Kinetic analysis based on
working surface interpolation

10.1 Introduction

In order to deduce unknown values of kinetic parameters (such as rate constants and equilibrium constants) from experimental data, a 'fitting' (optimisation) procedure must be used. For a given trial set of parameters, the experimental response may be predicted by the model. In order to minimise the deviation between the experimental and theoretically-predicted response, the trial parameters must be adjusted and the process repeated iteratively.

The repeated computation of physical problems in more than one dimension can be very resource-intensive in terms of CPU time and memory1,2. However, the partial differential equations describing many physical problems can be reduced to a dimensionless form so that the experimental observable may be represented as a sole function of a small number of dimensionless parameters which may be represented graphically as a working curve3 or surface4,5. Once this surface has been generated using a powerful workstation or supercomputer, the simulated response may be reconstructed from the stored surface using an appropriate interpolation algorithm.

In this chapter, the merits of bilinear, cubic convolution, bicubic spline and Artificial Neural Network interpolation methods are assessed using the working surfaces generated in Chapter 6. Working surface interpolation is shown to be a viable cost-effective alternative to optimisation by repeated simulation, especially for problems with mass transport in more than one dimension.

By using the Internet, the surfaces need not even be distributed and stored on experimentalists' personal computers - the data can be stored on a central World Wide Web server which can compute the dimensionless parameters, perform the interpolation and return the results. In the last section of this chapter this is realised. Initially, the dimensionless parameters which may be used to describe an electrochemical reaction mechanism are summarised.

Table 10.1: Characteristic timescales for common electrode geometries
time scale (tc = d2/D)
Rotating disc*
* Using a first-order approximation for the convection term.

10.2 Dimensionless parameters for common mechanisms & geometries

Table 10.1 shows the characteristic time scale parameters for a number of common electrode geometries which reach a steady-state. The quadratic dependence of the time scale on the electrode radius is the reason why microelectrodes can be used to study significantly faster processes than electrodes of traditional (mm) dimensions. Note the analogy between the hydrodynamic electrodes (under first-order convection approximations) - the Peclet number, Pe, is the ratio of mass transport by convection to that by diffusion (defined symbolically for each geometry in section 10.3).

In Appendices 1-4, the dimensionless parameters on which the current depends are derived for a number of electrode geometries. These, summarised in Tables 10.2 - 10.5, define the working curve, surface or hypersurface that completely characterises a particular system.

Table 10.2: Dimensionless parameters on which the steady-state transport-limited current depends
Hemispherical/spherical electrode-
Microdisc Electrode-
Rotating disc electrodeSc
Rotating disc electrode (high Schmidt no)-
Wall-jet electrode
1st order convection approximation
Wall-jet electrode (no radial diffusion)
1st order convection approximation
Channel (microband) electrodep1, p2
Channel (microband) electrode
Lévêque approximation
Channel electrode (no axial diffusion)
Lévêque approximation
The limiting currents for the geometries marked with a "-" may be calculated analytically.

Table 10.3: Dimensionless parameters due to chemical species
Different diffusion coefficients (N species)
[N-1 dimensionless parameters]
Reagent concentrations (N reagents)
[N-1 dimensionless parameters]
[B]bulk/[A]bulk .. [N]bulk/[A]bulk

Table 10.4: Dimensionless parameters due to kinetics
Reversible electron transferQ
Quasi-reversible electron transferQ, Ko
Irreversible homogeneous chemical reactionK
Irreversible heterogeneous chemical reactionKhet
Reversible homogeneous chemical reactionK, Keq

Table 10.5: Dimensionless parameters due to time dependence
Linear-sweep/Cyclic voltammetrys

Dimensionless time and rate constants may be formulated using the time scale and average diffusion layer thickness:

Khom = k.tc = kd2/D(10.1)
Khet = k(tc/D)1/2 = kd/D(10.2)
Ko = ko(tc/D)1/2 = kod/D(10.3)
t = t/tc = Dt/d2(10.4)

