On the non-dominance of counterions in the 1:z planar electrical double layer of point-ions

Abstract

According to the dominance prescription of point-ions in the non-linear Poisson-Boltzmann theory, proposed by Valleau and Torrie almost 40 years ago, the microscopic and thermodynamic properties of an asymmetric binary electrolyte converge asymptotically to those of a completely symmetric electrolyte, in the limit of an infinite surface charge density of a planar electrode, if the properties of the counterions are the same in both instances. By using the Grahame equation and the non-linear Poisson-Boltzmann theory, we show here that this prescription is certainly exact for the mean electrostatic potential at the electrode's surface and for the capacitive compactness. Contrastingly, analytical and numerical solutions of the non-linear Poisson-Boltzmann equation show that, in the limit of an infinite surface charge density of the planar electrode, it is possible to observe finite differences between the local mean electrostatic potentials and electric fields associated to a 1:1 and a 1:z electrolyte at places near the electrode's surface. Thus, we prove here that even in the absence of ion correlations and ionic excluded volume effects, the counterions do not fully dominate the structural properties in the entire electrical double layer in the non-linear Poisson-Boltzmann picture, which is confirmed through comparisons with new Monte Carlo simulations.

GRAPHICAL ABSTRACT

Keywords:

• Electrical double layer
• dominance of counterions
• the non-linear Poisson-Boltzmann theory

Acknowledgments

G.I. G.-G. acknowledges the SEP-CONACYT CB-2016 grant 286105, the PRODEP grant UASLP-PTC-652 511-6/2020-8585, and the 2019 Marcos Moshinsky Fellowship. The authors thankfully acknowledge computer resources, technical advice and support provided by the Laboratorio Nacional de Supercómputo del Sureste de México (LNS), a member of the CONACYT national laboratories, with project No. 202001024N; as well as the Centro Nacional of Supercómputo (CNS) for the computing time provided at THUBAT KAAL II. E.G.-T. acknowledges the CONACYT Grant No CF-2019-731759. G.I. G.-G. and E.G.-T. express their gratitude for the assistance from the computer technicians at the IF-UASLP. The authors thank the anonymous reviewer for his/her critical comments, which allowed to improve the present study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the CONACYT [grant numbers CB-2016-286105,CF-2019-731759], Fundación Marcos Moshinsky [grant number 2019], Laboratorio Nacional de Supercomputo del Sureste de Mexico (LNS) [grant number 202001024N], and PRODEP [grant number UASLP-PTC-652 511-6/2020-8585].