Quantitative Relation of the Frequency Dispersion of Double Layer Capacitances to Surface Roughness

Frequency dispersion of double layer (DL) capacitances, which can be represented by the power law of the frequency or the constant phase element, is modeled by the Arrhenius equation with the activation energy which has a linear relation with the free energy change in the orientation of solvent dipoles. The Arrhenius equation has a form of a differential equation of the number of oriented dipoles. The solution is the power law of the time, being equivalent to the DL capacitance with the power law of the frequency. The power number is associated with the surface roughness of the electrode on the assumption that a dipole is oriented with the help of interaction on a given local area of the electrode. Then it has an approximately linear relation with the surface roughness. Surface roughness of highly oriented pyrolytic graphite electrodes is varied unintentionally by peeling-off processes and intentionally by electrochemical oxidation. The power numbers determined by acimpedance techniques are compared with surface roughness obtained by scanning probe microscopy. They are approximately proportional to the surface roughness factor when the scanned domain on the surface is less than (40nm), which is much smaller than the domain for the fractal structure.