Spherical electrodes, current-time

Solving equation (1-8) (using Laplace transform techniques) yields the time evolution of the current of a spherical electrode ... 

The planar, cylindrical, and spherical forms of Fick s second law, and combinations of those forms, are sufficient to describe diffusion to most microelectrode geometries in use today. Just as was illustrated in Chapter 2, the appropriate form of Fick s second law is solved, subject to the boundary conditions that describe a given experiment, to provide the concentration profile information. The sought-after current-time or current-voltage relationship is then obtained by evaluating the flux at the electrode surface. 

Several approaches to solving this expression for various boundary conditions have been reported [25,26]. The solutions are qualitatively similar to the results at a hemisphere at very short times (i.e., when (Dt),y4 rD), the Cottrell equation is followed, but at long times the current becomes steady-state. Simple analytical expressions analogous to the Cottrell equation for macroplanar electrodes or Equation 12.9 for spherical electrodes do not exist for disk electrodes. For the particular case of a disk electrode inlaid in an infinitely large, coplanar insulator, the chronoamperometric limiting current has been found to follow [27] ... 

Fig. 2.15 (Solid line) Current-time curves for the application of a constant potential to a spherical electrode calculated from Eq. (2.142). D0 = Dr = 10-5 cm2 s 1, co = cr = 1 mM, rs = 0.001 cm, (E — E ) = -0.2 V, 7=298 K. (Dashed line) Current-time curves for the application of a constant potential to a planar electrode of the same area as the spherical one calculated from Eq. (2.28). (Dotted line) Steady-state limiting current for a spherical electrode calculated from Eq. (2.148). The inner figure corresponds to the plot of the current of the spherical electrode versus j ft...

The time evolution of the cathodic limiting current (Eq. 2.147) has been plotted in Fig. 2.15 together with that obtained for a planar electrode (Eq. 2.28) and the constant steady-state limiting current for a spherical electrode given by... 

In order to obtain values for the diffusion coefficient at spherical electrodes, a logarithmic plot of the current versus time would lead to nonlinear dependence (see Eq. 2.147). In this case a plot of the current versus 1 / fl is more appropriate (see inner curve in Fig. 2.15) and this plot also allows the determination of the electrode radius by combining the values of the slope (FAsc Q JD/n) and intercept (FAJ)c ())lrK... 

Fig. 4.1 Current density-time curves when both species are soluble in the electrolytic solution and only species O is initially present. Three electrode sizes are considered planar electrode (solid lines), spherical electrode with rs = 10 5 cm (dotted lines), and spherical ultramicroelectrode with rs = 10-5 cm (dashed lines), and three y values y = 0.5 (green curves), y = 1.0 (black curves), and y = 2.0 (red curves). The applied potential sequences are Ei -Ef -> -oo, E2 - E — +oo. n = T2 = 1 s, Cq = 1 mM, cR = 0, D0 = 10-5 cm2 s 1. Taken from [20] with permission...

The current-time curve corresponding to the application of the second potential is very sensitive to the presence of assimilation or amalgamation processes at spherical electrodes. In order to check this in Fig. 4.2, it can be seen the... 

From Eqs. (4.68)-(4.70), it is clear that the anodic normalized limiting currents (corresponding to e 2 —> oo) obtained for planar and spherical electrodes have an identical expression, and are only dependent on the ratio between the time lengths of the first and second potential pulse, r, and r2,... 

As discussed in Sect. 4.1, the solution for the first potential pulse in RPV corresponds to the well-known situation for a charge transfer process at a spherical electrode under limiting conditions, with the current-time expression given by Eq. (4.38). By following the mathematical procedure detailed in Appendix G, the following expression for the current-potential response at the second potential pulse under RPV conditions is obtained [44] ... 

Fig. 5.2 Current-time response calculated fromEq. (5.33), with phe = / phe/(FAs /DoCq/ /r), corresponding to the application of six potentials with abs — E f = 0.5 V. The values of the ratio (Cj)/Cq) are 0 and 1. The values of the radius of the spherical electrode (in microns) are 25, solid line 10, dashed line 5, dashed-dotted line 1, dotted line. Dq=D, Tl =. . . = T6 = t = 0.1 s...

The current response of a spherical electrode following a potential step thus contains both time-dependent and time-independent terms—reflecting the planar and spherical diffusional fields, respectively (Fig. 1.3)—becoming time independent at long timescales. As expected from Eq. (1.12), the change from one regime to another is strongly dependent on the radius of the electrode. 

The equations for the diffusion-limited current at planar and spherical electrodes are shown in Table 5.2 together with the expressions for the diffusion currents when the potential is not far from the equilibrium potential so that oxidation and reduction occur at the same time. 

It should be noted that the equation for the transition time at a spherical electrode is equal to that for a plane electrode. This result, perhaps unexpected, shows that it is only the current density that determines the transition time and not the curvature of the electrode surface. 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

The current is proportional to the rate at which X diffuses to the cathode, a function which can be expressed mathematically for a spherical electrode on which X converges from all directions. The situation is complicated by two factors. Because the mercury drop is steadily expanding, the surface area increases gradually and also the surface moves relative to the solution against the concentration gradient. The diffusion current at any time t during the life of the drop is given by an equation first derived by Ilkovid ... 

Diffusion Current at the Dropping Mercury Electrode To derive an equation for polarographic diffusion currents, we must take into account the rate of growth of the spherical electrode, which is related to the drop time in seconds, t the rate of flow of mercury through the capillary m in mg/s and the diffusion coefficient of the analyte D in cm-/s. These variables are taken into account in the Ilkovic equation ... 

General Current-Time Behavior at a Spherical Electrode... 

Short times At sufficiently short times, the thickness of the diffusion layer that is depleted of reactant is significantly smaller than the electrode radius and the spherical electrode appears to be planar to a molecule at the edge of this diffusion layen Under these conditions, the electrode behaves like a macroelectrode and mass transport is dominated by linear diffusion to the electrode surface as illustrated in Figure 4A. At short times the first part of eqn [14] becomes insignificant compared to the second part due to the dependence of the current. Therefore, the current decays over time in accordance with the Cottrell equation ... 

Current-time relationships for spherical electrodes have been available for years and are used by electrochemists when appropriate. Similarly, the corrections for edge effects at planar electrodes have been recognized, but these have seldom been used because it was easy to employ electrode... 

It is probably easiest to get a physical picture of this effect by considering the current-time response of a small, spherical electrode. The Cottrell equation for spherical diffusion has the form... 

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