Rotating disc electrode hydrodynamics

Levich124 has given the relationships between the limiting current i) and the bulk concentration C of the metal ion for plate electrodes, conical electrodes and rotated disc electrodes (RDEs) under hydrodynamic conditions anticipating his well known equations treated in Section 3.3.2.2 on hydrodynamic electrodes, we may assume the relationships concerned using the more general equation... 

The convective diffusion theory was developed by V.G. Levich to solve specific problems in electrochemistry encountered with the rotating disc electrode. Later, he applied the classical concept of the boundary layer to a variety of practical tasks and challenges, such as particle-liquid hydrodynamics and liquid-gas interfacial problems. The conceptual transfer of the hydrodynamic boundary layer is applicable to the hydrodynamics of dissolving particles if the Peclet number (Pe) is greater than unity (Pe > 1) (9). The dimensionless Peclet number describes the relationship between convection and diffusion-driven mass transfer ... 

None of the set-ups discussed so far provides stirring of the electrolyte for bubble removal or for enhancement of the reaction rates. A standard set-up developed to study kinetic electrode processes is the rotating disc electrode [11]. The electrode is a small flat disc set in a vertical axle. The hydrodynamic flow pattern at the disc depends on rotation speed and can be calculated. An additional ring electrode set at a different potential provides information about reaction products such as, for example, hydrogen. However, because this set-up is designed to study kinetic processes and is usually equipped with a platinum disc, it becomes inconvenient if silicon samples of different geometries have to be mounted. 

To appreciate that, experimentally, the best way to perform analyses at the rotated disc electrode (the most popular hydrodynamic electrode) is at a constant rotational frequency and with the face of the disc well below the surface of the liquid. 

The rotating disc electrode is constructed from a solid material, usually glassy carbon, platinum or gold. It is rotated at constant speed to maintain the hydrodynamic characteristics of the electrode-solution interface. The counter electrode and reference electrode are both stationary. A slow linear potential sweep is applied and the current response registered. Both oxidation and reduction processes can be examined. The curve of current response versus electrode potential is equivalent to a polarographic wave. The plateau current is proportional to substrate concentration and also depends on the rotation speed, which governs the substrate mass transport coefficient. The current-voltage response for a reversible process follows Equation 1.17. For an irreversible process this follows Equation 1.18 where the mass transfer coefficient is proportional to the square root of the disc rotation speed. 

As is thoroughly discussed in Chap. 2 of this volume, the convective diffusion conditions can be controlled under steady state conditions by use of hydrodynamic electrodes such as the rotating disc electrode (RDE), the wall-jet electrode, etc. In these cases, steady state convective diffusion is attained, becomes independent of time, and solution of the convective-diffusion differential equation for the particular electrochemical problem permits separation of transport and kinetics from the experimental data. 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

Hydrodynamic modulation has been performed almost exclusively at the rotating disc electrode. It has found use for analytical purposes at rotating and tubular electrodes owing to the fact that non-convectively dependent electrode processes are unaffected by the modulation [236]. 

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... 

In Sections 6.3-6.5 expressions for the analysis of the voltammograms corresponding to the simple electron transfer process O + ne-— R, obtained for uniformly accessible electrodes such as the rotating disc electrode, were presented. In this section these expressions will be applied to hydrodynamic electrodes in general. 

In this last case the use of a double hydrodynamic electrode, generating R on the upstream electrode and detecting it on the downstream electrode, may be easier and more sensitive. The rotating disc electrode has also been used with success to distinguish similar mechanisms with coupled homogeneous reactions (ECE, DISP1, and DISP2)5. 

Hydrodynamic electrodes permit the control of the diffusion layer thickness by imposing convection. This thickess can also be modulated. Implicit functions link the current, potential and convection modulation. For the rotating disc electrode... 

The last of these is the impedance which has been considered throughout this chapter. We now consider forced convection. For low frequencies the diffusion layer thickness due to the a.c. perturbation is similar to that of the d.c. diffusion layer in these cases convection effects will be apparent in the impedance expressions. For the rotating disc electrode these frequencies are lower than 40 Hz33. For higher frequencies where the two diffusion layers are of quite different thicknesses, the advantage of hydrodynamic electrodes is that transport is well defined with time, as occurs with linear sweep voltammetry. 

HMRDE hydrodynamically modulated rotating disc electrode... 

Miller and Bruckenstein [27,28] introduced the hydrodynamically modulated rotating disc electrode (HMRDE) in 1974. The steady-state current density at a rotating disc electrode is well-defined, given by the Levich equation (equation (10.15)) ... 

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. 

Velocity (v) — is a vector measure of the rate of change of the position of a point with respect to time. For cartesian space the velocity of a point (x) can be written as v = dx/dt and has units of m s-1 using the SI system. In polar coordinates a two-dimensional velocity can be represented by an angular velocity (to) and the distance to the origin (r), v = cor. Velocity is found widely within electrochemical analysis, for example, within hydrodynamic devices such as the rotating disc electrode where the solution velocity may often be approximated analytically [i, ii], permitting, via further analysis, cur-rent/voltage characteristics to be calculated. 

Altering the convective rate of transport, e.g. by changing the rotation frequency of a rotating-disc electrode. Experiments in which the convective rate of transport can be altered are known as hydrodynamic techniques. 

Working electrodes which have material reaching them by a form of forced convection are known as hydrodynamic electrodes. There is a wide range of hydrodynamic electrodes rotating-disc electrodes (Albery and Hitchman, 1971), in which the electrode rotates at a fixed frequency and sucks up material to its surface, and channel electrodes (Compton et al., 1993c), over which the electroactive species flows at a fixed volume flow rate, are the primary ones used in the work described in this review (Section 4). 

Examples of electrode reaction mechanisms consisting of extensive combinations of E and C steps 42 Hydrodynamic voltammetry 44 Rotating-disc electrodes 46 Channel electrodes 48 Wall jet electrodes 52 Electron-transfer processes 53... 

Both qualitative and quantitative insight can be garnered from transient X -i, i-t and r -t measurements in quiescent or stirred solutions, while measurements of steady-state behavior are best performed under well-defined hydrodynamic conditions. Typically, a rotating disc electrode (RDE), or a related method, is used to specify and/or modulate the hydrodynamic boundary layer thickness, 8. With an RDE the boundary layer is specified by... 

Fig. 1. The velocity pattern for a rotating-disc electrode. The functions F, G, and H describe the radial, angular, and normal components, respectively. The distance normal to the electrode, z, is normalised with the thickness of the hydrodynamic layer, ZH, where ZH is given by vft2itW) and v/cm2s-1 and W/Hz are the kinematic viscosity and rotation speed, respectively.

Let us first consider briefly how the use of mass transport as a variable can provide a guide to the reaction mechanism and give quantitative kinetic detail. As an illustration, we consider the behaviour of CE and EC processes (where E signifies electron transfer and C represents a chemical step) at a rotating disc electrode (RDE). This hydrodynamic system has already been discussed by Albery et al. and the reader is referred to Chap. 4 for details. CE and EC processes represent the simplest conceivable electrode reactions involving coupled homogeneous kinetics mechanistic examples of both types are shown in Table 1. In the discussion which follows, the electron-transfer reaction in the two mechanisms is considered to be a cathodic process the extension to the anodic case is trivial. 

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