Strong-coupling electrostatics in the presence of dielectric inhomogeneities

Electrostatic interactions between charge regulated spherical macroions

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte-thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

Charge fluctuations in charge-regulated systems: dependence on statistical ensemble

We investigate charge regulation of nanoparticles in concentrated suspensions, focusing on the effect of different statistical ensembles. We find that the choice of ensemble does not affect the mean charge of nanoparticles, but significantly alters the magnitude of its fluctuation. Specifically, we compared the behaviors of colloidal charge fluctuations in the semi-grand canonical and canonical ensembles and identified significant differences between the two. The choice of ensemble-whether the system is isolated or is in contact with a reservoir of acid and salt-will, therefore, affect the Kirkwood-Shumaker fluctuation-induced force inside concentrated suspensions. Our results emphasize the importance of selecting an appropriate ensemble that accurately reflects the experimental conditions when studying fluctuation-induced forces between polyelectrolytes, proteins, and colloidal particles in concentrated suspensions.

Impact of the inner solute structure on the electrostatic mean-field and strong-coupling regimes of macromolecular interactions

The structural diversity of the solute molecules involved in biomolecular processes necessitates the characterization of the forces between charged macromolecules beyond the point-ion description. From the field-theoretic partition function of an electrolyte confined between two anionic membranes, we derive a contact-value identity valid for general intramolecular solute structure and electrostatic coupling strength. In the electrostatic mean-field regime, the inner charge spread of the solute particles is shown to induce the twofold enhancement of the short-range Poisson-Boltzmann level membrane repulsion and a longer-range depletion attraction. Our contact theorem indicates that the twofold repulsion enhancement by solute size is equally present in the opposite strong-coupling regime of linear and spherical solute molecules. Upon the inclusion of the dielectric contrast between the electrolyte and the interacting membranes, the emerging polarization forces substantially amplify the solute specificity of the macromolecular interactions. Namely, the finite size of the dumbbell-like solute particles composed of similar terminal charges weakens the intermembrane repulsion. However, the extended structure of the solute molecules carrying opposite elementary charges such as ionized atoms and zwitterionic molecules enhances the membrane repulsion by several factors. We also show that these polarization forces can extend the range of the solute structure effects up to intermembrane distances exceeding the solute size by an order of magnitude. This radical alteration of the intermembrane interactions by the salt structure identifies the solute specificity as a key ingredient of the thermodynamic stability in colloidal systems.

Multivalent Ion-Mediated Attraction between Like-Charged Colloidal Particles: Nonmonotonic Dependence on the Particle Charge

Ion-mediated effective interactions are important for the structure and stability of charged particles such as colloids and nucleic acids. It has been known that the intrinsic electrostatic repulsion between like-charged particles can be modulated into effective attraction by multivalent ions. In this work, we examined the dependence of multivalent ion-mediated attraction between like-charged colloidal particles on the particle charge in a wide range by extensive Monte Carlo simulations. Our calculations show that for both divalent and trivalent salts, the effective attraction between like-charged colloidal particles becomes stronger with the increase of the particle charge, whereas it gradually becomes weakened when the particle charge exceeds a "critical" value. Correspondingly, as the particle charge is increased, the driving force for such effective attraction transits from an attractive electrostatic force to an attractive depletion force, and the attraction weakening by high particle charges is attributed to the transition of electrostatic force from attraction to repulsion. Our analyses suggest that the attractive depletion force and the repulsive electrostatic force at high particle charges result from the Coulomb depletion which suppresses the counterion condensation in the limited region between two like-charged colloidal particles. Moreover, our extensive calculations indicate that the "critical" particle charge decreases apparently for larger ions and smaller colloidal particles due to stronger Coulomb depletion and decreases slightly at higher salt concentrations due to the slightly enhanced Coulomb depletion in the intervening space between colloidal particles. Encouragingly, we derived an analytical formula for the "critical" particle charge based on the Lindemann melting law.

One-dimensional colloidal model with dielectric inhomogeneity

We consider a one-dimensional model allowing analytical derivation of the effective interactions between two charged colloids. We evaluate exactly the partition function for an electroneutral salt-free suspension with dielectric jumps at the colloids' position. We derive a contact relation with the pressure that shows there is like-charge attraction, whether or not the counterions are confined between the colloids. In contrast to the homogeneous dielectric case, there is the possibility for the colloids to attract despite the number of counterions (N) being even. The results are shown to recover the mean-field prediction in the limit N→∞.

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.

