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ISRN Physical Chemistry
Volume 2012 (2012), Article ID 745616, 12 pages
Approximate Analytical Solution of Nonlinear Reaction’s Diffusion Equation at Conducting Polymer Ultramicroelectrodes
1Department of Mathematics, The Madura College, Madurai-625011, India
2Nanomaterials Research and Education Center, University of South Florida, Tampa, FL, USA
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 Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A theoretical model of reaction/diffusion within conducting polymer microelectrodes is discussed. The model is based on the steady-state diffusion equation containing a nonlinear term related to the Michaelis-Menten kinetic of the enzymatic reaction. An analytical expression pertaining to the concentration of substrate and current is obtained using homotopy perturbation method for all values of diffusion and the saturation parameter. The substrate concentration profile and current response can be used in a large range of concentrations including the non-linear contributions. These approximate analytical results were found to be in good agreement with the previously reported limiting case results.
The advantages of the ultramicroelectrodes (UMEs) include steady-state current, rapid response time, minimal iR drop, lower detection limits, and sensitive analysis in a highly resistive medium [1–4]. Furthermore, due to their diminished surface area or radii of the electrodes, they are often employed as probes to monitor various chemical events occurring inside the living cell [5–8]. Ultramicroelectrodes possess many advantages for studying electrochemical kinetics and in electroanalytical applications, imaging, and surface modification . Microelectrodes modified with a polymer film find potential application in various sensing applications [10–12]. The working principle of polymer-modified ultramicroelectrodes occurs in the following manner: initially, the redox analyte interacts with the immobilized active receptor sites present in the polymer matrix, then at the underlying electrode surface. Briefly, we can say that the redox reaction is mediated by the polymeric layer. Further, due to the electroactive property of the polymer, charge can percolate through the polymer chain and thereby reaches the electrode/interface to give rise to a redox current and is directly 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 the substrate/product transformation will be governed by the properties of the mediating electroactive polymer film.
Recent advancements in ultramicroelectrodes modified with conducting polymers were reported elsewhere [13–18]. Further, the analytical applications of various polymer-modified sensors and the reaction/diffusion at the conducting polymer electrode (where the chemical reaction term is described by Michaelis-Menten kinetics) were reviewed extensively by various groups [19–23]. Recently, Lyons et al.  evaluated the analytical solutions corresponding to the steady-state substrate concentration profile and current observed at a conducting polymer microelectrode when the substrate concentration is low. For higher values of the substrate concentration, a kinetic rate law based on the Michaelis-Menten equation is more appropriate. Recently, Anitha et al.  derived the analytical expression for non-steady-state concentrations of substrate and mediator at a polymer modified ultramicroelectrodes using reduction of order method. However, to the best of our knowledge, there were no analytical results available till date that corresponds to the steady-state substrate concentration and current for all possible values of diffusion parameter and the saturation parameter . However, in general, analytical solutions of nonlinear differential equations are more interesting and useful numerical solutions, as they are used in various kinds of data analysis. Therefore, herein, we employ analytical method to evaluate the steady-state substrate concentration and current for all possible values of diffusion and saturation parameter.
2. Mathematical Formulation of the Boundary Value Problem and Analysis
The conducting polymer film will adopt a hemispherical geometry upon electrodepositing them on to a microelectrode support (Figure 1). If such a geometry is assumed then the substrate will exhibit spherical diffusion both in solution immediately adjacent to the polymer film and within the polymer film itself. We assume that the substrate exhibits Michaelis-Menten kinetics when it reacts at a site within the polymer film. The substrate exhibits first-order kinetics approximation when the substrate concentration is low. For higher values of the substrate concentration a kinetic rate law based on the Michaelis-Menten equation is more appropriate. We also assume that the partition coefficient for the substrate is unity.
