Limiting currents at hydrodynamic

Diffusion-limited currents at hydrodynamic electrodes under laminar flow conditions3... 

The method of resolution of (8.1) was indicated in Sections 5.7-5.9, showing as an example the calculation of the limiting current at the rotating disc electrode. In this chapter we discuss this and other hydrodynamic electrodes used in the study of electrode processes. The rotating disc electrode has probably been the hydrodynamic electrode... 

The expressions for the limiting currents at commonly used hydrodynamic electrodes are presented in Table 8.1. Nearly all of these have cylindrical symmetry, and the electrodes are embedded in surfaces of infinite extension. As explained in Chapter 5, calculation of /L begins with the velocity profile in solution1112. Thence, one obtains expressions for the velocity components close to the electrode surface and calculates /L. Streamlines and schematic profiles of solution movement for some configurations are shown in Fig. 8.2. The following points should be... 

Curve (b) of Figure 4 shows the same silent system as curve (a) but now upon a contracted current scale, while curve (c) shows the effect of ultrasonic irradiation upon curve (b), scanned at the same rate and in the oxidation direction only. Note that curves (b) and (c) are on the same current scale, both taken from ref. 31. Ultrasound has produced a 10-fold increase in maximum current. The plateau shape shows a limiting current at the extreme of oxidation potential reflecting hydrodynamic control independent of the voltammetric sweep rate. (This shape is also seen in other voltammetric procedures, e.g. when using rotating disk electrodes or microelectrodes.) In Figure 4 curve (c) this limiting current is found to be inde-... 

Ultrasound has converted a millielectrode into a microelectrode, and the current increase shows greatly enhanced mass transport within the cell. Limiting currents at both sizes of electrode increase with ultrasonic power within the ranges studied. This enhanced mass transport is further shown by rotating electrode studies. Here a situation of controlled hydrodynamics is achieved by spinning the disk micro or millielectrode. The limiting current in a silent system increases with rotation rate (Fig. 6), 0e to a maximum rotation at which the system becomes turbulent and the well-defined hydrodynamics are lost. This occurs typically at rpm. 

Similarly to the response at hydrodynamic electrodes, linear and cyclic potential sweeps for simple electrode reactions will yield steady-state voltammograms with forward and reverse scans retracing one another, provided the scan rate is slow enough to maintain the steady state [28, 35, 36, 37 and 38]. The limiting current will be detemiined by the slowest step in the overall process, but if the kinetics are fast, then the current will be under diffusion control and hence obey the above equation for a disc. The slope of the wave in the absence of IR drop will, once again, depend on the degree of reversibility of the electrode process. 

In an alternative design, the actual tip of the ultrasonic hom may be used as the working electrode after insertion of an isolated metal disc [77, 78 and 79]. With this electrode, known as the sonotrode, very high limiting currents are obtained at comparatively low ultrasound intensities, and diflflision layers of less than 1 pm have been reported. Furdiemiore, the magniPide of the limiting currents has been found to be proportional to D, enabling a parallel to be drawn with hydrodynamic electrodes. 

The term limiting-current density is used to describe the maximum rate at 100% current efficiency, at which a particular electrode reaction can proceed in the steady state. This rate is determined by the composition and transport properties of the electrolytic solution and by the hydrodynamic condition at the electrode surface. 

For a given hydrodynamic condition near the electrode in steady state, the maximum gradient is obtained when the concentration at the electrode is zero, or virtually zero. From the definition of limiting-current density, this situation corresponds to the limiting-current condition. 

Another survey by Ibl (13) in 1963 listed 13 mass-transfer correlations established by the limiting-current method, only four of which were derived from quantitative considerations. At the time of writing the total number of publications is more than 200. The majority of these concern flow conditions under which theoretical predictions are, at best, qualitative. More recently, an increasing number of publications deal with model hydrodynamic studies of more complex situations, for example, packed and fluidized beds. 

Analysis of nonstationary convective mass transfer, under well-defined hydrodynamic conditions, may be helpful to understand the way in which the limiting current is established at electrodes of appreciable dimensions. As referred to in the previous section, the transition time to steady-state... 

A rotating disk electrode (RDE) [7] is used to study electrode reactions, because the mass transfer to and from the electrode can be treated theoretically by hydrodynamics. At the RDE, the solution flows toward the electrode surface as shown in Fig. 5.22, bringing the substances dissolved in it. The current-potential curve at the RDE is S-shaped and has a potential-independent limiting current region, as in Fig. 5.6. The limiting current (A) is expressed by Eq. (5.33), if it is controlled by mass transfer ... 

A summary of limiting current expressions at many hydrodynamic electrodes is given in Table 3. 

At an appreciable fraction of the limiting current, it is usually not justified to neglect concentration variations— and resulting overpotential—near the electrode. In a general model we need to consider the electric field, kinetic limitations, and concentration variations. The problem is rendered more difficult by the need to know the system hydrodynamics, which, in turn, influence the concentration... 

Table 8,1. Limiting currents, /L, at hydrodynamic electrodes under laminar flow conditions". Adapted from Ref. 1...

If the charge transfer is so facile that every reacting species arriving at the electrode surface immediately reacts, then the concentration of the reacting species approaches zero, i.e., C (x = 0) = 0, the current becomes independent of potential, and reaches a maximum value which depends only on the actual hydrodynamic conditions. The maximum current is called the diffusion -> limiting current (ji) ... 

Finite diffusion — Finite (sometimes also called -> limited) diffusion situation arises when the -> diffusion layer, which otherwise might be expanded infinitely at long-term electrolysis, is restricted to a given distance, e.g., in the case of extensive stirring (- rotating disc electrode). It is the case at a thin film, in a thin layer cell, and a thin cell sandwiched with an anode and a cathode. Finite diffusion causes a decrease of the current to zero at long times in the - Cottrell plot (-> Cottrell equation, and - chronoamperometry) or for voltammetric waves (see also - electrochemical impedance spectroscopy). Finite diffusion generally occurs at -> hydrodynamic electrodes. 

It can be seen that - in contrast with the diffusion-limited current - the reaction-limited current does not depend on the hydrodynamical conditions. At intermediate overpotentials... 

Table 5 Expressions for the limiting current obtained at a range of hydrodynamic electrodes (see appendix for meaning of symbols).

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