Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions.

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

Differential capacitance of ionic liquids according to lattice-gas mean-field model with nearest-neighbor interactions.

The Bragg-Williams free energy is used to incorporate nearest-neighbor interactions into the lattice gas model of a solvent-free ionic liquid near a planar electrode. We calculate the differential capacitance from solutions of the mean-field consistency relation, arriving at an explicit expression in the limit of a weakly charged electrode. The two additional material parameters that appear in the theory-the degree of nonideality and the resistance to concentration changes of each ion type-give rise to different regimes that we identify and discuss. As the nonideality parameter, which becomes more positive for stronger nearest-neighbor attraction between like-charged ions, increases and the electrode is weakly charged, the differential capacitance is predicted to transition through a divergence and subsequently adopt negative values just before the ionic liquid becomes structurally unstable. This is associated with the spontaneous charging of an electrode at vanishing potential. The physical origin of the divergence and the negative sign of the differential capacitance is a nonmonotonic relationship between the surface potential and surface charge density, which reflects the formation of layered domains alternatingly enriched in counterions and coions near the electrode. The decay length of this layered domain pattern, which can be many times larger than the ion size, is reminiscent of the recently introduced concept of "underscreening."

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

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.

Differential capacitance of an electric double layer with asymmetric solvent-mediated interactions: mean-field theory and Monte Carlo simulations.

The differential capacitance of an electrical double layer is directly affected by properties of the electrolyte solution such as temperature, salt concentration, ionic size, and solvent structure. In the present work, we employ a mean-field approach and Monte Carlo simulations to investigate how the inclusion of asymmetric solvent-mediated ion-ion and ion-surface interactions affects the differential capacitance. We focus on a charged flat electrode immersed in an electrolyte solution of monovalent ions at physiological concentration in a uniform dielectric background. Solvent-mediated anion-anion, anion-cation and cation-cation interactions are modeled on the basis of Yukawa potentials with three independent strengths that add to Coulomb and excluded volume pair-potentials, the latter accounted for through a lattice gas approach. We use the three interaction strengths to produce and analyze asymmetric profiles of the differential capacitance as function of the electrode's surface charge density. While solvent-mediated anion-anion and cation-cation interactions mainly affect the behavior at medium charge densities of the electrode, anion-cation repulsion increases the differential capacitance of a weakly charged electrode. We present a simple phenomenological model to rationalize this finding. Most importantly, because the added solvent-mediated interaction potential is comparatively soft, our mean-field model is able to qualitatively - and in some cases quantitatively - reproduce all Monte Carlo simulation results, even at high surface charge densities of the electrode.

Role of ion hydration for the differential capacitance of an electric double layer.

The influence of soft, hydration-mediated ion-ion and ion-surface interactions on the differential capacitance of an electric double layer is investigated using Monte Carlo simulations and compared to various mean-field models. We focus on a planar electrode surface at physiological concentration of monovalent ions in a uniform dielectric background. Hydration-mediated interactions are modeled on the basis of Yukawa potentials that add to the Coulomb and excluded volume interactions between ions. We present a mean-field model that includes hydration-mediated anion-anion, anion-cation, and cation-cation interactions of arbitrary strengths. In addition, finite ion sizes are accounted for through excluded volume interactions, described either on the basis of the Carnahan-Starling equation of state or using a lattice gas model. Both our Monte Carlo simulations and mean-field approaches predict a characteristic double-peak (the so-called camel shape) of the differential capacitance; its decrease reflects the packing of the counterions near the electrode surface. The presence of hydration-mediated ion-surface repulsion causes a thin charge-depleted region close to the surface, which is reminiscent of a Stern layer. We analyze the interplay between excluded volume and hydration-mediated interactions on the differential capacitance and demonstrate that for small surface charge density our mean-field model based on the Carnahan-Starling equation is able to capture the Monte Carlo simulation results. In contrast, for large surface charge density the mean-field approach based on the lattice gas model is preferable.