American Journal of Analytical Chemistry, 2011, 2, 93-103 doi:10.4236/ajac.2011.22010 Published Online May 2011 (http://www.SciRP.org/journal/ajac)

Copyright © 2011 SciRes. AJAC

Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode

Reactions for CE Mechanism at a Spherical Electrode

Alagu Eswari, Seetharaman Usha, Lakeshmanan Rajendran* Department of Mathematics, The Madura College (Autonomous), Madurai, India

E-mail:* raj_sms@rediffmail.com Received October 18, 2010; revised January 16, 2011; accepted January 21, 2011

Abstract 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 approxi-mate 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. Keywords: Non-Linear Reaction/Diffusion Equation, Homotopy Perturbation Method, CE Mechanism,

Reduction of Order, Spherical Electrodes

1. Introduction Microelectrodes are of great practical interest for quanti-tative in vivo measurements, e.g. of oxygen tension in living tissues [1-3], because electrodes employed in vivo should be smaller than the unit size of the tissue of inter-est. Microelectrodes having the geometry of a hemi-sphere resting on an insulating plane are difficult to fab-ricate, but their behavior is easily predicted [4]. They also have advantages in electrochemical measurements of molten salts with high temperature [5]. Microelec-trodes of many shapes have been described [6]. Micro-electrodes 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 microelec-trodes 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 cur-rent at microelectrodes is one of the easiest and yet most powerful electrochemical methods for quantitative me-chanistic 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 essen-tial 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

A. ESWARI ET AL. 94

at hemispherical electrode for EC’ reaction, whereas Galceran’s et al. [14] evaluated shifted de facto expres-sion 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 re-cently, Molina and coworkers have derived the rigorous analytical solution for EC’, CE, catalytic processes at spherical electrodes [16]. Fleischmann et al. [17] dem-onstrate 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 homo-geneous reactions involves the elucidations of the me-chanism of reaction, as well as the determination of the rate constants. Earlier, the steady-state analytical expres-sions of the concentrations and current at microdisc elec-trodes in the case of first order EC’ and CE reactions were calculated [9]. However, to the best of our knowl-edge, till date there was no rigorous approximate solu-tions for the kinetic of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at spherical electrodes under non-steady- state conditions for all possible values of reactio- n/diffusion parameters E , S , 1S , 1E , 2S , ,1

2 and 2E have been reported. The purpose of this communication is to derive approximate analytical ex-pressions for the non-steady-state concentrations and current at spherical electrodes for all possible values of parameters using Homotopy perturbation method. 2. 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]:

1

2

2

HA H A1

H H2

k

k

e

(1)

where 1 and 2 are the rate constants for the forward and back reactiuons respectively and are related to an-other 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]:

k k

HA H A, ,D D D

2HA HA HA HA

HA 1 HA 2 H A2

2c c D cD k c

t r rr

2H H H H

H 1 HA2

2c c D cD k c

t r rr

2 H Ak c c (3)

2A A A A

A 1 HA2

2c c D cD k c

t r rr

2 A Hk c c (4)

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 follow-ing initial and boundary conditions:

H A, and D D DHA, H and A

, and c c c

k k

H H HA HA A A0 ; , , t c c c c c c (5)

H HA A; 0, 0, 0Sr r c dc dr dc dr

Ac

(6)

H H HA HA A ; , , r c c c c c (7)

where S is the radius of the spherical electrode. We introduce the following set of dimensionless variables:

r

HA H A

HA H A

HA H A1 22

HA HA

2 21 2 H A

HA HA HA

2 21 HA 2 A

1 S1HAHA H

2 21 HA 2 H

2 S2HAHA A

, , , ,

, , ,

,

, ,

,

S

S

S SE S

S SE

S SE

c c c ru v w

rc c c

D t D D

D Dr

k r k c c r

D D c

k c r k c r

DD c

k c r k c r

DD c

(8)

where , u ,v ,w and represent the dimen-sionless concentrations and dimensionless radial distance and dimensionless time parameters respectively.