2.2. Neglecting Substrate Concentration Polarization in Solution
Initially, the substrate diffusion in solution, the transport, and kinetic processes within the polymer film were all neglected. On the other hand, the reaction/diffusion equation under steady-state condition corresponding to the normalized substrate concentration within the polymer film can be expressed as  In (1), normalized substrate concentration , where denotes the substrate concentration within the polymer film and denotes the bulk concentration of substrate. The saturation parameter , where denotes the Michaelis constant. The reaction/diffusion parameter is given by , where represents the reaction rate constant, denotes the radius of the microelectrode. denotes the total catalyst concentration in the film, and is the diffusion coefficient of the substrate within the polymer film. The normalized distance parameter is given by (where represents the radial variable). The boundary conditions pertaining to the normalized form are
The boundary condition (2a) states that the substrate is electroinactive at the disk. The normalized current density is defined as
2.3. Unsaturated (First-Order) Catalytic Kinetics
We initially consider the situation where the substrate concentration in the film is less than the Michaelis constant . This situation will pertain when the product . Hence (1) reduces to By solving (4), we can obtain the expression for the normalized substrate concentration as follows: Also the expression of the normalized steady-state current density is given below
2.4. For Large Values of the Saturation Parameter and All Values of the Reaction/Diffusion Parameter
We now consider the limiting situation where the substrate concentration in the film is very much greater than the Michaelis constant . In this case and (1) reduces to We obtain the approximate expression of normalized concentration of substrate as Using (3), we obtain the expression of the normalized current density as The above approximation will be valid for all values of diffusion parameter and large values of the saturation parameter .
2.5. For Small and Medium Values of the Saturation Parameter and All Values of the Reaction/Diffusion Parameter
In recent days, homotopy perturbation method is often employed to solve several analytical problems. In addition, several groups demonstrated the efficiency and suitability of the HPM for solving nonlinear equations and other electrochemical problems [26–29]. He  used HPM to solve the Lighthill equation, the Duffing equation , and the Blasius equation . This method has also been used to solve nonlinear boundary value problems , integral equation [34–36], Klein-Gordon and Sine-Gordon equations , Emden-Flower-type equations , and several other problems [39–41]. Using homotopy perturbation method (refer to Appendix A), the approximate solution of (1) is The normalized concentration of the substrate satisfies the boundary conditions (2a) and (2b). The expression of the normalized current density becomes Equations (10) and (11) represent approximate expressions of normalized substrate concentration and current density for all small and medium values of the saturation parameter and all values of the diffusion parameter .
The kinetic response of a microelectrode depends on the concentration of substrate. The concentration of substrate depends on the following two factors . The diffusion parameter represents the ratio of the characteristic time of the enzymatic reaction to that of substrate diffusion. This parameter can be varied by changing either the radius of the microelectrode or the amount of catalyst in conducting polymer ultramicroelectrodes. This parameter describes the relative importance of diffusion and reaction in conducting polymer ultramicroelectrodes. When is small, the kinetics are dominant resistance; the uptake of substrate in the polymer film is kinetically controlled. Under these conditions, the substrate concentration profile across the microelectrode is essentially uniform. The overall kinetics are determined by the total amount of active catalyst . When the diffusion parameter is large, diffusion limitations are the principal determining factor. In both the unsaturated and saturated situations (small and large values of ), the current response increases as increases. This is to be expected as the reaction kinetics become more facile.
The approximate expressions of concentration of substrate and current density for various values of and are reported in Table 1. In the case when the substrate diffusion in the adjacent solution is neglected, the expression corresponding to the concentration of substrate (5) and current (6) was provided by Lyons et al.  (refer to Table 1).
Figure 2 represents the substrate concentration for various values of the reaction diffusion parameter and for . From Figure 2, it is evident that the normalized steady-state substrate concentration reaches the maximum value 1, when . Figure 3 indicates the values of substrate concentrations for large values of ) and all values of . From Figure 3, the value of concentration is inversely proportional to the value of the reaction diffusion parameter . When is small (), the substrate concentration profile across the microelectrode is uniform (refer to Figures 2 and 3).
Figure 4 indicates the normalized steady-state current for all values of . From Figure 4(a), it is noticed that our analytical results (9) and (11) agree with the limiting result of Lyons et al.  work. The normalized steady-state current for all large values of is calculated using (9) in Figure 4(b). A series of normalized current density for all values of is plotted in Figure 4(c). From Figures 4(b) and 4(c), it is evident that the value of the current decreases when increases as or radius of the electrode increases.