Consequences of Convex Nanopore Chemistry on Confined Water Dynamics

A fundamental understanding of confined water is crucial for developing selective ion transport and water purification membranes, yet the roles of nanopore geometry and functionality on confined water dynamics remain unresolved. We report the synthesis of perdeuterated ionic alkylsulfonate amphiphiles and their water-induced self-assembly into lyotropic liquid crystal (LLC) mesophases with well-defined, convex, sulfonate-lined nanopores. Quasielastic neutron scattering (QENS) measurements demonstrate that the water self-diffusion coefficients within these sulfonate-lined convex nanopores depend on the hydration level and amphiphile counterion identity (H+, K+, NMe4+). The consistency of the observed counterion-dependent water dynamics trends with those of carboxylate LLCs is rationalized on the basis of similarities in the counterion spatial distributions in the water-filled channels, which we deduce from electron density maps derived from small-angle X-ray scattering (SAXS) analyses. These findings indicate that water diffusion is systematically faster in sulfonate-lined nanopores as compared to carboxylate-lined pores due to weaker water interactions with the softer and more hydrophobic-SO3- functionalities. These molecular-level insights into the relationships between convex pore wall chemical functionalities, hydrated counterions, and confined water diffusion may inform future development of new nanoporous media.

Electric double layers with surface charge modulations: Exact Poisson-Boltzmann solutions

Poisson-Boltzmann theory is the cornerstone for soft matter electrostatics. We provide exact analytical solutions to this nonlinear mean-field approach for the diffuse layer of ions in the vicinity of a planar or a cylindrical macroion. While previously known solutions are for homogeneously charged objects, the cases worked out exhibit a modulated surface charge-or equivalently, surface potential-on the macroion (wall) surface. In addition to asymptotic features at large distances from the wall, attention is paid to the fate of the contact theorem, relating the contact density of ions to the local wall charge density. For salt-free systems (counterions only), we make use of results pertaining to the two-dimensional Liouville equation, supplemented by an inverse approach. When salt is present, we invoke the exact two-soliton solution to the 2D sinh-Gordon equation. This leads to inhomogeneous charge patterns, that are either localized or periodic in space. Without salt, the electrostatic signature of a charge pattern on the macroion fades exponentially with distance for a planar macroion, while it decays as an inverse power law for a cylindrical macroion. With salt, our study is limited to the planar geometry and reveals that pattern screening is exponential.

Ion-Specific Confined Water Dynamics in Convex Nanopores of Gemini Surfactant Lyotropic Liquid Crystals

The impact of pore geometry and functionality on the dynamics of water nanoconfined in porous media are the subject of some debate. We report the synthesis and small-angle X-ray scattering (SAXS) characterization of a series of perdeuterated gemini surfactant lyotropic liquid crystals (LLCs), in which convex, water-filled nanopores of well-defined dimensions are lined with carboxylate functionalities. Quasielastic neutron scattering (QENS) measurements of the translational water dynamics in these dicarboxylate LLC nanopores as functions of the surfactant hydration state and the charge compensating counterion (Na⁺, K⁺, NMe₄⁺) reveal that the measured dynamics depend primarily on surfactant hydration, with an unexpected counterion dependence that varies with hydration number. We rationalize these trends in terms of a balance between counterion-water attractions and the nanopore volume excluded by the counterions. On the basis of electron density maps derived from SAXS analyses of these LLCs, we directly show that the volume excluded by the counterions depends on both their size and spatial distribution in the water-filled channels. The translational water dynamics in the convex pores of these LLCs are also slower than those reported in the concave pores of AOT reverse micelles, implying that water dynamics also depend on the nanopore curvature.

A coupled effect of dehydration and electrostatic interactions on selective ion transport through charged nanochannels

Selective ion transport is an essential feature of biological ion channels. Due to the subnanometer size and negatively charged surface of ion channels, the ion selectivity is affected by both dehydration effects and electrostatic interactions. Their coupled effect on selective ion transport, however, has been elusive. Here, using molecular dynamics simulations, we study ion (Li⁺ and Mg²⁺) transport through subnanometer carbon nanotubes (CNTs) with varying charge densities. Our results indicate that the dehydration effect governs the ionic transport at low surface charge densities, hence the nanochannel shows a selectivity for Li⁺ ions. In contrast, the nanochannel switches to a selectivity for Mg²⁺ ions as the electrostatic interaction between the cations and the negatively charged wall dominates the transport at high surface charge densities.

Charged nanorods at heterogeneously charged surfaces

We study the spatial and orientational distribution of charged nanorods (rodlike counterions) as well as the effective interaction mediated by them between two plane-parallel surfaces that carry fixed (quenched) heterogeneous charge distributions. The nanorods are assumed to have an internal charge distribution, specified by a multivalent monopolar moment and a finite quadrupolar moment, and the quenched surface charge is assumed to be randomly distributed with equal mean and variance on the two surfaces. While equally charged surfaces are known to repel within the traditional mean-field theories, the presence of multivalent counterions has been shown to cause attractive interactions between uniformly charged surfaces due to the prevalence of strong electrostatic couplings that grow rapidly with the counterion valency. We show that the combined effects due to electrostatic correlations (caused by the coupling between the mean surface field and the multivalent, monopolar, charge valency of counterions) as well as the disorder-induced interactions (caused by the coupling between the surface disorder field and the quadrupolar moment of counterions) lead to much stronger attractive interactions between two randomly charged surfaces. The interaction profile turns out to be a nonmonotonic function of the intersurface separation, displaying an attractive minimum at relatively small separations, where the ensuing attraction can exceed the maximum strong-coupling attraction (produced by multivalent monopolar counterions between uniformly charged surfaces) by more than an order of magnitude.

