Hemispherical electrode—planar

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> ... 

Only three simple transport geometries are normally encountered planar, cylindrical, and spherical. These are shown in Fig. 13. In planar transport, the flux lines are parallel to each other and normal to the electrode. Cylindrical transport occurs with electrodes that are cylindrical, such as wires, or hemicyclindrical the flux lines converge in the plane which is normal to the cylinder axis but are parallel in planes which include the cylinder axis. Spherical transport is encountered with spherical or hemispherical electrodes, the flux lines being continuations of the radii... 

Fig. 5.8. Schematic diagram showing the uniform current density for planar and spherical/hemispherical electrodes, and the non-uniform accessibility of the disc...

Fig. 3. Representations of the diffusive fields at (a) a semi-infinite planar electrode, (b) a hemispherical electrode, and (c) a finite disc electrode.

In stagnant solution Planar disk electrode(r = radius) Spherical or hemispherical electrode (r = radius)... 

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,... 

Current lines and lines of equal concentration (or equal potential) in front of a (a) planar electrode, (b) convex hemispherical electrode. 

In addition, one must choose the most appropriate geometrical form for such an electrode. The most common forms for fast voltammetric techniques are the planar geometry and the spherical (or hemispherical) geometry. In this regard, we have seen (Chapter 1, Section 4.2.2) that the simplest theoretical relationships describing the kinetics of electrode processes are valid under conditions of linear diffusion (even if we have briefly discussed also radial diffusion). 

The DigiSim program enables the user to simulate cyclic voltanunetric responses for most of the common electrode geometries (planar, full and hemispherical, and full and hemicylindrical) and modes of diffusion (semiinfinite, finite and hydrodynamic diffusion), with or without inclusion of IR drop and double-layer charging. 

Neuroimaging techniques assessing cerebral blood flow (CBF] and cerebral metabolic rate provide powerful windows onto the effects of ECT. Nobler et al. [1994] assessed cortical CBE using the planar xenon-133 inhalation technique in 54 patients. The patients were studied just before and 50 minutes after the sixth ECT treatment. At this acute time point, unilateral ECT led to postictal reductions of CBF in the stimulated hemisphere, whereas bilateral ECT led to symmetric anterior frontal CBE reductions. Regardless of electrode placement and stimulus intensity, patients who went on to respond to a course of ECT manifested anterior frontal CBE reductions in this acute postictal period, whereas nonresponders failed to show CBF reductions. Such frontal CBF reductions may reflect functional neural inhibition and may index anticonvulsant properties of ECT. A predictive discriminant function analysis revealed that the CBF changes were sufficiently robust to correctly classify both responders (68% accuracy] and nonresponders (85% accuracy]. More powerful measures of CBF and/or cerebral metabolic rate, as can be obtained with positron-emission tomography, may provide even more sensitive markers of optimal ECT administration. 

In this section, we consider mass transport-controlled currents to disc and concentric ring electrodes on a planar spinning disc surface. For other less common rotating electrodes, e.g. rotating hemisphere, see Table 3. 

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... 

Fig. 7.2 (a) Dependence of current at macroscopic planar electrode placed in unstirred solution (b) current dependence for hemispherical microelectrode... 

Most electrodes are of the planar type and can be affected by such factors as convectional mass transport within the sample in which they are immersed. Any type of turbulence will tend to increase the supply of ions to an electrode surface, above and beyond that of diffusional supply. This can have an effect on the magnitude of sensor response and give rise to erroneous results. Diffusion to a large planar electrode can be approximated to be perpendicular to the electrode surface. However, when electrodes become very small, the diffusion profile is hemispherical, mass transport is greatly increased (Fig. 9) and diffusion becomes much less of a limiting factor in the sensor response [109]. Thus, turbulence has a much smaller effect and this means that very small microelectrodes display stir-independent responses. Also the small size of micro-... 

Microelectrodes often display superior properties to those of larger planar electrodes. They experience hemispherical diffusion profiles which can confer stir independence on the results, which may be important if measuring turbulent systems such as in a river or in the bloodstream. They also allow measurements in high-resistance media, which may be experienced if electrolyte concentrations are very low. 

Radial diffusion — Diffusion converging to a point is called radial diffusion, and is applied for diffusion at small microelectrodes when the radius of the electrode is much larger than the diffusion layer thickness estimated from (Dtj1/2. Genuine radial diffusion occurs at a spherical electrode. However, radial diffusion is sometimes used for diffusion at an edge of a planar electrode, which is also called lateral diffusion. See also -> hemispherical diffusion. 

Electrode geometry — Figure 3. Concentration profiles at an array of inlaid disks at different time in response to an electrochemical perturbation. a Semi-infinite planar diffusion at short times b hemispherical diffusion at intermediate times c semi-infinite linear diffusion due to overlap of concentration profiles at long times... 

During this stage of the growth of the deposit, the nuclei develop diffusion zones around themselves, and as these zones overlap, the hemispherical mass transfer gives way to linear mass transfer to an effectively planar surface. The current then falls, and the transient approaches that corresponding to the total electrode surface. 

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). 

Fundamentals. A microelectrode with a small diameter (e.g. 10-20 pm, such an electrode is sometimes also called ultramicroelectrode (UME) [112-116]) is exposed to an electrolyte solution containing an electrochemically active substance. The electrode potential is adjusted to a value sufficiently negative to drive the electrochemical reaction O -h riQ R under diffusion control. Diffusion of reactive species to the electrode surface is hemispherical instead of planar, as in the case of large electrodes. The current I flowing across the solid/electrolyte solution interface of the microelectrode tip quickly reaches a steady state value /xa = nFDcr with n as the number of electrons transferred in the electrochemical reaction step, F the Faraday constant, D the diffusion coefficient of the reacting species, c its concentration and r the tip radius. The experimental setup is pictured schematically in Fig. 7.10. 

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