Abstract
We analyse the problem of impedance for a diffusion controlled charge transfer process across an irregular interface. These interfacial irregularities are characterized as two class of random fractals: (i) a statistically isotropic self-affine fractals and (ii) a statistically corrugated self-affine fractals. The information about the realistic fractal surface roughness has been introduced through the band-limited power-law power spectrum over limited wave numbers. The details of power spectrum of such roughness can be characterized in term of four fractal morphological parameters, viz. fractal dimension (D H ), lower (ℓ), and upper (L) cut-off length scales of fractality, and the proportionality factor (μ) of power spectrum. Theoretical results are analysed for the impedance of such rough electrode as well as the effect of statistical symmetries of roughness. Impedance response for irregular interface is simplified through expansion over intermediate frequencies. This intermediate frequency expansion with sufficient number of terms offers a good approximation over all frequency regimes. The Nyquist plots of impedance show the strong dependency mainly on three surface morphological parameters i.e. D H , ℓ and μ. We can say that our theoretical results also provide an alternative explanation for the exponent in intermediate frequency power-law form.
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References
Cottis S, Turgoose S and Newman R 2000 Corrosion testing made easy: Impedance and noise analysis (NACE international, Houston, Tex)
Yang L and Li Y 2005 Biosens. and Bioelectron. 20 1407
Quanchao Z, Zuofeng C, Quanfeng D, Yanxia J, Ling H and Shigang S 2006 Chin. Sci. Bull. 51 1055
Urbain M, Rael S and Davat B 2007 Energetical modeling of lithium-ion batteries (Industry Application Conference, 42nd IAS Annual Meeting, IEEE)
Pfuch A, Heft A, Weidl R and Lang K 2006 Surf. Coat. Technol. 201 189
Sarac A S, Sezgin S, Ates M and Turhan C M 2008 Surf. Coat. Technol. 202 3997
Bevilaqua D, Acciari H A, Arena F A, Benedetti A V, Fugivara C S, Filho G T and Junior O G 2008 Miner. Eng.; doi:10.1016/j.mineng.2008.07.010
Yoon K H, Jang J H and Cho Y S 1998 J. Mater. Sci. Lett. 17 1755
Yaropolov A, Shleev S, Zaitseva E, Emnéus J, Marko-Varga G and Gorton L 2007 Bioelectrochemistry 70 199
Bard A J and Faulkner L R 1980 Electrochemical methods: Fundamentals and application (New York: Wiley)
Lasia A 1999 Modern aspects of electrochemistry (eds) B E Conway, J O’ M Bockris and R E White (New York: Kluwer Acad; Plenum) No. 32
Macdonald D D 2006 Electrochim. Acta 51 1376
De Gennes P G 1982 C.R. Acad. Sci. (Paris) Ser. II 295 1061
Kopelman R 1986 J. Stat. Phys. 42 185; 1988 Science 241 1620
Chaudhari A, Yan C-C S and Lee S-L 2002 Chem. Phys. Lett. 351 341
Dewey T G 1994 Proc. Natl. Acad. Sci USA 91 12101
Vandembroucq D, Boccaro A C and Roux S 1995 Europhys. Lett. 30 209
Sapoval B 1996 Fractal electrodes, fractal membranes and fractal catalyst in fractals and disordered systems (eds) A Bunde and S Havlin (Heidelberg: Springer-Verlag)
Gutfraind R and Sapoval B 1993 J. Phys. I France 3 1801
Nyikos L and Pajkossy T 1986 Electrochim. Acta 31 1347
Pajkossy T and Nyikos L 1989 Electrochim. Acta 34 171
Nyikos L and Pajkossy T 1990 Electrochim. Acta 35 1567
Pajkossy T 1991 J. Electroanal. Chem. 300 1
De Levie R 1990 J. Electroanal. Chem. 281 1
Ramesh P and Sampath S 2003 Anal. Chem. 75 6949
Kant R and Rangarajan S K 2003 J. Electroanal. Chem. 552 141
Kant R 1997 J. Phys. Chem. B101 3781
Kant R and Jha S K 2007 J. Phys. Chem. C111 14040
Jha S K, Sangal A and Kant R 2008 J. Electroanal. Chem. 615 180
Go J-Y and Pyun S-I 2005 Electrochim. Acta 50 3479
Ocon P, Herrasti P, Vazquez L, Salvarezza R C, Vara J M and Arvia A J 1991 J. Electroanal. Chem. 319 101
Go J-Y and Pyun S-I 2007 J. Solid State Electrochem. 11 323
Pajkossy T, Borosy A P, Imre A, Martemyanov S A, Schiller R and Nyikos L 1994 J. Electroanal. Chem. 366 69
Kant R 1993 Phys. Rev. Lett. 70 4094
Kant R and Rangarajan S K 1994 J. Electroanal. Chem. 368 1
Kant R and Rangarajan S K 1995 J. Electroanal. Chem. 396 285
Kant R and Jha S K Theory of partial diffusion-limited interfacial transfer/reaction on realistic fractals (Unpublished results)
Dassas Y and Duby P 1995 J. Electrochem. Soc. 142 4175
Imre A, Pajkossy T and Nyikos L 1992 Acta Metall. Mater. 40 1819
Palasantzas G 2005 Surf. Sci. 582 151
Rangarajan S K 1969 J. Electroanal. Chem. 22 89
Bisquert J and Compte A J 2001 J. Electroanal. Chem. 499 112
Ball R and Blunt M 1988 J. Phys. A: Math. Gen. 21 197
Kant R, Kumar R and Yadav V K 2008 J. Phys. Chem. C112 4019
Warburg E 1899 Ann. Physik 67 493
Maritan A, Stella A L and Toigo F 1989 Phys. Rev. B40 9269
Yordanov O I and Atanasov I S 2002 Euro Phys. J. B29 211
Niklasson G A et al 2000 Thin Solid Films 359 203
Abramowitz M and Stegan A (eds) 1972 Handbook of mathematical functions (New York: Dover Publications Inc)
Feder J 1988 Fractals (New York: Plenum)
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Dedicated to the memory of the late Professor S K Rangarajan
An erratum to this article is available at http://dx.doi.org/10.1007/s12039-010-0033-8.
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Kumar, R., Kant, R. Generalized Warburg impedance on realistic self-affine fractals: Comparative study of statistically corrugated and isotropic roughness. J Chem Sci 121, 579–588 (2009). https://doi.org/10.1007/s12039-009-0070-3
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DOI: https://doi.org/10.1007/s12039-009-0070-3