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RRDE Collection Efficiency Calculator

RRDE Collection Efficiency Calculator

This calculator predicts the theoretical collection efficiency (N) for any rotating ring-disk electrode (RRDE). Theoretical collection efficiency is a numerical value computed from three inputs: disk radius, ring inner radius, and ring outer radius. Below, use the calcaultor to compute the theoretical collection efficiency of an RRDE configuration by suppling these values. Later, you can also view the collection efficiency for a variety of current and legacy Pine Research RRDEs. We recommend reading our helpful support articles on the topic of rotating electrodes, including RRDE: Rotating Ring Disk Electrode Fundamentals and Comparing Two Competing Pathways by RRDE.

Calculate N for any RRDE

Dimensional drawing showing r<sub>1</sub>, r<sub>2</sub>, and r<sub>3</sub>.

Enter the disk and ring dimensions below (in mm) and click the calculate button. Refer to the image for the proper identification of disk radius (r1), ring inner radius (r2), and ring outer radius (r3).

Disk Radius
r1
Ring Inner Radius
r2
Ring Outer Radius
r3
Collection Efficiency (N)
ResetCalculate

View Collection Efficiency for Pine Research RRDE

From the dropdown below, select a Rotating Ring-Disk Electrode (RRDE) series to view dimensions and the Collection Efficiency (N).

Disk Radius
r1 (mm)
Ring Inner Radius
r2 (mm)
Ring Outer Radius
r3 (mm)
α
β
N

Empirical Determination of Collection Efficiency

It is critically important to empirically determine the collection efficiency of an RRDE, and to periodically verify this value. N is specific to an electrode. Even electrodes of the same series may have slightly different N, based on the history of use, care, and polishing sucess.

To empirically determine N, measure the limiting current at the disk and ring for a simple, well-behaved, and reversible electrochemical reaction where R is stable,
\displaystyle{O+ne\rightleftharpoons R} (disk)
\displaystyle{R\rightleftharpoons O + ne} (ring)

Once the limiting current has been measured, the relationship between ring and disk current is a constant for that particular RRDE, which is collection efficiency (N),

\displaystyle{N=\frac{-i_R}{i_D}}

Theory, Background, and Mathematics

The following description has been adapted from Bard, A.J., Faulkner, L.R., and White, H.S. Electrochemical Methods: Fundamentals and Applications. 3rd Edition. 2022. John Wiley & Sons, Ltd. Theoretical collection efficiency can be found numerically, requiring only disk radius (r1), ring inner diameter (r2), and ring outer diameter (r3). N is empirically calcualted as,

\displaystyle{N=1-F\left(\frac{\alpha}{\beta}\right)+\beta^{2/3}\left[1-F(\alpha)\right]-{(1+\alpha+\beta)}^{2/3}\left\lbrace1-F\left[\left(\frac{\alpha}{\beta}\right)(1+\alpha+\beta)\right]\right\rbrace}

where

\displaystyle{\alpha={\left(\frac{r_2}{r_1}\right)}^{3}-1}
and
\displaystyle{\beta=\left(\frac{r_3^3}{r_1^3}-\frac{r_2^3}{r_1^3}\right)}
and
\displaystyle{F(\theta)=\left(\frac{\sqrt{3}}{4\pi}\right)\ln{\left\lbrace\frac{{\left(1+\theta^{1/3}\right)}^3}{1+\theta}\right\rbrace}+\frac{3}{2\pi}\arctan{\left(\frac{2\theta^{1/3}-1}{3^{1/2}}\right)}+\frac{1}{4}}

While the math is rather straightforward, the functions are a bit cumbersome. This is why we have offered this calcualtor online - for our existing RRDE electrodes (current and legacy inventory), and for you to enter your own radii and determine empirical N from those values.

References

The theory of rotating ring-disk is quite interesting. In 2016 the world celebrated the semi-centennial (50th year) celebration of the rotating ring-disk. The General Manager of Pine Research, Frank Dalton, embarked on a historical journey to study and document the interesting history of the technique, which included personal interviews with folks around the world. He published this this work in ECS Transactions: Historical Origins of the Rotating Ring-Disk Electrode. There are thousands of high-quality research articles based on rotating ring-disk electrochemistry. In fact too many to list here. For a robust scientific background on the development of RRDE, readers are encouraged to review Albery's collection (tome) of papers on the topic:

