Planar Electric Double Layer Research Articles

A Gaussian field approach to the planar electric double layer structures in electrolyte solutions.

In this work, the planar, electric, double-layer structures of non-polarizable electrodes in electrolyte solutions are studied with Gaussian field theory. A response function with two Yukawa functions is used to capture the electrostatic response of the electrolyte solution, from which the modified response function in the planar symmetry is derived analytically. The modified response function is further used to evaluate the induced charge density and the electrostatic potential near an electrode. The Gaussian field theory, combined with a two-Yukawa response function, can reproduce the oscillatory decay behavior of the electric potentials in concentrated electrolyte solutions. When the exact sum rules for the bulk electrolyte solutions and the electric double layers are used as constraints to determine the parameters of the response function, the Gaussian field theory could at least partly capture the nonlinear response effect of the surface charge density. Comparison with results for a planar electrode with fixed surface charge densities from molecular simulations demonstrates the validity of Gaussian field theory.

Entropy characteristics in an electric double layer with off-centre charge ions. A statistical thermodynamics approach

Statistical thermodynamics methods are utilised to characterise aspects of the entropy of formation of a planar electric double layer. The model system consists of a primitive model electrolyte but with off-centre charge cations and centro-symmetric anions, next to a planar, uniformly charged electrode. An expression is developed for the entropy of off-centre charge ions. The theoretically predicted results of this entropy as a function of the electrode surface charge are compared with those from the corresponding Monte Carlo simulations. The entropy change with respect to the absolute value of the surface charge density is seen to be negative (or normal) if the displacement of the cation charge centre is zero or is independent of temperature. However, if the displacement is temperature dependent, the entropy can be positive (or anomalous). In each situation, the simulation data are well reproduced by the predicted results.

Entropy formation of an electrical double layer with divalent off-centre charge cations: Monte Carlo studies

Grand canonical Monte Carlo simulations are employed to study the entropy of formation of a planar electric double layer containing equi-sized valency ions. The cation charge is displaced from the ion-sphere centre towards its surface with the displacement being temperature dependent in general, while the monovalent anion charge is held fixed at the centre of the anion sphere. The results for the electrode-ion distributions, the charge density distributions, the differential capacitance, the temperature derivative of the electrode potential, and the entropy of formation of the double layer have been obtained as functions of the electrode charge. For zero cation charge displacement, or for temperature-independent cation charge displacements, a negative temperature dependence of the differential capacitance is observed, while for the temperature-dependent cation charge displacement at negative electrode charges, this temperature dependence can be positive. In the former case, the entropy of formation is negative decreasing on either side of the zero surface charge, while in the latter case the decrease slows down at negative charges reaching positive values at high negative charges. The positive temperature dependence of the differential capacitance is associated with an increase in entropy.

Structural and thermodynamic properties of the electrical double layer in slit nanopores: A Monte Carlo study.

Grand canonical Monte Carlo (GCMC) simulation techniques at a constant electrode-electrolyte potential drop are employed to study the differential capacitance of a planar electric double layer in slit nanopores. According to the technique, a single randomly selected ion is exchanged between a simulation box and a reservoir. The probability of this step is given by the GCMC algorithm. To preserve the electroneutrality of the system after the ion exchange, the electrode charge is adequately modified, which produces electrode charge fluctuations. The charge fluctuations are used to calculate the differential capacitance of the double layer. Results for the ion distributions, electrode surface charge density, and differential capacitance in slit nanopores are reported for a symmetric system of +1:-1 ionic valences with a common ionic diameter of 0.4 nm at electrolyte concentrations of 0.2M, 1.0M, and 2.5M, pore widths of 0.6 nm, 0.8 nm, and 1.2 nm, a potential drop of 0.05 V, a relative permittivity of 78.5, and a temperature of 298.15 K. These results are compared with the corresponding data for a +1:-2 valence asymmetric system and a size asymmetric system with ionic diameters of 0.4 nm and 0.3 nm. The results show that with increasing electrolyte concentration, the range of confinement effects decreases. For divalent anions, the width dependence of electrode charge and differential capacitance reveals a maximum. The differential capacitance curves show a camel shape to bell shape transition as the electrolyte concentration increases. Asymmetry in both ionic valences and diameters leads to asymmetric capacitance curves.

