Structure of electric double layers: A self-consistent weighted-density-functional approach

Pb2+ removal based on the confinement effect in polygonal carbon nanotubes: a molecular dynamics simulation

Heavy metal Pb²⁺ pollutants have become an important environmental problem, which threatens public health and ecosystems worldwide. In this study, to explore the effective treatment of trace Pb²⁺ pollution in water, molecular dynamics simulation combined with DFT calculations was used to study the transportation behavior of Pb²⁺ using polygonal carbon nanotubes (PCNT: P = 4, 5, 6, 8)/graphene composites (PCNTs/G). It is shown that due to the confinement effect of PCNTs, both H₂O and H₃O⁺ can form a hydrogen-bonding network and transport them in the form of proton exchange through the PCNT channels. The trajectory shows that with the help of a hydrogen-bonding network, the probability of Pb²⁺ passing through the 8N channel is enhanced. Then, upon the fluorine modification of PCNTs, mutual effects of both the hydrogen-bonding network and electrophilic attraction make Pb²⁺ get through the channel of 8F. It is indicated that with respect to 4CNT/G, 5CNT/G, and 6CNT/G, 8CNT/G is not accurate for Pb²⁺ interception at the outlets. In addition, the RDF, and HOMO-LUMO orbitals indicate that the affinity from the hydrogen-bonding network and PCNT walls both play important roles in particle transportation. This work can not only provide a basic understanding of Pb²⁺ transportation in PCNTs from the perspective of diffusion but also be helpful to guide the strategy on how to deal with Pb²⁺ pollution in waters.

Ionic Layering and Overcharging in Electrical Double Layers in a Poisson-Boltzmann Model

Electrical double layers (EDLs) play a significant role in a broad range of physical phenomena related to colloidal stability, diffuse-charge dynamics, electrokinetics, and energy storage applications. Recently, it has been suggested that for large ion sizes or multivalent electrolytes, ions can arrange in a layered structure inside the EDLs. However, the widely used Poisson-Boltzmann models for EDLs are unable to capture the details of ion concentration oscillations and the effect of electrolyte valence on such oscillations. Here, by treating a pair of ions as hard spheres below the distance of closest approach and as point charges otherwise, we are able to predict ionic layering without any additional parameters or boundary conditions while still being compatible with the Poisson-Boltzmann framework. Depending on the combination of ion valence, size, and concentration, our model reveals a structured EDL with spatially oscillating ion concentrations. We report the dependence of critical ion concentration, i.e., the ion concentration above which the oscillations are observed, on the counter-ion valence and the ion size. More importantly, our model displays quantitative agreement with the results of computationally intensive models of the EDL. Finally, we analyze the nonequilibrium problem of EDL charging and demonstrate that ionic layering increases the total charge storage capacity and the charging timescale.

Strong-coupling theory of counterions with hard cores between symmetrically charged walls

By a combination of Monte Carlo simulations and analytical calculations, we investigate the effective interactions between highly charged planar interfaces, neutralized by mobile counterions (salt-free system). While most previous analysis have focused on pointlike counterions, we treat them as charged hard spheres. We thus work out the fate of like-charge attraction when steric effects are at work. The analytical approach partitions counterions in two subpopulations, one for each plate, and integrates out one subpopulation to derive an effective Hamiltonian for the remaining one. The effective Hamiltonian features plaquette four-particle interactions, and it is worked out by computing a Gibbs-Bogoliubov inequality for the free energy. At the root of the treatment is the fact that under strong electrostatic coupling, the system of charges forms an ordered arrangement, that can be affected by steric interactions. Fluctuations around the reference positions are accounted for. To dominant order at high coupling, it is found that steric effects do not significantly affect the interplate effective pressure, apart at small distances where hard-sphere overlap are unavoidable, and thus rule out configurations.

