Double layer capacitance on a rough metal surface: Surface roughness measured by “Debye ruler”

The properties of the double layer capacity on rough electrodes are discussed in terms of the recently developed linear Poisson-Boltzmann theory [L.I. Daikhin, A.A. Kornyshev, M. Urbakh, Phys. Rev. E 53, 6192 (1996)]. This theory offers a concept of a “Debye-length κ dependent roughness factor”, ie roughness function, which determines the deviation of capacitance from the Gouy-Chapman result for a flat interface. Analytical expression for the roughness function is available for the case of weak Euclidean roughness. The two parameters—mean square height and correlation length—appear there. The way how the result changes in the case of moderate and strong roughness is analyzed on the basis of a numerical solution obtained for a model roughness profile; an efficient interpolation formula which covers the limits of weak and strong roughness is suggested. The role of anisotropy of the roughness profile is investigated. The predicted effects could be screened by the crystallographic inhomogeneity of a rough surface not taken into account here. Tentatively ignoring such (often important) complications we discuss the results in the context of a possible method for in situ characterization of surface roughness of metal electrodes, based on the double layer capacity measurements in solutions of variable concentration.