Differential capacitance of uniformly charged hard-sphere ions in planar electric double layers

The theoretical model, which is based both on the modified fundamental-measure theory for the hard-sphere contribution and on the mean-spherical approximation for the cross correlation between the hard-sphere contribution and Coulomb interaction, has been developed to study the differential capacitance of uniformly charged hard-sphere ions. The charge distribution of ions, electronic valence, bulk concentration and ion size are affected on the differential capacitance and the camel-to-bell shape transition. The charge distribution of ions enhances the differential capacitance and tends to delay the camel-to-bell shape transition. The low bulk concentration predicts higher differential capacitance. The transition from a camel-shaped to bell-shaped curve occurs when the bulk concentration rises to an appropriate value. The camel-to-bell shape transition occurs at a large ion size with increasing the bulk concentration. The high electrical valence tends to delay the transition and raise the transition concentration. The increase of electronic valence shifts the maximum to the low surface charge density. The decrease of , after passing through the maximum , results from the packing effect due to the formation of counterion layers.

Off-centre charge model of the planar electric double layer for asymmetric 2:1/1:2 valencies

ABSTRACTGrand canonical Monte Carlo simulation results are reported for the structure and capacitance of a planar electric double layer containing off-centre charged rigid sphere cations and centrally charged rigid sphere anions. The ion species are assigned asymmetric valencies, +2:−1 and +1:−2, respectively, and set in a continuum dielectric medium (solvent) characterised by a single relative permittivity. An off-centre charged ion is obtained by displacing the ionic charge from the centre of the sphere towards its surface, and the physical double layer model is completed by placing the ionic system next to a uniformly charged, non-penetrable, non-polarizable planar electrode. Structural results such as electrode-ion singlet distribution functions, ionic charge density and orientation profiles are complemented by differential capacitance results at electrolyte concentrations of 0.2 mol/dm3 and 1 mol/dm3, respectively, and for various displacements of the cationic charge centre. The effect of asymmetry due to off-centre cations and valency asymmetry on the double layer properties is maximum for divalent counterions and when the cation charge is closest to the hard sphere surface.

Capacitance of electrolytes with hydration-mediated interaction in planar electric double layers

ABSTRACTThe theoretical model, which is based on the density functional approach, has been developed for studying the electrostatic properties of electrolytes with the soft, hydration-mediated ion–ion and ion–surface interactions. The theory approximates the hydration-mediated ion–ion interaction through the mean-field approximation. The results show that the theory leads to a good agreement with the simulation results and predicts the camel-to-bell shape transition for the charged hard-core Yukawa ions with the hydration-mediated interaction. The high is observed at a low bulk concentration of ions and the camel-to-bell shape transition occurs when the bulk concentration rises to an appropriate value. The increase of an ion size shifts the maximum differential capacitance () to a low surface charge density. The addition of a repulsive hydration interaction reduces the , whereas the attractive hydration interaction enhances the . The increase of ion–surface and ion–ion interactions decreases the and shifts the maximum to a higher surface charge density. The increase of hydration anion–anion repulsion decreases the of a positively charged electrode and shifts its maximum to more positively charged surfaces.

Off-center charge model revisited: Electrical double layer with multivalent cations.

The off-center charge model of ions is a relatively simple model for introducing asymmetry in Coulomb interaction while retaining the simplicity and convenience of the spherical hard core geometry. A Monte Carlo simulation analysis of the planar electric double layer formed by this ionic model for 1+:1- valence systems [S. Lamperski et al., Langmuir 33, 11554-11560 (2017)] is extended to include solutions of multivalent (2+, 3+) hard spherical cations and single valence (1-) hard spherical anions near a uniformly charged, planar electrode. The solvent is modelled as a uniform dielectric continuum with a dielectric constant equal to that of the pure solvent, viz., the primitive model. Results are reported for the ion density, the cation charge profile, and the electrostatic potential profile at 1 mol/dm3 salt concentration. Additionally, the double layer potential drop, that is, the electrode potential, and the integral and the differential capacitances are computed as functions of the electrode surface charge density. The latter two quantities show an expected asymmetry as long as the cation valence is not too great and the charge of the off-center ion cannot approach too close to the electrode surface. It is unusual that the integral and differential capacitances are negative for high valence cations and a negatively charged electrode when the off-center charge is large and can be very near the surface of the electrode. The corresponding electrode potential versus surface charge density curve becomes non-monotonic and shows a change of slope, and thus the resultant integral and differential capacitances can become negative. This nonphysical result is the result of an incipient singularity when a large positive charge is too near a negatively charged electrode. Overall, the off-center charge model suggests a useful recipe to model electrical asymmetry within the broader context of the primitive model provided that the off-center charge is not too near the surface of the electrode.

