Short communication
New analytical method for solving nonlinear equation in rotating disk electrodes for second-order ECE reactions

https://doi.org/10.1016/j.jelechem.2020.114106Get rights and content

Highlights

  • Second-order non-linear steady-state system of equations arising in RDE is studied.

  • The hyperbolic function and the Taylor series method are used to solve the equation.

  • Pade approximation technique is used to increase the convergence level.

  • The chemical rate constant is determined.

Abstract

The analytical and numerical solution of nonlinear diffusion equations are performed to study the chronoamperometric limiting current generated from the electrochemical reaction in a rotating disk electrode for second-order ECE reactions when the chemical step is irreversible. Simple and closed-form of expressions for the concentration of reactant and the current response are obtained as a function of the rotation rate and diffusion coefficients. The effect of various physical parameters on concentration and current have been computed and shown graphically. Also, the concentration/current expressions here derived using hyperbolic function and padé approximation technique are in satisfactory agreement with numerical results and limiting case results.

Introduction

Rotating disc electrode is a powerful tool employed in electrochemistry to control the transport of electroactive species. RDE systems are relatively easy to use and easy to build. The rate of fabric transport depends hugely well on the driver's rotation speed. Many attempts have been made in recent decades to develop analytical methods to solve these nonlinear equations in rotating disk electrodes.

Two electrode reactions are coupled by a chemical reaction if the product of the first electrode reaction is the reactant of chemical reaction and the product of the latter is a reactant of the second electrode reaction [[1], [2], [3], [4], [5], [6]]. For steady-state conditions, Levich [7] obtained the analytical expression for limiting the current of the rotating electrode under the assumption of infinite Schmidt numbers (Sc). Compton et al. [8] obtained the chronoamperometry current for ECE, DISP1, DISP2,EC and CE reaction by solving the convective diffusion equation using Hales method. Lin et al. [9] derived the catalytic current at a rotating disk electrode using the perturbation method. Bartlett et al. [10] derived the approximate analytical expression of flow at a rotating disc electrode for ECE reactions for various limiting cases. Saravanakumar et al. [11] obtained the non-steady state current at a rotating disk electrode for all time by solving the convection-diffusion equation analytically.

Jansi Rani et al. [12] reported current at a rotating disk electrode under transient and steady-state conditions using the homotopy perturbation method. Chitra Devi et al. [13] derived the approximate analytical expression for the velocity component in the rotating disk electrode. Rajendran et al. [14] obtained a two-point Padé approximation of mass-transfer rate for a rotating disk electrode for all Schmidt numbers.

Recently Visuvasam et al. [15] derived an analytical expression of the current generated from the electrochemical reaction in a porous rotating disk electrode (PRDE). Diard and Montella [16] obtained the steady-state concentration of species near a uniformly accessible rotating disk electrode, using both symbolic and numerical methods. Oldham et al. [17] reviewed the advantages and disadvantages of electrochemistry with little or no supporting electrolyte. The kinetics of hydrogen peroxide reduction reaction is analyzed by Amirfakhri et al. [18] using a rotating disk electrode. An extensive study on homogeneous catalysis of electrochemical reactions for the catalytic (EC') mechanism via rotating disk electrode made by Compton et al. [19, 20]. Do et al. [21] described a one dimensional mathematical model for a porous enzymatic electrode with a direct electron transfer mechanism. Recently, Dolinska et al. [22] proposed a new approach to study the electrocatalytic oxidation of glucose on a rotating disk electrode.

He recently solved the one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes [23]. In a microgravity space [24] and fluid mechanics [25] He applied a variational fractal theory. Also, He developed the Taylor series solution for fractal Bratu-type equations arising in the electrospinning process [26]. He recently proposed the exp-function method for solving the non-linear equations [27]. More recently, some effective methods with fractal derivatives are reviewed for the fourth-order non-linear integral boundary value problems [28].

In this communication, the analytical expression for concentration profile and current are derived using a hyperbolic function method, which is a particular case of the exp-function method. Also we obtained the concentration using the Taylor series method and Padé approximation technique [23]. The analytical expression for concentration and current is validated by comparing it with simulation results (Matlab program).

