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International Scholarly Research NetworkISRN Physical ChemistryVolume 2012, Article ID 745616, 12 pagesdoi:10.5402/2012/745616

Research Article

Approximate Analytical Solution of Nonlinear Reaction’sDiffusion Equation at Conducting Polymer Ultramicroelectrodes

Anitha Shanmugarajan,1 Subbiah Alwarappan,2 and Rajendran Lakshmanan1

1 Department of Mathematics, The Madura College, Madurai-625011, India2 Nanomaterials Research and Education Center, University of South Florida, Tampa, FL, USA

Correspondence should be addressed to Rajendran Lakshmanan, raj [email protected]

Received 14 November 2011; Accepted 22 December 2011

Academic Editors: R. Gómez and H. Luo

Copyright © 2012 Anitha Shanmugarajan et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A theoretical model of reaction/diffusion within conducting polymer microelectrodes is discussed. The model is based on thesteady-state diffusion equation containing a nonlinear term related to the Michaelis-Menten kinetic of the enzymatic reaction. Ananalytical expression pertaining to the concentration of substrate and current is obtained using homotopy perturbation methodfor all values of diffusion and the saturation parameter. The substrate concentration profile and current response can be used ina large range of concentrations including the non-linear contributions. These approximate analytical results were found to be ingood agreement with the previously reported limiting case results.

1. Introduction

The advantages of the ultramicroelectrodes (UMEs) includesteady-state current, rapid response time, minimal iR drop,lower detection limits, and sensitive analysis in a highly re-sistive medium [1–4]. Furthermore, due to their diminishedsurface area or radii of the electrodes, they are often em-ployed as probes to monitor various chemical events occur-ring inside the living cell [5–8]. Ultramicroelectrodes pos-sess many advantages for studying electrochemical kineticsand in electroanalytical applications, imaging, and surfacemodification [9]. Microelectrodes modified with a polymerfilm find potential application in various sensing applications[10–12]. The working principle of polymer-modified ultra-microelectrodes occurs in the following manner: initially, theredox analyte interacts with the immobilized active receptorsites present in the polymer matrix, then at the underlyingelectrode surface. Briefly, we can say that the redox reactionis mediated by the polymeric layer. Further, due to theelectroactive property of the polymer, charge can percolatethrough the polymer chain and thereby reaches the elec-trode/interface to give rise to a redox current and is di-rectly proportional to the concentration of the analyte.The electron transfer occurs between the substrate and the

catalytic receptor site and as a result the kinetics of thesubstrate/product transformation will be governed by theproperties of the mediating electroactive polymer film.

Recent advancements in ultramicroelectrodes modifiedwith conducting polymers were reported elsewhere [13–18]. Further, the analytical applications of various polymer-modified sensors and the reaction/diffusion at the conduct-ing polymer electrode (where the chemical reaction termis described by Michaelis-Menten kinetics) were reviewedextensively by various groups [19–23]. Recently, Lyons etal. [24] evaluated the analytical solutions corresponding tothe steady-state substrate concentration profile and currentobserved at a conducting polymer microelectrode when thesubstrate concentration is low. For higher values of thesubstrate concentration, a kinetic rate law based on theMichaelis-Menten equation is more appropriate. Recently,Anitha et al. [25] derived the analytical expression for non-steady-state concentrations of substrate and mediator at apolymer modified ultramicroelectrodes using reduction oforder method. However, to the best of our knowledge, therewere no analytical results available till date that correspondsto the steady-state substrate concentration and current for allpossible values of diffusion parameter γ and the saturationparameter α. However, in general, analytical solutions of

2 ISRN Physical Chemistry

Conducting wire

Insulating sheathUMD electrode

Polymer film

Figure 1: Schematic representations of the geometry adopted by the polymer-coated microelectrode and the expected diffusion profileadopted by the analytes.

Table 1: Approximate expressions of concentration of substrate and current under the boundary conditions (2a) and (2b).