Counterion-only electrical double layers: An application of density functional theory

Within the framework of density functional theory, a self-consistent approach of weighted correlation approximation is developed to give an accurate account of the cross correlations between the Coulombic interaction and the hard-sphere exclusion in the counterion-only electrical double layers. Application of the approach to the cases of practical interest, against the Monte Carlo simulations, shows that it is excellent in describing the structural properties and the pressures of the confined solutions involving both mono- and divalent counterions between two planar charged walls. In particular, the study suggests that the relative importance of electrostatic correlations in comparison to the effects of ionic excluded volume and direct Coulomb interactions depends on the valency of the counterions and the surface charge density. In a clay system with mixed counterions, the competition between the mono- and divalent ions results in a large swelling when the fraction of surface charge compensated by monovalent counterions is greater than 30%. In the opposite situation involving mostly divalent counterions, a limited swelling is found and the attraction between the clay particles favors the formation of stacks incorporating a water layer of about 1.0 nm. These findings are consistent with experimental observations, giving insight into some mechanisms governing the stability of colloidal clay in salt-free or dilute solutions.

Structure of spherical electric double layers with fully asymmetric electrolytes: a systematic study by Monte Carlo simulations and density functional theory

A systematic investigation of the spherical electric double layers with the electrolytes having size as well as charge asymmetry is carried out using density functional theory and Monte Carlo simulations. The system is considered within the primitive model, where the macroion is a structureless hard spherical colloid, the small ions as charged hard spheres of different size, and the solvent is represented as a dielectric continuum. The present theory approximates the hard sphere part of the one particle correlation function using a weighted density approach whereas a perturbation expansion around the uniform fluid is applied to evaluate the ionic contribution. The theory is in quantitative agreement with Monte Carlo simulation for the density and the mean electrostatic potential profiles over a wide range of electrolyte concentrations, surface charge densities, valence of small ions, and macroion sizes. The theory provides distinctive evidence of charge and size correlations within the electrode-electrolyte interface in spherical geometry.

Ionic density distributions near the charged colloids: spherical electric double layers

We have studied the structure of the spherical electric double layers on charged colloids by a density functional perturbation theory, which is based both on the modified fundamental-measure theory for the hard spheres and on the one-particle direct correlation functional (DCF) for the electronic residual contribution. The contribution of one-particle DCF has been approximated as the functional integration of the second-order correlation function of the ionic fluids in a bulk phase. The calculated result is in very good agreement with the computer simulations for the ionic density distributions and the zeta potentials over a wide range of macroion sizes and electrolyte concentrations, and compares with the results of Yu et al. [J. Chem. Phys. 120, 7223 (2004)] and modified Poisson-Boltzmann approximation [L. B. Bhuiyan and C. W. Outhwaite, Condens. Matter Phys. 8, 287 (2005)]. The present theory is able to provide interesting insights about the charge inversion phenomena occurring at the interface.

Weighted correlation approach: an extended version with applications to the hard-sphere fluid

The purpose of this study is to extend the weighted correlation approach (WCA) for inhomogeneous fluids. It now introduces a generic expression to evaluate the single-particle direct correlation function in terms of a series of pair direct correlation functions weighted by different correlation-weight functions and adjustable weight factors. When applied for practical use, however, approximations of the pair direct correlation functions have to be made, together with appropriate definitions of the weighted densities and the choices of the correlation-weight functions. The WCA approach would, then, not only help us to connect and compare different strategies and their underlying assumptions in the density functional approaches, but also enable us to propose and apply density functional theory methods to predict the density profile of, e.g., the hard-sphere fluid confined between a pair of parallel planar hard walls. Numerical results of the extended WCA approach, against the Monte Carlo (MC) simulations in a range of surface separations and bulk densities, suggest that it is capable of representing the fine features of the hard-sphere density distributions. The WCA results also agree well with the calculations from the fundamental measure theory. In addition, the thermodynamic self-consistency of the WCA approach is confirmed by its fairly good agreement with the MC fitted data for the surface tension of a hard-sphere fluid at a planar hard wall. All these tests show that a pure WCA approach can be constructed to investigate the states of ionic hard-sphere fluids.

Comparison of Molecular Dynamics with Classical Density Functional and Poisson-Boltzmann Theories of the Electric Double Layer in Nanochannels

Comparisons are made among Molecular Dynamics (MD), Classical Density Functional Theory (c-DFT), and Poisson-Boltzmann (PB) modeling of the electric double layer (EDL) for the nonprimitive three component model (3CM) in which the two ion species and solvent molecules are all of finite size. Unlike previous comparisons between c-DFT and Monte Carlo (MC), the present 3CM incorporates Lennard-Jones interactions rather than hard-sphere and hard-wall repulsions. c-DFT and MD results are compared over normalized surface charges ranging from 0.2 to 1.75 and bulk ion concentrations from 10 mM to 1 M. Agreement between the two, assessed by electric surface potential and ion density profiles, is found to be quite good. Wall potentials predicted by PB begin to depart significantly from c-DFT and MD for charge densities exceeding 0.3. Successive layers are observed to charge in a sequential manner such that the solvent becomes fully excluded from each layer before the onset of the next layer. Ultimately, this layer filling phenomenon results in fluid structures, Debye lengths, and electric surface potentials vastly different from the classical PB predictions.

