The excess current in cyclic voltammetry arising from the presence of an electrode edge

Abstract

The popular inlaid disc electrode suffers from an edge effect that is usually, and sometimes unwarrantedly, ignored in analyzing transient voltammograms. This study addresses the role played by the edge in linear scan and cyclic voltammetries when the electron transfer is reversible or quasi-reversible. A simulation models the concentrations, current densities and currents in two circumstances—when the edge in important and when it is absent—simultaneously, and thereby the evolving edge current is quantified. Special attention is paid to the effect that the edge has on the heights and positions of the voltammetric peaks. It is demonstrated that disregarding the edge may lead to the bogus classification of a reversible electrode reaction as quasi-reversible.

Appendices

Appendix A

Here, we demonstrate that the approximations represented by the terms on right-hand side of Eq. (4:1) may be justified by assuming that quadratic equations link the dependent variable c _P to each of the spatial variables, r and z.

Consider the dependence of the product concentration c _P on the vertical coordinate z, at constant values of the r and t variables. Assume that, in the immediate vicinity of a particular value of z, say within the region between Z − Δ and Z + Δ, the relationship between concentration and distance may be represented by the quadratic equation

$$ {c}_{\mathrm{P}}\left(r,z\;\mathrm{near}\;Z,t\right)={w}_0+{w}_1z+{w}_2{z}^2 $$

(A:1)

with w ₀, w ₁ and w ₂ being local constants. Note the distinction between Z, a particular value of the coordinate, and z, which here is any value of the coordinate close to Z. In addition to

$$ {c}_{\mathrm{P}}\left(r,\;Z,\;t\right)={w}_0+{w}_1Z+{w}_2{Z}^2, $$

(A:2)

two similar equations also apply; these differ from A:2 in having Z replaced by Z − Δ or Z + Δ. The three w coefficients may now be found by solving the trio of equations simultaneously. When the results of this exercise are substituted back into Eq. (A:1), the relation

$$ \begin{array}{c}\hfill {c}_{\mathrm{P}}\left(r,\;z\;\mathrm{near}\;Z,\;t\right)=\left[1-\frac{{\left(z-Z\right)}^2}{\varDelta^2}\right]{c}_{\mathrm{P}}\left(r,Z,t\right)\hfill \\ {}\hfill +\left[\frac{z-Z}{2\varDelta }+\frac{{\left(z-Z\right)}^2}{2{\varDelta}^2}\right]{c}_{\mathrm{P}}\left(r,Z+\varDelta, t\right)-\left[\frac{z-Z}{2\varDelta }-\frac{{\left(z-Z\right)}^2}{2{\varDelta}^2}\right]{c}_{\mathrm{P}}\left(r,Z-\varDelta, t\right)\hfill \end{array} $$

(A:3)

emerges. This fitted quadratic equation describes how c _P depends on z in the vicinity of z = Z.

Equation (A:3) may be differentiated once:

$$ \begin{array}{c}\hfill \frac{\partial }{\partial z}{c}_{\mathrm{P}}\left(r,z\;\mathrm{near}\;Z,t\right)=-\frac{2\left(z-Z\right)}{\varDelta^2}{c}_{\mathrm{P}}\left(r,Z,t\right)\hfill \\ {}\hfill +\left[\frac{1}{2\varDelta }+\frac{z-Z}{\varDelta^2}\right]{c}_{\mathrm{P}}\left(r,Z+\varDelta, t\right)-\left[\frac{1}{2\varDelta }-\frac{z-Z}{\varDelta^2}\right]{c}_{\mathrm{P}}\left(r,Z-\varDelta, t\right)\hfill \end{array}, $$

(A:4)

and twice:

$$ \frac{\partial^2}{\partial {z}^2}{c}_{\mathrm{P}}\left(r,z\;\mathrm{near}\;Z,t\right)=-\frac{2}{\varDelta^2}{c}_{\mathrm{P}}\left(r,Z,t\right)+\frac{1}{\varDelta^2}{c}_{\mathrm{P}}\left(r,Z+\varDelta, t\right)+\frac{1}{\varDelta^2}{c}_{\mathrm{P}}\left(r,Z-\varDelta, t\right). $$

(A:5)

Equations (A:4) and (A:5) apply for any value of z close to Z including, of course, when z = Z. In the latter eventuality, the equations reduce to

$$ \frac{\partial }{\partial z}{c}_{\mathrm{P}}\left(r,z,t\right)=\frac{c_{\mathrm{P}}\left(r,z+\varDelta, t\right)-{c}_{\mathrm{P}}\left(r,z-\varDelta, t\right)}{2\varDelta }, $$

(A:6)

and

$$ \frac{\partial^2}{\partial {z}^2}{c}_{\mathrm{P}}\left(r,z,t\right)=\frac{c_{\mathrm{P}}\left(r,z+\varDelta, t\right)+{c}_{\mathrm{P}}\left(r,z-\varDelta, t\right)-2{c}_{\mathrm{P}}\left(r,z,t\right)}{\varDelta^2}. $$

(A:7)

This derivation of Eq. (A:7) is a vindication of the final term in approximation 4:1. Likewise, the penultimate term in 4:1 is justified by the radial equivalent of Eq. (A:6).