Conditions Concentrations and currents Figures

For very small values of α u =sinh(√γρ)ρ sinh(√γ) (5)

Figures2(a)–2(c)

ψ = √γ coth(√γ)− 1 (6) Figure 4(a)

For large values of α and allvalue of γ

u = 1 + γ(12α2 − 6α− γ)36α3

(ρ2 − 1) + γ2

120α3(ρ4 − 1) (8) Figure 3

ψ = γ(30α2 − 15α− γ)45α3

(9)Figures 4(b)

and 4(c)

For small and mediumvalues of α and all values of γ

u = sinh(√γρ)

ρ sinh(√γ) −αγ sinh(√γρ)ρsinh3(√γ) +

αγ

sinh2(√γ) (10) Figure 2

ψ = √γ coth(√γ)−1+αγ cosech2(√γ)(√γ coth(√γ)−1) (11) Figures 4(a)and 4(c)

Table 2: Approximate expressions of concentration of substrate, and current under the boundary conditions (12a)–(12c).

Conditions Concentrations and currents Figures

For very small values of αu = sinh(

√γρ)

ρ{

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ])}

(17)Figures 5 and

6

ψ =√γ cosh[√γ]− sinh[√γ]

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ]) (18) Figure 8

For large values of α (α ≥ 1)and all values of γ

u = 1 + γ(12α2v − 6αv − γv − 2γ)

36α3v

(ρ2 − 1− 2

v

)

+γ2

120α3

(ρ4 − 1− 4

v

) (20) Figure 7

ψ = γ(30α2v − 15αv − γv − 5γ)

45α3v(21) Figure 9

For small and mediumvalues of α and all values of γ

u = sinh(√γρ)

ρ{

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ])}

− αγ sinh(√γρ)

ρ{

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ])}3

+αγ{

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ])}2

(22)Figures 5 and

6

ψ =√γ cosh[√γ]− sinh[√γ]

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ])

− αγ(√γ cosh[√γ]− sinh[√γ])(

sinh[√γ] + 1v

(√γ cosh[√γ]− sinh[√γ]))3

(23) Figure 8

ISRN Physical Chemistry 3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized distance ρ

(i)

(ii)

(iii)

(iv)

Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4


(i)

(ii)

(iii)

(iv)

Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(b)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(i)

(ii)

(iii)

(iv)

(c)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(i)

(ii)

(iii)

(iv)

(d)

Figure 2: Plot of normalized substrate concentration “u”at a microelectrode modified with a polymer film as a function of “ρ”. Theconcentrations were computed for various values of the reaction diffusion parameter “γ” when (a) α = 0.001, (b) α = 0.01, (c) α = 0.1, and(d) α = 1. (solid line) (10); (dotted line) (5). Solid lines are compared with points. In this figure (i) γ = 0.1, (ii) γ = 1, (iii) γ = 10, and(iv) γ = 100.

nonlinear differential equations are more interesting anduseful numerical solutions, as they are used in various kindsof data analysis. Therefore, herein, we employ analyticalmethod to evaluate the steady-state substrate concentrationand current for all possible values of diffusion and saturationparameter.

2. Mathematical Formulation of the BoundaryValue Problem and Analysis

2.1. Assumptions. The conducting polymer film will adopta hemispherical geometry upon electrodepositing them onto a microelectrode support (Figure 1). If such a geometryis assumed then the substrate will exhibit spherical diffusionboth in solution immediately adjacent to the polymer filmand within the polymer film itself. We assume that thesubstrate exhibits Michaelis-Menten kinetics when it reacts

at a site within the polymer film. The substrate exhibits first-order kinetics approximation when the substrate concentra-tion is low. For higher values of the substrate concentrationa kinetic rate law based on the Michaelis-Menten equationis more appropriate. We also assume that the partitioncoefficient for the substrate is unity.

