Electrode radius

This expression is the sum of a transient tenu and a steady-state tenu, where r is the radius of the sphere. At short times after the application of the potential step, the transient tenu dominates over the steady-state tenu, and the electrode is analogous to a plane, as the depletion layer is thin compared with the disc radius, and the current varies widi time according to the Cottrell equation. At long times, the transient cunent will decrease to a negligible value, the depletion layer is comparable to the electrode radius, spherical difhision controls the transport of reactant, and the cunent density reaches a steady-state value. At times intenuediate to the limiting conditions of Cottrell behaviour or diffusion control, both transient and steady-state tenus need to be considered and thus the fiill expression must be used. Flowever, many experiments involving microelectrodes are designed such that one of the simpler cunent expressions is valid. 

If ignition is assumed to occur in a hotspot formed at the electrode, the local release of potential energy W = QV/2 is directly proportional to charge while independent of electric field except in the immediate vicinity of the electrode, since this determines the local change in potential. The electric field near the electrode becomes increasingly uniform as electrode radius increases and eventually, uniform field breakdown conditions are approached (C-2.5.3). These concepts allow first approximations for effective energy to be made. First it is assumed that air breakdown occurs at 30 kV/cm and that the electric field is approximately uniform between about 2-5 mm from the electrode (C-2.5.2). Second, it is assumed that minimal ignition occurs in a 3 mm diameter hotspot formed sufficiently far from the electrode to minimize heat losses. This distance is assumed to be about 2 mm. 

The deposition rate is not uniform over the grounded electrode. A common requirement for large-scale reactors is a 5% difference in deposition rate over a certain electrode area, preferably the complete electrode. Using this, the 2D model would predict a useful electrode area of about. 0%, i.e., an electrode radius of about 4 cm. 

FIG. 4 Schematic of SECM feedback at an ITIES with the co-ordinate system used for the theoretical model. The co-ordinates r and z are measured from the center of the UME in the radial and normal directions, respectively. The UME is characterized by an electrode radius, a, while is the distance from the center of the electrode to the edge of the surrounding insulating sheath. The ITIES is located at a distance, d, from the UME in the z direction. Species Red] and Oxj are confined to phase 1, while species Red2 and 0x2 are present in phase 2. 

Although there are differences in the approach curves with the constant-composition model, it would be extremely difficult to distinguish between any of the K cases practically, unless K was below 10. Even for K = 10, an uncertainty in the tip position from the interface of 0. d/a would not allow the experimental behavior for this rate constant to be distinguished from the diffusion-controlled case. For a typical value of Z)Red, = 10 cm s and electrode radius, a= 12.5/rm, this corresponds to an effective first-order heterogeneous rate constant of just 0.08 cm s. Assuming K,. > 20 is necessary... 

Diffusion of electroactive species to the surface of conventional disk (macro-) electrodes is mainly planar. When the electrode diameter is decreased the edge effects of hemi-spherical diffusion become significant. In 1964 Lingane derived the corrective term bearing in mind the edge effects for the Cotrell equation [129, 130], confirmed later on analytically and by numerical calculation [131,132], In the case of ultramicroelectrodes this term becomes dominant, which makes steady-state current proportional to the electrode radius [133-135], Since capacitive and other diffusion-unrelated currents are proportional to the square of electrode radius, the signal-to-noise ratio is increased as the electrode radius is decreased. 

Here F is the Faraday constant C = concentration of dissolved O2, in air-saturated water C = 2.7 x 10-7 mol cm 3 (C will be appreciably less in relatively concentrated heated solutions) the diffusion coefficient D = 2 x 10-5 cm2/s t is the time (s) r is the radius (cm). Figure 16 shows various plots of zm(02) vs. log t for various values of the microdisk electrode radius r. For large values of r, the transport of O2 to the surface follows a linear type of profile for finite times in the absence of stirring. In the case of small values of r, however, steady-state type diffusion conditions apply at shorter times due to the nonplanar nature of the diffusion process involved. Thus, the partial current density for O2 reduction in electroless deposition will tend to be more governed by kinetic factors at small features, while it will tend to be determined by the diffusion layer thickness in the case of large features. 

