Spherical microelectrodes

Microelectrodes with several geometries are reported in the literature, from spherical to disc to line electrodes each geometry has its own critical characteristic dimension and diffusion field in the steady state. The difhisional flux to a spherical microelectrode surface may be regarded as planar at short times, therefore displaying a transient behaviour, but spherical at long times, displaying a steady-state behaviour [28, 34] - If a... 

Galceran, J., Puy, J., Salvador, J., Cecilia, J. and van Leeuwen, H. P. (2001). Voltammetric lability of metal complexes at spherical microelectrodes with various radii, J. Electroanal. Chem., 505, 85-94. 

The simplest specific example to examine is for a microelectrode with a spherical geometry [53]. The resistance (R) of a spherical microelectrode was given in Chapter 7. However, for a sphere whose radius is much smaller than the distance from the sphere to the counter electrode, the resistance is... 

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. 

Special attention should be paid to spherical geometry, since the mathematical treatment of spherical microelectrodes is the simplest and exemplifies very well the attainment of the steady state observed at microelectrodes of more complex shapes. Indeed, spherical or hemispherical microelectrodes, although difficult to manufacture, are the paragon of mathematical model for diffusion at microelectrodes, to the point that the behavior of other geometries is always compared against them. 

Concerning the determination of kinetic parameters of the voltammograms of quasi-reversible and irreversible electrode processes, Fig. 3.10b shows the existence of different linear zones in a similar way to that observed for planar electrodes (see Fig. 3.6). For practical purposes, it is helpful to use spherical microelectrodes, for which a broader linear region is obtained under steady-state conditions, since the process behaves as more irreversible as the radius decreases. For fully irreversible charge transfers, Eq. (3.74) simplifies to... 

Equations (3.105)-(3.107) point out the existence of three different polarization causes. So, 7km is a kinetically controlled current which is independent of the diffusion coefficient and of the geometry of the diffusion field, i.e., it is a pure kinetic current. The other two currents have a diffusive character, and, therefore, depend on the geometry of the diffusion field. I((((s corresponds to the maximum current achieved for very negative potentials and I N is a current controlled by diffusion and by the applied potential which has no physical meaning since it exceeds the limiting diffusion current 7 ss when the applied potential is lower than the formal potential (E < Ef"). This behavior is indicated by Oldham in the case of spherical microelectrodes [15, 20, 25]. 

So, the half-wave potentials for these mechanisms at spherical microelectrodes are... 

The theoretical study of other electrode processes as a reduction followed by a dimerization of the reduced form or a second-order catalytic mechanism (when the concentration of species Z in scheme (3.IXa, 3.IXb) is not too high) requires the direct use of numerical procedures to obtain their voltammetric responses, although approximate solutions for a second-order catalytic mechanism have been given [83-85]. An approximate analytical expression for the normalized limiting current of this last mechanism with an irreversible chemical reaction is obtained in reference [86] for spherical microelectrodes, and is given by... 

In Fig. 4.18, the influence of the kinetic parameters (k°, a) on the ADDPV curves is modeled at a spherical microelectrode l /Dr /r, = 0.2). In general terms, the peak currents decrease and the crossing and peak potentials shift toward more negative values as the electrode processes are more sluggish (see Fig. 4.18a). For quasireversible systems (k° 10-2 — 10 4 cm s ). the peak currents are very sensitive to the value of the heterogeneous rate constant (k°) whereas the variation of the crossing potential is less apparent. On the other hand, for totally irreversible... 

As stated in Sect. 5.2.3.4, there is always a potential difference generated by the flow of faradaic current I through an electrochemical cell, which is related to the uncompensated resistance of the whole cell (Ru). This potential drop (equal to IRU) can greatly distort the voltammetric response. At microelectrodes, the ohmic drop of potential decreases strongly compared to macroelectrodes. The resistances for a disc or spherical microelectrode of radius rd or rs are given by (see Sect. 1.9 and references [43, 48-50]). 

A voltammetric experiment in a microelectrode array is highly dependent on the thickness of the individual diffusion layers, <5, compared with the size of the microelectrodes themselves, and with the interelectrode distance and the time experiment or the scan rate. In order to visualize the different behavior of the mass transport to a microelectrode array, simulated concentration profiles to spherical microelectrodes or particles calculated for different values of the parameter = fD Ja/r s can be seen in Fig. 5.17 [57] when the separation between centers of... 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

For this reaction scheme, only the condition /Dfk < 10rs has to be fulfilled for the kinetic reaction to be detected in a spherical microelectrode, since the catalytic mechanism presents the reaction layer, whose expression is also given by Eq. (6.93). The influence of both reaction and diffusion layers is shown in Fig. 6.12 for a CE and a catalytic mechanism. 