The dimensionless parameters in the tables above may be summed to predict on which dimensionless parameters the measured parameter (usually Ilim, Neff or DE1/2) depends. For example, consider the chronoamperometric study of a chemically irreversible, electrochemically reversible EC' reaction:


at a wall-jet electrode (at low Peclet numbers where radial diffusion cannot be ignored), assuming the oxidised and reduced forms of the catalyst have equal diffusion coefficients which are different from that of the substrate. If the potential was not high enough to give the limiting current at steady-state, the dimensionless parameters would be:

t, Pe, K, DZ/DA, [Z]bulk/[A]bulk, Q(10.7)

Clearly such a high-dimensional hypersurface would require a large number of simulated values to define it. Suppose 20 points were enough to define each co-ordinate - 64 million points would be required! Fortunately, many experiments are conducted under conditions where there are fewer degrees of freedom. A more likely wall-jet experiment would be measuring steady-state limiting current at flow rates where radial diffusion effects were negligible. The current would then be a function of just 3 parameters:

K, DZ/DA, [Z]bulk/[A]bulk(10.8)

requiring a more acceptable 8000 points to be simulated and stored.

10.3 Equivalence between hydrodynamic electrodes

From Table 10.1, it is evident that the dimensionless parameters for hydrodynamic electrodes, under a 1st order convection approximation, are all related to the (shear rate) Peclet number, Pe. Throughout this thesis, a shear-rate Peclet number, denoted Ps, has been used which is proportional, but not necessarily equal to, Pe. The constants of proportionality arise from normalisation by electrode diameter vs. radius, factors of p, empirical hydrodynamic constants etc. The definitions of Ps (and K) employed were chosen to maintain consistency with other definitions in the literature3,6. Here these constants are summarised, allowing the characteristic timescale and all the associated parameters (10.1)-(10.5), to be calculated from the Ps values reported in this work. To begin, definitions of Ps are summarised in Table 10.6 together with P - the 'raw' Peclet number in terms of the experimental variables, with any constants of proportionality removed. This provides a 'baseline' from which to relate Pe and Ps. Note (comparing Table 10.1 with Table 10.6) that the time scale for the uniformly accessible RDE and MJE electrodes is independent of the electrode radius.

Table 10.6: Peclet numbers used for hydrodynamic electrodes
Peclet number used'Raw' Peclet number
Rotating disca
a For the rotating disc electrode the Reynolds number is given by7 Re=re2f/n; the Schmidt number, Sc = n/D. b For the wall-jet electrode the hydrodynamic constant CWJE is given by8 CWJE = [125M3/(216n5)]1/4; The constant M is given by M = kc4vf3/(2p3rjet2); kc is an experimentally determined constant. c For the micro-jet electrode the hydrodynamic constant CMJE is given by9 CMJE = 0.54 [(rjet/zjet)-0.054 (vf/rjet3)1/2]3 n-1/2.

Next some of the equations from the previous section are re-written to allow for the constants of proportionality:

Ps = kp P Pe = k Ps(10.9)
d = kd xeP-1/3 d = kdp xePs-1/3(10.10)
K = kk (kxe2/D) P-2/3 K = kkp (kxe2/D) Ps-2/3(10.11)
Nu = kn P1/3 Nu = knp Ps1/3(10.12)

Substituting Ps into the above expressions gives the relationships:

knp = knkp-1/3 and kdp = kdkp1/3 and kk = kkpkp-2/3(10.13)


Nu = xe/d
Nu = Pe1/3
k = knp3

Hence a 'conversion table' may be created for each electrode geometry, shown in Table 10.7.

Table 10.7: Constants for each electrode geometry

By equating the (average) diffusion layer thickness of various electrode geometries, a set of equivalence relationships can be generated3,6. The 'equivalence' factors between the geometries are given in Table 10.8 and Table 10.9.

Table 10.8: 'Equivalence' between shear-rate Peclet numbers for hydrodynamic electrodes
Geometry relationshipPeclet Number DifferenceValue
Channel/RDElog10PsCHE - log10PsRDE0.854
Wall-jet/RDElog10PsWJE - log10PsRDE0.060
Micro-jet/RDElog10PsMJE - log10PsRDE0.932

