7
 
Rotating disc electrode

 

A great deal of electrochemical simulation work has been done on the rotating disc electrode (RDE), as is evident from Table 7.1. This is because the RDE is the most popular hydrodynamic electrode and it is also relatively easy to model, requiring the simulation of only a single spatial dimension. The system is thus free from any spatial singularities and converged simulations could be conducted on a uniform mesh using simple explicit methods and relatively primitive computers. Also if a first-order approximation is used for the convection term (see below), approximate analytical solutions may be obtained for simple mechanisms.

Since these early days, a number of improved simulation methods have been introduced utilising Hale's conformal mapping1 or an expanding grid and more efficient and stable implicit algorithms. One of the most recent2 is Feldberg et al's simulation using Rudolph's FIFD algorithm (see Chapter 2) with an exponentially expanding grid which forms the basis of the RDE module of the commercial cyclic voltammetry simulator, DigiSim 2.0(tm). While this work and the commercial software is directed at cyclic voltammetry, many of the early analytical and numerical simulations focused on the steady-state response (since this is the most common RDE experiment and also easier to simulate) producing working curves for various mechanisms.

In this chapter an RDE module is developed for the general simulator outlined in Chapter 5, based on PKS methods. This uses Hale's conformal mapping for efficient simulations with a 9-term convection approximation. Due to the modular design of the simulator, simulations of all the various experiment types and mechanistic possibilities become immediately available for the RDE. This has allowed the generation of working curves and surfaces of the steady-state limiting current/shift in half-wave potential for a number of common mechanisms. In this chapter, these are compared with the working curves published using approximate analytical and early numerical simulations and are shown to be slightly different. The earlier methods were inaccurate either due to the approximations made in the analytical solution or the convergence of the numerical methods. The working surfaces presented here form the basis of the RDE section of the steady-state data analysis service described in Chapter 10.


Table 7.1: Theory for voltammetry at the rotating disc electrode

7.1 Hydrodynamics of the RDE

The mass transport equation for the RDE is:

(7.1)

The function vz has been evaluated numerically by Cochran3. Close to the electrode surface vz may be approximated by the first term in the series expansion in z, namely:

vz = cz2 where c = 8.032f3/2n-1/2(7.2)

For a more accurate description of the convection, vz may be expressed using the higher terms in the series expansion:

(7.3)

where z is the dimensionless z coordinate for an RDE, given by z=f1/2n-1/2z. Feldberg et al.4 corrected and used Levart and Schuhmann's equations5 to calculate the first nine terms of ak (shown in Table 1 of [4]). Later Feldberg et al.2 used the expression:

(7.4)

where ??= 0.88447-1 and

+ 0.16214z6 - 0.07143z7 + 0.019911z8 - 0.0030405z9 + 0.00019181z10
S = 0.51023z2 - 0.32381z3 + 0.34474z4 - 0.25972z5 (7.5)

and this forms the basis of the hydrodynamic functions used in the commercial simulator DigiSim 2.0(tm).

7.2 Simulation method

The Hale transformation1 is given by:

where and (7.6)

For easy incorporation into the Hale space transformation, equation (7.3) is used to represent the convection term. Then, as shown in Appendix 3, the mass-transport equation transforms into:

where (7.7)

and the coefficients in the series k1, k2, ... are given by

(7.8)

This was then discretised using central finite differences (or optionally upwind differencing for the 1st derivative) to give a steady-state 3-point finite difference equation (which may be modified as described in Chapter 5 to include time-dependence):

(7.9)

where

and (7.10)

This results in a tridiagonal matrix onto which the kinetic relationships are summed as outlined in Chapter 5. In order to evaluate Z at points along ?, the differential equation must be integrated for Z:

(7.11)

This was done by Compton and Unwin6 using a 3rd-order Runga-Kutta integration. Here the NAG library subroutine D02CJF was used (based on an Adams method - see section 3.4.1).

7.3 Results

7.3.1 Schmidt number correction of the Levich equation

Under the first-order convection approximation, the (dimensionless) current, for a simple transport-limited electron-transfer is given by the Levich equation7:

Nu = 0.621.Ps1/3or I = 0.621nF[A]bulk?re2D2/3v1/6?-1/2(7.12)

where the Peclet number is given by:

Ps = Sc.Re3/2 and (is the Reynolds number)(7.13)

Figure 7.1: Breakdown of the Levich equation at low Schmidt numbers.

This approximation breaks down with decreasing Schmidt number as shown in Figure 7.1. The error in the transport-limiting current for some of the lowest Schmidt numbers encountered experimentally are approximately 3% for the oxidation of FeCp2 in MeCN and approximately 3.5% for the reduction of H+ in H2O. For convection approximations higher than first-order, the current is an (unknown) function of Sc, plus any kinetic parameters.


Figure 7.2: Working curve for a simple electron-transfer

This means for the simple transport-limited electron transfer the dimensionless current may be completely characterised by a working curve of Nu vs. Sc as shown in Figure 7.2. The Levich and simulated responses using second and third-order approximations of the convection are also shown. Newman's third order correction8 to the Levich equation, given by:

or (7.14)

is also shown on Figure 7.1 and can be seen (as reported by Feldberg2) to be accurate to within 0.1% for Sc > 100 and is therefore adequate for the description of a transport-limited electrolysis even in solvents of low viscosity such as acetonitrile.