Fig. 6.33 Catalytic CV curves for spherical microelectrodes calculated from Eq. (6.225) for different values of A (shown in the curves). These curves have been calculated for A = 0.01 mV, a scan rate v = O.IVs-1, and three electrode radii rs (in pm) (a) 50, (b) 20, and (c) 10. Reproduced with permission from [76]...

In the case of spherical microelectrodes under steady-state conditions, Eq. (7.29) becomes... 

Electrical bulk properties of ionic solids can be rather inhomogeneous (Sec. 3.1). In the following it is shown that microelectrodes are a very useful tool to gain spatially resolved information on the conductivity of such inhomogeneous solids. Let us first consider the case of a spherical microelectrode (radius rme) atop a sample with homogeneous bulk conductivity Ubuik- The bulk resistance R between the microelectrode and a hemispherical counter-electrode of radius rce (Fig. 12a) can be calculated by integrating the infinitesimal resistances of hemispherical shells according to... 

The current density on the tip of a protrusion, /tip, is determined by k, hence by the shape of the protrusion. If A —>- 0, y)ip — j (see (7)) and if k -> oo, jt p -> j0(fc - fa) > j. The electrochemical process on the tip of a sharp needle-like protrusion can be under pure activation control outside the diffusion layer of the macroelectrode. Inside it, the process on the tip of a protrusion is under mixed control, regardless it is under complete diffusion control on the flat part of the electrode for k -> 0. If k = 1, hence for hemispherical protrusion, y tip will be somewhat larger than j, but the kind of control will not be changed. It is important to note that the current density to the tip of hemispherical protrusion does not depend on the size of it if k = 1. This makes a substantional difference between spherical microelectrodes in bulk solution and microelectrodes inside diffusion layer of the macroelectrode.16 In the first case, the limiting diffusion current density depends strongly on the radius of the microelectrode. 

Electrochemical Behavior of Self-Assembled Spherical Gold Microelectrodes. The electrochemical behavior of these spherical microelectrodes... 

It can be concluded from the above discussion that the improvements gained by utilizing self-assembled spherical microelectrodes as SECM... 

Unfortunately the use of planar and (hemi)spherical electrodes is not always appropriate or possible in electrochemical studies. Electrodes with large areas lead to problems derived from large ohmic drop and capacitive effects, and the fabrication of (hemi)spherical microelectrodes is difficult. Consequently microdisc electrodes are ubiquitous in electrochemical experiments since they allow for the reduction of the above undesirable effects and are easy to manufacture and clean. This is also true in the case of band electrodes and electrodes with heterogeneous surfaces due to the non-... 

For spherical microelectrodes in the limit of low scan rates T 1), the diffusive mass transport is able to keep the surface gradient constant with time and the steady state is attained. Under these conditions, a sigmoidal response is obtained in cyclic voltammetry with the current reaching a... 

The stationary current response at spherical microelectrodes is not dependent on the scan rate, but on the applied potential according to the following expression ... 

Fig. 4.8. The half-wave potential, 1/2, of the steady-state voltammetry of a spherical microelectrode as it varies with for cr = 10 .

Figure 7.3 shows the chronoamperograms and the profile of the potential drop at a spherical microelectrode under weakly supported conditions (C g p = 0.1) for different za values and an applied potential of —0.5 V. This corresponds to mass-transport-limited current conditions in large excess of supporting electrolyte. 

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] ... 

Simultaneously, the cathodic current density on the spherical microelectrode, spher, is given by ... 

Dr. Archer conrmented that mass transport to spherical microelectrodes was much more efficient than semi-infinite linear diffusion to planar macroelectrodes, and asked whether this advantage was lost if the microparticles were embedded in a polymer matrix, as described in one part of Professor Bard s talk. Professor Bard replied that this was the case, and added that mass transport to and from, and within, polymer matrix films is now the limiting factor controlling reaction rates in such systems. It will probably be necessary to imitate photobiological systems in the use of ultra-thin membranes to avoid these mass-transport limitations. 

Lovric, M. (1999) Differential pulse voltammetry on spherical microelectrodes. Electroanalysis, 11,1089-1093. 

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