General Correction for Electrode Sphericity in Voltammetry of General Correction for Electrode Sphericity in Voltammetry of Nernstian Systems Nernstian Systems

The current is considered at a stationary reversible spherical electrode whose potential E ( t ) as a function of time is given, such that E ( t ) determines the ratio of oxidized to reduced species at the electrode surface. Writing the current as that for planar geometry, I 0 , plus corrections for sphericity, we derive formulae for the corrections. The first two are expressed as integrals over I 0 , with no explicit dependence on the potential, for any form of E ( t )


Introduction
To compute the current at a reversible stationary spherical microelectrode with changing potential E(t), one must solve the diffusion equation for oxidized and reduced species in the solution, coupled by the Nernst equilibrium conditions on the concentrations at the electrode surface. The equations for linear diffusion (planar symmetry), being simpler than those for other geometries, an often used strategy [1][2][3][4][5] is to solve the linear diffusion problem and then generate corrections to the current for deviations of the electrode from planarity. However, some authors [2,6] have solved the problem for spherical diffusion directly in certain cases, and the diffusion problems themselves have been studied in detail [4,7,8].
We present here a general procedure for obtaining the leading corrections to the planar result for any variation of electrode potential E with time, for any ratio of diffusion coefficients of oxidized and reduced species, and for diffusion of reduced species into the electrode or into the solution. In all cases, the corrections are expressed as integrals over the current corresponding to planar symmetry (eqns. 24 and 29). If oxidized and reduced species have equal diffusion coefficients, the integrals can be evaluated explicitly (eqn. 30). A numerical evaluation is necessary otherwise, and this is done for one case, to check results of other authors.
The basic idea is a perturbative approach with the perturbation parameter being [End of page 33] 1/r 0 , where r 0 is the radius of the spherical microelectrode. The planar electrode corresponds to infinitely large r 0 . After reviewing the background and the equation to be solved in section (I), we introduce the perturbation procedure in section (II). Several special cases and illustrative applications are discussed in section (Ill).

(I) Theory
We consider first a reduction of species O to species R, both present in solution. (Amalgam formation, with R diffusing into the electrode, is considered later.) The diffusion equation [1,7,8]  where is the transform of c o and a is a constant. In order to introduce the current, which is [9] where F is the Faraday constant and A the electrode area, we differentiate eqn. Similarly, for the reduced species. The Nernst condition relates the concentrations of oxidized and reduced species on the electrode surface at each instant of time, if the system is reversible: where E 0 is the formal potential and E(t) the electrode potential at any time. On substituting c O (r 0 , t) and C R (r 0 , t) from eqns. (7) and (8), and the gradients of these quantities in terms of the current from eqn. (5), we obtain For r 0 infinite, terms in erfc disappear and eqn. (10) becomes the planar diffusion equation which may be written [10] in the Abel form with where γ 2 = D O /D R . Here, the subscript on I 0 identifies it as the current for planar diffusion (1/r 0 → 0). The general solution [11]

