Steady-state current sphere

By comparing Eq. (5.68) for the CV steady-state currents and Eq. (2.159) for NPV under transient conditions, it can be deduced that at those electrodes where a true steady-state response can be attained (spheres and discs) it holds that... 

Because this relationship contains r, the current depends on time therefore it is not a steady-state limit such as we found for the sphere and the disk. Even so, time appears only as an inverse logarithmic function, so that the current declines rather slowly in the longtime limit. It can still be used experimentally in much the same way that steady-state currents are exploited at disks and spheres. In the literature, this case is sometimes called the quasi-steady state. 

Indeed, a sphere or a hemisphere is a good model for the diffusion zone that surrounds these electrodes, and because of this enhanced mass transport to the electrode, a steady-state current is rapidly achieved after a potential pulse is applied to the electrode. 

Alfred LCR, Oldham KB (1996) Steady-state currents at sphere-cap microelectrodes and electrodes of related geometry. J Phys Chem 100 2170-2177... 

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. 

In general, because of non-uniform accessibility of the electrode, equations for current—voltage curves have to be solved either by using approximations or numerically. For historical reasons, the dropping mercury electrode was the first to be treated theoretically [146-149] and the equations obtained depend on whether the steady-state, expanding-plane, or expanding-sphere models are utilised [150]. 

Here, r0 is the radius of the hemispherical electrode A = Anr for a sphere and A = 27t/q for a hemisphere. The first term on the right-hand side of (7.18) is the Cottrell term (7.17) and the second is the correction for radial diffusion to the microelectrode. With time, the first term becomes negligible compared to the second. The time te required for the current to reach the steady-state value depends on the desired accuracy (e%) and on the diameter of the electrode d = 2ro (in Am). It can be estimated by making the first term in (7.18) negligible against the second term, according to the formula... 

The forward and reverse currents i/rf and i//( of the square wave voltammograms corresponding to Fig. 7.5c are shown in Fig. 7.6a for microelectrodes of the four electrode geometries considered. From these curves, it can be seen that both currents present a sigmoidal shape and they are separated by 2Esw in the case of spheres and discs. This behavior clearly shows that the steady state has been attained. On the other hand, in the case of cylinders and bands, y/f and i/// show a transient behavior under these conditions. From Fig. 7.6b, c, it can be verified that a decrease in the radius, ((w/2) = rc = 0.1 pm) and that of both radius and frequency (Fig. 7.6c, (w/2) = rc = 0.1 pm and/= 10 Hz) do not lead to a stationary SWV response at cylinder and band microelectrodes. 

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]. 

Figure 6.29 shows some example linear sweep voltammograms assuming different scan rates (Osrrefers to the dimensionless scan rate Osr = F/RT)(yrl/D)). As the experimental time scale decreases, the diffusional behavior changes from near-steady-state to near-planar diffusion. With respect to the different shapes of microparticles, the mass transport-limiting current was found to be fairly consistent that is, a difference of less than 2% for sphere and hemispheres of equal surface area. 

Hence, t2 boundary velocity is time dependent, it changes so slowly when the creeping-flow limit is applicable (i.e., Re/S 1) that the velocity and pressure fields at each instant are identical to the steady-state velocity and pressure fields for a sphere moving at constant velocity at the current, instantaneous value of U (t ). 

Since these relations hold at any time along the current decay, for sampled voltammetry we can replace t by the sampling time r. Likewise, for the steady-state regime at a sphere, equations (5.4.63) and (5.4.64) can be rearranged and reexpressed as... 

The plateau currents of steady-state voltammograms can also provide the critical dimension of the electrode (e.g., tq for a sphere or disk). When a new UME is constructed, its critical dimension is often not known however, it can be easily determined from a single voltammogram recorded for a solution of a species with a known concentration and diffusion coefficient, such as Ru(NH3)5 [D = 5.3 X 10 cm /s in 0.09 M phosphate buffer, pH 7.4 (8)]. 

For the early transient regime, 8 1 [see Section 5.4.2(a)], and (5.5.42) becomes (5.2.10), which is inverted to the Cottrell equation, (5.2.11). For the steady-state regime, 8 1 and (5.5.42) collapses to a form that is easily inverted to the relationship for the steady-state limiting current (5.3.2). Actually, (5.5.42) can be inverted directly to the full diffusion-controlled current-time relationship at a sphere, (5.2.18). All of these relationships also hold for a hemisphere of radius tq, which has half of both the area and the current for the corresponding sphere. 

The electrochemical processes on microelectrodes in bulk solution can be under activation control at overpotentials which correspond to the limiting diffusion current density plateau of the macroelectrode. The cathodic limiting diffusion current density for steady-state spherical diffusion, /l,sphere is given by ... 

Let us consider now diffusion inside a sphere neglecting the diffusion gradient outside the sphere. Such a case might be observed for hydrogen absorption or Li intercalation into spherical particles. Diffusion inside the sphere can go only to the sphere center and is called finite-length internal spherical diffusion. In the steady state in which the impedance measurements are carried out, dc concentration inside the sphere is uniform, and no dc current is flowing. Ac perturbation causes oscillations of concentration at the sphere surface, which diffuse inside the sphere. In such a case, two boundary conditions in Eq. (4.94) are changed ... 

According to Eq. (12) the time for achieving the nonzero current limit due to the radial diffusion diminishes in decreasing radius ro of the spherical electrode. On microdrops of mercury formed at the tip of a polarographic capillary (of less than 100 j,m internal diameter) a steady state can be achieved within the drop-time, id < Is (cf. section 3). Unfortunately, even with the conical shroud of the capillary tip, only a fraction of the whole sphere is active [21]. 

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