2

2

2E S

u u uu v

w (9)

21

1 12

2E S

v v vu v

1 w (10)

22

2 22

2E S

w w wu v

2 w (11)

where E , S , 1E , 1S , 2E and 2S are the di-mensionless reaction/diffusion parameters and 1 , 2 are dimensionless diffusion coefficients. The initial and boundary conditions are represented as follows:

0, 1; 1; 1u v w (12)

1, 0; 0; 0v u w (13)

k c c (2) , 1; 1; 1u v w (14)

The dimensionless current at the microdisc electrode

Copyright © 2011 SciRes. AJAC

A. ESWARI ET AL. 95 can be given as follows:

H 1=S SI nFAD r dv d

(15)

3. 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 [20-25]. 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 [26-32] is used to give the approximate solutions of coupled non-linear reaction/diffusion Equations (9) to (11). The dimensionless reaction diffusion parameters E , S ,

1E , 1S , 2E and 2S 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).

2

1

1

1

1

2

1

11( , ) 1 exp

1 4π

1 exp 1 exp

2

1 + exp 1 exp

1 2

11 exp

4 π

E

E

u

erfc

erfc

(16)

1

2

1

11

1 1, 1

2

11 1 1 exp

4 2 π

E

v erfc

(17)

2

21 2

1 2 2

2

11

22

1 2

2 1 2 1

2

11, 1 exp

4 π

11 exp

4 π

+ exp 1 exp

1

2

E

E

w

erfc 2

(18)

The Equations (16)-(18) satisfies the boundary condi-tions (12) to (14). These equations represent the new approximate dimensionless solution for the concentration profiles for all possible values of parameters E , S ,

1E , 1S , 2E , 2S , 1 and 2 . e

From Eqdim nsionless curr

is as follows:

uationsent, w

(15) and (17), we can obtain the hich

H H

1

1 1

0.28217 0.564191

S S

E

I r nFD AC

(19)

Equation (19) represents the new approximate expres-sio

Fleischm n et al. h ri

s:

n for the current for small and medium of parameters.

4. Comparison with Fleischmann Work [9]

an [9] ave de ved the analytical ex-pressions of dimensionless steady-state concentrations u and v as follow

1 1

11 1

1 1E S

1 1

1 1 exp 1

EE

E S

u

(20)

1 1

1 11 1

1 exp 1EE S

E S

v

1 E

(21)

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 concentra-tion profiles of w is

H H

1 1 1 1

1 1

1

S S

E E E S

E S

I r nFD AC

(22)

When 1 1E S the above equation becomes

1 1

11 exp 1 E Eu 1

(23)

1 1

11 exp 1 E Ev

(24)

The normalized current is given by

H H 11S S E EI r nFD AC 1 (25)

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 concend curre rodisc electrodes were calcul

Copyright © 2011 SciRes. AJAC

A. ESWARI ET AL.

Copyright © 2011 SciRes. AJAC

96

sented an approximate solution for the non-steady state concentrations and current.

5. Discussions

Equations (16)-(18) are the new and simple approximate so isomers calculated using Homotopy perturbation method for thboundary 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 1E . Moreover when

1E and 1 , the conce ains the steady- res 2(a)-(d alized c n-isomers v us values pa-

ntr), the fo

ation attnorm

r variostate value. In Figutration profiles of

onceof

rameters are plotted. From these figures, it is inferred that the concentration v increases abruptly and reaches the steady-state value when 5 . In Figures 1(a)-(d)

a)-(d), values of dimensionless concentrations u and v for various values of

and 2( the

E , E and and for

1 1 are reported and a satisfactory agreement with the available [9] estimates of Fleischmann et al. is noticed when is large. Figures 3(a)-(d) show the normalized dimensional concentration pro f w in file o space cal-culated using Equation (18). The plot was constructed for various values 2 0.1, 1E 1 1and . Fr m these

s it is confirmed that the value of the concentration profile of w increases when

ofigure

and 2E increases. Al-so from the Figures 1(a)-(d) and 2(a)-(d), it is evident that the concentration of species HA and H increases when the radius of the electrode ( Sr ) decreases. There-fore, the use of the f the sm dius is clearly advangeous for the study of CE reaction mechanism. The concentration of specie A de eases w n the radius of the electrode decreases. It reaches the steady state value when 1

electrode o

c

all

h

r

e

a

r

. The dimensionless current log versus for various values of 1E is given in Figure 4. From these figure, it is evident that the value of the current

decreases abruptly and reaches the steady-state value.

