorientation of a nematic liquid crystal (5CB in this case) can be used to tune photonic modes of a microdisc resonator, in which embedded InAs quantum dots serve as emitters feeding the optical modes of the GaAs-based photonic cavity [332],... 

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 ... 

It has been verified numerically that, when DQ =/ DR, the stationary current-potential response of a microdisc presents the same half wave potential as that observed for a microsphere, which is given by Eq. (2.167) (see [70] and Appendix C). Therefore, the stationary I-E response can be written as (Table 2.3)... 

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. 

So, in the case of microdiscs under steady-state conditions, the following general expression for the current can be written ... 

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).

The current for a reversible EE mechanism can achieve a stationary feature when microelectrodes are used since in these conditions the function fG(t, qa) that appears in Eq. (3.150) transforms into fG,micro given in Table 2.3 of Sect. 2.6. For microelectrode geometries for which fo.micro is constant, the current-potential responses have a stationary character, which for microdiscs and microspheres can be written as [16] ... 

The analysis of the EC and CE mechanisms under steady-state conditions at other microelectrode geometries is much more complex. In the case of microdiscs,... 

Note that the peak current densities (Aippp = A/ppp /Ao) of microspheres and microdiscs of the same radius fulfill Aipppyphe peak = (w/4)Aippp lsc peak. 

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). 

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