2.2. Neglecting Substrate Concentration Polarization in Solu-tion. Initially, the substrate diffusion in solution, the trans-port, and kinetic processes within the polymer film wereall neglected. On the other hand, the reaction/diffusionequation under steady-state condition corresponding to thenormalized substrate concentration within the polymer filmcan be expressed as [24]

d2u

dρ2+

2ρ

du

dρ− γu

1 + αu= 0. (1)


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(i)

(ii)

(iii)

(iv)

(v)

(a)

0 0.2 0.4 0.6 0.8 1


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu (i)

(ii)

(iii)

(iv)

(v)

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

(b)

0 0.2 0.4 0.6 0.8 1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

(i)(ii)

(iii)

(iv)

(v)


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(c)

Figure 3: Plot of normalized substrate concentration “u” at a microelectrode modified with a polymer film as a function of “ρ”. Theconcentrations were computed using (8) for various values of the reaction diffusion parameter “γ” when (a) α = 10 for (i) γ = 0.1, (ii)γ = 1, (iii) γ = 10, (iv) γ = 20, and (v) γ = 30, (b) α = 50 for (i) γ = 0.1, (ii) γ = 1, (iii) γ = 10, (iv) γ = 50, and (v) γ = 100, and (c) whenα = 100 for (i) γ = 0.1, (ii) γ = 1, (iii) γ = 50, (iv) γ = 100, and (v) γ = 200.

In (1), normalized substrate concentration u = s/s∞, wheres denotes the substrate concentration within the polymerfilm and s∞denotes the bulk concentration of substrate. Thesaturation parameter α = s∞/KM , where KM denotes theMichaelis constant. The reaction/diffusion parameter γ isgiven by γ = kccΣa2/KMDS, where kc represents the reactionrate constant, a denotes the radius of the microelectrode.cΣ denotes the total catalyst concentration in the film, andDS is the diffusion coefficient of the substrate within thepolymer film. The normalized distance parameter ρ is givenby r/a (where r represents the radial variable). The boundaryconditions pertaining to the normalized form are

ρ = 0, dudρ= 0, (2a)

ρ = 1, u = 1. (2b)

The boundary condition (2a) states that the substrate iselectroinactive at the disk. The normalized current density isdefined as

ψ = ianFADSs∞

=(du

dρ

)ρ=1

. (3)

2.3. Unsaturated (First-Order) Catalytic Kinetics. We initiallyconsider the situation where the substrate concentration inthe film is less than the Michaelis constant KM . This situationwill pertain when the product αu� 1. Hence (1) reduces to

d2u

dρ2+

2ρ

du

dρ− γu = 0. (4)


0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

10

Normalized diffusion parameter

Nor

mal

ized

curr

ent

(i)

(a)

0 20 40 60 80 1000

2

4

6

8

10

12

(iii)

(ii)

(i)


Nor

mal

ized

curr

ent

(b)

0 20 40 60 80 100


Nor

mal

ized

curr

ent

0

1

2

3

4

5

6

7

8

9

10

(iv)

(iii)

(ii)

(i)

(c)

Figure 4: Plot of normalized current “ψ” versus the reaction/diffusion parameter “γ” at a microelectrode modified using a conductingpolymer film. (a) (i) α = 0.1, 1, the current was calculated using (6) and (11) for all small and medium values of α. (b) (i) α = 5, (ii) α = 10,(iii) α = 100, the current was calculated using (9) for all large values of “α”. (c) (i) α = 0.01, 0.1, 1 (ii) α = 10, (iii) α = 50, and, (iv) α = 100,the current was calculated using (9) and (11) (solid line) (11); (dotted line) (6). Solid lines are compared with points.

By solving (4), we can obtain the expression for the normal-ized substrate concentration as follows:

u(ρ, γ) = sinh

(√γρ)

ρ sinh(√

γ) . (5)

Also the expression of the normalized steady-state currentdensity is given below

ψ(γ) = √γ coth(√γ)− 1. (6)

2.4. For Large Values of the Saturation Parameter α and AllValues of the Reaction/Diffusion Parameter γ. We now con-sider the limiting situation where the substrate concentration

in the film is very much greater than the Michaelis constantKM . In this case αu� 1 and (1) reduces to

d2u

dρ2+

2ρ

du

dρ− γα

+γ

α2u= 0. (7)

We obtain the approximate expression of normalized con-centration of substrate as

u(ρ, γ,α

) = 1 + γ(12α2 − 6α− γ)

36α3(ρ2 − 1) + γ

2

120α3(ρ4 − 1).

