Rotating disc electrode: derivation

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°. 

As already mentioned before, the diffusion of the redox species can be enhanced by disturbing the solution. The most well-defined mass transport is obtained by using a rotating disc electrode as described in Section 4.2.3). As derived at first by Levich, the... 

The equations derived for the rotating-disc electrode are limited for laminar flow. The Reynolds number was introduced as a measure for the transition to turbulent flow (Section 5.1). For the rotating disc the Reynolds number is given by the equation... 

The exact solution of the convection-diffusion equations is complicated, because the theoretical treatments involve solving a hydrodynamic problem, that is, the determination of the solution flow velocity profile by using Navier-Stokes equation. For the calculation of a velocity profile, the solution viscosity, densities, rotation rate, or stirring rate, as well as the shape of the electrode should be considered. Exact solution has been derived for the rotating disc electrode (RDE) ... 

The rotating ring—disc electrode (RRDE) is probably the most well-known and widely used double electrode. It was invented by Frumkin and Nekrasov [26] in 1959. The ring is concentric with the disc with an insulating gap between them. An approximate solution for the steady-state collection efficiency N0 was derived by Ivanov and Levich [27]. An exact analytical solution, making the assumption that radial diffusion can be neglected with respect to radial convection, was obtained by Albery and Bruckenstein [28, 29]. We follow a similar, but simplified, argument below. 

This derivation makes a number of assumptions. Firstly, we assume that there is no disruption to the laminar flow pattern due to a finite disc surface, finite cell size, or eccentricity in disc rotation. To what extent design factors affect measured currents will be discussed further in the section on electrode construction. It is sufficient at this point to say that the criteria for negligible disruption can be met. 

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