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

u

1.005

1

0.995

0.99

0.98

τ = 1

τ = 0.5

τ = 0.1

γE = 0.1, ε1 = 0.01

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

u 1

0.96

0.94τ = 1

τ = 0.5

τ = 0.198

1 1.5 2 2.5 3 3.5 4 4.5 5

5

0.98

0.975

0.97

τ = 10, 100

1 1.5 2 2.5 3 3.5 4 4.5 5

τ = 10, 100 γE = 0.1, ε1 = 0.5

0.

0.92

0.9

0.88

0.86

(a) (b)

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

u

1 1.5 2 2.5 3 3.5 4 4.5 5

1

0.9

0.95

τ = 1

τ = 10, 50

τ = 0.5 τ = 0.1

γE = 1, ε1 = 0.010.85

0.8

0.75

0.7

Dimensionless distance ρ

Dim

ensi

onle τ = 10, 100

0.7

ss c

once

ntra

tion

u

1 1.5 2 2.5 3 3.5 4 4.5 5

1

0.9

0.95

τ = 1

τ = 0.5

τ = 0.1

γE = 1, ε1 = 0.5

0.85

0.8

0.75

0.65

0.6

0.55

(c) (d)

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 1 E when the reaction/diffusion parameter and dimensionless diffusion coef-ficient (a) , 10.1 0.01 E (b) , 10.1 0.5 E (c) 1 , 11 0.0 E (d) 5, 11 0. E . The key to the graph: ( __ ) represen +ts Equation (16) and ( ) represents Equation (23) [9].

A. ESWARI ET AL. 97

Dim

ensi

onle

ss c

once

ntra

tion

v

1 1.1 1.2 1.3 1.4

1 τ = 1

τ = 10, 50

τ = 0.5

τ = 0.1

γE1 = 0.1, ε1 = 0.01

0.8

0.6

0.4

0.2

0

Dimensionless distance ρ

1.5 1.6 1.7 1.8 1.9 2

Dimensionless distance ρ (a) (b)

Dim

ensi

onle

ss c

once

ntra

tion

v

3 3.5 4 4.5 5

1τ = 1

τ = 10, 50

τ = 0.5

τ = 0.1

γE1 = 0.1, ε1 = 0.5

0.8

0.6

0.4

0.2

1 1.5 2 2.5 0

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

v

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

1 τ = 1

τ = 10, 100

τ = 0.5

τ = 0.1

γE1 = 1, ε1 = 0.01

0.8

0.6

0.4

0.2

0

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

v

1 1.5 2 2.5 3 3.5 4

1

τ = 1

τ = 10, 100

τ = 0.5

τ = 0.1

γE1 = 1, ε1 = 0.5

0.8

0.6

0.4

0.2

0

(c) (d)

Figure 2. Normalized concen r various values of

tration v at microelectrode. The concentrations were computed using Equation (17) fo and for some fixed small value of E when the reaction/diffusion parameter and dimensionless diffusion coef-

ficient (a) , 1 10.1 0.01 E (b) , 1 10.1 .50 E (c) 1 , 1 11 0.0 E (d) .5, 1 11 0 E . The key to the graph: ( __ )

represen +) rep

ts Equation (17) and ( resents Equation (24) [9].

