Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode Reactions for CE Mechanism at a Spherical Electrode

A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential equation that describes the homogeneous processes coupled to electrode reaction. In this paper the approximate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These approximate results are compared with the available analytical results and are found to be in good agreement.


Introduction
Microelectrodes are of great practical interest for quantitative in vivo measurements, e.g. of oxygen tension in living tissues [1][2][3], because electrodes employed in vivo should be smaller than the unit size of the tissue of interest.Microelectrodes having the geometry of a hemisphere resting on an insulating plane are difficult to fabricate, but their behavior is easily predicted [4].They also have advantages in electrochemical measurements of molten salts with high temperature [5].Microelectrodes of many shapes have been described [6].Microelectrodes of simple shapes are experimentally preferable because they are more easily fabricated and generally conformed to simpler voltammetric relationships.Those shapes with restricted size in all superficial dimensions are of special interest because many of these reach true steady-state under diffusion control in a semi infinite medium [7].Nevertheless, there is interest in microelectrodes of more complicated shapes, only because the shapes of small experimental electrodes may not always be quite as simple as their fabricators intended.Moreover, and ironically, complex shapes may sometimes be more easily modeled than simpler ones [8].However, many applications of microelectrodes of different shapes are impeded by lack of adequate theoretical description of their behavior.
As far back as 1984, Fleischmann et al. [9,10] used microdisc electrodes to determine the rate constant of coupled homogeneous reactions (CE, EC', ECE, and DISPI mechanisms).Fleischmann et al. [9] obtained the steady-state analytical expression of the concentration of the species HA and H by assuming the concentration of the specie A is constant.Also measurement of the current at microelectrodes is one of the easiest and yet most powerful electrochemical methods for quantitative mechanistic investigations.The use of microelectrodes for kinetic studies has recently been reviewed [11] and the feasibility demonstrated of accessing nano second time scales through the use of fast scan cyclic voltammetry.However, these advantages are earned at the expense of enhanced theoretical difficulties in solving the reaction diffusion equations at these electrodes.Thus it is essential to have theoretical expressions for non steady state currents at such electrodes for all mechanisms.
The spherical EC' mechanism was firstly solved by Delmastro and Smith [12].In electrochemical context Diao et al. [13] derived the chronoamperometric current at hemispherical electrode for EC' reaction, whereas Galceran's et al. [14] evaluated shifted de facto expression and shifted asymptotic short-time expression for disc electrodes using Danckwerts relation.Rajendran et al. [15] derived an accurate polynomial expression for transient current at disc electrode for an EC' reaction.More recently, Molina and coworkers have derived the rigorous analytical solution for EC', CE, catalytic processes at spherical electrodes [16].Fleischmann et al. [17] demonstrate that Neumann's integral theorem can be used to numerically simulate CE mechanism at a disc electrode.Dayton et al. [18] also derived the spherical response using Neumann's integral theorem.In this literature steady-state limiting current is discussed in [19] In general, the characterization of subsequent homogeneous reactions involves the elucidations of the mechanism of reaction, as well as the determination of the rate constants.Earlier, the steady-state analytical expressions of the concentrations and current at microdisc electrodes in the case of first order EC' and CE reactions were calculated [9].However, to the best of our knowledge, till date there was no rigorous approximate solutions for the kinetic of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at spherical electrodes under non-steadystate conditions for all possible values of reaction/diffusion parameters  have been reported.The purpose of this communication is to derive approximate analytical expressions for the non-steady-state concentrations and current at spherical electrodes for all possible values of parameters using Homotopy perturbation method.

Mathematical Formulation of the Problems
At a range of Pt microelectrodes, the electroreduction of acetic acid, a weak acid, is strutinized by as in a usual CE reaction scheme.This reaction is known to proceed via the following reaction sequence [9]: where 1 and 2 are the rate constants for the forward and back reactiuons respectively and are related to another by the known equilibrium constant for the acid dissociation [9].The initial boundary value problems for different diffusion coefficients ( ) can be written in the following forms [9]: where HA are the diffusion coefficient of the species , 1 and 2 are the rate constant for the forward and back reactions respectively and HA H A are the concentration of the species HA, H and A. These equations are solved for the following initial and boundary conditions:  and  c c  c k k where S is the radius of the spherical electrode.We introduce the following set of dimensionless variables: where , u , v , w  and  represent the dimensionless concentrations and dimensionless radial distance and dimensionless time parameters respectively.
where are dimensionless diffusion coefficients.The initial and boundary conditions are represented as follows: The dimensionless current at the microdisc electrode can be given as follows:

Analytical Expression of Concentrations and Current Using HPM
Recently, many authors have applied the HPM to various problems and demonstrated the efficiency of the HPM for handling non-linear structures and solving various physics and engineering problems [20][21][22][23][24][25].This method is a combination of homotopy in topology and classic perturbation techniques.The set of expressions presented in Equations ( 9)-( 14) defines the initial and boundary value problem.The homotopy perturbation method [26][27][28][29][30][31][32] is used to give the approximate solutions of coupled non-linear reaction/diffusion Equations ( 9) to (11).The dimensionless reaction diffusion parameters E  , S  ,  are related to one another, since the bulk solution is at equilibrium in the non-steady state.Using HPM (see Appendix A and B), we can obtain the following solutions to the Equations ( 9) to (11).
The Equations ( 16)-( 18) satisfies the boundary conditions (12) to (14).These equations represent the new approximate dimensionless solution for the concentration profiles for all possible values of parameters n for the current for small and medium of parameters.

Comparison with Fleischmann Work [9]
an [9] ave de ved the analytical expressions of dimensionless steady-state concentrations u and v as follow not arrived upon in the third specie A. The normalized current is given by Fleischmann assumed that the concentration profiles of w is constant.So the definite solution for concentration profiles of w is When The normalized current is given by Previously, mathematical expressions pertaining to steady-state analytical expres trations an nt at mic ated by Fleischmann et al. [9].In addition, we have also pre-sions of the concen d curre rodisc electrodes were calcul sented an approximate solution for the non-steady state concentrations and current.

Discussions
Equations ( 16)-( 18) are the new and simple approximate so isomers calculated using Homotopy perturbation method for th boundary conditions Equations ( 12)- (14).approxim lution of the concentrations of the e initial and The closed ate solution of current is represented by the Equation (19).The dimensionless concentration profiles of u versus dimensionless distance  are expressed in Figures 1(a)-(d).From these figures, we can infer that the value of the concentration decreases when  and distance  increases when

Conclusions
The time dependent non-linear reaction/diffusion equations for spherical microelectrodes for CE mechanism has been formulated and solved using HPM.The primary result of this work is simple approximate calculation of concentration profiles and current for all values of fundamental parameters.We have presented approximate solutions corresponding to the species HA, H and A i terms of the parameters of

ppendix A: Solution of the Equations (9) to 1) Using Homotopy Perturbation Method
this Appendix, we indicate how Equations ( 16) to (18) this paper are derived.To find the solution of Equaons (9) to (11) we first construct a Homotopy as folws: 2 0 and the initial approximations are as follows: Substituting Equation (A10) into Equations (A1) and (A2) and (A3) and arranging the coefficients of powers uations.
Subjecting Equations (A11) to (A16) to Laplace transformation with respect to results in Now the initial and boundary conditions become where s is the Laplace variable and an overbar ind a Laplace-t to icates ransformed quantity.Solving equations (A17) (A22) using reduction of order (see Appendix-B) for initial and boundary conditions (A26) to (A28), we can find the following results solving the Equation (A20), and using the After putting Equations (A29) and (A30) into Equation (A35) and Equations (A31) and (A32) into Equation (A36) and Equations (A33) and (A34) into Eq (A37).Using inverse Laplace transform [33], the final results can be described in Equations ( 16) to (18) in the text.The remaining components of and According to the HPM, we can conclude that x is deter-be completely determined such th term mined by the previous term.

Appendix B
In this Appendix, we derive the solution of Equation (A20) by using reduction of order.To illustrate the basic concepts of reduction of order, we consider the equation at each where P, Q, R are function of r.Equation (A20) can be lified to Using reduction of order, we have Substituting the value of P in the above Equation (A7) become The given Equation (B3) reduces to Substituting (B8) in (B7) we obtain, Integrating Equation (B9) twice, we obtain

1 E 1 Figure 1 . 1 Figure 3 . 1 andFigure 4 .
Figure 1.Normalized concentration u at microelectrode.The concentrations were computed using Equation (16) for various values of  and for some fixed small value of hnalytical Chemistry, Marcel Dekker, New York, Vol.11, 1972, p. 85.I. A. Silver, "Microelectrodes and trodes used in Biology," In: D. J. G. Ives and G. J. Jane is our pleasure to thank the referees for their valuable comments.

DDD
) and (B10) in (B4) we have, conditions Equations (A27) and ), we can obtain the value of the constants A and B. Substituting the value of the constants A Equation (B11) we obtain the Equation (A we can solve the other differential Equations (A17), Diffusion coefficient of the species HA (cm 2 sec -1 ) H Diffusion coefficient of the species H (cm 2 sec -1 )A Diffusion coefficient of the species A (cm 2 sec -1 )DDiffusion coefficient (cm 2 sec -1 )Area of the spherical electrode (cm 2 )Number of the electron (dimensionless)