Effects of counterion size on the attraction between similarly charged surfaces

Interaction between similarly charged surfaces can be attractive at high electrostatic coupling constants Ξ = l(B)Z(2)/μ(GC), where l(B) is the Bjerrum length, μ(GC) the Gouy-Chapman length, and Z the valency of counterions. While this effect has been studied previously in detail, as a function of surface charge density and valency of the pointlike counterions, much less is known about the effect of counterion size. We apply the Wang-Landau sampling Monte Carlo (MC) simulation method to compute the free energy F as a function of the scaled distance between the plates D̃=D/μ(GC) for a range of Ξ and scaled counterion radii R̃=R/μ(GC). We find that for large Ξ and small ion radius, there is a global equilibrium distance D̃=D̃(eq)=2(1+R̃), correctly giving the expected value at the point counterion limit. With increasing R̃ the global minimum in F(D̃) changes to a metastable state and finally this minimum vanishes when R̃ reaches a critical value, which depends on Ξ. We present a state diagram indicating approximate boundaries between these three regimes. The Wang-Landau MC method, as it is applied here, offers a possibility to study a wide spectrum of extended problems, which cannot be treated by the use of contact value theorem.

Structure of spherical electric double layers containing mixed electrolytes: a systematic study by Monte Carlo simulations and density functional theory

The structure of spherical electric double layers in the presence of mixed electrolytes is studied using Monte Carlo simulation and density functional theory within the restricted primitive model. The macroion is modeled as an impenetrable charged hard sphere carrying a uniform surface charge density, surrounded by the small ions represented as charged hard spheres, and the solvent is taken as a dielectric continuum. The density functional theory uses a partially perturbative scheme, where the hard-sphere contribution to the one-particle correlation function is evaluated using weighted density approximation and the ionic interactions are calculated using a second-order functional Taylor expansion with respect to a bulk electrolyte. The Monte Carlo simulations have been performed in canonical ensemble. The system is studied at varying ionic concentrations, at different concentration ratios of mono- and multivalent counterions of mixed electrolytes, at different diameters of hard spheres, at different macroion radius, and at varying polyion surface charge densities. The theoretical predictions in terms of the density profiles and the mean electrostatic potential profiles are found to be in good agreement with the simulation results. This model study shows clear manipulations of ionic size and charge correlations in dictating a number of interesting phenomena relating to width of the diffuse layer and charge inversion under different parametric conditions.

Molecular solvent model of spherical electric double layers: a systematic study by Monte Carlo simulations and density functional theory

The structure of spherical electric double layers is studied using Monte Carlo simulation and density functional theory by considering solvent as the third component. In this molecular solvent model (MSM), ions and solvent molecules are considered as charged and neutral hard spheres, respectively, having equal diameter. The macroion is considered as an isolated hard sphere having uniform surface charge density surrounded by the electrolyte and the solvent. The theory is partially perturbative as the hard-sphere contribution to the one particle correlation function is evaluated using suitably averaged weighted density, and the ionic part is obtained through a second-order functional Taylor expansion around the bulk electrolyte. The Monte Carlo simulations have been performed in a canonical ensemble. The system is studied at varying concentrations of electrolytes, and the solvent molecules, at different valences of the electrolyte, at different macroion radii, and at varying surface charge densities. The theory is found to be in good agreement with the simulation results over a wide range of parametric conditions. The excluded volume effects due to the molecular nature of the solvent are shown to have much richer features in diffuse layer phenomena like layering and charge inversion.

