Analytical study and parameter-sensitivity analysis of catalytic current at a rotating disk electrode

A convective-diffusion equation with semi-infinite boundary conditions for rotating disk electrodes under the hydrodynamic conditions is discussed and analytically solved for electrochemical catalytic reactions. The steady-state catalytic current of the rotating disk electrode is derived for various possible values of parameters by using a new approach of the homotopy perturbation method. The theoretical approach in this paper is described, for the first time, on the basis of convection–diffusion equations for the kinetics of Fenton's reagent using a platinum rotating disk electrode. The obtained approximate analytical expression for the concentrations of ferric and ferrous ions for steady-state conditions are shown to be highly accurate when compared to numerical results and other approximations found in the literature. A sensitive analysis of parameters of the current and concentration is presented.

x Distance from the RDE (cm) y dimensionless distance from the surface of electrode

Introduction
The rotating disc electrode (RDE), also known as the hydrodynamic working electrode, is the hydrodynamic electroanalytical technique used to limit the diffusion layer thickness. RDE, whose steady-state current is determined by a solution flow rather than a diffusion, is becoming one of the most powerful methods for studying both diffusions in electrolytic solutions and kinetics of fast electrode reaction [1,2]. Since the sixties of the last century, RDE has been an effective tool for investigating electrochemical reactions when Levich [3] suggested an empirical relationship between diffusion limiting current and rotation rate (ω) by solving the convective-diffusion equation. Nikolic et al [4] calculated an important hydrodynamic parameter in RDE, namely the diffusion coefficient. Compton et al [5] applied the Hale approach for the calculation of the chronoamperometric responses arising from potential-step experiments at rotating disk electrodes. Newman [6], Gregory et al [7], and Diard and Montella [8] obtained different expressions of the convection-diffusion equation in rotating disc electrode for finite and infinite Schmidt number (Sc) using symbolic and numerical computation procedures. Rajendran et al [9] derived a two-point Padé approximation of mass-transfer rate for rotating disc electrodes for various values of Schmidt numbers. Rani et al [10] solved the mathematical problem corresponding to a one-electron reversible electron transfer at a rotating disk electrode under transient and steady-state conditions by using the homotopy perturbation method. Daniel Okuonghae et al [11] employed the discontinuous Galerkin finite element method for solving the rotating disk electrode for various reaction mechanisms.
To our knowledge, analytical expressions for the concentrations of the ferric ion and ferrous ion, and the corresponding current response are not available for all possible values of parameters under steady-state conditions. In addition, the theory and experimental results of rotating disk electrodes are available only for the case of boundary conditions in a finite domain. However, the discussion in this manuscript is concerned with infinite domain boundary condition, that when the concentration of both ferric ion and ferrous ion are known when the distance from the surface of electrode tends to infinity. Approximate analytical expressions for the steady-state concentrations and the current for rotating disk electrodes for the catalytic reaction mechanism will be driven.

Mathematical formulation of the problem
The kinetics of catalytic reactions occurring at the rotating disk electrode can be written as follows: where in the case of Fenton's reaction, A, R, C, B and D respectively denote ferric ion, reactant, ferrous ion, hydrogen peroxide and radical. k f and k b are, respectively, the forward and backward reaction rate constants.
This catalytic model is considered under the following assumptions [12]: I. Second order rate constant in the forward direction is greater than the second order rate constant in the backward direction.
II. Concentration of the catalyzing species B is much larger than that of the reactant and product of the electrochemical reaction, A and C.
III. All other species have uniform concentration throughout the diffusion layer.
IV. Diffusion coefficients of reacting species are equal (D A =D C =D).
V. Mass transfer of any electroactive species takes place only by means of diffusion and convection, and that concentration changes due to any other effects such as migration are negligible. Figure 1 shows a schematic diagram of the rotating disk electrode for the catalytic reaction mechanism [13].
The mass balance equations for the above reaction scheme may be written as follows [12] - and ¢ k f respectively denote ferric ion, ferrous ion, the hydrogen peroxide, radical and the product of the forward rate constant, respectively. In addition, the velocity of the solution is x 3 2 1 2 2 In the bulk solution, the ferric and ferrous ion concentrations are both assumed constants. At the electrode surface (x=0), the ferric ion concentration is zero because of the limiting current condition. Also the flux of ferric and ferrous ion near the disk electrode are assumed equal. Therefore, we have the following set of boundary conditions:

Dimensionless form of convection-diffusion equations
In dimensionless form, equations (3) and (4) take the form

