Current microdisc

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. 

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. 

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 ... 

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] ... 

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. 

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). 

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 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. 

Table 6.1 Linear sweep and cyclic voltammetry characteristics associated with the four categories (see text), where 5 is the size of the diffusion zone, Rb is the microdisc radius, d is the center-to-center separation, /p is the peak current, lum is the limiting current, and V is the scan rate [35].

The above discussion applies to spherical geometry. Unfortunately, no analytical solution can be found for the finite disc configuration however, we saw earlier that, for simple reversible electron transfer reactions, the limiting current density at the microdisc electrode was identical to that at a sphere of radius nrm/4. Fleischmann et al. [21] have investigated whether this analogy can be extended to systems with coupled chemical reactions. The system chosen for this study was the oxidation of anthracene in very dry acetonitrile at platinum electrodes. This reaction is thought to proceed by the ece mechanism [22]... 

A microdisc electrode is a micron-scale flat conducting disc of radius r. that is embedded in an insulating surface, with the disc surface flush with that of the insulator. It is assumed that electron transfer takes place only on the surface of the disc and that the supporting smlace is completely electroinactive under the conditions of the experiment. These electrodes are widely employed in electrochemical measurements since they offer the advantages of microelectrodes (reduced ohmic drop and capacitive effects, miniaturisation of electrochemical devices) and are easy to fabricate and clean for surface regeneration. In Chapter 2, we considered a disc-shaped electrode of size on the order of 1 mm. In that case we could approximate the system as one-dimensional because the electrode was large in comparison to the thickness of the diffusion layer, such that the current was essentially uniform across the entire electrode surface. Due to the small size of the microdisc, this approximation is no longer valid so we must work in terms of a three-dimensional coordinate system. While the microdisc can... 

As discussed for the case of (hemi)spherical microelectrodes in Chapter 4, the response in cyclic voltammetry at microdiscs varies from a transient, peaked shape to a steady-state, sigmoidal one as the electrode radius and/or the scan rate are decreased, that is, as the dimensionless scan rate, a = Y r lv/TZTD, is decreased. The following empirical expression describes the value of the peak current of the forward peak for electrochemically reversible processes [11] ... 

In terms of computational implementation, the container that stores the concentration grid may still be rectangular one simply sets the initial concentration of all species at all points in this exclusion zone to zero. In addition, any coefficients for the Thomas algorithm (ogj, fij) that refer to spatial points inside this zone are also set to zero, and the discretised boundary conditions derived from (10.42) are applied in the appropriate places. The current is calculated in exactly the same manner as for a microdisc electrode. 

Fig. 5 Theoretical limitations on ultrafast cyclic voltammetry. The shaded area between the slanted lines represents the radius that a microdisc must have if the ohmic drop is to be less than 15 mV and distortions due to nonplanar diffusion account for less than 10% of the peak current.

The comparison of the current response to the same dopamine concentration for a bare microdisc electrode and for the modified electrode is shown in Figure 10.2. It clearly shows that the current at the bare electrode is the sum of two oxidation processes (AA) and (DA). On the other hand, with the modified electrode a direct detection of DA, without interfering oxidation reaction of ascorbic acid, is possible. 

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