A simple model for charge storage in a nanotube

We consider a narrow nanotube in contact with an ionic liquid; the sizes of the ions are supposed to be so different that only one kind of ion can enter the tube. For a given number of N equal ions in the tube, the distance between the ions is determined by Coulomb repulsion. Using results obtained previously from density functional theory, we calculate the energy of an ensemble of equal ions, including vibrations about the equilibrium positions. For large tubes, in the thermodynamic limit, the resulting capacity diverges at zero charge for entropic reasons. It then increases rapidly, passes through a maximum, and becomes smaller as the distance between the ions decreases with increasing charge. Small nanotubes exhibit discrete charging steps reminiscent of quantum dots.