Hemispherical electrodes equations

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> ... 

In the above equations, a is the conical aspect ratio, r/h 7 is the ratio of the cone or hemisphere radius to the interelectrode distance, r/d and I, the dimensionless faradaic current (either Icon or hsph > t e rati° between the one-dimensional current contribution, ifLC ant t le limiting current for an isolated hemispherical electrode, i gph (see Eq. 5)(64) ... 

Chronoamperometry at a hemispherical electrode illustrates the consequences of diffusion geometry on the electrochemical response. The expression obtained by solution of Equation 12.8 for the chronoamperometric limiting... 

Suppose that a pair of hemispherical electrodes makes contact with the surface of a semi-infinite sample in the way shown in Fig. 5.21(a). If the separation of the electrodes d is much greater than the hemispherical radius ro, analogy with the field equations for charged spheres (Moon and Spencer, 1961) shows that the measured resistance R between the electrodes (neglecting contact resistances) will be given by... 

The results in this section have been derived for spherical geometry thus they apply rigorously only for spherical and hemispherical electrodes. Because disk UMEs are important in practical applications, it is of interest to determine how well the results for spherical systems can be extended to them. As we noted in Section 5.3, the diffusion problem at the disk is considerably more complicated, because it is two-dimensional. We will not work through the details here. However, the literature contains solutions for steady state at the disk showing that the key equations, (5.4.55), (5.4.56), (5.4.63), and (5.4.64), apply for reversible systems (26, 27). The limiting current is given by (5.3.11). 

For careful measurements of the diffusion coefficient, it is critical that the measured current be truly at steady state, otherwise the current will contain a planar diffusion contribution that will produce a larger apparent value of D. At a hemispherical electrode, the Cotffell equation is used to determine the relative contributions of planar and spherical diffusion ... 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

The maximum diffusion current density is similarly affected by the size of the electrode. For a hemispherical electrode of radius a, the local current density i(r) decreases according to Equation 1.125 with increasing distance r from the electrode surface due to the diverging current lines within the electrolyte (see Figure 1.24b). 

For a hemispherical pit the diffusion-limited local current density f p changes according to Equation 7.18, which is a simple combination of Pick s first diffusion law and Faraday s law for the hemispherical electrode geometry with the same variables as used for Equation 7.17 ... 

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] ... 

The equivalency found between the behavior of hemisphere and that of disk electrodes also exists between cylinder and band electrodes [29]. Diffusion to a cylinder electrode is linear and described by Equation 12.7, while diffusion to a band is nonlinear. A plane of symmetry passes through the center of the band and normal to its surface, so the nonlinear diffusion process can be broken down into two planar components, one in the direction parallel to the electrode surface, x, and the other in the direction perpendicular to the electrode surface, y. So Fick s second law for a band electrode is... 

In the range of electrode potential more positive (more anodic) than the pitting potential, the pitting corrosion occurs in the presence of chloride ions and the metal dissolution at a pit, initially hemispherical, proceeds through the mode of electropolishing, in which concentrated chloride salts in an occluded pit solution will control the pit dissolution. It is likely that the polishing mode of metal dissolution proceeds in the presence of a metal salt layer on the pit surface in the salt-saturated pit solution. It was experimentally found with stainless steels in acid solution [54] that the pit dissolution current density, pit, is an exponential function of the electrode potential, E (Tafel equation) ... 

Equations of type (1), as appropriate to the microdisc electrode, are often insoluble analytically. Solutions are only afforded numerically or by making analogy to the uniformly accessible hemispherical micro-electrode (see Chap. 5). 

Equations (1.3.64) and (1.3.67) are valid only for planar electrodes of infinite size. For spherical electrodes or microelectrodes - where hemispherical diffusion conditions exist - the solution of the diffusion equation leads to the following equation ... 

These equations can be solved for semi-infinite external diffusion, where both Red and Ox forms are in the solution outside the sphere (diffusion to a spherical or hemispherical hanging mercury electrode, metallic solid spherical electrode), or they may diffuse inside the sphere (amalgam formation at mercury electrode, intercalation of Li into particles, hydrogen absorption into spherical hydrogenabsorbing particles). 

Thin layers of electrochemically deposited metals and thin polymer layers deposited on electrode surfaces can be conveniently studied by ellipsometry combined with other electrochemical experiments. Electrocrystallization of nickel was studied by Abyaneh, Visscher, and Barendrecht with ellipsometry and simultaneous amperometric measurements. The initial changes in A and ij/ showed nonlinear variations with the deposition time (Fig. 12), which is apparently abnormal, indicating a marked deviation of the optical properties of the deposited film from the bulk metal properties. The observed trend was explained by theoretical calculations using equations of effective medium theory (see Section IV.4 for effective medium theory) for hemispherical growth of the nucleation centers. The observed ellipsometry data clearly demonstrate that in the initial stage of nonuniform deposition the measured parameters, ij/ in particular, can change in a... 

Using the Stokes-Einstein equation " and an expression for diffusive flux at a disk electrode under hemispherical diffusion, one can estimate the expected frequency of collisions for spherical particles at a UME ... 

Spherical and hemispherical microelectrodes are the simplest cases, as the diffusion equation is Tj 2, he., same as that described for conventional spherical electrodes. Thus, the first term (i.e., the Cottrell equation) dominates at short times, where the diffusion layer is thin with respect to the electrode radius. At longer times, the second term dominates and the diffusion layer grows much larger than ro-... 

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