Eur. Phys. J. E (2018) 41: 113
https://doi.org/10.1140/epje/i2018-11723-7

Regular Article

Modeling the camel-to-bell shape transition of the differential capacitance using mean-field theory and Monte Carlo simulations

¹ Department of Physics, North Dakota State University, 58108-6050, Fargo, ND, USA
² Department of Physics, São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, 15054-000, São José do Rio Preto, SP, Brazil
³ Faculty of Health Sciences, University of Ljubljana, Poljanska 26a, 1000, Ljubljana, Slovenia

^* e-mail: sylvio.may@ndsu.edu

Received: 11 July 2018
Accepted: 30 August 2018
Published online: 27 September 2018

Abstract

Mean-field electrostatics is used to calculate the differential capacitance of an electric double layer formed at a planar electrode in a symmetric 1:1 electrolyte. Assuming the electrolyte is also ion-size symmetric, we derive analytic expressions for the differential capacitance valid up to fourth order in the surface charge density or surface potential. Our mean-field model accounts exclusively for electrostatic interactions but includes an arbitrary non-ideality in the mixing entropy of the mobile ions. The ensuing criterion for the camel-to-bell shape transition of the differential capacitance is analyzed using commonly used mixing models (one based on a lattice gas and the other based on the Carnahan-Starling equation of state) and compared with Monte Carlo simulations. We observe a reasonable agreement between all our mean-field models and the simulation data for the camel-to-bell shape transition. The absolute value of the differential capacitance for an uncharged (or weakly charged) electrode is, however, not reproduced by our mean-field approaches, not even upon introducing a Stern layer with a thickness equal of the ion radius. We show that, if a Stern layer is introduced, its thickness dependence on the ion size is non-monotonic or, depending on the salt concentration, even inversely proportional.