**Modeling and Quantification of Electrochemical Reactions in RDE (Rotating Disk Electrode) and IRDE (Inverted Rotating Disk Electrode) Based Reactors**

Lucía Fernández Macía, Heidi Van Parys, Tom Breugelmans, Els Tourwé and Annick Hubin *Electrochemical and Surface Engineering Group, Vrije Universiteit Brussel Belgium*

#### **1. Introduction**

20 Electrochemical Cells – New Advances in Fundamental Researches and Applications

Srivastav, S. & Kant, R. (2011) Anomalous Warburg Impedance: Influence of

Sutton, L.; Gavaghan, D.J. & Hahn, C.E.W. (1996). Numerical Simulation of the Time-

Thompson, M.; Wildgoose, G.G. & Compton, R.G. (2006). The Theory of Non-Cottrellian

Thompson, M. & Compton R.G. (2006). Fickian Diffusion Constrained on Spherical Surfaces:

Thurzo, I.; Gmucová, K.; Orlický, J. & Pavlásek, J. (1999). Introduction to a Kinetics-Sensitive

Ward, K.R.; Lawrence, N.S.; Hartshorne, R.S. & Compton, R.G. (2011) Cyclic Voltammetry of

*Phys. Chem. C.* Vol.115, No.22, (June 2011), pp. 11204–11215, ISSN 1932-7447 Welch, Ch. M. & Compton, R.G. (2006) The Use of Nanoparticles in Electroanalysis: A

Wildgoose, G.G.; Banks, C.E.; Leventis, H.C. & Compton, R.G. (2006). Chemically Modified

Zhao, X.; Lu, X.; Tze, W.T.I. & Wang P. (2010). A Single Carbon Fiber Microelectrode with

*Bioelectronics* Vol.25, No.10, (July 2010), pp. 2343–2350, ISSN 09565663 Zhou, Y.-G.; Campbell, F.W.; Belding, S.R. & Compton, R.G. (2010) Nanoparticle Modified

pp. 12232–12242, ISSN 1932-7447

No.6, (June 2006), pp. 1328–1336, ISSN 1439-7641

(January 2006), pp. 187-214, ISSN 1436-5073

(September 2010), pp. 200–204, ISSN 0009-2614

6657

1439-7641

1618-2650

3723-3734, ISSN 0034-6748

Uncompensated Solution Resistance. *J. Phys. Chem C.* Vol.115, No.24, (June 2011),

Dependent Current to Membrane-Covered Oxygen Sensors. Part IV. Experimental Verification that the Switch-on Transient is Non-Cottrellian for Microdisc Electrodes. *J. Electroanal. Chem.* Vol.408, No.1-2, (May 1996), pp. 21–31, ISSN 1572-

Diffusion on the Surface of a Sphere or Truncated Sphere. *ChemPhysChem.* Vol.7,

Voltammetry. *ChemPhysChem.* Vol.7, No.9, (September 2006), pp. 1964–1970, ISSN

Double-Step Voltcoulometry. *Rev. Sci. Instrum,* Vol.70, No.9, (September 1999), pp.

the EC' Mechanism at Hemispherical Particles and Their Arrays: The Split Wave. *J.* 

Review. *Anal. Bioanal. Chem.* Vol.384, No.3, (February 2006), pp. 601–619, ISSN

Carbon Nanotubes for Use in Electroanalysis. *Microchimica Acta,* Vol.152, No.3-4,

Branching Carbon Nanotubes for Bioelectrochemical Processes. *Biosensors and* 

Electrodes: Surface Coverage Effects in Voltammetry Showing the Transition from Convergent to Linear Diffusion. The Reduction of Aqueous Chromium (III) at Silver Nanoparticle Modified Electrodes. *Chem. Phys. Letters.* Vol.497, No.4-6, Whether it is for the design of a new electrochemical reactor or the optimization of an existing electrochemical process, it is of primordial importance to have the possibility to predict the behavior of a system. For example, the process of electrogalvanization of steel on an industrial scale would not be possible without knowing the main and side reactions taking place during the deposition of zinc on the metallic surface. However, understanding their mechanism is only the first point. The characteristic parameters of the reaction need to be identified and quantified in order to obtain a correct reactor design and to achieve the optimal operating conditions. Nowadays, part of the technological know-how still relies on best practice guidance, most often gained from years of experience with trial and error. Condensing those findings into empirical models may help to control some of the process parameters and make predictions of the system behavior within a small operation window. The problem, however, is that such a model acts as a black box and that a profound comprehension of the physical and electrochemical phenomena will fail to come.