(8)

Using (3), we obtain the expression of the normalized currentdensity as

ψ(γ,α) = γ

(30α2 − 15α− γ)

45α3. (9)



Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

0 0.2 0.4 0.6 0.8 10.7

0.75

0.8

0.85

0.9

0.95

1

(i)

(ii)

(iii)

(iv)(v)

(vi)

(a)

0 0.2 0.4 0.6 0.8 10.7

0.75

0.8

0.85

0.9

0.95

1

(i)

(ii)

(iii)

(iv)(v)

(vi)


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(b)


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

0 0.2 0.4 0.6 0.8 10.7

0.75

0.8

0.85

0.9

0.95

1

(i)

(ii)

(iii)

(iv)(v)

(vi)

(c)

Figure 5: Plot of normalized substrate concentration “u” at a microelectrode modified with a conducting polymer film as a function of“ρ”. The concentrations were computed for various values of Biot number “v” when (a) α = 0.001, γ = 0.1, (b) α = 0.01, γ = 0.1, and (c)α = 0.1, γ = 0.1. (solid line) (22); (dotted line) (17). Solid lines are compared with points. In this figure (i) v = 0.1, (ii) v = 0.5, (iii) v = 1,(iv) v = 5, (v) v = 10, (vi) v = 50.

The above approximation will be valid for all values of dif-fusion parameter γ and large values of the saturation pa-rameter α.

2.5. For Small and Medium Values of the Saturation Parameterα and All Values of the Reaction/Diffusion Parameter γ.In recent days, homotopy perturbation method is oftenemployed to solve several analytical problems. In addition,several groups demonstrated the efficiency and suitabilityof the HPM for solving nonlinear equations and otherelectrochemical problems [26–29]. He [30] used HPM tosolve the Lighthill equation, the Duffing equation [31], andthe Blasius equation [32]. This method has also been usedto solve nonlinear boundary value problems [33], integralequation [34–36], Klein-Gordon and Sine-Gordon equations[37], Emden-Flower-type equations [38], and several other

problems [39–41]. Using homotopy perturbation method(refer to Appendix A), the approximate solution of (1) is

u(ρ, γ,α

) = sinh(√

γρ)

ρ sinh(√

γ) − αγ sinh

(√γρ)

ρ sinh3(√

γ) + αγ

sinh2(√

γ) .(10)

The normalized concentration of the substrate satisfies theboundary conditions (2a) and (2b). The expression of thenormalized current density becomes

ψ(γ,α) = √γ coth(√γ)− 1

+ αγ cosech2(√γ)⌊√

γ coth(√γ)− 1⌋.

(11)

Equations (10) and (11) represent approximate expressionsof normalized substrate concentration and current density



Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(i)(ii)

(iii)

(vi)

(iv)

(v)

(a)


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(i)(ii)

(iii)

(vi)

(iv)

(v)

(b)

0 0.2 0.4 0.6 0.8 1


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(i)(ii)

(iii)

(iv)

(v)

(vi)

(c)

Figure 6: Normalized substrate concentration “u” at a microelectrode modified with a conducting polymer film as a function of “ρ”. Theconcentrations were computed for various values of the Biot number “v” when (a) α = 0.001, γ = 50, (b) α = 0.01, γ = 50, and (c)α = 0.1, γ = 50. (solid line) (22); (dotted line) (17). Solid lines are compared with points. In this figure (i) v = 0.1, (ii) v = 0.5, (iii) v = 1,(iv) v = 5, (v) v = 10, (vi) v = 50.

for all small and medium values of the saturation parameterα and all values of the diffusion parameter γ.

2.6. Discussion. The kinetic response of a microelectrodedepends on the concentration of substrate. The concen-tration of substrate depends on the following two factorsγ and α. The diffusion parameter γ represents the ratio ofthe characteristic time of the enzymatic reaction to that ofsubstrate diffusion. This parameter can be varied by chang-ing either the radius of the microelectrode or the amountof catalyst in conducting polymer ultramicroelectrodes. Thisparameter describes the relative importance of diffusion andreaction in conducting polymer ultramicroelectrodes. Whenγ is small, the kinetics are dominant resistance; the uptakeof substrate in the polymer film is kinetically controlled.Under these conditions, the substrate concentration profile

across the microelectrode is essentially uniform. The overallkinetics are determined by the total amount of active catalystcΣ. When the diffusion parameter γ is large, diffusionlimitations are the principal determining factor. In both theunsaturated and saturated situations (small and large valuesof α), the current response increases as γ increases. This is tobe expected as the reaction kinetics become more facile.