Dimensionless distance ρ

1 1.5 2 2.5 3 3.5

1

τ = 1

τ = 0.01

τ = 0.5

τ = 0.1

γE2 = 0.1, ε1 = 0.01

1.035

1.03

1.025

1.02

1.015

1.01

1.005

0.995 4 4.5 5

Dim

ensi

onle

ss c

once

ntra

tion

w

men

sion

less

con

cent

ratio

n w

D

i

Dimensionless distance ρ 1 1.5 2 2.5 3 3.5

1

τ = 1

τ = 0.01

τ = 0.5

τ = 0.1

γE2 = 0.1, ε1 = 0.5

1.003

1.0025

1.002

1.0015

1.001

1.0005

(a) (b)

0.9995

4 4.5 50.999

Copyright © 2011 SciRes. AJAC

A. ESWARI ET AL.

Copyright © 2011 SciRes. AJAC

98

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

w

1.08

1.07

1 1.5 2 2.5 3 3.5 4 4.5 5

1

τ = 1

τ = 0.01

τ = 0.1

γE2 = 1, ε1 = 0.01

0.99

1.06

1.05

1.04

1.03

1.02

1.01

Dimensionless distance ρ

Dim

ensi

onle

ss c

once

ntra

tion

w

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

1

τ = 1

τ = 0.01

τ = 0.1

γE2 = 1, ε1 = 0.5

1.045

1.04

1.005

0.995

1.035

1.03

1.025

1.02

1.015

1.01

(c) (d)

Figure 3. Normalized concentration w at microelectrode. The concentrations were computed using Equation (18) for various values of

, 2 1

and for some fixed value of the reaction/diffusion parameter and dimensionless diffusion coefficient (a) 0.1 0.01 E (b) , 10.1 0.52 (c) , 2 11 0.01 E (d) , 2 11 0.5 E . E

Dimensionless time τ

Dim

ensi

onle

ss C

urre

nt lo

g ψ

0 1 2 3 4 5 6 7 8 9 10

ε1 = 0.01

104

γE1 = 10

103

102

101

100

γE1 = 1

γE1 = 0.5γE1 = 0.1

Dimensionless time τ

Dim

ensi

onle

ss C

urre

nt lo

g ψ

0 1 2 3 4 5 6 7 8 9 10

ε1 = 0.5

γE1 = 10

103

102

101

100

γE1 = 1 γE1 = 0.5

γE1 = 0.1

a

(a) (b)

Figure 4. V riation of normalized non-steady-state current response log as a function of the dimensionless time for various values 1 ) 1 0.5 1 0.01 (bof E and fo a) r the fixed values of ( . The curv were

he key to the represents Equation (19) and (+) resen tion (25) [9].

when the values of

es computed using Equation (19). ts EqT

graph: ( __ ) rep ua

1 0.1E . Also, the value of the current

2S , 1 , 2 and e

and cr va

dgements

pporDSr. Mo

hiagatics, The Madura

based on the Homotopy perturba-his m thod can be easily extended to find

the conce trations urrent for all mechanism for all s fo rious complex boundary condi-

e

rk s su ted by the Department of Science gy ( T) Government of India. The au-

ank M . S. Meenakshisundaram, Secre-ura C llege Board, T. V. Krishnamoorthy, S. T rajan Head of the Department of

College, Madurai, for their

tion m

mitions.

This andthors alstary

e

croelectro

7. Acknow

wo Tech

o, Th

pal ana

thod. Tn

de

l

wanolo th

e Madd

increa e reaction diffusion pa-rameter

ses when th

1E increa 6. Conclusions The time dependent non-linear reaction/diffusion equa-tions 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 fun-damental parameters. We have presented approximate solutions corresponding to the species HA, H and A iterms of the parameters of

ses

n PrinciMathem, S , 1E , 1S , 2E , E

A. ESWARI ET AL. 99 constant encouragement. It

hnalytical Chemistry, Marcel Dekker,

New York, Vol. 11, 1972, p. 85.

I. A. Silver, “Microelectrodes andtrodes used in Biology,” In: D. J. G. Ives and G. J. Jane

is our pleasure to thank the referees for their valuable comments.

8. References

[1] J. Koryta, M. Brezina and J. Pradacova, “Laboratory Tec iques in Electroanalytical Chemistry,” In: A. J. Bard, Ed., Electroan

[2] D. B. Cater and Elec-

Eds., Reference Electrodes Theory and Practical, Aca-demic Press, New York, 1961, p. 464.

