    # Rotating Disk Electrode (RDE) Theory

Last Updated: 9/1/20 by Alex Peroff

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1. General Theory of Rotating Disk Electrochemistry

### 1General Theory of Rotating Disk Electrochemistry

The general theory describing mass transport at a rotating disk electrode (RDE) was developed by Benjamin Levich at the Institute of Electrochemistry at the Academy of Sciences of the USSR.  Levich described the theory in his landmark book, Physiochemical Hydrodynamics, originally published in Russian in 1952.  Ten years later, Levich’s book was translated from Russian to English, and the RDE became more widely known to Western researchers. In the early 1960’s, Stanley Bruckenstein at the University of Minnesota (and his students Dennis Johnson and Duane Napp) and Ronnie Bell at Oxford University (and his student John Albery) began working with rotating electrodes.  Subsequent generations of researchers expanded on this initial work, and the rotating disk electrode has since grown into a mature tool for probing electrochemical reaction kinetics. The laminar flow at a rotating disk electrode conveys a steady stream of material from the bulk solution to the electrode surface.  While the bulk solution far away from the electrode remains well-stirred by the convection induced by rotation, the portion of the solution nearer to the electrode surface tends to rotate with the electrode.  Thus, if the solution is viewed from the frame of reference of the rotating electrode surface, then the solution appears relatively stagnant.  This relatively stagnant layer is known as the hydrodynamic boundary layer, and its thickness δH can be approximated, $\displaystyle{\delta_H = 3.6 (\nu/\omega)^{1/2}}$ (1)
in terms of the kinematic viscosity (ν) of the solution and the angular rotation rate (ω = 2πf/60, where f is the rotation rate in revolutions per minute).  In an aqueous solution at a moderate rotation rate (~1000 RPM), the stagnant layer is approximately 300 to 400 µm thick.

Net movement of material to the electrode surface can be described mathematically by applying general convection-diffusion concepts from fluid dynamics.  Mass transport of material from the bulk solution into the stagnant layer occurs by convection due to the stirring generated by the rotating electrode.  But after the material enters the stagnant layer and moves closer to the electrode surface, convection becomes less important and diffusion becomes more important.  The final movement of an ion or molecule to the electrode surface is dominated by diffusion across a very thin layer of solution immediately adjacent to the electrode known as the diffusion layer.

The diffusion layer is much thinner than the hydrodynamic layer.  The diffusion layer thickness δF can be approximated as follows, $\displaystyle{\delta_F = 1.61 {D_F^{1/3}}\nu^{1/6}\omega^{-1/2}}$ (2)
in terms of the diffusion coefficient DF of the molecule or ion.  For a molecule or ion with a typical diffusion coefficient (DF ≈ 10-5 cm2/s) in an aqueous solution, the diffusion layer is about twenty times thinner than the stagnant layer (δF ≈ 0.05 δH).

The first mathematical treatment of convection and diffusion towards a rotating disk electrode was given by Levich.  Considering the case where only the oxidized form of a molecule (or ion) of interest is initially present in the electrochemical cell, the cathodic limiting current (iLC) observed at a rotating disk electrode is given by the Levich equation,  $\displaystyle{i_{LC} = 0.620 n F A D_O^{2/3}\nu^{-1/6}\omega^{1/2}C_O}$ (3)
in terms of the concentration of the oxidized form in the solution (CO), the electrode area (A), the Faraday constant (F = 96485 C/mol), the kinematic viscosity of the solution (ν), the diffusion coefficient (DO) of the oxidized form, and the angular rotation rate (ω).  Alternatively, when the solution initially contains only the reduced form, the equation for the anodic limiting current (iLA) can be written as $\displaystyle{i_{LA} = 0.620 n F A D_O^{2/3}\nu^{-1/6}\omega^{1/2}C_R}$ (4)
where the concentration and diffusion coefficient terms (CR and DR) refer to the reduced form rather than the oxidized form.

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