This appendix is not presented for its own sake; most readers will accept approximations 4:1 without algebraic justification. Rather, the above algebra provides a preamble to the derivation that follows.

Appendix B

Equation (A:3) was derived by fitting a quadratic through the three points z = Z − Δ, z = Z and z = Z + Δ. If we repeat the procedure, but instead fit the quadratic to the three unevenly spaced points $z=0,\kern0.24em z={\scriptscriptstyle \frac{1}{2}}\varDelta \kern0.24em \operatorname{and}\kern0.24em z={\scriptscriptstyle \frac{3}{2}}\varDelta,$ the analogue of Eq. (A:3) turns out to be

$$ \begin{array}{c}\hfill {c}_{\mathrm{P}}\left(r,\;0\le z\le {\scriptscriptstyle \frac{3}{2}}\varDelta,\;t\right)=\frac{3z\varDelta -2{z}^2}{\varDelta^2}{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)\hfill \\ {}\hfill -\frac{z\varDelta -2{z}^2}{3{\varDelta}^2}{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)+\left[1-\frac{8z\varDelta -4{z}^2}{3{\varDelta}^2}\right]{c}_{\mathrm{P}}\left(r,0,t\right)\hfill \end{array} $$

(B:1)

The first and second derivatives are

$$ \frac{\partial }{\partial z}{c}_{\mathrm{P}}\left(r,0\le z\le {\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)=\frac{3\varDelta -4z}{\varDelta^2}{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)-\frac{\varDelta -4z}{3{\varDelta}^2}{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)-\frac{8\varDelta -8z}{3{\varDelta}^2}{c}_{\mathrm{P}}\left(r,0,t\right), $$

(B:2)

and

$$ \frac{\partial^2}{\partial {z}^2}{c}_{\mathrm{P}}\left(r,0\le z\le {\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)=-\frac{4}{\varDelta^2}{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)+\frac{4}{3{\varDelta}^2}{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)+\frac{8}{3{\varDelta}^2}{c}_{\mathrm{P}}\left(r,0,t\right). $$

(B:3)

These are the equations that we use to replace (A:6) and (A:7) close to z = 0 boundary.

When that boundary is with the electrode, Eq. (B:3) provides a means of discretizing the final term in Eq. (2:2). Thereby, a replacement for Eq. (4:1), valid for $0\le z\le {\scriptscriptstyle \frac{3}{2}}\varDelta,$ is

$$ \begin{array}{c}\hfill \frac{c_{\mathrm{P}}\left(r,z,t+\delta \right)-{c}_{\mathrm{P}}\left(r,z,t\right)}{D\delta}\approx \frac{c_{\mathrm{P}}\left(r+\varDelta, z,t\right)+{c}_{\mathrm{P}}\left(r-\varDelta, z,t\right)-2{c}_{\mathrm{P}}\left(r,z,t\right)}{\varDelta^2}\hfill \\ {}\hfill \frac{c_{\mathrm{P}}\left(r+\varDelta, z,t\right)-{c}_{\mathrm{P}}\left(r-\varDelta, z,t\right)}{2r\varDelta }+\frac{4{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)+8{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)-12{c}_{\mathrm{P}}\left(r,0,t\right)}{3{\varDelta}^2}\hfill \end{array}. $$

(B:4)

After choosing z = ${\scriptscriptstyle \frac{1}{2}}\varDelta,$ discretization and rearrangement, Eq. (5:3) of the main text results.

On the insulator surface, the flux of product is zero, enabling the z = 0 instance of Eq. (B:2) to yield the identity

$$ {c}_{\mathrm{P}}\left(r>a,0,t\right)={\scriptscriptstyle \frac{9}{8}}c\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)-{\scriptscriptstyle \frac{1}{8}}c\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right). $$

(B:5)

On conversion to the discrete notation, this becomes Eq. (5:7) of the main text.

Equation (B:2) provides one expression, namely

$$ \frac{\partial }{\partial z}{c}_{\mathrm{P}}\left(r,0,t\right)=\frac{3}{\varDelta }{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)-\frac{1}{3\varDelta }{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)-\frac{8}{3\varDelta }{c}_{\mathrm{P}}\left(r,0,t\right), $$

(B:6)

for the gradient of the product concentration at z = 0; the boundary condition on the electrode surface, Eq. (2:7), provides another. Combining these two expressions to eliminate the concentration gradient leads to the result

$$ \left[\frac{8}{3\varDelta }+\frac{k^{\mathrm{o}}\left[1+\xi (t)\right]}{D{\xi}^{\alpha }(t)}\right]{c}_{\mathrm{P}}\left(r,0,t\right)-\frac{3}{\varDelta }{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{1}{2}}\varDelta, t\right)+\frac{1}{3\varDelta }{c}_{\mathrm{P}}\left(r,{\scriptscriptstyle \frac{3}{2}}\varDelta, t\right)=\frac{k^{\mathrm{o}}\xi (t)}{D{\xi}^{\alpha }}{c}^{\mathrm{b}} $$

(B:7)

that interrelates three contemporaneous concentrations. Equation (5:6) of the main text is the discrete version of this equation, after rearrangement.