[3] R. S. Pickard, “A Review of Printed Circuit Microelec-trodes and Their Production,” Journal of Neuroscience Method, Vol. 1, No. 4, 1979, pp. 301-318. doi:10.1016/0165-0270(79)90019-0

[4] K. B. Oldham, “Comparison of Voltammetric Steady States at Hemispherical and Disc Microelectrodes,” Journal of Electroanalytical , 1988, pp. 11-19. oi:10.1016/0022-0728(88)85002-2

Chemistry, Vol. 256, No. 1d

[5] G. J. Hills, D. Inman and J. E. Oxley, “Progress in Reac-tion Kinetics,” In: I. S. Longmuir, Ed., Advances in Po-larography, Pergamon Press, Oxford, Vol. 3, 1960, p. 982.

[6] Rcroelectrod

. M. Wightman and D. O. Wipf, “Voltammetry at Mi-es,” In: A. J. Bard, Ed., Electroanalytical

New York, Vol. 15, 1989, p. 26

ochemical Microscopy,” In:

Chemistry, Marcel Decker,

[7] C. Amatore, “Scanning Electr

7.

I. Rubinstein, Ed., Physical Electrochemistry: Principles, Methods And Applications, Marcel Dekker, New York, 1995, p. 131.

[8] A. M. Bond, K. B. Oldham and C. G. Zoski, “Steady State Voltammetry,” Analytical Chemica Acta, Vol. 216, No. 16, 1989, pp. 177-230. doi:10.1016/S0003-2670(00)82009-7

[9] M. Fleischmann, F. Lasserre, J. Robinson and D Swan, “The Application of MicroelectrodHomogeneous Processes Coupled to

es to the Study of Electrode Reactions:

nd CE Reactions,” Journal of Electroana-try, Vol. 177, No. 1-2, 1984, pp. 97-114.

doi:10.1016/0022-0728(84)80215-6

Part I. EC’ Alytical Chemis

urnal of Electroanalyti-

[10] M. Fleischmann, F. Lasserre and J. Robinson, “The Ap-plication of Microelectrodes to the Study of Homogene-ous Processes Coupled to Electrode Reactions: Part II. ECE And DISP 1 Reactions,” Jocal Chemistry, Vol. 177, No. 1-2, 1984, pp. 115-127. doi:10.1016/0022-0728(84)80216-8

[11] M. I. Montenegro, “Application of Microelectrodes in Kinetics,” In: R. G. Compton and G. Hancock Eds., Re-

here Bound-t Polarography,” Jo-

search in Chemical Kinetics, Elsevier, Amsterdam, 1994.

[12] J. R. Delmastro and D. E. Smith, “Methods for Obtaining Approximate Solutions to the Expanding-Spary Value Problem in Direct Curren urnal of Physical Chemistry, Vol. 71, No. 7, 1967, pp. 2138-2149. doi:10.1021/j100866a026

[13] G. Daio and Z. Zhang, “The Theory of Catalytic Elec-

trode Processes at a Hemispherical Ultramicroelectrode nd Itedu

728(96)05022-X

a s Application for the Catalytic Behavior of the Sixth R ction Wave of Fullerence, C60,” Journal of Electro-analytical Chemistry, Vol. 429, No. 1-2, 1997, pp. 67-74. doi:10.1016/S0022-0

tate

[14] J. Galceran, S. L. Taylor and P. N. Bartlett, “Steady-SCurrents at Inlaid and Recessed Microdisc Electrodes forFirst-Order EC’ Reactions,” Journal of Electroanalytical Chemistry, Vol. 476, No. 2, 1999, pp. 132-147. doi:10.1016/S0022-0728(99)00378-2

[15] L. Rajendran and M. V. Sangaranarayanan, “Diffusion at Ultramicro Disk Electrodes: Chronoamperometric Cur-rent for Steady-State EC’ Reaction Using Scattering Analogue Techniques,” Journal of Physical Chemistry B, Vol. 103, No. 9, 1999, pp. 1518-1524. doi:10.1021/jp983384c