Simulational and theoretical study of the spherical electrical double layer for a size-asymmetric electrolyte: the case of big coions

Monte Carlo simulations of a spherical macroion, surrounded by a size-asymmetric electrolyte in the primitive model, were performed. We considered 1:1 and 2:2 salts with a size ratio of 2 (i.e., with coions twice the size of counterions), for several surface charge densities of the macrosphere. The radial distribution functions, electrostatic potential in all the space and at the Helmholtz surfaces, and integrated charge are reported. We compare these simulational data with original results obtained from the Ornstein-Zernike integral equation, supplemented by the hypernetted chain-hypernetted chain (HNC-HNC) and hypernetted chain-mean spherical approximation (HNC-MSA) closures, and with the corresponding calculations using the modified Gouy-Chapman and unequal-radius modified Gouy-Chapman theories. The HNC-HNC and HNC-MSA integral equations formalisms show good concordance with Monte Carlo "experiments," whereas the notable limitations of point-ion approaches are evidenced. Most importantly, the simulations confirm our previous theoretical predictions of the nondominance of the counterions in the size-asymmetric spherical electrical double layer [J. Chem. Phys. 123, 034703 (2005)], the appearance of anomalous curvatures at the outer Helmholtz plane, and the enhancement of the charge reversal and screening at high colloidal surface charge densities due to the ionic size asymmetry.

Density functional theory for planar electric double layers: closing the gap between simple and polyelectrolytes

We report a nonlocal density functional theory (NLDFT) for polyelectrolyte solutions within the primitive model; i.e., the solvent is represented by a continuous dielectric medium, and the small ions and polyions by single and tangentially connected charged hard spheres, respectively. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for hard-sphere repulsion, an extended first-order thermodynamic perturbation theory for chain connectivity, and a quadratic functional Taylor expansion for electrostatic correlations. With the direct and cavity correlation functions of the corresponding monomeric systems as inputs, the NLDFT predicts the segment-level microscopic structures and adsorption isotherms of polyelectrolytes at oppositely charged surfaces in good agreement with molecular simulations. In particular, it faithfully reproduces the layering structures of polyions, charge inversion, and overcharging that cannot be captured by alternative methods including the polyelectrolyte Poisson-Boltzmann equation and an earlier version of DFT. The NLDFT has also been used to investigate the influences of the small ion valence, polyion chain length, and size disparity between polyion segments and counterions on the microscopic structure, mean electrostatic potential, and overcharging in planar electric double layers containing polyelectrolytes.

Structure of spherical electric double layers: A density functional approach

A density functional theory is presented for the structure of spherical electric double layers within the restricted primitive model, where the macroion is considered as a hard sphere having uniform surface charge density, the small ions as charged hard spheres, and the solvent is taken as a dielectric continuum. The theory is partially perturbative as the hard-sphere contribution to the one-particle correlation function is evaluated using suitably averaged weighted density and the ionic part is obtained through a second-order functional Taylor expansion around the uniform fluid. The theory is in quantitative agreement with Monte Carlo simulation for the density profiles and the zeta potentials over a wide range of macroion sizes and electrolyte concentrations. The theory is able to provide interesting insights about the layering and the charge inversion phenomena occurring at the interface.

Simple correlation-corrected theory of systems described by screened coulomb interactions

We present a simple correlation-corrected density functional treatment of dispersions containing macroions, where these are assumed to interact via screened Coulomb potentials, as given by Debye-Hückel theory. A straightforward mean-field description even fails to qualitatively capture important correlation effects displayed by such systems. However, if an effective, correlation-corrected potential is adopted at short range, then the predictions are in qualitative and semiquantitative agreement with simulated results. The correlation corrections are evaluated in a manner that is completely analogous to those recently presented in correlation-corrected Poisson-Boltzmann theory (Forsman, J. J. Phys. Chem. B 2004, 108, 9236). The accuracy of the theory is evaluated by comparison with simulation data on systems displaying correlation-generated packing effects and stratification forces.

Effect of attractions on the structure of polymer solutions confined between surfaces: A density functional approach

A density functional theory is presented to study the effect of attractions on the structure of polymer solutions confined between surfaces. The polymer molecules have been modeled as a pearl necklace of freely jointed hard spheres and the solvent as hard spheres, both having Yukawa-type attractions and the mixture being confined between attractive Yukawa-type surfaces. The present theory treats the ideal gas free energy functional exactly and uses weighted density approximation for the hard chain and hard sphere contributions to the excess free energy functional. The attractive interactions are calculated using the direct correlation function obtained from the polymer reference interaction site model theory along with the mean spherical approximation closure. The theoretical predictions on the density profiles of the polymer and the solvent molecules are found to agree quite well with the Monte Carlo simulation results for varying densities, chain lengths, wall separations, and different sets of interaction potentials.