Analytical expressions of concentrations and current
Many analytical methods have been developed in recent years to find approximate solutions for nonlinear boundary value problems that arise in various fields of sciences and engineering such as fluid mechanics, plasma physics, optical fibre, solid-state physics, and biological and chemical sciences. These methods include, but not limited to, the residual method [14], series method [15], hyperbolic function method [16], variation iteration method [17,18], The Adomian decomposition method [19], the differential transformation method [20], Green's function based methods [21][22][23] and Taylor series method [24].
One of the methods that has received a great deal of attention is the homotopy perturbation method (HPM), which uses the embedding parameter as a small parameter. The HPM often reaches an asymptotic solution with a few iterations. First proposed by Ji-Haun He [25], HPM has been intensively used by researchers, in all fields of sciences and engineering, for solving various nonlinear problems. In recent years, HPM has gone through many modifications to make its implementation easier and to achieve faster convergence even for strong nonlinear differential equations [26][27][28][29][30]. In this paper, a modified form of the HPM is employed to obtain the following concentrations of ferric ion (θ A ) and ferrous ion (θ C ) (see details in appendix): The analytical homotopy perturbation solution for equations (7) and (8) is expressed in the following series forms (14) and (15) respectively into equations (12) and (13), then equating like powers of p lead to the nonlinear system for p 0

Substituting equations
and so on up to the desired power of p. For equations (7) and (8), we will find a series solution up to the second power of p. So, by solving system (16)-(23), we obtain The two-term HPM solution is now obtained from the relations q q q The limiting current of ferric ion reduction at the RDE is computed by ) is the potential.

Previous analytical results
The theory and experimental results of rotating disk electrodes that are available in the literature are only for boundary conditions in the finite domain. For example, a system of a convection-diffusion equation in a finite domain modeling an EC-catalytic mechanism at the rotating disk electrode [31] and a system of convectiondiffusion equations representing a pseudo-first-order EC-catalytic process at a rotating disc electrode in a finite domain [32]. Lin et al [12] used the perturbation technique to derive the following approximate distributions of ferric ion and ferrous ion, respectively This result is valid for low rotation rates or when hydrodynamic boundary layer thickness (δ) is large. In addition, Haberland et al [33] discussed the same nonlinear diffusion equations for the case when

Results and discussion
In this section, we test the accuracy of the obtained analytical expressions by the means of numerical simulations. The derived equations (30) and (31)) represent new simple analytical expressions for the concentrations of ferric ion and ferrous ion for all possible values of parameters. These concentrations are dependent on the dimensionless parameters  and  , which in return are dependent on the parameters and k f , k b , C B , C D , D and δ. The MATLAB function pdex4 is used to solve the boundary value problems for the two convection-diffusion equations (7) and (8). In tables 1-6, the analytical expressions for the concentrations of Table 1. Comparison between the results of the proposed HPM and perturbation method [12] for the concentration of ferric ion θ A assuming =  1 and =  3. ferric and ferrous ions, obtained in section 4, are compared with numerical simulation results and other analytical results in the literature for various values of the dimensionless parameters  and  . In all tables, the average relative error of the proposed method is significantly lower than the one obtained in [12]. In tables 1-3, it is observed that the concentration of the ferric ion is depleted at the surface and more flux to the surface and hence a higher cathode current is noticed. Equation  figure 2(a), it is observed that the current begins its course decreasing until it reaches a minimum value then increase over the domain for all values of D. This behavioour of the current seems to be identical for all other parameters as evident in figures 2(c)-(d). Table 3. Comparison between the results of the proposed HPM and perturbation method [12] for the concentration of ferric ion θ A assuming =  2 and =  1.  Table 4. Comparison between the results of the proposed HPM and perturbation method [12] for the concentration of ferrous ion θ C assuming =  1 and =  3.  Figure 3 shows the current versus the hydrodynamic boundary layer thickness for stationary and rotating disk electrodes (δ is directly proportional to w -1 2 ). It is inferred from figure 3 that the current decreases to its minimum value (approx. 1.531 82) before it starts to increase again whereas the current for stationary electrode attains the steady-state value. Figures 4(a) and (b) confirm that the catalytic current for steady-state voltammetry tends to a limiting value at a high potential value, which increases dramatically as the thickness of boundary layer δ is raised. The dependence of voltammetry current on rotation rate or boundary layer thickness is an evidence that a convection and diffusion control the process.
By the sensitivity analysis, we study the effect of each parameter on the current density. By evaluating the slope of the current density with respect to a parameter, we can determine the influence of that parameter on the current. The percentage of change in current with respect to D, δ and m are found to be 99.9%, 0.08% and 0.02%, respectively. As depicted in figure 5, it is concluded that the diffusion coefficient almost has the entire effect on the current density compared to δ and m. Table 5. Comparison between the results of the proposed HPM and perturbation method [12] for the concentration of ferrous ion θ C assuming =  5 and =  0.5.  Table 6. Comparison between the results of the proposed HPM and perturbation method [12] for the concentration of ferrous ion θ C assuming =  2 and =  1.

Conclusion
In this paper, the system of steady-state nonlinear convection-diffusion equations in the kinetics of the catalytic reaction system is solved analytically. The model investigated the influence of parameters on the response of catalytic current. Approximate closed-form analytical expressions pertaining to the concentrations of the ferric ion and ferrous ion in addition to the current for all possible values of the diffusion and kinetic parameters are obtained using a modified approach of the HPM. The obtained analytical results are believed to be useful in describing the fundamental feature of hydrogen peroxide action by the ferric and ferrous ion concentration. The ideas and discussions in this paper can be extended to other nonlinear problems in rotating disc electrodes and rotating ring-disc electrodes.