Steady-state current hemisphere

Thus, the behavior of hemispherical and disk electrodes with equivalent radii are analogous that is, both show steady-state currents at large values of (Dt)lyi/r0. Again speaking phenomenologically, one could say that a molecule at a distance of 10ro or so from the electrode surface, that is, at the outer reaches... 

It is intuitively obvious that at longer times, when the diffusion layer thickness far exceeds the radius of a disk or hemisphere (for small P), or of the width of a band or the hemicylinder, currents at flat electrodes (disk, band) must resemble those at round electrodes (hemisphere, hemicylinder). Some relations between these have been established. Oldham found [427] that the steady-state currents at a microdisk and microhemisphere are the same if their diameters along the surfaces are the same. This means that for a microdisk of radius a, the steady-state current is the same as that at a microhemisphere of radius 2a/ir. At band or hemicyclindrical electrodes, there is... 

At low V, the transport to and from an ultramicroelectrode is best described as hemispherical diffusion, which results in a faradaic current that greatly exceeds that expected for linear diffusion [142,179] (see Sec. II.D.l. and Fig. 21). An important feature of the voltammogram shown in Fig. 21(d) is the absence of a peak. Instead, the current reaches a plateau indicating that a steady-state has been obtained. The steady-state current for an ultramicroelectrode inserted in a large insulating shaft [Fig. 20(c)] is given by Eq. (61), where r is the radius of the electrode surface [180]. The effective transport resulting from hemispherical diffusion also results in an electrode system that is relatively insensitive to natural convection. 

Several advantages of the inlaid disk-shaped tips (e.g., well-defined thin-layer geometry and high feedback at short tip/substrate distances) make them most useful for SECM measurements. However, the preparation of submicrometer-sized disk-shaped tips is difficult, and some applications may require nondisk microprobes [e.g., conical tips are useful for penetrating thin polymer films (18)]. Two aspects of the related theory are the calculation of the current-distance curves for a specific tip geometry and the evaluation of the UME shape. Approximate expressions were obtained for the steady-state current in a thin-layer cell formed by two electrodes, for example, one a plane and the second a cone or hemisphere (19). It was shown that the normalized steady-state, diffusion-limited current, as a function of the normalized separation for thin-layer electrochemical cells, is fairly sensitive to the geometry of the electrodes. However, the thin-layer theory does not describe accurately the steady-state current between a small disk tip and a planar substrate because the tip steady-state current iT,co was not included in the approximate model (19). 

Method ofMirkin and Bard (33). If the voltammetry is based on steady-state currents, one can analyze a quasireversible wave very conveniently in terms of two differences, Eu — and IE3/4 - Ey. Mirkin and Bard have published tables correlating these differences with corresponding sets of and a hence one can evaluate the kinetic parameters by a look-up process. Reference 33 contains a table for uniformly accessible electrodes, which applies to a spherical or hemispherical UME. A second table is given for voltammetry at a disk UME, which is not a uniformly accessible electrode. 

One of the most remarkable features of microelectrode measurements is that one can observe steady state redox currents in relatively short time domains. When the current response at a microelectrode is in a steady state, a hemispherical diffusion region of the electrogenerated species is formed around the microelectrode probe (Fig. la). The size of the diffusion region largely relies on the probe radius and the steady state current is expressed by the following equation ... 

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. 

Figure 5. Steady state at ultramicroelectrodes. (a) Normalized theoretical concentration profiles for ferrocene chronoamperometric oxidation at a hemispherical electrode (r = 1 fim) under conditions where a steady state current is observed. From top to bottom, t = 0, 10, 10", 0.1 and 1 s (b) Theoretical simulation of ECL at a double band assembly, showing the current and ECL intensities (i jj and ECL jj are the limits at infinite time).