Assessing the accuracy of three classical density functional theories of the electrical double layer.

Classical density functional theory (DFT) is a useful tool to compute the structure of the electrical double layer because it includes ion-ion correlations due to excluded-volume effects (i.e., steric correlations) and ion screening effects (i.e., electrostatic correlations beyond the electrostatic mean-field potential). This paper systematically analyzes the accuracies of three different electrostatic excess free-energy functionals, as compared to Monte Carlo (MC) simulations of the planar electrical double layer, over a large parameter space. Specifically, we tested the reference fluid density (RFD) [Gillespie et al., J. Phys.:Condens. Matter 14, 12129 (2002)10.1088/0953-8984/14/46/317], functionalized mean spherical approximation (fMSA) [Roth and Gillespie, J. Phys.: Condens.Matter 28, 244006 (2016)10.1088/0953-8984/28/24/244006], and bulk fluid (BF) [Kierlik and Rosinberg, Phys. Rev. A 44, 5025 (1991)10.1103/PhysRevA.44.5025; Y. Rosenfeld, J. Chem. Phys. 98, 8126 (1993)10.1063/1.464569] functionals. Previous work compared these DFT methods to MC simulations only for a small set of parameters. Here, a total of twelve different cations were studied, with valences of +1, +2, and +3 and ion diameters of 0.15, 0.30, 0.60, and 0.90 nm at bulk concentrations between 1 μM and 1 M. The anion always had valence -1 and diameter 0.30 nm. The structure of the double layer of these charged, hard-sphere ions was computed for surface charges ranging from 0 to -0.50C/m^{2}. All the DFTs were compared against each other for all the parameters, as well as to 378 MC simulations. Overall, RFD was the best of the three functionals, while BF was the least accurate. fMSA performed significantly better than BF, making it a reasonable choice that is less computationally expensive than RFD. For monovalent cations, all three functionals worked reasonably well, except BF was qualitatively different from MC at very low surface charges. For multivalent cations, BF underestimated charge inversion while fMSA overestimated it. All DFTs performed poorly for small multivalent ions.

A systematic Monte Carlo simulation study of the primitive model planar electrical double layer over an extended range of concentrations, electrode charges, cation diameters and valences

The purpose of this study is to provide data for the primitive model of the planar electrical double layer, where ions are modeled as charged hard spheres, the solvent as an implicit dielectric background (with dielectric constant ϵ = 78.5), and the electrode as a smooth, uniformly charged, hard wall. We use canonical and grand canonical Monte Carlo simulations to compute the concentration profiles, from which the electric field and electrostatic potential profiles are obtained by solving Poisson’s equation. We report data for an extended range of parameters including 1:1, 2:1, and 3:1 electrolytes at concentrations c = 0.0001 − 1 M near electrodes carrying surface charges up to σ = ±0.5 Cm−2. The anions are monovalent with a fixed diameter d− = 3 Å, while the charge and diameter of cations are varied in the range z+ = 1, 2, 3 and d+ = 1.5, 3, 6, and 9 Å (the temperature is 298.15 K). We provide all the raw data in the supplementary material.

Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory

The modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the earlier MPB4 and MPB5 formulations and clarifies the relationship between these. The MPB4, MPB5, and a new formulation of the theory are employed in an analysis of the structure and charge reversal phenomenon in asymmetric 2:1/1:2 valence electrolytes. Furthermore, polarization induced surface charge amplification is studied in 3:1/1:3 systems. The results are compared to the corresponding Monte Carlo simulations. The theories are seen to predict the "exact" simulation data to varying degrees of accuracy ranging from qualitative to almost quantitative. The results from a new version of the theory are found to be of comparable accuracy as the MPB5 results in many situations. However, in some cases involving low electrolyte concentrations, theoretical artifacts in the form of un-physical "shoulders" in the singlet ionic distribution functions are observed.

Monte Carlo Study of a Planar Electric Double Layer Formed by Ions with Off-Center Charge.

