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 mapping^{1} or an expanding
grid and more efficient and stable implicit algorithms. One of the most recent^{2}
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

The mass transport equation for the RDE is:

(7.1) |

The function v_{z} has been evaluated numerically by Cochran^{3}. Close to the electrode surface v_{z} may be approximated by the first term in the series expansion in z, namely:

v_{z} = cz^{2} where c = 8.032f^{3/2}n^{-1/2} | (7.2) |

For a more accurate description of the convection, v_{z} 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=f^{1/2}n^{-1/2}z. Feldberg et al.^{4} corrected and used Levart and Schuhmann's equations^{5} to calculate the first nine terms of a_{k} (shown in Table 1 of [4]). Later Feldberg et al.^{2} used the expression:

(7.4) |

where ??= 0.88447^{-1 }and

S = 0.51023z^{2} - 0.32381z^{3} + 0.34474z^{4} - 0.25972z^{5} | (7.5) |

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

The Hale transformation^{1} 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 k_{1}, k_{2}, ... are given by

(7.8) |

This was then discretised using central finite differences (or optionally upwind
differencing for the 1^{st} 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 Unwin^{6} using a
3^{rd}-order Runga-Kutta integration. Here the
NAG library subroutine D02CJF was used (based on an Adams method - see section
3.4.1).

Under the first-order convection approximation, the (dimensionless) current, for a simple transport-limited electron-transfer is given by the Levich equation^{7}:

Nu = 0.621.P_{s1/3}or I = 0.621nF[A]_{bulk}?r_{e2}D^{2/3}v^{1/6}?^{-1/2} | (7.12) |

where the Peclet number is given by:

P_{s} = Sc.Re^{3/2} 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 FeCp_{2} in MeCN and approximately 3.5% for the reduction of H^{+} in H_{2}O. 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 correction^{8} to the Levich equation, given by:

or | (7.14) |

is also shown on Figure 7.1 and can be seen (as reported by Feldberg^{2}) 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.

For common electrochemical mechanisms (see Table 1.1) with a single kinetic parameter (transport-limited irreversible ECE, EC_{2}E, DISP1, DISP2; shift in half-wave potential for irreversible EC, EC_{2}), the observable (N_{eff} or DQ_{1/2}) is a sole function of the dimensionless rate constant under the 1^{st} 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-layer^{9} approximation:

(7.15) |

and Tong's^{10} more sophisticated analytical approximation:

(7.16) |

Figure 7.4: Limiting current ratio (N

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 DQ_{1/2} 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 approximation^{1,11}^{ }:

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 approximation^{12}:

(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 Feldberg^{13} 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 N

Figure 7.5 shows the complete set of PKS-simulated working curves of N_{eff} vs. Log K for ECE, EC_{2}E, DISP1 and DISP2 reactions. The working curves of DQ_{1/2} vs. Log K for EC and EC_{2} 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 N_{eff} may be summarised as a working surface of log N_{eff} vs. log K vs. log r, shown in Figure 7.7.

Figure 7.6: Working curves of DQ

The working curves for ECE, EC_{2}E, DISP1, DISP2,
EC and EC_{2} were all computed using a 9 term
convection approximation over the dimensionless rate constant range 10^{-4}
- 10^{8}. If a more accurate convection approximation
is used, the experimental observable (N_{eff} or
DQ_{1/2}) becomes a function
of both Sc and K. Therefore working surfaces were simulated by generating working
curves for Schmidt numbers from 100 to 10^{7} 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 log

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