Theory of Potential Step Chronoamperometry at a Microband Electrode: Complete Explicit Semi-Analytical Formulae for the Faradaic Current Density and the Faradaic Current
Theory of potential step chronoamperometry under limiting current conditions and for purely diffusional transport at a microband electrode has been a subject of several studies. However, no complete and explicit expressions for the Faradaic current density and the Faradaic current have been reported...
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| Published in | Electrochimica acta Vol. 178; pp. 25 - 33 |
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| Format | Journal Article |
| Language | English |
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01.10.2015
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| Abstract | Theory of potential step chronoamperometry under limiting current conditions and for purely diffusional transport at a microband electrode has been a subject of several studies. However, no complete and explicit expressions for the Faradaic current density and the Faradaic current have been reported thus far. In the present study such expressions are derived using a novel theoretical approach. The microband is considered as a limiting case of an elliptic cylinder, when the length of the smallest diameter of the elliptic cross-section tends to zero. Solution to the problem of heat conduction around an elliptic cylinder, due to Tranter [Quart. J. Mech. Appl. Math. 4 (1951) 461], is utilised. Following Tranter, the method of separation of variables in the Laplace space is used, resulting in two Mathieu differential equations. The concentration of the depolarizer, the Faradaic current density, and the Faradaic current, are then expressed as inverse Laplace transforms of certain infinite series involving appropriate Mathieu functions. The series are amenable to further analytical examinations. In particular, it is proven that a quasi-steady state develops at large time. It is also demonstrated how the popular idea, of an hemicylinder electrode “equivalent” to a microband, has to be understood to be correct. Numerical evaluation of the series provides unprecedentedly highly accurate solution values. Hence, the present solutions should be preferred over formerly used low-accurate formulae, for the purposes of experimental data analysis, and for the testing of modelling/simulation techniques. |
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| AbstractList | Theory of potential step chronoamperometry under limiting current conditions and for purely diffusional transport at a microband electrode has been a subject of several studies. However, no complete and explicit expressions for the Faradaic current density and the Faradaic current have been reported thus far. In the present study such expressions are derived using a novel theoretical approach. The microband is considered as a limiting case of an elliptic cylinder, when the length of the smallest diameter of the elliptic cross-section tends to zero. Solution to the problem of heat conduction around an elliptic cylinder, due to Tranter [Quart. J. Mech. Appl. Math. 4 (1951) 461], is utilised. Following Tranter, the method of separation of variables in the Laplace space is used, resulting in two Mathieu differential equations. The concentration of the depolarizer, the Faradaic current density, and the Faradaic current, are then expressed as inverse Laplace transforms of certain infinite series involving appropriate Mathieu functions. The series are amenable to further analytical examinations. In particular, it is proven that a quasi-steady state develops at large time. It is also demonstrated how the popular idea, of an hemicylinder electrode “equivalent” to a microband, has to be understood to be correct. Numerical evaluation of the series provides unprecedentedly highly accurate solution values. Hence, the present solutions should be preferred over formerly used low-accurate formulae, for the purposes of experimental data analysis, and for the testing of modelling/simulation techniques. |
| Author | Bieniasz, L.K. |
| Author_xml | – sequence: 1 givenname: L.K. surname: Bieniasz fullname: Bieniasz, L.K. email: nbbienia@cyf-kr.edu.pl organization: Faculty of Physics, Mathematics, and Computer Science, Cracow University of Technology, ul. Warszawska 24, 31-155 Cracow, Poland |
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| Keywords | Equivalent hemicylinder Microband electrode Laplace transformation Chronoamperometry Mathieu equation Limiting current |
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| SubjectTerms | Chronoamperometry Equivalent hemicylinder Laplace transformation Limiting current Mathieu equation Microband electrode |
| Title | Theory of Potential Step Chronoamperometry at a Microband Electrode: Complete Explicit Semi-Analytical Formulae for the Faradaic Current Density and the Faradaic Current |
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