to eqn. (11) is[End of page 35]
For instance, the current for the linear sweep potential E = E iυt is where In B = nF(E i -E 0 )/RT , as shown by other authors [1,9].
We now consider the case of diffusion of the reduced species into the electrode. The condition on c R (∞, t), analogous to eqn. (3), is no longer relevant, and is replaced by a condition at the origin The Laplace transformation of the equations and boundary conditions proceeds as before. The difference is that a solution such as eqn. (4) is not acceptable for , the Laplace transform in time of c R . The proper solution, satisfying eqn. (15), is where α R is a constant.
We differentiate as previously to introduce the current, now equal to nFAD R (∂c R /∂r) r=ro . Replacing eqn. (8) we find where L(r, t) is the inverse Laplace transform of The expansion on the right, although it may be obtained by successive differentiation of the function on the left with respect to x = r -1 s -1/2 D R 1/2 and evaluation of derivatives at x = 0, is not a true power series expansion. In fact, it corresponds to replacing the hyperbolic cotangent by unity, which is valid for a large argument. It has been noted previously [12] that there are two kinds of spherical corrections in the case of diffusion into the electrode: the correction for the shape of the electrode and the correction for its finite size. For most applications [12,13], the latter us much less important. The replacement of the cotangent by unity corresponds to neglecting this correction.  (12). We now turn to the terms in 1/r o , which give the first spherical correction I 1 . Setting the coefficient of 1/r o in eqn. (21) equal to zero, we have Since I 0 (s) is known, this may be rearranged into an equation of the Abel form, like eqn. (11), except that the unknown function is I 1 (t) and Q is replaced by Note that P(0) = 0; thus the solution according to the formula (13) is where I 0 is given by eqn. (13) and The first correction for sphericity is thus calculable, regardless of the form of E(t), in terms of the current for a planar electrode I 0 . If I 0 is known either analytically or numerically for the problem of interest, I 1 and hence the current corrected for the sphericity of the electrode may be obtained by a single integration. Some simplification is possible; after introduction of eqn. (13), the integration over s may be performed, giving where the upper sign is for diffusion of the reduced species into the solution and the lower sign for diffusion into the electrode (amalgamation).
For example, consider a voltage pulse: E = E 1 for t < t o and E = E 1 + ΔE for t > t 0 . This has been treated by Birke [5]. Equation (28) then becomes an Abel-type integral equation [11] for the second-order correction I 2 (t). It may be treated as we did the equation for the first-order correction I 1 (t), by comparing with eqns. (11) and (13). Then I 2 may be calculated, if the sizes of I 0 and I 1 indicate it is warranted, as an integral over the zero-order (planar symmetry) current I 0 .
(III) Applications The general problem of calculating spherical corrections to the current for a planar electrode has now been solved. For any particular application, one would have to evaluate an integral, according to eqn. (25), as we illustrate below. There is also one situation for which an extreme simplification occurs.
If we assume with Reinmuth and others [1,6,14] that γ, may be taken as unity for the calculation of the correction when diffusion is into the solution, so F(t)= 1, the corrections take on an exceedingly simple form. Inserting I 0 from eqn. (13) into eqn. (24), All the integrals have thus been performed, and the correction for sphericity in this case is, on inserting Q according to eqn. (12), This is an explicit expression for the correction to the time-dependent current, valid regardless of the dependence of E on t.
The second-order correction simplifies even more drastically. Using eqn. (29) for the right-hand member of eqn. (28), one has

[End of page 40]
The second correction I 2 /r 0 vanishes identically, regardless of f(t). Reinmuth [1] has in fact shown that, in this case, the current is exactly equal to I 0 + I 1 /r 0 , where I 1 is given by eqn. (23). Obviously, γ = 1 leads to no simplification in the amalgamation case, since it does not make F equal to unity.
For a voltage pulse, considered by Birke [5], suppose E is equal to E 1 for t < T and equal to The spherical corrections for a potential scan or ramp, E = E 1 -υt, and for a triangular wave, have been given in a semi-empirical form by Beyerlein and Nicholson [13] for the case of amalgamation. For the potential scan, where In Θ = nF(E i -E 0 )/RT and a = nFυ/RT. Then eqn. (25) is (with no initial concentration of R) where S = at. For comparison with the results of ref. 13, we put γ = 1 and assume the first correction suffices (ϕ < 1 in the notation of ref. 13). Then ß, tabulated in ref. 13, should be given by For large S, the terms Θ e -T and Θ e -U may be dropped (because the factor (S --T) -1/2 weights values of S near T heavily), and ß may be directly evaluated to give unity. Otherwise, the singular factor (T--U) -1/2 may be dealt with by the substitution U = T sin 2 ϕ so that

[End of page 42]
and ß evaluated by numerical quadratures. Table 1 shows that the results are quite close to those of Beyerlein and Nicholson [12]. Here, ΔE = E --E 0 and nF/RT= 38.92 V -1 . The results are independent of Θ for sufficiently large Θ, as should be the case.

Conclusions
The most important results of the present article are given by eqns. (20) and (25) and in the prescription for obtaining the second-order correction I 2 . Equation (25) gives a general expression for the first-order correction I 1 for any variation of the potential (see eqn. 9). It is valid whether the reduced species diffuses into the solution or the amalgam, and for all values of the ratio D 0 /D R . In any particular case, specific approximations may be warranted to simplify eqn. (25).