Figure 6.3J.7 (A) Steady-state current-distance curves for mercury/Pt (25 jun diameter) hemispherical tip (line) as compared with theoretical behavior of a planar disk ( ) and hemispherical ( ). [Reproduced with permission from J. Mauzeroll, E. A. Hueske, A. J. Bard, Anal. Ghent. 75, 3880-3889 (2003). Copyright 2003, American Chemical Society.] (B) Steady-state current-distance curves for a lagooned tip over a planar conductive substrate corresponding to different values of the parameter l/a (where I is the depth of metal recession and a is the tip radius) and the analogous working curve for a disk-shaped tip. l/a 10 ( ), 5 ( ), 1 ( ), 0.5 (O), and 0.1 (A). The upper curve was computed for a disk-shaped tip from equation Sll of reference (2). [Reproduced with permission from Y. Shao, M. V. Mirkin, G. Fish, S. Kokotov, D. Palanker, A. Lewis, Anal. Ghent. 69, 1627 (1997). Copyright 1997, American Chemical Society.]...

The change in geometry from a disk to a hemisphere can be observed by the change in limiting current in voltammograms of Ru(NH3)g + (Eigure 6.3.8.3a). This follows theoretical equations of the steady-state current at microelectrodes (see Section 6.1 in Chapter 6 of this handbook) where the ratio of the limiting current of a disk UME (12)... 

To study the kinetics of electron transfer, SECM can be operated in two modes a feedback mode (53, 69) and a substrate generation-tip collection mode (53). In the feedback mode, a mediator redox couple is required to probe the interface. The tip is poised to a potential well above Eredox- The steady-state current observed due to hemispherical diffusion of O toward the tip is expressed as (Figure 9.29a)... 

As long as the microelectrodes in the array do not interact with each other, a steady-state current is monitored at long times. This regime is a characteristic of the spherical or hemispherical diffusion achieved at individual electrodes of the array. 

Chronoamperometry at hemispherical UME Measure steady-state current D =... 

Much more accurate measurements of diffusion coefficients can be obtained with LSV or CV using UMEs. These measurements are much less dependent on the electrochemical reversibility of the redox couple. Measurement of the diffusion-limited current from a voltammogram recorded at a microelectrode is demonstrated in Figure 19.3c. The concept is identical to that already discussed for chronoamperometry at UMEs at slow scan rates (i.e., long times) the current becomes steady state as long as the potential is well past Ey2-The dependence of D on the steady-state current is given by equation (19.3) for a hemispherical UME and equation (19.4) for a disk UME. The time considerations for CV are the same as those discussed above for chronoamperometry, except that the time is estimated from the scan rate and the difference between the final potential and Ey . 

On the other hand, when the thickness of the diffusion layer becomes sufficiently thicker than the radius of the electrode, the diffusion layer spreads hemispherically around the electrode and the mode of diffusion becomes radiative. As a result, as a steady-state diffusion layer of the solute concentration that is inversely proportional to the distance from the electrode surface is formed, the current also becomes steady state. This steady-state current value (limiting current value) is expressed as follows ... 

Many applications of MBs are based on steady-state currents, and therefore Table 15.2 shows the equations that predict steady-state currents for both spherical and hemispherical MEs (T22 and T23, respectively). Moreover, an equation has been derived that allows establishing the time (C) needed to achieve a steady state within a e % closeness for a spherical ME ... 

Davis et. al. (64) have calculated the steady-state thin-layer current component for a series of electrode geometries. In their derivation, these authors have assumed that the flux between the electrodes is one-dimensional (perpendicular to the plane). Particularly relevant to the STM geometry are the equations for the current in a conical electrode/planar electrode TLC, Icon, and those for a hemispherical electrode/planar electrode TLC, Xhsph (64> ... 

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] ... 

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... 

Another type of nondisk-shaped SECM tips are UMEs shaped as spherical caps. They can be obtained, for example, by reducing mercuric ions on an inlaid Pt disk electrode or simply by dipping a Pt UME into mercury [15]. An approximate procedure developed for conical geometry was also used to model spherical cap tips [12]. Selzer and Mandler performed accurate simulations of hemispherical tips using the alternative direction implicit final difference method to obtain steady-state approach curves and current transients [14]. As with conical electrodes, the feedback magnitude deceases with increasing height of the spherical cap, and it is much lower for a hemispherical tip than for the one shaped as a disk. 

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