Fig. 16. Logarithmic plot of im(o2) vs. t for different values of microdisk electrode radius r (equation 64). Adapted from ref. 128.

In the high-scan-rate range, another valuable approach to minimize ohmic drop is to use very small electrodes, down to micrometric sizes. Decreasing the electrode radius, ro, the resistance Ru increases approximately as 1/ro, but the current decreases proportionally to r. Overall the ohmic drop decreases proportionally to r0. The double-layer charging time constant, RuCd, also... 

Table 1 Typical tabulation of cyclic voltammetric data. [Fe(f-C5H )2)] solution (1.0 X 10 3 mol dm 3) in CH2CI2. Supporting electrolyte [NBu4][PF6] (0.2 mol dm 3). Platinum disc electrode radius = 1 mm. Potential values are referred to SCE...

Figure 2.21 shows the dependence of dimensionless net peak currents of ferrocene and ferricyanide on the sphericity parameter (note that A0p = AT], andy = p)-The SWV experiments were performed at three different gold inlaid disk electrodes (ro = 30, 12.5 and 5 pm) and the freqnencies were changed over the range from 20 to 2000 Hz [26]. For ferrocene the relationship between AT], and p is linear A Fp = 0.88 + 0.74p. This indicates that the electrode reaction of ferrocene is elec-trochemically reversible regardless of the frequency and the electrode radius over the range examined. For ferricyanide the dependence of AT], on p appears in sequences. Each seqnence corresponds to a particular value of the parameter The results obtained with the same freqnency, but at different microelectrodes, are cormected with thin, broken lines. The difference in the responses of these... 

Distance from the centre of electrode, m Electrode radius, m Gas constant, J mol" ... 

Fig. 5.23 Influence of the scan rate on the cyclic voltammograms of ferrocene at an ultramicroelectrode. Electrode radius 5 pm supporting electrolyte 0.6 M Et4NCI04 in AN.

Fig. 8.20 Effect of scan rate on the cyclic voltammogram for the oxidation of anthracene (2.4 mM) at a platinum ultramicroelectrode in 0.6 M Et4NCI04-AN. Electrode radius 5 pm [62],...

In a cell with spherical symmetry, the resistance between the large auxiliary electrode (radius ra) and a small working electrode (radius rw) is given by... 

The steady state will be approached when the diffusion-layer thickness becomes much larger than the radius of the disk, roughly speaking, when (Dt)1/2/r 1, where D is the diffusion coefficient, t is the duration of the experiment, and r is the electrode radius. In the opposite extreme, where (Dt)l/2/ r 1, a normal peak-shaped response will be obtained that is characteristic of planar diffusion with little contribution of diffusion from the solution at the periphery of the disk. 

When using microelectrodes to obviate resistance problems, it is convenient to develop a procedure to determine what conditions are required to reduce the error to an acceptable level. The results of such a procedure applied to disk electrodes are shown in Figure 16.6 [45]. In this and the remaining discussion, the technique of cyclic voltammetry is considered, as it is one of the most widely used voltammetric methods. The region of practical working conditions of electrode radius and scan rate is defined by the area set off by lines A, B, and C. 

The flow sensitivity of the electrode has the same origin, as has been pointed out previously. A stagnant (Prandtl) boundary layer of thickness 5 forms around the spherical electrode (radius ro) placed in the liquid of kinematic viscosity v which is moving with linear velocity U. 

If the depletion layer is completely inside the stagnant layer, the current is not affected by the change of flow. From (7.18), we know that this happens when the electrode radius becomes small. For Clark-type electrodes, the flow insensitivity is obtained even for larger diameters of the electrode, because of the additional confining effect of the membrane which has lower oxygen transmissivity, DmSm, than that of the solution. 