Section snippets

Nomenclature

SymbolsNameUnitValues
 [A]Concentration of species [A]mol cm−3
 [B]Concentration of species  [B]mol cm−3
 [C]Concentration of species  [C]mol cm−3
 [D]Concentration of species  [D]mol cm−3
 [E]Concentration of species  [E]mol cm−3
[A]z→∞Bulk concentration of species [A]mol cm−33 × 10−3
[C]z→∞Bulk concentration of species [B]mol cm−33 × 10−3
kSecond order homogeneous rate constantcm2s−1109 − 1012
zDistance from the electrode surfacecm
DDiffusion coefficientcm2s−15 × 10−6
nNumber of the electrons

Mathematical formulation of the problem

A second-order ECE reaction mechanism may be written as follows [10]:A±eBB+CkDD±eEwhere k is the second-order homogeneous rate constant, and the homogeneous chemical reaction in that model is entirely irreversible. We assume that product B of the first electron transfer reaction is not electroactive. Then B undergoing a homogeneous chemical reaction with C to produce an electroactive product D. Also, in this model, the influence of disproportionation/comproportionation reactions has not

An analytical expression for the concentrations in the second-order ECE reaction mechanism using a new analytical method

In recent decades, many attempts have been made to develop analytical methods for solving such nonlinear equations. Recently many analytical methods have been focused on constructing an analytical solution such as, homotopy perturbation method (HPM) [[29], [30], [31], [32]], variational iteration method (VIM) [33,34], homotopy analysis method (HAM) [[35], [36], [37]], Akbari-Ganji Method (AGM Method) [38], Li-He's variational principle methods [[39], [40], [41]], exp-function method [42] and

An analytical expression for the concentrations in second-order ECE reaction mechanism using Taylor's series and Padé approximation technique

He [23,29] suggests a Taylor's series and Padé approximation technique to solve the Lane-Emden equation. This method also extended to all non-linear differential equation in fractional calculus [43,44]. We can also obtain concentrations u( χ), v( χ) and w( χ) using Taylor's series and Padé approximation technique (Appendix D) as follows:uχ=bχ+2κb3γbγ+1χ21κγbγ+16χ2vχ=γuχ+1γ1χwχ=1γγbγ+1γbγχvχwhere ‘b’ can be obtained from the equation2κγb2+b32κγ+2κ3=0

If a higher order approximate is

Numerical simulation

The importance of accurate reference data and the testing/validation of electrochemical modelling/simulation algorithms is recognized by Britz et al. [[45], [46], [47]]. The non-linear differential Eqs. (14), (15), (16), (17), (18) have been solved numerically using MATLAB software [48]. A respective script pdex4 is provided in Appendix-E. The analytical expressions of concentration u (χ) obtained from hyperbolic function method and Taylor series and Padé approximation method is compared with

Discussion

Eqs. (20), (21), (22) and Eqs. (26), (27), (28) are the new analytical expressions for the concentration in second-order ECE reaction mechanism at the rotating disk electrode for the steady-state condition. In this section, the effect of the ratio between the rate of the chemical step and the rate of mass transport across the diffusion layer are discussed. Fig. 2, Fig. 3, Fig. 4 shows the dimensionless steady-state concentration for various species u (χ), v (χ) and w (χ) involved in the ECE

Conclusions

The system of non-linear second-order differential equations is solved using the hyperbolic function method and Taylors series method. This method has been established for studying an electrode reaction coupled with a homogeneous chemical reaction. The method presented in this paper is powerful since it yields a simplified treatment of complicated reaction-diffusion equations. The effects of numerous essential parameters such as diffusion rates, potential and the homogeneous rate constant on

CRediT authorship contribution statement

J. Visuvasam: Data curation, Software, Formal analysis, Writing - original draft. A. Meena: Visualization, Supervision, Validation. L. Rajendran: Conceptualization, Methodology, Resources, Writing - review & editing, Investigation, Project administration.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This work was supported by the Department of Science and Technology, SERB-DST (EMR/2015/002279) Government of India. The Authors are very grateful to the reviewers for their careful and meticulous reading of the paper. The Authors are also thankful to Shri J. Ramachandran, Chancellor, Col. Dr. G. Thiruvasagam, Vice-Chancellor, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai, for their constant encouragement.

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