[16] A. Molina and I. Morales, “Comparison Between Deriva-

nal Journal of Electroche- .

nalytical Chemistry,

tive and Differential Pulse Voltammetric Curves of EC, CE and Catalytic Processes at Spherical Electrodes and Microelectrodes,” Internatiomical Science, Vol. 2, No. 5, 2007, pp. 386-405

[17] M. Fleischmann, D. Pletcher, G. Denuault, J. Daschbach and S. Pons, “The Behavior of Microdisk and Microring Electrodes: Prediction of The Chronoamperometric Re-sponse of Microdisks and of The Steady State for CE and EC Catalytic Reactions by Application of Neumann’s In-tegral Theorem,” Journal of ElectroaVol. 263, No. 2, 1989, pp. 225-236. doi:10.1016/0022-0728(89)85096-X

[18] M. A. Dayton, A. G. Ewing and R. M. Wightman, “Re-sponse of Microvoltammetric Electrodes to Homogene-ous Catalytic and Slow Heterogeneous Charge-TransfReactio-Ns,” Analytical Chemistry, V

er ol. 52, No. 14, 1980,

pp. 2392-2396. doi:10.1021/ac50064a035

[19] G. S. Alberts and I. Shain, “Electrochemical Study of Kinetics of a Chemical Reaction Coupled between Two Charge Transfer Reactions Potentiostatic Reduction of p-Nitrosophenol,” Analytical Chemistry, Vol. 35, No. 12, 1963, pp. 1859-1866. doi:10.1021/ac60205a019

[20] Q. K. Ghori, M. Ahmed and A. M.Siddiqui, “Application of Homotopy Perturbation Method to Squeezing Flow of a Newtonian Fluid,” International Journal of Nonlinear Science and Numerical Simulation, Vol. 8, No. 2, 2007, pp. 179-184. doi:10.1515/IJNSNS.2007.8.2.179

[21] T. Ozis and A. Yildirim, “A Comparative Study of He’s Homotopy Perturbation Method for Determining Fre-quency-Amplitude Relation of a Nonlinear Oscillater with Discontinuities,” International Journal of Nonlinear Science and Numerical Simulation, Vol. 8, No. 2, 2007, pp. 243-248. doi:10.1515/IJNSNS.2007.8.2.243

[22] S. J. Li and Y. X. Liu, “An Improved Approach to Non-linear Dynamical System Identification Using PID Neural Networks,” International Journal of Nonlinear Science and Numerical Simulation, Vol. 7, No. 2, 2006, pp. 177-182. doi:10.1515/IJNSNS.2006.7.2.177

[23] M. M. Mousa and S. F. Ragab, “Application of the Ho-motopy Perturbation Method to Linear and Nonlinear Schrödinger Equations,” Zeitschrift fur Naturforschung,

Copyright © 2011 SciRes. AJAC

A. ESWARI ET AL.

Copyright © 2011 SciRes. AJAC

100

Vol. 63, 2008, pp. 140-144.

[24] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3

[25] J. H. He, “Homotopy Perturbation Method: A New Non-linear Analytical Technique,” Applied Mathematics and Computation, Vol. 135, No. 1, 2003, pp. 73-79. doi:10.1016/S0096-3003(01)00312-5

[26] J. H. He, “A Simple Perturbation Approach to Blasius Equation,” Applied Mathematics and Computation, Vol. 140, No. 2-3, 2003, pp. 217-222. doi:10.1016/S0096-3003(02)00189-3

[27] J. H. He, “Homotopy Perturbation Method for Solving Boundary Value Problems,” Physics Letter A, Vol. 350, No. 1-2, 2006, pp. 87-88. doi:10.1016/j.physleta.2005.10.005

[28] J. H. He, “Some Asymptotic Methods for Strongly Non-linear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796

[29] A. Eswari and L. Rajendran, “Analytical Solution of Steady State Current at a Micro Disk Biosensor,” Journal

of Electroanalytical Chemistry, Vol. 641, No. 1-2, 2010, pp. 35-44. doi:10.1016/j.jelechem.2010.01.015

atical Modeling of

try, Vol.