This effect is demonstrated in Fig. 7.6, in which the effect of liquid velocity is compared for several sensors of different dimensions of the membrane and of the electrode radius. 

It would be expected that the speed of response scales again with the radius of the electrode. However, it has been found (Vacek et al., 1986) that the fastest speed of response for a hemicylindrical Clark electrode is obtained with an electrode radius of approximately 5-10 urn. This is due to the fact that as the radius decreases, the effect of the layers that are closer to the electrode surface becomes relatively more important than those that are farther away. In fact, with the further decrease of the radius the time response becomes longer than for the corresponding planar electrodes. 

Fig. 1.24 Cell resistance a function of the distance between the working and reference electrodes. k = 20 mS cm-1. The values of the electrode radius are...

In Fig. 2.12, the analytical current-time curves under anodic and cathodic limiting current conditions calculated from Eq. (2.137) (Fig. 2.12a and b, respectively) when species R is soluble in the electrolytic solution (solid curves) and when species R is amalgamated in the electrode (dotted lines) are plotted. In Fig. 2.12a, the amalgamation effect on the anodic limiting current has been analyzed. As expected, when species R is soluble in the electrolytic solution, the absolute value of the current density increases when the electrode radius decreases because of the enhancement of... 

Fig. 2.11 Temporal evolution of the surface concentrations of oxidized species calculated from Eq. (2.140). Two electrode radius values are considered r = 5 x 10 3 cm (solid curves) and rs = 5 x 10-4 cm (dashed curves), and different y values indicated in the figure. E-Ef = -0.05 V, Cq = 4 = 1 mM, D0 = 10 5 cm2 s 1...

The analytical equations obtained allow us to study the anodic-cathodic wave. The current-potential curves for y = 0.7 are plotted in Fig. 2.13 for three values of the electrode radius and for two different initial conditions when species O is the only one present (Fig. 2.13a), and when both species are present in the system (Fig. 2.13b). 

In the first case, when only species O is initially present in the electrolytic solution (Fig. 2.13a), it is observed that the amalgamation of species R leads to a shift of the wave to more negative potential values, and this shift is greater the more spherical the electrode, i.e., when the duration of the experiment increases or the electrode radius decreases. In the second case (Fig. 2.13b), both species are initially present in the system so we can study the anodic-cathodic wave. In the anodic branch of the wave, the amalgamation produces a decrease in the absolute value of the current. As is to be expected, the null current potential, crossing potential, or equilibrium potential ( Eq) is not affected by the diffusion rates (D0 and Z)R), by the electrolysis time, by the electrode geometry (rs), nor by the behavior of species R... 

Fig. 2.12 Influence of the electrode radius on the current-time curves under anodic (a) and cathodic (b) limiting conditions (Eq. 2.137) when species R is soluble in the electrolytic solution (solid curves) and when it is amalgamated in the electrode (dashed curves). The electrode radius values (in cm) are rs = 5 x 1CT2 (red curves), rs = 1CT2 (blue curves), and rs = 5 x 10-3 (green curves). c 0 = c R= 1 mM, D0 = Dr = 1CT5 cm2 s-1. (The dashed green curve has been calculated numerically for t > 0.5 s). Reproduced with permission [52]...

When compared with the linear concentration profiles of Fig. 2.1a, it can be observed that, in agreement with Eq. (2.146) for spherical electrodes, the Nemst diffusion layer is, under these conditions, independent of the potential in all the cases. As for the time dependence of the profiles shown in Fig. 2.14b, it can be seen that the Nemst diffusion layer becomes more similar to the electrode size at larger times. Analogous behavior can be observed when the electrode radius decreases. 

From this figure, it can be seen that the current decays with time as in the planar case although this decrease leads to a constant value, / lc ss, different from zero, which will be achieved sooner as the electrode radius diminishes. The current for times close to zero is identical to that obtained in a planar electrode given to the prevalence of the term 1 / over the inverse of the radius. For longer times, the opposite happens and the term l/rs is dominant. 

---