Single-File Charge Storage in Conducting Nanopores

Charging of a conducting tubular nanopore in a nanostructured electrode is treated using an exactly solvable 1D lattice model, including ion correlations screened by ion-image interactions. Analytical expressions are obtained for the accumulated charge and capacitance as a function of voltage. They show that the mechanism of charge storage, and the qualitative form of the capacitance-voltage curve, are sensitive to how favorable it is for ions to occupy the unpolarized pore, and the pore radius. Qualitative predictions of the theory are corroborated by Monte Carlo simulations. These results highlight the effect of ion affinity to unpolarized pores on the charge and energy storage in supercapacitors. Furthermore, they suggest that the question of the occupancy of unpolarized pores could be answered by measuring the capacitance-voltage dependence.

Figure 1

1D lattice model for charge storage in a cylindrical nanopore. $R$ is the pore radius and $2a$ is the lattice constant.

Figure 2

Ionophilic pores: charge storage occurs via removing co-ions at low voltages and filling the pore with counterions at large voltages. (a) The main panel shows the dimensionless surface charge density $(2ap/e)\sigma$ with increasing voltage, plotted for various values of transfer energy $w$. The insets show the number density of cations and anions ${\rho }_{\pm }=(1/Z)(\partial Z/\partial {\mu }_{\pm })$, when $w=0$ and the total density of ions ${\rho }_{\text{total}}={\rho }_{+}+{\rho }_{-}$ with increasing voltage, both normalized with respect to the maximum density. (b) Maximum differential capacitance increases as the transfer energy increases and the pore size decreases. The differential capacitance is symmetric about $u=0$, and the $R<a$ case corresponds to elongated (e.g., elliptical) ions. All curves are plotted for $J=1.6$ ($R=1.02a$), except of the inset in (b).

Figure 3

Ionophobic pores: the main panel shows that the differential capacitance emerges as two peaks when the transfer energy becomes more negative. The insets show how the total density varies with voltage for different transfer energies, and how the population of cation, anion and void vary at zero applied potential with transfer energy. In all curves $J=1.6$ ($R=1.02a$).

Figure 4

Ionophobic pores: the main panel shows the differential capacitance obtained via Monte Carlo simulation for $(R,a)=(5,6)\AA$ ($J=2.6$), and $(R,a)=(8,8.1)\AA$ ($J=0.92$). The inset shows the capacitance predicted by the theory. The curves are plotted for $w=-20$.

Figure 5

A main panel shows a map of different charging regimes with negative transfer energy. With $w\ll 0$, ions need to first overcome the energy penalty associated with entering the pore (“filling”), and, subsequently, packing themselves close together. For less negative values $w$, the two charging processes merge together. The inset shows how the stored energy density varies with applied voltage for different values of $w$. The figures are plotted with $a=5\AA$.