Density-functional theory and Monte Carlo simulation study on the electric double layer around DNA in mixed-size counterion systems

A density-functional approach and canonical Monte Carlo simulations are presented for describing the ionic microscopic structure around the DNA molecule immersed in mixed-size counterion solutions. In the density-functional approach, the hard-sphere contribution to the Helmholtz energy functional is obtained from the modified fundamental measure theory [Y.-X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)], and the electrostatic contribution is evaluated through a quadratic functional Taylor expansion. The new theory is suitable to the systems containing ions of arbitrary sizes and valences. In the established canonical Monte Carlo simulation, an iterative self-consistent method is used to evaluate the long-range energy, and another iterative algorithm is adopted to obtain desired bulk ionic concentrations. The ion distributions from the density-functional theory (DFT) are in good agreement with those from the corresponding Monte Carlo (MC) simulations. It is found that the ratio of the bulk concentrations of two species of counterions (cations) makes significant contribution to the ion distributions in the vicinity of DNA. Comparisons with the electrostatic potential profiles from the MC simulations show that the accuracy of the DFT becomes low when a small divalent cation exists. Both the DFT and MC simulation results illustrate that the electrostatic potential at the surface of DNA increases as the anion diameter or the total cation concentration is increased and decreases as the diameter of one cation species is increased. The calculation of electrostatic potential using real ion diameters shows that the accuracy of DFT predictions for divalent ions is also acceptable.

Structures and adsorption of binary hard-core Yukawa mixtures in a slitlike pore: Grand canonical Monte Carlo simulation and density-functional study

The grand canonical ensemble Monte Carlo simulation and density-functional theory are applied to calculate the structures, local mole fractions, and adsorption isotherms of binary hard-core Yukawa mixtures in a slitlike pore as well as the radial distribution functions of bulk mixtures. The excess Helmholtz energy functional is a combination of the modified fundamental measure theory of Yu and Wu [J. Chem. Phys. 117, 10156 (2002)] for the hard-core contribution and a corrected mean-field theory for the attractive contribution. A comparison of the theoretical results with the results from the Monte Carlo simulations shows that the corrected theory improves the density profiles of binary hard-core Yukawa mixtures in the vicinity of contact over the original mean-field theory. Both the present corrected theory and the simulations suggest that depletion and desorption occur at low temperature, and the local segregation can be observed in most cases. For binary mixtures in the hard slitlike pore, the present corrected theory predicts more accurate surface excesses than the original one does, while in the case of the attractive pore, no improvement is found in the prediction of a surface excess of the smaller molecule.

The electrical double layer for a fully asymmetric electrolyte around a spherical colloid: An integral equation study

The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation for a totally asymmetric primitive model electrolyte around a spherical macroparticle is obtained and solved numerically in the case of size-asymmetric systems. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular-dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential versus charge relationship, radial distribution functions, mean electrostatic potential, and cumulative reduced charge for representative examples of 1:1 and 2:2 salts with a size-asymmetry ratio of 2. Our results are collated with those of the modified Gouy-Chapman (MGC) and unequal radius modified Gouy-Chapman (URMGC) theories and with those of HNC/MSA in the restricted primitive model (RPM) to assess the importance of size-asymmetry effects. One of the most striking characteristics found is that, contrary to the general belief, away from the point of zero charge the properties of an asymmetric electrical double layer (EDL) are not those corresponding to a symmetric electrolyte with the size and charge of the counterion, i.e., counterions do not always dominate. This behavior suggests the existence of a new phenomenology in the EDL that genuinely belongs to a more realistic size-asymmetric model where steric correlations are taken into account consistently. Such novel features cannot be described by traditional mean-field theories such as MGC, URMGC, or even by enhanced formalisms, such as HNC/MSA, if they are based on the RPM.