The approximate expressions of concentration of sub-strate and current density for various values of α and γ arereported in Table 1. In the case when the substrate diffusionin the adjacent solution is neglected, the expression corre-sponding to the concentration of substrate (5) and current(6) was provided by Lyons et al. [24] (refer to Table 1).

Figure 2 represents the substrate concentration u forvarious values of the reaction diffusion parameter γ and forα ≤ 1. From Figure 2, it is evident that the normalized



Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

0 0.2 0.4 0.6 0.8 10.86

0.88

0.9

0.92

0.94

0.96

0.98

1

(i)

(ii)

(iii)(v)

(vi)(iv)

(a)

0 0.2 0.4 0.6 0.8 10.93

0.94

0.95

0.96

0.97

0.98

0.99

1

(i)

(ii)

(iii)

(iv)(v)

(vi)


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(b)

0 0.2 0.4 0.6 0.8 10.985

0.99

0.995

1

(i)

(iii)

(iv)

(ii)

(v)

(vi)


Nor

mal

ized

subs

trat

eco

nce

ntr

atio

nu

(c)

Figure 7: Plot of Normalized substrate concentration “u” at a microelectrode modified with a polymer film as a function of “ρ”. Theconcentrations were computed using (20) for various values of the Biot number “v” when γ = 0.1 and for (a) α = 5, (b) α = 10 and (c)α = 50. In this figure (i) v = 0.1, (ii) v = 0.5, (iii) v = 1, (iv) v = 5, (v) v = 10, (vi) v = 50.

steady-state substrate concentration u reaches the maximumvalue 1, when ρ = 1. Figure 3 indicates the values of substrateconcentrations for large values of α (α ≥ 10) and all valuesof γ. From Figure 3, the value of concentration u is inverselyproportional to the value of the reaction diffusion parameterγ. When γ is small (γ ≤ 1), the substrate concentrationprofile across the microelectrode is uniform (refer to Figures2 and 3).

Figure 4 indicates the normalized steady-state current ψfor all values of α. From Figure 4(a), it is noticed that ouranalytical results (9) and (11) agree with the limiting result ofLyons et al. [24] work. The normalized steady-state currentfor all large values of α is calculated using (9) in Figure 4(b).A series of normalized current density for all values of αis plotted in Figure 4(c). From Figures 4(b) and 4(c), it isevident that the value of the current decreases when α in-creases as γ or radius of the electrode increases.

3. Problem Resolution including SubstrateConcentration Polarization in Solution

Here, we include the substrate diffusion in the solutionadjacent to the polymer film. In this case transport andkinetics are described by (1), but the boundary conditionsare given by [24]

du

dρ= 0 at ρ = 0, (12a)

u = u1 at ρ = 1, (12b)

du

dρ= v(1− u1) at ρ = 1, (12c)


0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

(i)(ii)

(iii)

(iv)

(v)

(vi)

(vii)


Nor

mal

ized

curr

ent

(a)

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

(i)(ii)

(iii)

(iv)

(v)

(vi)

(vii)


Nor

mal

ized

curr

ent

(b)

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

(i)(ii)

(iii)

(iv)

(v)

(vi)

(vii)


Nor

mal

ized

curr

ent

(c)

Figure 8: Normalized current “ψ” versus the reaction/diffusion parameter “γ” at a microelectrode modified using a conducting polymerfilm. The current was calculated for various values of the Biot number “v” when (a) α = 0.001, (b) α = 0.01, and (c) α = 0.1. (solid line)(23); (dotted line) (18). Solid lines are compared with points. In this figure (i) v = 0.1, (ii) v = 0.5, (iii) v = 1, (iv) v = 5, (v) v = 10, (vi)v = 50, and (vii) v = 100.