3. Problem Resolution including Substrate Concentration Polarization in Solution
Here, we include the substrate diffusion in the solution adjacent to the polymer film. In this case transport and kinetics are described by (1), but the boundary conditions are given by where the Biot number has been introduced where represents the diffusional rate constant of the substrate in solution and is the value that represents the transport of substrate within polymer film. The diffusional rate constant to a microelectrode is given by Using (13), we obtain
3.1. Unsaturated (First-Order) Catalytic Kinetics
Initially, we considered a situation where the substrate concentration in the film is less than the Michaelis constant . This situation will pertain when the product . Hence (1) reduces to
By solving (16) using the boundary condition (12a)–(12c), we can obtain the analytical expression of normalized concentration of substrate as follows: Also the expression of the normalized current density is shown below The above analytical expression of substrate concentration and current is identical to Lyons et al.  work.
3.2. For Large Values of the Saturation Parameter and All Values of the Reaction/Diffusion Parameter
We now consider the limiting situation where the substrate concentration in the film is very much greater than the Michaelis constant . In this case and (1) reduces to By solving (19), we obtain the approximate expression of the normalized concentration of substrate as Also we can obtain the expression of the normalized current density as The above approximation will be valid for all values of diffusion parameter and large values of saturation parameter .
3.3. For Small and Medium Values of the Saturation Parameter and All Values of the Reaction/Diffusion Parameter
Using this homotopy perturbation method, we can obtain the solution of (1) The above equation satisfies the boundary conditions (12a)–(12c). The expression of the normalized current density becomes Equations (22) and (23) represent a new closed form of approximate expressions of normalized substrate concentration and current density for small and medium of parameters and all values of .
In the case when the substrate concentration is very low, the expression corresponding to the concentration of substrate (17) and current (18) was provided by Lyons et al.  (refer to Table 2). Figure 5 represents the normalized steady-state substrate concentration at a polymer-coated microelectrode. The concentration of substrate was calculated for all small values of the saturation parameter . From Figure 5, it is inferred that the concentration increases when increases. Also for any fixed values of and small values of and , the concentration is uniform throughout the film.
The normalized steady-state substrate concentration is plotted for all small values of the saturation parameter in Figure 6. From Figure 6, it is evident that when the values of the Biot number increase, the values corresponding to the substrate concentration also increase when . Our analytical results agree with the limiting result of Lyons et al.  work. Figure 7 indicates the values of substrate concentrations for large values of and all values of the Biot number . From Figure 7, it is inferred that when and .
Figure 8 represents the normalized current density for all values of the Biot number . In addition, from Figure 8, we noticed that the normalized current density increases as the Biot number increases. Normalized current density versus for various values of the Biot number and for large values of is plotted using (21) in Figure 9. From Figure 9, it is evident that the value of the current increases when the Biot number increases.
The steady-state amperometric response for a conducting polymer microelectrode system which exhibits Michaelis-Menten kinetics has been discussed. We have presented a mathematical model of reaction and diffusion within a conducting polymer film which is deposited on a support surface of micrometer dimensions. Approximate analytical solutions of the nonlinear reaction diffusion equation have been derived. Analytical expressions of substrate concentration within the polymer film are derived for all values of the diffusion parameter and the saturation parameter using homotopy perturbation method. The analytical results derived therein may be used to predict the steady-state sensor response on experimental values, and the theoretical value of surface concentration for which the amperometric response is nonlinear.
Solution of (1) Using Homotopy Perturbation Method
In this appendix, we indicate how (10) in this paper is derived. To find the solution of (1), we first construct a homotopy as follows: The approximate solution of (A.1) is given by Substituting (A.2) into (A.1) and comparing the coefficients of like powers of , we get The initial approximations are as follows: Upon solving (A.3) and (A.4) and using the boundary conditions (A.5) and (A.6), we get is valid only when and are small. According to the HPM, we can conclude that Using (A.7) and (A.8) in (A.9), we obtain the final result as described in (10). Similarly (22) can also be obtained.
This work is supported by the Council of Scientific and Industrial Research (CSIR), Government of India. The authors are thankful to the Secretary, the Principal, The Madura College, Madurai, India for their constant encouragement. They thank the reviewers for their valuable comments to improve the quality of the paper.
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