The aim of a kinetic study is the determination of the mechanism of the electrochemical reaction and the quantification of its characteristic parameters: charge transfer parameters (rate constants and transfer coefficients) and mass transfer parameters (diffusion coefficients). Nevertheless, determining kinetic parameters accurately from the experimental results remains complex.

Linear sweep voltammetry (LSV) in combination with a rotating disk electrode (RDE) is a widely used technique to study electrode kinetics. Different methods exist to extract the values of the process parameters from polarization curves. The Koutecky-Levich graphical method is frequently used to determine the mass transfer parameters (Diard et al., 1996) : the slope of a plot of the inverse of the limiting current versus the inverse of the square root of the rotation speed of the rotating disk electrode is proportional to the diffusion coefficient. If more than one diffusing species is present, this method provides the mean diffusion coefficient of all species. The charge transfer current density is determined from the inverse of the intercept. In practical situations, however, the experimental observation of a limiting current

Although the reaction and transport models are defined more precisely, the lack of a fitting

<sup>23</sup> Modeling and Quantification of Electrochemical Reactions in RDE (Rotating Disk Electrode) and IRDE (Inverted Rotating Disk Electrode) Based Reactors

A quantitative, accurate and statistically founded modeling approach of electrochemical reactions has been the focus of an extensive work in our research group (Aerts et al., 2011; Tourwé et al., 2007; 2006; Van Parys et al., 2008). It is a generally applicable method to model an electrochemical reaction and to determine its mass and charge transfer parameters quantitatively. The reliability of the model parameters and the accuracy of the parameter fitting are key-elements of the method. A plausible reaction mechanism and the characteristic parameters of the electrochemical reaction are extracted from LSV experiments with a rotating disk electrode. Compared to others, this method offers the advantage that it uses one integrated expression that accounts for mass and charge transfer steps, and this without simplifying their mathematical expressions. The whole polarization curve is considered, rather than just some part in which only mass or charge transfer are supposed to be rate

In this paper, we explain throughly this modeling methodology for the rotating disk electrode (RDE) and the inverted rotating disk electrode (IRDE) configurations. The modeling and quantification of the electrochemical parameters are applied to redox reactions with one electron transfer mechanism: the ferri/ferrocyanide system and the hexaammineruthenium

Linear sweep voltammetry with a rotating disk electrode (LSV/RDE) is a powerful technique for providing information on the mechanism and kinetics of an electrochemical reaction. Since the current density is a measure for the rate of an electrochemical reaction, LSV provides a stationary method to measure the rate as a function of the potential. In other words, the technique is used to distinguish between the elementary reactions taking place at the electrode as a function of the applied potential. Different elementary steps are often coupled, however, the overall current is determined by the slowest process (rate determining step). As a steady state technique, linear sweep voltammetry can only give mechanistic information about rate

To determine a quantitative model for an electrochemical process, first a plausible reaction model is proposed and afterwards combined with a transport model. The combination of both models enables the formulation of the mass balances of the species and the conservation laws, which results in a set of non-linear partial differential equations, where the electrochemical reactions constitute a boundary condition at the electrode. While the reaction model is proper to the reaction under study, the transport model is merely determined by the mass transport of the species in the electrochemical reactor. As a result, it is possible to direct an electrochemical investigation in an adapted experimental reactor (electrochemical cell) under conditions for which the description of the transport phenomena can be simplified, without a

For controlling the mass transport contribution to the overall electrochemical kinetics, a rotating disk electrode possesses favorable features. The RDE configuration provides analytical equations to describe the mass transport and hydrodynamics in the electrochemical cell. It is known that a simplified transport model can be used if an RDE and diluted solutions are used in the experimental set-up. The hydrodynamic equations and the

**2. Linear sweep voltammetry in combination with a rotating disk electrode**

tool does not allow a reliable determination of the model parameters.

determining.

