Oscillations in the capacitance of a nanopore containing an electrolyte due to pore width and nonzero size ions

Jiang, Jin, and Wu (Nano Lett., 11 (2011), pp. 5373-5377) have reported the results of a density functional theory (DFT) study of the capacitance of a nanopore containing an electrolyte consisting of charged hard spheres of equal diameter and charge. They find that the capacitance of the nanopore and electrolyte oscillates. The 'period' of the oscillations is of the order of the ionic diameter. Intuitively, the capacitance should tend to zero when the pore diameter is too small to accommodate the electrolyte ions and, thus, would contains no charge. A superficial glance at their table of contents figure might lead one to think that this does not occur in their calculation. However, they do not report results for exceedingly small pore diameters. In order to gain insight into their results, the Poisson-Boltzmann (PB) theory, which does not account for the diameter, is examined briefly. Unsurprisingly, the PB capacitance decreases monotonically to zero as the pore diameter decreases. The effect of a nonzero ion diameter is included in a semi-empirical manner by appealing to the results of the mean spherical approximation (MSA). The resulting capacitance oscillates and is qualitatively similar to the DFT results; it is zero at small pore diameter.