Grand canonical Monte Carlo simulation results are reported for an electric double layer (EDL) modeled by a planar charged hard wall, hard sphere cations with an off-center charge, and spherical anions with a charge at the center of the sphere. The ion charge numbers are Z+ = +1 and Z- = -1, and the diameter, d, of a hard sphere is the same for anions and cations. The ions are immersed in a solvent mimicked by a continuum dielectric medium at standard temperature. The results are obtained for three values of charge displacement, s+0 = d/16, d/4, 7d/16 from the center of the sphere and the following electrolyte concentrations: 0.5, 1.0, 2.0, and 3.0 M. The profiles of electrode-ion singlet distributions, cation reduced charge density, angular function, and mean electrostatic potential are reported for an electrode surface charge density σ = -0.30 C m-2, whereas the electrode potential and the differential capacitance of EDL are shown as functions of the electrode charge density varying from -1.00 to +1.00 C m-2. At negative electrode charges and with increasing values of the charge separation, the differential capacitance curve rises. As the electrolyte concentration increases, the shape of the differential capacitance curve changes from that of a minimum surrounded by two maxima into that of a distorted single maximum.

Density functional theory of electrolyte solutions in slit-like nanopores I. The RFD/WCA approach extended to non-restricted primitive model

An extended reference fluid density approach/weighted correlation approximation (RFD/WCA) of density functional theory (DFT) is tested for the description of the non-restricted primitive model of electrolyte solutions in slit-like nanopores. The RFD/WCA approach of modeling size-asymmetric ions is validated by reproducing the density and electrostatic potential profiles of planar electrical double layer systems in the presence of mixtures of mono- and multivalent ions. The results from the DFT agree quite well with Monte Carlo simulations and satisfy the contact density sum rules of ionic fluid under a wide range of conditions. These findings suggest that a generic RFD/WCA approach can be formulated within the framework of the restricted and non-restricted primitive model to further investigate the swelling and ion exchange of clay minerals.

Structure and capacitance of an electric double layer of an asymmetric valency dimer electrolyte: A comparison of the density functional theory with Monte Carlo simulations

Abstract The density functional theory is applied to a study of the structure and differential capacitance of a planar electric double layer formed by a valency asymmetric mixture of charged dimers and monomers. The dimer consists of two tangentially tethered hard spheres of equal diameters of which one is charged and the other is neutral, while the monomer is a charged hard sphere of the same size. The dimer electrolyte is next to a uniformly charged, smooth planar electrode. The electrode-particle singlet distributions, the mean electrostatic potential, and the differential capacitance for the model double layer are evaluated for a 2:1/1:2 valency electrolyte at a given concentration. Important consequences of asymmetry in charges and in ion shapes are (i) a finite, non-zero potential of zero charge, and (ii) asymmetric shaped 2:1 and 1:2 capacitance curves which are not mirror images of each other. Comparisons of the density functional results with the corresponding Monte Carlo simulations show the theoretical predictions to be in good agreement with the simulations overall except near zero surface charge.

Double layer for hard spheres with an off-center charge

Simulations for the density and potential profiles of the ions in the planar electrical double layer of a model electrolyte or an ionic liquid are reported. The ions of a real electrolyte or an ionic liquid are usually not spheres; in ionic liquids, the cations are molecular ions. In the past, this asymmetry has been modelled by considering spheres that are asymmetric in size and/or valence (viz., the primitive model) or by dimer cations that are formed by tangentially touching spheres. In this paper we consider spherical ions that are asymmetric in size and mimic the asymmetrical shape through an off-center charge that is located away from the center of the cation spheres, while the anion charge is at the center of anion spheres. The various singlet density and potential profiles are compared to (i) the dimer situation, that is, the constituent spheres of the dimer cation are tangentially tethered, and (ii) the standard primitive model. The results reveal the double layer structure to be substantially impacted especially when the cation is the counterion. As well as being of intrinsic interest, this off-center charge model may be useful for theories that consider spherical models and introduce the off-center charge as a perturbation.

Structure and capacitance of an electric double layer formed by fused dimer cations and monomer anions: a Monte Carlo simulation study

ABSTRACTCanonical Monte Carlo simulation results are obtained for the structure and differential capacitance of a planar electric double layer consisting of fused dimer cations and monomer anions in a dielectric continuum (solvent). The physical double layer model is mimicked by a uniformly charged, non-polarisable, planar electrode next to an electrolyte consisting of positively charged fused dimers and negatively charged monomer rigid spheres in a continuum dielectric. The fused dimer comprises two same sized hard spheres that are fused into one another, one of the hard spheres being charged and the other neutral. The various electrode-particle species singlet distribution functions and the mean electrostatic potential profiles are obtained for 1:1, 2:2, 2:1, and 1:2 valency systems for different degrees of fusion. The differential capacitance as a function of the electrode surface charge density is evaluated for a 1:1 system. Comparison of these results are made with those corresponding to the situation when the dimer spheres are simply tangentially tethered. It is seen that the fused dimer influences the double layer structure and capacitance substantially for a negatively charged electrode but less so for a positively charged electrode.

