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1944
Contribution of Surface Roughness to Constant-Phase Element (CPE) Behavior

While CPEs are commonly used to improve the fit to experimental impedance spectra, the explanation for their origin has been largely conjectural. In early experiments on solid electrodes, micro-scale surface roughness was believed to contribute to the non-ideality of electrochemical measurements.^{1} Borisova and Ershler were the first to observe that roughness influenced electrochemical measurements.^{2} They found that the extent of frequency dispersion was reduced by melting a metal electrode and letting it cool to form into a droplet, suggesting that the smoother surface led to a more ideal response.

Following the development of fractal theory, there was an attempt to correlate the fractal dimensions of the surface to the CPE exponent.^{3} Fractal geometry was shown to cause frequency dispersion, however a correlation between the fractal dimension and variance from ideality could not be found. Pajkossy showed experimentally that annealing can reduce the degree of frequency dispersion even though the roughness of the surface remained the same. He thereby concluded that the frequency dispersion cannot be due to the geometric effect solely but may also have a contribution of atomic scale heterogeneities, distinguishing the effect of a distribution of solution resistance and a distribution of surface properties.^{4}

Thus, while roughness was initially considered to cause CPE behavior, recent work has questioned this premise. The objective of this work is to explore, by use of finite-element models, whether surface roughness could provide a valid physical explanation for CPE behavior associated with a surface distribution of time constants. An essential question in this analysis is whether electrochemical impedance spectroscopy measurements encompass the frequency at which roughness is expected to influence the impedance response.

Finite-element simulations were used to calculate the influence of surface roughness on the impedance of ideally polarized disk electrodes. The characteristic length associated with roughness was found to depend on the width or period of the roughness as well as the roughness factor. This work shows that for small roughness factors, while roughness causes frequency dispersion, the frequency dispersion is seen only at frequencies that are much higher than those associated with the disk geometry. Thus, a constant-phase element associated with a surface distribution of time constants cannot be attributed to surface roughness.

**References**

R. de Levie, Electrochemical Response of Porous and Rough Electrodes, in: P. Delahay (Ed.),

*Advances in Electrochemistry and Electrochemical Engineering, Interscience Publishers*, 1967, 329-395.T. Borisova and B. Ershler, Determination of the Zero Voltage Points of Solid Metals from Measurements of the Capacity of the Double Layer,

*Zhurnal Fizicheskoi Khimii***24**(1950), 337-344.L. Mehaute and G. Crepy, Introduction to Transfer and Motion in Fractal Media: The Geometry of Kinetics,

*Solid State Ionics***9**(1983), 17-30.T. Pajkossy, Impedance of Rough Capacitive Electrodes,

*Journal of Electroanalytical Chemistry*,**364**(1994), 111-125.