where the Biot number v has been introduced

v = k′D

kD, (13)

where k′D represents the diffusional rate constant of thesubstrate in solution and kD is the value that represents thetransport of substrate within polymer film. The diffusionalrate constant kD to a microelectrode is given by

kD = 4DSπa

. (14)

Using (13), we obtain

v = 4π

(D′SDS

). (15)

3.1. Unsaturated (First-Order) Catalytic Kinetics. Initially, weconsidered a situation where the substrate concentration inthe film is less than the Michaelis constant KM . This situationwill pertain when the product αu� 1. Hence (1) reduces to

d2u

dρ2+

2ρ

du

dρ− γu = 0. (16)

By solving (16) using the boundary condition (12a)–(12c), we can obtain the analytical expression of normalizedconcentration of substrate as follows:

u(ρ, γ, v

) = sinh(√

γρ)

ρ{

sinh(√

γ)

+1v

[√γ cosh

(√γ)− sinh

(√γ)]} .

(17)


0 20 40 60 80 1000

1

2

3

4

5

6

7

(i)

(ii)

(iii)

(iv)(v)


Nor

mal

ized

curr

ent

Figure 9: Normalized current “ψ” versus the reaction/diffusionparameter “γ” at a microelectrode modified using a conductingpolymer film. The current was calculated using (21) for variousvalues of the Biot number “v” when α = 10. In this figure (i) v = 0.5,(ii) v = 1, (iii) v = 5, (iv) v = 10, and (v) v = 100.

Also the expression of the normalized current density isshown below

ψ(γ, v) =

√γ cosh

⌊√γ⌋− sinh

⌊√γ⌋

sinh(√

γ)

+1v

[√γ cosh

(√γ)− sinh

(√γ)] .

(18)

The above analytical expression of substrate concentrationand current is identical to Lyons et al. [24] work.

3.2. For Large Values of the Saturation Parameter α and AllValues of the Reaction/Diffusion Parameter γ. We now con-sider the limiting situation where the substrate concentrationin the film is very much greater than the Michaelis constantKM . In this case αu� 1 and (1) reduces to

d2u

dρ2+

2ρ

du

dρ− γα

+γ

α2u= 0. (19)

By solving (19), we obtain the approximate expression of thenormalized concentration of substrate as

u(ρ, γ,α, v

) = 1 + γ(12α2v − 6αv − γv − 2γ)

36α3v

(ρ2 − 1− 2

v

)

+γ2

120α3

(ρ4 − 1− 4

v

).

(20)

Also we can obtain the expression of the normalized currentdensity as

ψ(γ,α, v

) = γ(30α2v − 15αv − γv − 5γ)

45α3v. (21)

The above approximation will be valid for all values of dif-fusion parameter γ and large values of saturation parameterα.

3.3. For Small and Medium Values of the Saturation Parameterα and All Values of the Reaction/Diffusion Parameter γ. Usingthis homotopy perturbation method, we can obtain thesolution of (1)

u(ρ, γ,α, v

)

=sinh

(√γρ)

ρ{

sinh(√

γ)

+1v

[√γ cosh

(√γ)− sinh

(√γ)]}

−αγ sinh

(√γρ)

ρ{

sinh(√

γ)

+1v

[√γ cosh

(√γ)− sinh

(√γ)]}3

+αγ{

sinh(√

γ)

+1v

[√γ cosh

(√γ)− sinh

(√γ)]}2 .

(22)

The above equation satisfies the boundary conditions (12a)–(12c). The expression of the normalized current densitybecomes

ψ(γ,α, v

) =√γ cosh

⌊√γ⌋− sinh

⌊√γ⌋

sinh(√

γ)

+1v

[√γ cosh

(√γ)− sinh

(√γ)]

−αγ(√

γ cosh⌊√

γ⌋− sinh

⌊√γ⌋)

{sinh

(√γ)

+1v

[√γ cosh

(√γ)−sinh

(√γ)]}3 .

(23)

Equations (22) and (23) represent a new closed form ofapproximate expressions of normalized substrate concentra-tion and current density for small and medium of parametersα and all values of γ.

