Analytical Solution of Modified Poisson–Boltzmann Equation and Application to Cylindrical Nanopore Supercapacitor Energy Storage

• Published in: Colloid journal of the Russian Academy of Sciences 2022-04, Vol.84 (2), p.222-242
• Main Authors: Zhou, S., Zhang, L.-T.
• Format: Article
• Language: English
• Subjects: Boltzmann transport equation; Capacitance; Charge density; Chemistry; Chemistry and Materials Science; Computer simulation; Energy storage; Exact solutions; Ion concentration; Polymer Sciences; Pore size; Supercapacitors; Surface charge; Surfaces and Interfaces; Thin Films
• ISSN: 1061-933X; 1608-3067
• DOI: 10.1134/S1061933X22020107

An approximate and analytical solution is obtained for a modified Poisson−Boltzmann (MPB) equation describing + z /− z electrolyte confined inside a cylindrical pore. Three expressions are obtained for the ration of surface charge density, , to surface potential, , electrical potential distribution, and ion concentration profiles. The expression is in near-perfect agreement with the numerical solution for any values of and pore radius non-dimensionalized by screening constant . The expression is applied to study electrical capacitance and energy storage behavior of the cylindrical pore supercapacitor. The MPB calculations accord with previous results from computer simulation and other theories. Main new findings are summarized as follows. (i) Influence of bulk salt mole concentration is mainly manifested in that high value lowers the saturation energy storage density and the threshold electrode potential , beyond which the is just reached, and reduces the width of the zero energy domain around the zero charge potential. The effect of value is only obvious in narrow pore. (ii) Although an increase in size of the ion reduces the value, it simultaneously reduces the value. Moreover, it brings a transition from camel shaped to bell shaped differential capacitance curve. (iii) High ion valency helps in raising both the and values. Moreover, the -enhancing effect by the pore size reduces with the ion valency increasing. (iv) Large size pore raises both the and the values.