Table 10.9: 'Equivalence' between dimensionless rate constants at various electrode geometries
Geometry relationshipDifference in KValue
Disc/Spherelog10KDisc - log10Ksphere-3 log10(4/p)
RDE/Spherelog10KRDE - log10Ksphere-0.384
ChE/Spherelog10KChE - log10Ksphere0.185
WJE/Spherelog10KWJE - log10Ksphere-0.310
MJE/Spherelog10KMJE - log10Ksphere0.237
Channel/RDElog10KCHE - log10KRDE0.569
Wall-jet/RDElog10KWJE - log10KRDE0.074
Micro-jet/RDElog10KMJE - log10KRDE0.621

For non-uniformly accessible electrodes, the 'equivalence' is only approximate since it is based on an average diffusion layer thickness (therefore the deviation gives a measure of the degree of non-uniformity). For uniformly accessible hydrodynamic electrodes this equivalence is valid within the 1st-order convection approximation. Since the effects of radial diffusion are negligible for the MJE above log10Ps of 4.5 the RDE analogy, originally proposed by Albery and Bruckenstein10, may be employed. This allows the analysis of steady-state voltammograms at the MJE using the working curves computed for the RDE in Chapter 7.

10.4 Automated simulation methods

In order to generate comprehensive working curves/surfaces, simulations must be conducted over a wide range of conditions (both in terms of mass-transport and kinetics). If the values interpolated from the surface are to be accurate, a reasonably high resolution mesh of points is required to define the surface. Therefore the simulation methods used to generate such a surface should ideally be able to run without manual optimisation of the parameters at each point. Below is a summary of the key features of the automated simulations used to generate working curves/surfaces throughout this thesis.

10.4.1 Spherical, microdisc and rotating disc electrode simulations using PKS methods

10.4.2 Channel microband and wall-jet electrodes

For the channel microband electrode, the simulation geometry needs to be modulated as a function of Peclet number. At high Peclet numbers, the diffusion layer is constrained close to the electrode surface and there is little propagation of axial diffusion upstream or downstream of the electrode. The opposite is true at low Peclet numbers: a large space upstream and downstream is necessary to accommodate axial diffusion. The 2-D channel simulation is slow to converge due to the boundary singularity on the upstream edge of the electrode, so it is not possible simply to adopt the maximum space values and push up the number of nodes to remedy this problem.

To allow this simulation to run automatically, subroutines were written to 'search' for the edge of the diffusion layer. A coarse mesh problem was run first across the entire channel height. The resulting concentration profile was then searched, locating the point at which the normalised concentration departed from unity. A percentage error bound was added, and this was fixed as the fraction of the channel to be simulated. A similar search was conducted upstream of the electrode, starting from a maximum value of 200 electrode lengths. Since it was found in Chapter 6 that the upstream and downstream contributions from axial diffusion were approximately equal, it is possible to allow the same amount of propagation space downstream as upstream, thus only a single parameter needs to be optimised. This was also found to be optimal empirically.

For the wall-jet, two similar parameters require optimisation - the space above the electrode and the space downstream. The former is addressed by the conformal space, since the diffusion layer thickness scales with the h co-ordinate - hence a value of hmax = 1 appears to be suitable down to Ps=1. In the case of the space downstream, ld, an alternative strategy was adopted. Simulations were manually optimised across a broad range of Peclet numbers (1 - 1010). A simple quadratic fitted as a function of log(Ps) over the range 0-4 above which ld=0.5 was sufficient, namely:


so that intermediate values could be interpolated.

10.4.3 Channel electrode working curves (using the BI method)

For a channel macroelectrode, the effects of axial diffusion can be ignored and the current may be expressed as a sole function of dimensionless kinetic parameters, as shown in Table 10.2. Therefore the working surfaces for common reaction mechanisms (such as those simulated in sections 6.2.1 and 6.2.2) collapse to working curves which may be simulated efficiently using a frontal strategy such as the BI method.

Compton et al.11-14 have published working curves for a number of common mechanisms using the BI method with a Cartesian mesh. Inevitably working curves published graphically are inconvenient for quantitative analysis, since a value must be interpolated for each measured current (the 'traditional' remedy to this problem is to use approximate functions to the curve or surface, based on regression15 or asymptotic formulae16). These were therefore re-simulated using the BI method and stored in an electronic format at a high enough resolution (steps of 0.05 in log K) to allow accurate linear interpolation. In order to improve on the accuracy of the published data, and generate working curves over a wider range of rate constants, the following measures were taken:

Simulations were conducted under conditions where the Lévêque approximation is valid (small electrode length of 0.1mm, high flow rate of 0.1cm3s-1) using a minimum of 1000x1000 nodes and rectangular integration.

