### Overview

A Rotating Ring-Disk Electrode (RRDE) is a special type of working electrode used for analytical purposes in an electrochemical cell. Spinning a disk at high speed in an electrolyte solution sets up a well-defined mass transport flow condition. Material is initially transported by the solution flow to the disk electrode and then subsequently past the ring electrode.

### Collection Efficiency

The theoretical collection efficiency can be computed from the three principle diameters describing the RRDE geometry: the disk outer diameter ($d_1$), the ring inner diameter ($d_2$), and the ring outer diameter ($d_3$). This somewhat tedious computation is made easier by normalizing the ring diameters with respect to the disk diameter

${\sigma}_{OD} = \frac{d_3}{d_1}$

${\sigma}_{ID} = \frac{d_2}{d_1}$

and by defining three additional quantities in terms of the normalized diameters

${\sigma}_{A} = {\sigma}_{ID}^3 - 1$

${\sigma}_{B} = {\sigma}_{OD}^3 - {\sigma}_{ID}^3$

${\sigma}_{C} = \frac{{\sigma}_{A}}{{\sigma}_{B}}$

and by defining a complex function, $G(x)$, as follows:

$G(x) = \frac{1}{4} + \left(\frac{\sqrt{3}}{4{\pi}}\right) ln \left[{\frac{(x^{1/3} + 1)^3}{x+1}}\right] + \frac{3}{2{\pi}} {\arctan}\left[{\frac{2x^{1/3}-1}{\sqrt{3}}}\right]$

In terms of the normalized quantities and complex function above, the theoretical collection efficiency ($N_{theoretical}$) for a rotating ring disk electrode is given by the following equation:

$N_{theoretical} = 1-{\sigma}_{OD}^2+{\sigma}_B^{2/3}-G({\sigma}_C)-{\sigma}_B^{2/3}G({\sigma}_A)+{{\sigma}_{OD}^2}G({\sigma}_C{\sigma}_{OD}^3)$

### References

Allen J. Bard and Larry R. Faulkner, Electrochemical Methods: Fundamentals and Applications, New York: Wiley, 2001, 2nd ed., Chapter 9.