Figure 7.3: Shift in dimensionless half-wave potential vs. dimensionless rate constant for an EC reaction.

7.3.2 Working curves and surfaces for kinetic analysis

For common electrochemical mechanisms (see Table 1.1) with a single kinetic parameter (transport-limited irreversible ECE, EC2E, DISP1, DISP2; shift in half-wave potential for irreversible EC, EC2), the observable (Neff or DQ1/2) is a sole function of the dimensionless rate constant under the 1st order convection approximation. Therefore the behaviour may be characterised by a working curve.

Figure 7.3 shows the working curve for an EC process. The simulated response is shown together with the simple reaction-layer9 approximation:

(7.15)

and Tong's10 more sophisticated analytical approximation:

(7.16)

Figure 7.4: Limiting current ratio (Neff) vs. dimensionless rate constant for an ECE reaction.

Early numerical simulations by Unwin et al.6 using an explicit method in Hale space agreed with Tong's results, yet the fully implicit steady-state simulations show a small but nevertheless marked difference. Explicit simulations were re-run using Unwin's program and it was found that it agreed with the PKS simulations if a tighter tolerance was set for the asymptotic approach to steady-state, as shown in the figure. The published values appeared to have a lower DQ1/2 since the limiting current was read when it was further from steady-state than the currents near the half-wave potential. This illustrates one of the advantages of using more sophisticated methods such as PKS and multigrid to simulate steady-state response with a well-defined measure of the residual error. Tong's analytical treatment is in the limit of zero convection rate. This may be mimicked in the Hale space finite difference simulation by making Siver's approximation1,11 :

for small Z (corresponding to small z or small w)(7.17)

which reduces the term f(Z) to 1/A. In this case the simulation overlays Tong's curve.

Figure 7.4 shows the simulated response for an ECE reaction compared with Karp's analytical approximation12:

(7.18)

The simulated response is a little higher. Again Karp's response may be mimicked by Siver's approximation, essentially switching off the convection. Marcoux, Adams and Feldberg13 reported that their explicit finite difference simulations were 'virtually superimposable' with Karp's simulated response. Therefore it is likely that these early simulations were subject to inaccuracies similar to those found in Unwin's.


Figure 7.5: Working curves of Neff vs. log K for ECE, EC2E, DISP1 and DISP2 reactions.

Figure 7.5 shows the complete set of PKS-simulated working curves of Neff vs. Log K for ECE, EC2E, DISP1 and DISP2 reactions. The working curves of DQ1/2 vs. Log K for EC and EC2 processes are shown in Figure 7.6.

An EC' reaction involves a second dimensionless parameter - the ratio of the substrate concentration to that of the catalyst, r. Thus at high Schmidt numbers Neff may be summarised as a working surface of log Neff vs. log K vs. log r, shown in Figure 7.7.


Figure 7.6: Working curves of DQ1/2 vs. log K for EC and EC2 reactions.

The working curves for ECE, EC2E, DISP1, DISP2, EC and EC2 were all computed using a 9 term convection approximation over the dimensionless rate constant range 10-4 - 108. If a more accurate convection approximation is used, the experimental observable (Neff or DQ1/2) becomes a function of both Sc and K. Therefore working surfaces were simulated by generating working curves for Schmidt numbers from 100 to 107 at which point the curves tend to the 'classical' limit. These form the basis of the RDE section of the data analysis service reported in Chapter 10.


Figure 7.7: Working surface for an EC' reaction. The contour values are log10 Neff.

References

1 J.M. Hale, J. Electroanal. Chem., 6, (1963), 187.
2 S.W. Feldberg, C.I. Goldstein and M. Rudolph, J. Electroanal. Chem., 413, (1996), 25.
3 W.G. Cochran, Proc. Cambr. Phil. Soc., 30, (1934), 365.
4 S.W. Feldberg, M.L. Bowers and F.C. Anson, J. Electroanal. Chem., 215, (1986), 11.
5 E. Levart, D. Schuhmann, Int. J. Heat Mass Transfer, 17, (1974), 555.
6 R.G. Compton, M.E. Laing, D. Mason, R.J. Northing, P.R. Unwin, Proc. R. Soc. Lond. A, 418, (1988), 113.
7 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Engelwood Cliffs, New Jersey, (1962).
8 J. Newman, J. Phys. Chem., 70, (1966), 1327.
9 Z. Galus, R.N. Adams, J. Electroanal. Chem., 4, (1962), 248.
10 L.K.J. Tong, K. Linag, W.R. Ruby, J. Electroanal. Chem., 13, (1967), 245.
11 Y.G. Siver, Zh. Fiz. Khim., 34, (1960), 577.
12 S. Karp, J. Phys. Chem., 72, (1967), 1082.
13 L.S. Marcoux, R.N. Adams, S.W. Feldberg, J. Phys. Chem. , 73, (1969), 2611.