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

A. Eswari, S. Usha, L. Rajendran
DOI: 10.4236/ajac.2011.22010                

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 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.

Keywords

Non-Linear Reaction/Diffusion Equation, Homotopy Perturbation Method, CE Mechanism, Reduction of Order, Spherical Electrodes

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Koryta, M. Brezina and J. Pradacova, “Laboratory Techniques in Electroanalytical Chemistry,” In: A. J. Bard, Ed., Electroanalytical Chemistry, Marcel Dekker, New York, Vol. 11, 1972, p. 85.
[2]	D. B. Cater and I. A. Silver, “Microelectrodes and Electrodes used in Biology,” In: D. J. G. Ives and G. J. Jane Eds., Reference Electrodes Theory and Practical, Academic Press, New York, 1961, p. 464.
[3]	R. S. Pickard, “A Review of Printed Circuit Microelectrodes 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 Chemistry, Vol. 256, No. 1, 1988, pp. 11-19. doi:10.1016/0022-0728(88)85002-2
[5]	G. J. Hills, D. Inman and J. E. Oxley, “Progress in Reaction Kinetics,” In: I. S. Longmuir, Ed., Advances in Polarography, Pergamon Press, Oxford, Vol. 3, 1960, p. 982.
[6]	R. M. Wightman and D. O. Wipf, “Voltammetry at Microelectrodes,” In: A. J. Bard, Ed., Electroanalytical Chemistry, Marcel Decker, New York, Vol. 15, 1989, p. 267.
[7]	C. Amatore, “Scanning Electrochemical Microscopy,” In: 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 Microelectrodes to the Study of Homogeneous Processes Coupled to Electrode Reactions: Part I. EC’ And CE Reactions,” Journal of Electroanalytical Chemistry, Vol. 177, No. 1-2, 1984, pp. 97-114. doi:10.1016/0022-0728(84)80215-6
[10]	M. Fleischmann, F. Lasserre and J. Robinson, “The Application of Microelectrodes to the Study of Homogeneous Processes Coupled to Electrode Reactions: Part II. ECE And DISP 1 Reactions,” Journal of Electroanalytical 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., Research in Chemical Kinetics, Elsevier, Amsterdam, 1994.
[12]	J. R. Delmastro and D. E. Smith, “Methods for Obtaining Approximate Solutions to the Expanding-Sphere Boundary Value Problem in Direct Current Polarography,” Jo- 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 Electrode Processes at a Hemispherical Ultramicroelectrode and Its Application for the Catalytic Behavior of the Sixth Reduction Wave of Fullerence, C60,” Journal of Electroanalytical Chemistry, Vol. 429, No. 1-2, 1997, pp. 67-74. doi:10.1016/S0022-0728(96)05022-X
[14]	J. Galceran, S. L. Taylor and P. N. Bartlett, “Steady-State Currents at Inlaid and Recessed Microdisc Electrodes for First-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 Current 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 Derivative and Differential Pulse Voltammetric Curves of EC, CE and Catalytic Processes at Spherical Electrodes and Microelectrodes,” International Journal of Electroche- mical 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 Response of Microdisks and of The Steady State for CE and EC Catalytic Reactions by Application of Neumann’s Integral Theorem,” Journal of Electroanalytical Chemistry, Vol. 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, “Response of Microvoltammetric Electrodes to Homogeneous Catalytic and Slow Heterogeneous Charge-Transfer Reactio-Ns,” Analytical Chemistry, Vol. 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 Frequency-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 Nonlinear 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 Homotopy Perturbation Method to Linear and Nonlinear Schr?dinger Equations,” Zeitschrift fur Naturforschung, 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 Nonlinear 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 Nonlinear 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
[30]	A. Meena and L. Rajendran, “Mathematical Modeling of Amperometric and Potentiometric Biosensors and System of Non-Linear Equations-Homotopy Perturbation Appro-Ach,” Journal of Electroanalytical Chemistry, Vol. 644, 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 Electroenzymatic Processes Involved in a PPO-Rotating-Disk-Bioelec- trode,” Natural Science, Vol. 3, No. 1, 2011, pp. 1-7. doi:10.4236/ns.2011.31001
[32]	V. Marget and L. Rajendran, “Analytical Expression of Non Steady-State Concentration Profiles at Planar Electrode 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 Mathematical Functions,” Dover publications, Inc., New York, 1970.