(III)/(II) system.

loss of precision.

determining elementary reactions.

can sometimes be masked by other reactions, e.g., in (Gattrell et al., 2004), and in that case Koutecky-Levich method cannot be used.

Also, to calculate the charge transfer parameters, a plot of the natural logarithm of the charge transfer current density as a function of potential, known as a Tafel plot, is often constructed. In the linear region of this curve, the transfer coefficient can be deduced from the slope and the rate constants from the intercept. The Tafel method is well established for simple reaction mechanisms (Bamford & Compton, 1986; Diard et al., 1996), but it becomes much more complicated for complex mechanisms (Gattrell et al., 2004; Wang et al., 2004). When there are significant diffusional or ohmic effects in the electrolyte, or additional electrode reactions, the Tafel plot deviates from linearity (Yeum & Devereux, 1989) and the charge transfer parameters cannot be determined.

Besides these well-known graphical methods, some authors suggest other methods to extract the kinetic parameters from an LSV experiment. They usually involve the fitting of theoretical expressions to the experimental data. In (Rocchini, 1992) the charge transfer parameters are estimated by fitting experimental polarization curves with exponential polynomials. Obviously, this method is only valid if the reaction rate is determined by charge transfer alone. Caster et al. fit convolution potential sweep voltammetry experiments with equations for a reversible charge transfer reaction with only one reactant present initially and under conditions of planar diffusion (Caster et al., 1983). No other steps are allowed to occur either before or after the electrode reaction. Yeum and Devereux propose an iterative method for fitting complex electrode polarization curves (Yeum & Devereux, 1989). They split up the total current density into contributions from the partial reactions and use simplified expressions for the current-potential relations. With these expressions they try to find the parameters that optimize the correlation between model and experimental data by minimizing a least squares cost function. This optimization is done by trial-and-error. In (Rusling, 1984) tabulated dimensionless current functions are fitted to linear sweep voltammograms. Therefore, a least squares cost function is minimized; however, no details on the minimizing algorithm are given.

In a series of papers, Harrison describes a hardware/software system for the complete automation of electrode kinetic measurements (Aslam et al., 1980; Cowan & Harrison, 1980a;b; Denton et al., 1980; Harrison, 1982a;b; Harrison & Small, 1980a;b). This involves the fitting of the data using a library of reaction schemes to determine the model parameter values. A quasi-Newton method is used to minimize the modulus of the differences between experiment and theory or the sum of the weighted squares of the differences. Although it is emphasized that care has to be taken in weighting the observations, no information on the determination of the weighting factors is given. Moreover, no criterion to decide whether the fitting is acceptable or not is discussed.

In (Bortels et al., 1997; Van den Bossche et al., 1995; 2002; Van Parys et al., 2010) a numerical approach is developed in order to define the underlying reaction mechanism . By using the MITReM (Multiple Ion Transport and Reaction Model) model, mass transport by convection, diffusion and migration but also the presence of homogeneous reactions in the electrolyte, are accounted for. The related model parameters such as diffusion coefficients, rate constants and transfer coefficients are adjusted in order to improve the agreement between experimental and simulated polarization curves. Thus, the best parameter values, corresponding to the best simulated curve, are selected by a *chi-by-eye* approach, without a statistical evaluation. 2 Electrochemical Cells

can sometimes be masked by other reactions, e.g., in (Gattrell et al., 2004), and in that case