10.5 Extrapolation of DQ1/2 to the reaction-layer limit

The rate constants in both the EC and EC2 mechanisms may be characterised by measuring the shift in half-wave potential as a function of mass transport, then comparing this with a working curve of dimensionless shift vs. dimensionless rate constant. At high rate constants this behaviour limits to that predicted by simple reaction-layer theory - the shift in half-wave potential becomes a linear function of the log of the dimensionless rate constant. Reaction layer expressions have been derived for spherical18,19 and rotating disc electrodes (see Table 7.1) and are summarised Table 10.10.

The 'equivalence' factors, given in the previous section, allow reaction layer limits to be applied to other electrode geometries, though as shown in Chapter 9, the 'equivalence' relationship will not necessarily work for the full working curve, especially at lower rate constants. The transformations are applied to the EC and EC2 working curves in Figure 10.1 which were computed in the previous chapters. Note that the small difference between the WJE and RDE is about the same for both EC and EC2 processes - contradicting earlier work by Compton et al.6, where a marked difference was observed for the EC2 process. One might expect second-order kinetics to pronounce the non-uniform concentration distribution at the electrode surface. This effect does not appear to be 'felt' in the shift in the half-wave potential. The effect on the limiting current (measured as Neff) is discussed in the conclusions chapter.

Figure 10.1: Working curves, superimposed by equivalent diffusion layer thickness for (a) EC mechanism, and (b) EC2 mechanism.

Table 10.10: Reaction layer expressions for EC and EC2 reactions
MechanismShift in dimensionless HWP

Due to the thin reaction layer at high rate constants, the (stiff) systems are computationally expensive to simulate accurately. Once the reaction layer limit has been reached, a linear extrapolation may be used to predict the behaviour at high rate constants. However one must be confident that one is not extrapolating using simulation data before the fully linear region, or that the spatial convergence is not slipping towards high rate constants (due to the thinning reaction layer). The safest way to perform the extrapolation is to use one of the reaction layer expressions from Table 10.10.

10.6 A comparison of interpolation methods

The working surfaces for E, ECE and EC2E reactions at a (microband) channel electrode were used as test cases. These are representative of many of the log-scale working curves and surfaces found in voltammetry - the response curve is either sigmoidal or an increasing gradient up to a linear limit.

10.6.1 Method

Interpolation was conducted on the decadic logarithm of the current so that the percentage error from interpolation would be approximately uniform across the surface. For ECE and EC2E mechanisms, the interpolation was conducted on the Neff value rather than on its logarithm, since Neff spans a narrow range of magnitude (1-2).

Four methods were investigated. The first, bilinear interpolation (using the INTERPOLATE routine in IDL 4(r)20) was chosen because it is the simplest and most easily programmed two-dimensional interpolation method. Interpolation using bicubic convolution was also assessed (using the /CUBIC option of the IDL INTERPOLATE routine).

Bicubic spline interpolation was conducted using the NAG library routines E02DAF to fit the spline surface and E02DBF to evaluate it at the sought co-ordinates. These routines were used with the parameters suggested in the NAG FORTRAN library manual21 to produce 'interpolating splines'. These give more accurate reconstruction of values near the ends of the range than a natural bicubic spline interpolation such as the routine E01ACF21 (where the second derivatives are zero at the ends of the ranges).

Figure 10.2: An Artificial Neural Network with an input layer of 2 nodes, 1 or more hidden layers of internal nodes, and an output layer with a single node.

The last method investigated was an Artificial Neural Network (ANN) based on sigmoidal transfer functions22. A neural network with 2 inputs, 1 output and a number of internal nodes (shown in Figure 10.2) serves as a non-linear regression method in which the variable coefficients (connection weightings) are optimised for the data set via the simple back-propagating training procedure. For a fully-connected network (which was used in all cases) with 2 inputs and 1 output, the number of connections are given by:

3l1for one hidden layer
2l1 + l1.l2 + l2for two hidden layers
2l1 + l2(l1+l3) + l3for three hidden layers

where l1, l2 and l3 are the numbers of nodes in the first, second and third hidden layers, respectively.

