Fractal dimension of Li insertion electrodes studied by diffusion-controlled voltammetry and impedance spectroscopy

Physical Review B - Condensed Matter and Materials Physics

Volumn 54, Issue 5, 1996, Pages 2968-2971

  Stromme Mattsson, M ^a   Niklasson, G ^a   Granqvist, C ^a  

^a UPPSALA UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000789768     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.54.2968     Document Type: Article

References (19)

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19. Equation (2) was derived for a self-similar surface, but it can easily be shown to hold for a self-affine surface if a certain condition holds, viz. that the bottleneck region (cf. Ref. 2)—where the current density lines lie closest to each other—is located at a distance from the WE surface larger than the distance corresponding to the outer lateral cutoff. In this case the resistance scales as (Formula presented) when the system size is increased by a factor r, and Eq. (2) is obtained as described by L. Nyikos and T. Pajkossy [Electrochim. Acta 30, 1533 (1985)]. If the bottleneck region is located at a distance from the WE surface smaller than the outer cutoff, the resistance scales as (Formula presented) /(Formula presented) . The Hölder exponent χ is related to the local fractal dimension via χ=3-(Formula presented) [cf. A.-L. Barabási and H. E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, England, 1995)]. Following Nyikos and Pajkossy we now obtain n=3-(Formula presented), which is the same expression as the one found from models based on transmission lines by R. de Levie [J. Electro- anal. Chem. 281, 1 (1990)]. Applying the above relation to the n values in Fig. 4 gives a surface dimension of 2.227 and 2.170 for the Sn oxide and Ti oxide sample, respectively. The surface dimensions derived by using Eq. (2) are closer than the ones obtained here to the (Formula presented) 's reached by voltammetry. Thus we conclude that for our samples, self-similar or self-affine, the bottleneck region is located outside the outer cutoff of the fractal region. The morphology of the Ti oxide film suggests that the scaling in the direction perpendicular to the surface and along the surface are equal, i.e., that the surface probably is self-similar. The Sn oxide film, however, is most likely self-affine, taking into consideration its small surface roughness and the large value of the outer cutoff.