[30] A. Meena and L. Rajendran, “MathemAmperometric and Potentiometric Biosensors and System of Non-Linear Equations-Homotopy Perturbation Ap-pro-Ach,” Journal of Electroanalytical Chemis644, No. 1, 2010, pp. 50-59. doi:10.1016/j.jelechem.2010.03.027

[31] G. Varadharajan and L. Rajendran, “Analytical Solution of the Concentration and Current in the Processes Involved in a PPO-Rota

Electroenzymatic ting-Disk-Bioelec-

trode,” Natural Science, Vol. 3, No. 1, 2011, pp. 1-7. doi:10.4236/ns.2011.31001

[32] V. Marget and L. RajendNon Steady-State Concentration Profil

ran, “Analytical Expression of es at Planar Elec-

trode for the CE Mechanism,” Natural Science, Vol. 2, No. 11, 2010, pp. 1318-1325. doi:10.4236/ns.2010.211160

[33] M. Abramowitz and I. A. Stegun, “Handbook of Mathe-matical Functions,” Dover publications, Inc., New York, 1970.

A. ESWARI ET AL. 101

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 Equa-ons (9) to (11) we first construct a Homotopy as fol-ws:

A(1 Inintilo

2

2

2

2

21

20E S

d u du dup

d dd

d u du dup u

d dd

vw

(A1)

2

11 2

21

1 12

21

20E S

d v dv dvp

d dd

d v dv dvp u

d dd

1vw

(A2)

2

22 2

21

d w dw dwp

d dd

(A32

) 2

2 2 22

20E S

d w dw dwp u vw

d dd

and the initial approximations are as follows:

0 0 00; 1; 0; 1u v w (A4)

0 0 01; 0; 0, 0v du d dw d (A5)

0 0 0; 1; 0; 1u v w (A6)

0; 0; 0; 0i i iu v w (A7)

1; 0; 0, 0i i iv du d dw d (A8)

; 0; 0; 0 1, 2,i i iu v w i (A9)

and

(A10)

Substituting Equation (A10) into Equations (A1) and (A2) and (A3) and arranging the coefficients of powers

uations.

2 30 1 2 3

2 30 1 2 3

2 30 1 2 3

u u pu p u p u

v v pv p v p v

w w pw p w p w

p , we can obtain the following differential eq2

0 0 0 02

2:

d u du dup

d dd 0 (A11)

21 1 1 1

0 0 0S v w (A12)

20 0 0 01

1 2

2: 0

d v dv dvp

d dd

(A13)

21 1 1 1 1

1 1 02

2: 0E S

d v dv dvp u

d dd 1 0 0v w

(A14)

and 2

0 0 0 022 2

2: 0

d w dw dwp

d dd

(A15)

21 1 2 1 1

2 2 02

2: 0E S

d w dw dwp u

d dd

2 0 0v w (A16)

Subjecting Equations (A11) to (A16) to Laplace transformation with respect to results in

20 0

02

21 0

d u dusu

dd (A17)

20 0

021 1

2 10

d v dv sv

dd (A18)

20 0

022 2

2 10

d w d w sw

d

2

2: 0E

d u du dup u

d dd

and

d (A19)

1 121 1

12

2 1

0

s

ES

d u du esu

d s s sd

(A20)

1 1211 1 1

121 1 1

2 1s

SEd v dv s ev

d s s sd

(A21)

0

1 1221 1 2

122 2 2

2 1

0

s

SEd w d w s ew

d s s sd

(A22)

Now the initial and boundary conditions become

0 0 00; 1; 0; 1u v w (A23)

0 0 01; 0; 0, 0v du d dw d (A24)

0 0 0; 1 ; 1 ; 1u s v s w s (A25)

0; 0; 0; 0i i iu v w (A26)

0idw d1; 0; 0, i iv du d (A27)

; 0; 0; 0 1, 2,i i iu v w i (A28)

where s is the Laplace variable and an overbar inda Laplace-tto

icates ransformed quantity. Solving equations (A17)

