Diffusion layer, rotating disc

The diffusion layer widtli is very much dependent on tire degree of agitation of tire electrolyte. Thus, via tire parameter 5, tire hydrodynamics of tire solution can be considered. Experimentally, defined hydrodynamic conditions are achieved by a rotating cylinder, disc or ring-disc electrodes, for which analytical solutions for tire diffusion equation are available [37, 4T, 42 and 43]. 

The dissolution rate of a solid from a rotating disc is governed by the controlled hydrodynamics of the system, and it has been treated theoretically by Levich [104]. This theory considers only forced convection due to rotation and ignores natural convection, which may occur at low speeds of rotation. Figure 16 shows the solvent flow held near the surface of the rotating disc. The apparent thickness, h, of the diffusion layer next to the surface of the disc is given by... 

At a rotating disc the thickness of the diffusion layer decreases with increasing rotation rate according to ... 

In contrast to the rotating disc electrode, mass transport to the ring is nonuniform. Nevertheless, the thickness of the diffusion layer Spj, which depends on the coordinate x in the direction of flow, and the rate of mass transport can be calculated. We consider a simple redox reaction, and rewrite Eq. (14.5) in the form ... 

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 ... 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

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... 

Disc electrodes are commonly used in voltammetry as stationary and as rotating electrodes. The diffusion of electroactive species towards the surface of these electrodes is linear, as shown in Fig. 1.6a. The advantage of the second configuration is that rotation of the electrode causes convection in solution that compensates for the increase of the diffusion layer thickness with time after a period of about 200 ms. This results in a limiting current instead of a peak-shaped current (see also section 3.2) according to the Levich equation3 ... 

According to V.G. Levich299 (see also Refs 300, 302, 304), the thickness of the diffusion boundary layer at the rotating disc surface is determined by the equation... 

Another example for the HMRRD electrode is given in Fig. 9 for Fe in alkaline solutions [12, 27]. The square wave modulation of the rotation frequency co causes the simultaneous oscillation of the analytical ring currents. They are caused by species of the bulk solution. Additional spikes refer to corrosion products dissolved at the Fe disc. This is a consequence of the change of the Nemst diffusion layer due to the changes of co. This pumping effect leads to transient analytical ring currents. Besides qualitative information, also quantitative information on soluble corrosion products may be obtained. The size of the spikes is proportional to the dissolution rate at the disc, as has been shown by a close relation of experimental results and calculations [28-30]. As seen in Fig. 7, soluble Fe(II) species are formed in the po-... 

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. 

This type of diffusion/reaction mechanism has been treated semi-analyti-cally by Albery et al. [42, 44, 45], under steady-state conditions and its applications to amperometric chemical sensors has been described by Lyons et al. [46]. In both models, only diffusion and reaction within a boundary layer is considered, while the effect of concentration polarisation in the solution is neglected. Thus, to apply the model to an experimental system it is necessary to be able to accurately determine the concentration of substrate at the polymer/solution interface. Assuming that the system is in the steady state, the use of the rotating disc electrode allows simple determination of the substrate concentration at the interface from the bulk concentration and the experimentally determined flux using [47]... 

For a -> rotating disc electrode (RDE) the - convective-diffusion equations can be solved which gives the dependence of the diffusion layer thickness on the angular velocity of the rotation (on)... 

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. 

If the reaction has been studied at a rotating disc catalyst spinning at a speed f (Sect. 1.6.3), the diffusion layer thickness Sj is given by the Levich equation (49) so that... 

For details and an exact derivation of the reader is referred to ref. [13]. The derivation also shows that Z is in series with as shown in Fig. 4.13a. Typically, the Warburg impedance leads to a linear increase of Z with rising Z" and the slope is 45° as also shown in Fig, 4.13a. In this case, Z has been calculated assuming an infinite thickness of the diffusion layer. Any convection of the liquid limits the thickness of the diffusion layer. The latter is limited to a well defined value when a rotating disc electrode is used (see Section 4.2.3). In this case, the impedance spectrum is bent off at low frequencies as shown in Fig. 4.13b. The Z branch i.s only linear at its high frequency end where it shows a slope of 45°. 

Fig. 3. (a) A rapid transition in the number of electrons transferred (ntS) is observed at the uniformly accessible rotating disc electrode when the diffusion layer becomes sufficiently thin for the intermediate (shown stippled) to survive long enough to cross it (b) A more gradual transition is seen at the channel electrode, where some intermediate escapes (upstream electrode edge) and some is always trapped (downstream electrode edge), due to the shape of the diffusion layer. 

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