The commercial package ARD Propagator(r) V1.023 was used for ANN interpolation. In all cases a validation data set (the points which were to be interpolated) was used when training the neural network in order to prevent over-training. In order to be used with the neural network, the data was normalised to values suitable for the network transfer functions. Sigmoidal transfer functions were used, taking values between zero and 1, so the data was normalised to values between 0.05 and 0.95 before use with the ANN.

10.6.2 Results

In order to evaluate the performance of various interpolation methods, a pair of surface data sets was simulated for each mechanism, the second (validation) surface being offset by half an interval in both the x and y directions from the first (training) data set, as shown in Table 10.11. The observable at the co-ordinates corresponding to the validation data set could then be interpolated using the training data set and the error measured. The parameters used in ARD Propagator(r) for Neural Network fitting are shown in Table 10.12. Suitable learning rate and momentum factors were deduced empirically.

Table 10.11: Training and validation data sets for interpolation
ET traininglog10p1-2.420.2log10p2260.2
ET validationlog10p1-
ECE traininglog10Ps-44.40.4log10KECE-320.2
ECE validationlog10Ps-
EC2E traininglog10Ps-44.40.4log10KEC2E-320.2
EC2E validationlog10Ps-

Table 10.12: Parameters for ARD Propagator(r) neural network
Transfer functionsSigmoidal
Random seed0
Learning RuleGeneralised Delta
Learning Rate0.4
Momentum Factor0.5
Training patterns orderRandom
Training Cycles100,000
Initial Weights-1 to 1
Input Noisenone

Figure 10.3: Number of connections in an Artificial Neural network with an equal number of nodes per layer (where the number of nodes was not exactly divisible by the number of layers, nodes were dropped to maximise the number of connections).

The interpolation errors for various network architectures are shown in Table 10.13. The first column corresponds to the number of nodes in each layer - two in the input layer and one in the output layer in all cases, but a variation in the number of hidden layers and nodes in the hidden layers. As one would expect, fitting improves and training time increases with the overall number of hidden-layer nodes as the number of variable parameters (weights) increases. Also perhaps as would be expected, the mean error decreases as the number of connections (and hence 'fitting' parameters) increases. The optimal architecture appears to be with two hidden layers as this gives rise to the maximum number of connections for a given number of nodes, as shown in Figure 10.3, though three hidden layers give a lower maximum error with a large number of nodes.

Table 10.13: Interpolation errors on each of the three surfaces for various ANN architectures
2 10 1300.554%3.485%0.119%0.405%0.036%0.187%
2 5 5 1400.371%2.464%0.067%0.297%0.023%0.162%
2 3 4 3 1330.433%4.171%0.082%0.404%0.034%0.192%
2 20 1600.549%3.457%0.099%0.464%0.034%0.197%
2 10 10 11300.371%2.200%0.049%0.273%0.022%0.161%
2 6 7 7 11110.370%2.414%0.048%0.311%0.019%0.172%
2 40 11200.664%3.811%0.105%0.419%0.044%0.158%
2 20 20 14600.316%2.18%0.0382%0.265%0.021%0.150%
2 13 14 13 14030.327%2.03%0.0429%0.236%0.021%0.142%

Table 10.14 shows the percentage error in current for interpolation on each of the three surfaces using the three methods. The success of simple bilinear interpolation on the electron transfer surface can be rationalised by the planarity of the surface in the region where the Levich equation holds.

Table 10.14: Interpolation errors on each of the three surfaces for various interpolation methods
Interpolation methodMeanMaxMeanMaxMeanMax
Cubic convolution0.525%5.925%0.066%0.465%0.030%0.279%
Bicubic Spline0.043%0.894%0.012%0.225%0.007%0.134%
Neural Network*0.316%2.18%0.0382%0.265%0.021%0.142%
*Using optimal architecture from Table 10.13.