  1. Albery, W. J. Ring-Disc Electrodes. Part 1.—A New Approach to the Theory. Trans. Faraday Soc. 1966, 62, 1915–1919.
  2. Albery, W. J.; Bruckenstein, S. Ring-Disc Electrodes. Part 2.—Theoretical and Experimental Collection Effciencies. Trans. Faraday Soc. 1966, 62, 1920–1931.
  3. Albery, W. J.; Bruckenstein, S.; Napp, D. T. Ring-Disc Electrodes. Part 3. - Current-Voltage Curves at the Ring Electrode with Simultaneous Currents at the Disc Electrode. Trans. Faraday Soc. 1966, 62, 1932–1937.
  4. Albery, W. J.; Bruckenstein, S.; Johnson, D. C. Ring-Disc Electrodes. Part 4.—Diffusion Layer Titration Curves. Trans. Faraday Soc. 1966, 62, 1938–1945.
  5. Albery, W. J.; Bruckenstein, S. Ring-Disc Electrodes. Part 5.—First-Order Kinetic Collection Effciencies at the Ring Electrode. Trans. Faraday Soc. 1966, 62, 1946–1954.
  6. Albery, W. J.; Bruckenstein, S. Ring Disc Electrodes. Part 6.—Second-Order Reactions. Trans. Faraday Soc. 1966, 62, 2584–2595.
  7. Albery, W. J.; Bruckenstein, S. Ring-Disc Electrodes. Part 7.—Homogeneous and Heterogeneous Kinetics. Trans. Faraday Soc. 1966, 62, 2596–2606.
  8. Albery, W. J. Ring-Disc Electrodes. Part 8.—Transient Currents and First-Order Kinetics. Trans. Faraday Soc. 1967, 63, 1771–1781.
  9. Albery, W. J.; Hitchman, M. L.; Ulstrup, J. Ring-Disc Electrodes. Part 9.—Application to First-Order Kinetics. Trans. Faraday Soc. 1968, 64, 2831–2840.
  10. Albery, W. J.; Hitchman, M. L.; Ulstrup, J. Ring-Disc Electrodes. Part 10.—Application to Second-Order Kinetics. Trans. Faraday Soc. 1969, 65, 1101–1112.
  11. Albery, W. J. Ring-Disc Electrodes. Part 11.—General Theory of Transient Currents. Trans. Faraday Soc. 1971, 67, 153–160.
  12. Albery, W. J.; Drury, J. S.; Hitchman, M. L. Ring-Disc Electrodes. Part 12.—Application to Ring Current Transients. Trans. Faraday Soc. 1971, 67, 161–165.
  13. Albery, W. J.; Drury, J. S.; Hitchman, M. L. Ring-Disc Electrodes. Part 13.—The Laplace Transformation of Transients. Trans. Faraday Soc. 1971, 67, 166–169.
  14. Albery, W. J.; Drury, J. S.; Hitchman, M. L. Ring Disc Electrodes. Part 14.—Kinetic and Transient Parameters. Trans. Faraday Soc. 1971, 67, 2162–2166.
  15. Albery, W. J.; Drury, J. S.; Hutchinson, A. P. Ring Disc Electrodes. Part 15.—Alternating Current Measurements. Trans. Faraday Soc. 1971, 67, 2414–2418.
  16. Albery, W. J.; Drury, J. S. Ring-Disc Electrodes. Part 16.—A Comparison of Analytical and Numerical Solutions. J. Chem. Soc., Faraday Trans. 1 1972, 68, 456–464.
  17. Albery, W. J.; Bruckenstein, S.; Tokuda, K. Ring–Disc Electrodes. Part 17.—Ring response to periodic disc electrode forcing functions. J. Chem. Soc., Faraday Trans. 1 1977, 73, 823-829.
  18. Albery, W. J.; Compton, R. G.; Hillman, A. R. Ring–Disc Electrodes. Part 18.—Collection Efficiency for High Frequency a.c. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1007–1019.
  19. Albery, W. J.; Hillman, A. R. Ring–Disc Electrodes. Part 19.—Adsorption Studies at Low Frequency A.C. J. Chem. Soc., Faraday Trans. 1 1979, 75, 1623–1634.
  20. Albery, W. J.; Boutelle, M. G.; Colby, P. J.; Hillman, A. R. Ring–Disc Electrodes. Part 20.—A General Procedure for Deducing the Faradaic Component of a Disc-Current Transient from a Ring-Current Transient. J. Chem. Soc., Faraday Trans. 1 1982, 78(9), 2757–2763.
  21. Albery, W. J.; Calvo, E. J. Ring–Disc Electrodes. Part 21.—pH Measurement with the Ring. J. Chem. Soc., Faraday Trans. 1 1983, 79(11), 2583–2596.
  22. Albery, W. J.; Mount, A. R. Ring–Disc Electrodes. Part 22.—Theory of the Measurement of Proton Fluxes at the Disc. J. Chem. Soc., Faraday Trans. 1 1989, 85(5), 1181–1188.
  23. Albery, W. J.; Mount, A. R. Ring–Disc Electrodes. Part 23.—Studies of Proton Fluxes at a Thionine-Coated Electrode. J. Chem. Soc., Faraday Trans. 1 1989, 85(5), 1189–1198.
  24. Albery, W. J.; Mount, A. R. Ring–Disc Electrodes. Part 24.—Studies of Counterion Fluxes at a Thionine-Coated Electrode. J. Chem. Soc., Faraday Trans. 1 1989, 85(11), 3717–3724.

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