Effect of attractive interactions on the structure of polymer melts confined between surfaces: A density-functional approach

A density-functional theory is presented to study the structure of polymers, having attractive interactions, confined between attractive surfaces. The theory treats the ideal-gas free-energy functional exactly and uses weighted density approximation for the hard-chain contribution to the excess free-energy functional. The bulk interactions of freely jointed hard spheres are obtained from generalized Flory equation of state and the attractive interactions are calculated using the direct correlation function obtained from the polymer reference interaction site model theory along with the mean spherical approximation closure. The theoretical predictions are found to be in quite good agreement with the Monte Carlo simulation results for varying densities, chain lengths, and different interaction potentials. The results confirm important implications of using different approximations for the hard-sphere and attractive interactions.

Density functional theory of adsorption of mixtures of charged chain particles and spherical counterions

We propose a microscopic density functional theory to describe nonuniform ionic fluids composed of chain molecules with charged "heads" and spherical counterions. The chain molecules are modeled as freely jointed chains of hard spheres, the counterions are oppositely charged spheres of the same diameter as all segments of chain molecules. The theory is based on the approach of Yu and Wu [J. Chem. Phys. 117, 2368 (2002)] of adsorption of chain molecules and on theory of adsorption of electrolytes [O. Pizio, A. Patrykiejew, and S. Sokolowski, J. Chem. Phys. 121, 11957 (2004)]. As an application of the proposed formalism we investigate the structure and adsorption of fluids containing segments of different length in a slitlike pore.

Structure of inhomogeneous polymer solutions: A density functional approach

The structure of polymer solutions confined between surfaces is studied using a density functional theory where the polymer molecules have been modeled as a pearl necklace of freely jointed hard spheres and the solvent as hard spheres. The present theory uses the concept of universality of the free energy density functional to obtain the first-order direct correlation function of the nonuniform system from that of the corresponding uniform system, calculated through the Verlet-modified type bridge function. The uniform bulk fluid direct correlation function required as input has been calculated from the reference interaction site model integral equation theory using the Percus-Yevick closure relation. The calculated results on the density profiles of the polymer as well as the solvent are shown to compare well with computer simulation results.

Density functional study on the structures and thermodynamic properties of small ions around polyanionic DNA

A density functional theory (DFT) is presented for describing the distributions of small ions around an isolated infinitely long polyanionic DNA molecule in the framework of the restricted primitive model. The hard-sphere contribution to the excess Helmholtz energy functional is derived from the modified fundamental measure theory, and the electrostatic interaction is evaluated through a quadratic functional Taylor expansion. The predictions from the DFT are compared with integral equation theory (IET), the nonlinear Poisson-Boltzmann (PB) equation, and computer simulation data for the ionic density profiles, electrostatic potentials, and charge compensation functions at varieties of solution conditions. Good agreement between the DFT and computer simulations is achieved. The charge inversion phenomena of DNA are observed in a moderately concentrated solution of 2:1 and 2:2 electrolytes using the DFT, IET, and computer simulation, but can never be predicted from the PB equation. The predictions of charge inversion from the DFT prove to be more accurate than those from the IET when compared with computer simulation data. The preferential interaction coefficients from the DFT are also compared with those from the PB equation and Monte Carlo simulation, and it is shown that the DFT is superior to the PB equation.

Density-functional theory of spherical electric double layers and ζ potentials of colloidal particles in restricted-primitive-model electrolyte solutions

A density-functional theory is proposed to describe the density profiles of small ions around an isolated colloidal particle in the framework of the restricted primitive model where the small ions have uniform size and the solvent is represented by a dielectric continuum. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-sphere repulsion and a quadratic functional Taylor expansion for the electrostatic interactions. The theoretical predictions are in good agreement with the results from Monte Carlo simulations and from previous investigations using integral-equation theory for the ionic density profiles and the zeta potentials of spherical particles at a variety of solution conditions. Like the integral-equation approaches, the density-functional theory is able to capture the oscillatory density profiles of small ions and the charge inversion (overcharging) phenomena for particles with elevated charge density. In particular, our density-functional theory predicts the formation of a second counterion layer near the surface of highly charged spherical particle. Conversely, the nonlinear Poisson-Boltzmann theory and its variations are unable to represent the oscillatory behavior of small ion distributions and charge inversion. Finally, our density-functional theory predicts charge inversion even in a 1:1 electrolyte solution as long as the salt concentration is sufficiently high.