Bicubic spline interpolation is clearly the superior method, though linear and ANN interpolation both give highly respectable results. For the ECE and EC2E surface the 'maximum' error occurs at very low Ps (<-3.8) where there is a minor turning point in the data due to slightly imperfect convergence (maximum error of 3-4% in this region compared with <1% elsewhere). This is due to the extremely high number of nodes required to converge the Cartesian grid in the x direction under these conditions where there is a very large amount of axial diffusion relative to the size of the electrode. With 'perfect' (infinitely converged) data the results from bicubic spline and linear interpolation would be considerably better, however rather than rejecting data from this region, the data with a realistic imperfection was chosen evaluate the interpolation methods.

10.6.3 Conclusions

Working surfaces, together with an appropriate interpolation algorithm, allow experimental responses to be predicted accurately and instantaneously by the non-specialist. Of the interpolation methods investigated, bicubic spline interpolation was shown to be superior to both bilinear interpolation and Artificial Neural Networks. Cubic convolution interpolation performed relatively poorly in all cases. Bilinear interpolation was found to give highly satisfactory results for minimal programming effort and CPU time and is therefore the choice method for data sets of this resolution. For lower resolution surfaces bicubic spline interpolation would probably be the optimal method. For problems where the experimental observable must be represented by a higher-dimension (hyper) surface, the choice is more restricted - only linear interpolation and ANNs can readily be extended to higher dimensions.

10.7 A data analysis service via the World Wide Web

The working curves and surfaces generated in this thesis form a mechanism 'library' for a number of common geometries. These were used as the basis of a data analysis program24,25, available via the World Wide Web (WWW) at http://physchem.ox.ac.uk:8000/wwwda. The user simply selects a geometry and mechanism, enters the cell parameters along with the limiting current/half-wave potential as a function of radius/rotation speed/flow rate. Data may be entered directly via the WWW, or uploaded as a file by FTP. For each experimental data set, the theoretical response is predicted for the chosen mechanism and plotted as a function of mass transport appropriate to the electrode geometry.

In order to reduce the number of dimensionless parameters to a manageable number (Table 10.15):

Table 10.15: Dimensionless parameters characterising the functional dependence of the observable for each mechanism at each geometry.
Spherical electrodeconst.KK, r
Microdisc electrodeconst.KK, r
Rotating disc electrode
(1st order convection)
const.KK, r
Rotating disc electrode
(9th order convection)
ScSc, KSc, K, r
Wall-jet electrode
(No radial diffusion)
const.KK, r
Wall-jet electrodePsPs, K1Ps, K, r
Channel electrode
(Lévêque approximation)
const.K2K, r
Channel (microband) electrode
(Lévêque approximation)
PsPs, K3Ps, K, r
Channel (microband) electrode (parabolic flow)p1, p2p1, p2, Kp1, p2, K, r
Grey cells indicate combinations not used (due to a number of dimensions greater than 2, or because a more accurate alternative is available). 1 For ECE,EC2E, DISP1 and DISP2 mechanisms 2 For EC,EC2,DISP1 and DISP2 mechanisms 3 For ECE and EC2E mechanisms

A measure of the 'goodness of fit' is computed between the experimental and theoretical sets of data. The least squares error, given by26:


(where x denotes the observable) would seem a natural choice to measure the goodness of fit, but it is biased by outlying experimental data points. Therefore the more robust mean absolute deviation (MAD)26 is used:


This is scaled when treating limiting currents or Neff values, so that the relative error is minimised:


in order not to bias the error by the magnitude of the current. For half-wave potentials, where the theoretical E1/2 value may be very close to zero, the absolute error is minimised.

Figure 10.4: Error surface fitting a rate constant and diffusion coefficient for an ECE mechanism. The experimental Neff measured at a channel microband electrode array for para-bromonitro-benzene, with photo-excitation using a 250W mercury lamp. Full experimental details can be found in [29].

This error may be minimised to fit (i.e. optimise) unknown parameters such as rate constants or diffusion coefficients. A Golden Section Search26 is used to find a local minimum in one dimension. Downhill Simplex minimisation26 is used for multidimensional searches. One-, two- and multi-dimensional sampling routines are also available so that the error curve or surface may be visualised, for example see Figure 10.4, allowing the global minimum to be discerned from local minima. This service has been used as part of a number of mechanistic investigations to analyse data gathered at an array of channel electrodes, in order to distinguish between candidate mechanisms and fit diffusion coefficient and rate constant values to experimental data27-29.

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