Also, to calculate the charge transfer parameters, a plot of the natural logarithm of the charge transfer current density as a function of potential, known as a Tafel plot, is often constructed. In the linear region of this curve, the transfer coefficient can be deduced from the slope and the rate constants from the intercept. The Tafel method is well established for simple reaction mechanisms (Bamford & Compton, 1986; Diard et al., 1996), but it becomes much more complicated for complex mechanisms (Gattrell et al., 2004; Wang et al., 2004). When there are significant diffusional or ohmic effects in the electrolyte, or additional electrode reactions, the Tafel plot deviates from linearity (Yeum & Devereux, 1989) and the charge transfer parameters

Besides these well-known graphical methods, some authors suggest other methods to extract the kinetic parameters from an LSV experiment. They usually involve the fitting of theoretical expressions to the experimental data. In (Rocchini, 1992) the charge transfer parameters are estimated by fitting experimental polarization curves with exponential polynomials. Obviously, this method is only valid if the reaction rate is determined by charge transfer alone. Caster et al. fit convolution potential sweep voltammetry experiments with equations for a reversible charge transfer reaction with only one reactant present initially and under conditions of planar diffusion (Caster et al., 1983). No other steps are allowed to occur either before or after the electrode reaction. Yeum and Devereux propose an iterative method for fitting complex electrode polarization curves (Yeum & Devereux, 1989). They split up the total current density into contributions from the partial reactions and use simplified expressions for the current-potential relations. With these expressions they try to find the parameters that optimize the correlation between model and experimental data by minimizing a least squares cost function. This optimization is done by trial-and-error. In (Rusling, 1984) tabulated dimensionless current functions are fitted to linear sweep voltammograms. Therefore, a least squares cost function is minimized; however, no details on the minimizing algorithm are

In a series of papers, Harrison describes a hardware/software system for the complete automation of electrode kinetic measurements (Aslam et al., 1980; Cowan & Harrison, 1980a;b; Denton et al., 1980; Harrison, 1982a;b; Harrison & Small, 1980a;b). This involves the fitting of the data using a library of reaction schemes to determine the model parameter values. A quasi-Newton method is used to minimize the modulus of the differences between experiment and theory or the sum of the weighted squares of the differences. Although it is emphasized that care has to be taken in weighting the observations, no information on the determination of the weighting factors is given. Moreover, no criterion to decide whether the

In (Bortels et al., 1997; Van den Bossche et al., 1995; 2002; Van Parys et al., 2010) a numerical approach is developed in order to define the underlying reaction mechanism . By using the MITReM (Multiple Ion Transport and Reaction Model) model, mass transport by convection, diffusion and migration but also the presence of homogeneous reactions in the electrolyte, are accounted for. The related model parameters such as diffusion coefficients, rate constants and transfer coefficients are adjusted in order to improve the agreement between experimental and simulated polarization curves. Thus, the best parameter values, corresponding to the best simulated curve, are selected by a *chi-by-eye* approach, without a statistical evaluation.

Koutecky-Levich method cannot be used.

cannot be determined.

given.

fitting is acceptable or not is discussed.

Although the reaction and transport models are defined more precisely, the lack of a fitting tool does not allow a reliable determination of the model parameters.

A quantitative, accurate and statistically founded modeling approach of electrochemical reactions has been the focus of an extensive work in our research group (Aerts et al., 2011; Tourwé et al., 2007; 2006; Van Parys et al., 2008). It is a generally applicable method to model an electrochemical reaction and to determine its mass and charge transfer parameters quantitatively. The reliability of the model parameters and the accuracy of the parameter fitting are key-elements of the method. A plausible reaction mechanism and the characteristic parameters of the electrochemical reaction are extracted from LSV experiments with a rotating disk electrode. Compared to others, this method offers the advantage that it uses one integrated expression that accounts for mass and charge transfer steps, and this without simplifying their mathematical expressions. The whole polarization curve is considered, rather than just some part in which only mass or charge transfer are supposed to be rate determining.

In this paper, we explain throughly this modeling methodology for the rotating disk electrode (RDE) and the inverted rotating disk electrode (IRDE) configurations. The modeling and quantification of the electrochemical parameters are applied to redox reactions with one electron transfer mechanism: the ferri/ferrocyanide system and the hexaammineruthenium (III)/(II) system.