The planar electric double layer capacitance for the solvent primitive model electrolyte.

The planar electric double layer capacitance of the solvent primitive model electrolyte is studied using simulation and two versions of the modified Poisson-Boltzmann theory. At small values of the surface charge and varying electrolyte concentration, the capacitance has a behaviour analogous to that of the restricted primitive model electrolyte. As the electrolyte concentration is increased at a fixed total packing fraction, the minimum at zero surface charge changes to a maximum. This qualitative change is predicted by both simulation and the modified Poisson-Boltzmann theories. The transition envelope, separating the change in the capacitance from a minimum to a maximum at zero surface charge, is shifted from the restricted primitive model value to higher ion densities at fixed total packing fractions.

Structure of an electric double layer containing a 2:2 valency dimer electrolyte

The structure of a planar electric double layer formed by a 2:2 valency dimer electrolyte in the vicinity of a uniformly charged planar hard electrode is investigated using density functional theory and Monte Carlo simulations. The dimer electrolyte consists of a mixture of charged divalent dimers and charged divalent monomers in a dielectric continuum. A dimer is constructed by two tangentially tethered rigid spheres, one of which is divalent and positively charged and the other neutral, whereas the monomer is a divalent and negatively charged rigid sphere. The density functional theory reproduces well the simulation results for (i) the singlet distributions of the various ion species with respect to the electrode, and (ii) the mean electrostatic potential. Comparison with earlier results for a 2:1/1:2 dimer electrolyte shows that the double layer structure is similar when the counterion has the same valency.

Structure of a planar electric double layer containing size-asymmetric ions: density functional approach

We have studied the structure of the electrolytes with asymmetries in charge and size near a charged planar electric double layer by a density functional theory. In the present theory, the hard-sphere contribution has been approximated as the direct pair correlation function with the coupling parameter, whereas the electronic contribution has been approximated as the mean-spherical approximation in the bulk phase. This theoretical approach for the size-symmetric and size-asymmetric electrolytes displays a good agreement with the simulation results over a wide range of surface charge densities and electrolyte concentrations. However, the accuracy between the present theory and the simulation results slightly deteriorates for the highly size-asymmetric electrolytes and the multivalent electrolytes. In these cases, the performance of the present theory is comparable to those of the simplified extension of the Poisson–Boltzmann theory and the modified Poisson–Boltzmann theory. The calculated result indicates that the surface charge distribution function, which was introduced as an indicator for studying the charge reversal, layering effect, and surface charge amplification in a planar electric double layer, describes the electronic properties of a planar electric double layer well.

Restoring the consistency with the contact density theorem of a classical density functional theory of ions at a planar electrical double layer.

Classical density functional theory (DFT) of fluids is a fast and efficient theory to compute the structure of the electrical double layer in the primitive model of ions where ions are modeled as charged, hard spheres in a background dielectric. While the hard-core repulsive component of this ion-ion interaction can be accurately computed using well-established DFTs, the electrostatic component is less accurate. Moreover, many electrostatic functionals fail to satisfy a basic theorem, the contact density theorem, that relates the bulk pressure, surface charge, and ion densities at their distances of closest approach for ions in equilibrium at a smooth, hard, planar wall. One popular electrostatic functional that fails to satisfy the contact density theorem is a perturbation approach developed by Kierlik and Rosinberg [Phys. Rev. A 44, 5025 (1991)PLRAAN1050-294710.1103/PhysRevA.44.5025] and Rosenfeld [J. Chem. Phys. 98, 8126 (1993)JCPSA60021-960610.1063/1.464569], where the full free-energy functional is Taylor-expanded around a bulk (homogeneous) reference fluid. Here, it is shown that this functional fails to satisfy the contact density theorem because it also fails to satisfy the known low-density limit. When the functional is corrected to satisfy this limit, a corrected bulk pressure is derived and it is shown that with this pressure both the contact density theorem and the Gibbs adsorption theorem are satisfied.