Capacitance of electrical double layer formed inside a single infinitely long cylindrical pore

By using classical density functional theory, we investigate the capacitance of a primitive model and an extended primitive model electrical double layer (EDL) formed in a single infinitely long cylindrical pore. We obtain dependencies of the EDL differential capacitance C-d on the parameters characterizing the electrolyte-electrode system. It is shown that (i) high electrical valence and small size of counter-ions help to raise the C-d and the two factors have a synergistic effect; the C-d increases with the co-ion size at high bulk concentrations, but is not sensitive to the co-ion size at lower concentrations. (ii) The curve of the C d versus sigma (surface charge density) moves upwards completely with the cylindrical pore radius regardless of the electrolyte types and bulk concentrations and whether the C-d - sigma curve is a camel-shaped or bell-shaped curve. (iii) At low concentrations, the C-d - sigma curve is camel shaped regardless of the electrolyte types and pore sizes, and a transition from a camel-shaped to bell-shaped curve will eventually occur when the bulk concentration rises to an appropriate value. Moreover, the high electrical valence tends to delay the transition and raise the transition concentration. (iv) In the case of the camel-shaped curve, the C-d tends to decrease with the vertical bar sigma vertical bar after the C-d - sigma curve goes beyond its maximal value; the surface charge strength (corresponding to the C-d maxima) increases with the counter-ion valence and pore size, but decreases with the bulk concentration. However, the concentration effect is very small in the presence of only one valence counter-ion, whereas it will become obvious with the presence of two valence counter-ions. (v) In determining the shape of the C-d - sigma curve, increasing the strength of an attractive interionic neutral and non-HS interaction has the same effect as decreasing the bulk concentration, i.e. it makes the characteristics of the camel-shaped curve more obvious. Conversely, the incorporation of a repulsive interionic neutral and non-HS interaction transforms an originally not obvious camel-shaped C-d - sigma curve into a bell-shaped curve. The above observations can be explained self-consistently by analyzing the adsorption of salt ions and the resulting screening effect and their influencing factors.