3.4. Discussion. In the case when the substrate concentrationis very low, the expression corresponding to the concentra-tion of substrate (17) and current (18) was provided by Lyonset al. [24] (refer to Table 2). Figure 5 represents the nor-malized steady-state substrate concentration u at a polymer-coated microelectrode. The concentration of substrate wascalculated for all small values of the saturation parameter α.From Figure 5, it is inferred that the concentration increaseswhen v increases. Also for any fixed values of v and smallvalues of α and γ, the concentration is uniform throughoutthe film.

The normalized steady-state substrate concentration u isplotted for all small values of the saturation parameter α inFigure 6. From Figure 6, it is evident that when the valuesof the Biot number v increase, the values correspondingto the substrate concentration u also increase when γ ≥50. Our analytical results agree with the limiting result of


Lyons et al. [24] work. Figure 7 indicates the values of sub-strate concentrations for large values of α (α ≥ 5) and allvalues of the Biot number v. From Figure 7, it is inferred thatu ≈ 1 when v ≥ 50 and γ ≤ 0.1.

Figure 8 represents the normalized current density ψ forall values of the Biot number v. In addition, from Figure 8,we noticed that the normalized current density increases asthe Biot number v increases. Normalized current density ψversus γ for various values of the Biot number v and for largevalues of α is plotted using (21) in Figure 9. From Figure 9,it is evident that the value of the current increases when theBiot number v increases.

4. Conclusions

The steady-state amperometric response for a conductingpolymer microelectrode system which exhibits Michaelis-Menten kinetics has been discussed. We have presented amathematical model of reaction and diffusion within a con-ducting polymer film which is deposited on a support surfaceof micrometer dimensions. Approximate analytical solutionsof the nonlinear reaction diffusion equation have been de-rived. Analytical expressions of substrate concentration with-in the polymer film are derived for all values of the diffusionparameter and the saturation parameter using homotopyperturbation method. The analytical results derived thereinmay be used to predict the steady-state sensor response onexperimental values, and the theoretical value of surface con-centration for which the amperometric response is nonlin-ear.

Appendix

Solution of (1) Using HomotopyPerturbation Method

In this appendix, we indicate how (10) in this paper is de-rived. To find the solution of (1), we first construct a homo-topy as follows:

(1− p)

[d2u

dρ2+

2ρ

du

dρ− γu

]

+ p

[(1 + αu)

(d2u

dρ2+

2ρ

du

dρ

)− γu

]= 0.

(A.1)

The approximate solution of (A.1) is given by

u = u0 + pu1 + p2u2 + p3u3 + · · · . (A.2)

Substituting (A.2) into (A.1) and comparing the coefficientsof like powers of p, we get

p0 :d2u0dρ2

+2ρ

du0dρ

− γu0 = 0, (A.3)

p1 :d2u1dρ2

+2ρ

du1dρ

− γu1 + αu0 d2u0dρ2

+2ραu0

du0dρ

= 0.(A.4)

The initial approximations are as follows:

u0(ρ = 1) = 1,

(du0dρ

)ρ=0

= 0, (A.5)

ui(ρ = 1) = 0,

(duidρ

)ρ=0

= 0, ∀i = 1, 2, 3, . . . .(A.6)

Upon solving (A.3) and (A.4) and using the boundary con-ditions (A.5) and (A.6), we get

u0(ρ) = sinh

(√γρ)

ρ sinh(√

γ) , (A.7)

u1(ρ) = −αγ sinh

(√γρ)

ρsinh3(√

γ) + αγ

sinh2(√

γ) . (A.8)

u1(ρ) is valid only when α and γ are small. According to theHPM, we can conclude that

u(ρ) = lim

p→ 1u(ρ) ≈ u0 + u1. (A.9)

Using (A.7) and (A.8) in (A.9), we obtain the final result asdescribed in (10). Similarly (22) can also be obtained.

Acknowledgments

This work is supported by the Council of Scientific and In-dustrial Research (CSIR), Government of India. The authorsare thankful to the Secretary, the Principal, The MaduraCollege, Madurai, India for their constant encouragement.They thank the reviewers for their valuable comments toimprove the quality of the paper.

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