(A22) using reduction of order (see Appendix-B) for

Copyright © 2011 SciRes. AJAC

A. ESWARI ET AL. 102

initial and boundary conditions (A26) to (A28), we can find the following results

solving the Equation (A20), and using the

0 1 u s (A29)

1

1

11 2 2

1

1 1

1 1

1 1

1 1

1 1 1

s

S SE

s s

S S

eu

s ss

e e

s s s

(A30)

and

1 1

0

1s

ev

s s

(A31)

1

1

1

1 11 11 2 2 2 2

1

1

1

1 12

s

S SE E

s

S

ev

s s s s

e

s

(A32)

and

0 1w s (A33)

1

2

2

1

2 2 121 2 2

2

1

2 1 2

2 2 1

1

2 1 2

s

S SE

s

S

s

S

ew

s s s

e

s

e

s s

(A36)

After putting Equations (A29) and (A30) into Equa-tion (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

1

2 2 1

According to the HPM, we can conclude that

0 11

limp

u u u u

(A35)

v

(A34)

0 11

limp

v v v

0 11

limp

w w w

w (A37)

uation

nu x nv 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

2

2

d d

dd

c cP Qc

R

where P, Q, R are function of r. Equation (A20) can be lified to

(B1)

simp

1 11

0s

E e

2

1 112

d d2

dd S

u usu

s s s

Using reduction of order, we have

(B2)

2; P Q

s

and 1 1

1s

ES

eR

s s s

(B3)

Let u cv (B4)

Substitute (B4) in (B1), if u is so chosen that

d2 0

d

cPc

(B5)

Substituting the value of P in the above Equation (A7) become

1c (B6)

The given Equation (B3) reduces to

1 1v Q v R (B7)

where 2

1 10, 4 2

P P RQ Q R

c

(B8)

Substituting (B8) in (B7) we obtain,

1 1s

SES

ev sv

s s s

(B9)

Integrating Equation (B9) twice, we obtain

2

SS Ev A e B es

1 1 1

s

S e

(B10)

g (B6) and (B10) in (B4) we have,

21 1

Ss s

Substitutin

Copyright © 2011 SciRes. AJAC

A. ESWARI ET AL.

Copyright © 2011 SciRes. AJAC

103

1

12

s

SS

e

2

1

1

1

SSEA e B e

us

s s

(B11)

ing the boundary 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 (Awe can solve the other differential Equations (A17), (A and or Appendix C

N

Us(A28

and B in the 30). Similarly

18), (A19), (A21) (A22) using the reduction of der method.

omenclature

Symbols

HA

Hc Concentration of the species H (mole

c Concentration of the species HA

cm–3)

Concentration of the spec

Bulk concentration of the species HA (molecm–3)

Bulk concentration of the species H –3

(molecm–3)

(molecm–3)

Ac ies A (molecm–3)

HAc

Hc (molecm )

Bulk concentration of the species A Ac

HAD Diffusion coefficient of the species HA (cm2sec–1)

HD Diffusion coefficient of the species H (cm2sec–1)

AD Diffusion coefficient of the species A (cm2sec–1)

D Diffusion coefficient (cm2sec–1)

R Radial distance(cm)

e (s)

Rate constant for the forward reactions (cm3/molesec)

r

u, v, w s concentrations (dimensionless)

T Tim

1k

2k

Rate constant for the backward reactions 3(cm /molesec)

Sr Radius of spherical electrode (cm)

Distance in the radial direction (cm)

Dimensionles

Dimensionless radial distance (dimensionless)

Dimensionless time (dimensionless)

SI Current density at a sphere (ampere/cm2)

A

F Faraday constant (Cmole–1)

n

Greek s b

Area of the spherical electrode (cm2)

Number of the electron (dimensionless)

ym ols

1 Dimensionless diffusion coefficient (dimensionless)

2 Dimensionless diffusion coefficient (dimensionless)

1 1

2 2

, ,

, ,

,

E S

E S

E S

Dimensionless reaction/diffusion parameters (dimensionless)