Strong-coupling theory of counterions between symmetrically charged walls: from crystal to fluid phases

We study thermal equilibrium of classical pointlike counterions confined between symmetrically charged walls at distance d. At very large couplings when the counterion system is in its crystal phase, a harmonic expansion of particle deviations is made around the bilayer positions, with a free lattice parameter determined from a variational approach. For each of the two walls, the harmonic expansion implies an effective one-body potential at the root of all observables of interest in our Wigner strong-coupling expansion. Analytical results for the particle density profile and the pressure are in good agreement with numerical Monte Carlo data, for small as well as intermediate values of d comparable with the Wigner lattice spacing. While the strong-coupling theory is extended to the fluid regime by using the concept of a correlation hole, the Wigner calculations appear trustworthy for all electrostatic couplings investigated. Our results significantly extend the range of accuracy of analytical equations of state for strongly interacting charged planar interfaces.

Counterion-Dependent Access to Low-Symmetry Lyotropic Sphere Packings of Ionic Surfactant Micelles

The water-driven self-assembly of homologous dianionic surfactants into lyotropic liquid crystals (LLCs) is investigated, with a focus on understanding how surfactant headgroup and counterion identities guide supramolecular spherical mesophase selection. Using temperature-dependent small-angle X-ray scattering (SAXS), we demonstrate that 2-alkylmalonate surfactants (C_nMal-M₂) with n = 8 (octyl) or 10 (decyl) and M = K⁺, Cs⁺, or (CH₃)₄N⁺ form both simple and complex micelle packings. Observed spherical morphologies include body-centered cubic (BCC), hexagonally closest-packed (HCP), and tetrahedrally closest-packed Frank-Kasper (FK) A15 and σ phases (Pm3(-)n and P4₂/mnm symmetries, respectively). Previously observed in only one other minimally hydrated surfactant, the σ phase is a rare LLC morphology comprising a low-symmetry unit cell containing 30 sub-2-nm quasispherical micelles, each of which belongs to one of five symmetry-equivalent classes with discrete aggregation numbers. Temperature versus water concentration phase maps for C_nMal-M₂ LLCs reveal that σ-phase formation depends sensitively on the size and polarizability of the surfactant counterion and the length of the surfactant alkyl tail. These observations are rationalized in terms of a delicate interplay between global packing symmetry and local particle symmetry, and the extent to which counterion-headgroup correlations enforce the latter structures in these LLC phases.

Interactions between charged particles with bathing multivalent counterions: experiments vs. dressed ion theory

We compare the recent experimentally measured forces between charged colloidal particles, as well as their effective surface potentials (surface charge) in the presence of multivalent counterions in a bathing monovalent salt solution, with the predictions of the dressed ion theory of strongly charged colloidal systems. The benchmark for comparison is provided by the DLVO theory and the deviations from its predictions at small separations are taken as an indication of the additional non-DLVO attractions that can be fitted by an additional phenomenological exponential term. The parameters characterizing this non-DLVO exponential term as well as the dependencies of the effective potential on the counterion concentration and valency predicted by the dressed ion theory are well within the experimental values. This suggests that the deviations from the DLVO theory are probably caused by ion correlations as formalized within the dressed ion theory.

Ion Structure Near a Core-Shell Dielectric Nanoparticle

A generalized image charge formulation is proposed for the Green's function of a core-shell dielectric nanoparticle for which theoretical and simulation investigations are rarely reported due to the difficulty of resolving the dielectric heterogeneity. Based on the formulation, an efficient and accurate algorithm is developed for calculating electrostatic polarization charges of mobile ions, allowing us to study related physical systems using the Monte Carlo algorithm. The computer simulations show that a fine-tuning of the shell thickness or the ion-interface correlation strength can greatly alter electric double-layer structures and capacitances, owing to the complicated interplay between dielectric boundary effects and ion-interface correlations.

Mean-field beyond mean-field: the single particle view for moderately to strongly coupled charged fluids

In a counter-ion only charged fluid, Coulomb coupling is quantified by a single dimensionless parameter. Yet, the theoretical treatment of moderately to strongly coupled charged fluids is a difficult task, central to the understanding of a wealth of soft matter problems, including biological systems. We show that the corresponding coupling regime can be remarkably well described by a single particle treatment, which, at variance with previous works, takes due account of inter-ionic interactions. To this end, the prototypical problem of a planar charged dielectric interface is worked out. Testing our predictions against Monte Carlo simulation data reveals an excellent agreement.

Bundling in semiflexible polymers: A theoretical overview

Supramolecular assemblies of polymers are key modules to sustain the structure of cells and their function. The main elements of these assemblies are charged semiflexible polymers (polyelectrolytes) generally interacting via a long(er)-range repulsion and a short(er)-range attraction. The most common supramolecular structure formed by these polymers is the bundle. In the present paper, we critically review some recent theoretical and computational advances on the problem of bundle formation, and point a few promising directions for future work.

Ion-mediated interactions between net-neutral slabs: Weak and strong disorder effects

We investigate the effective interaction between two randomly charged but otherwise net-neutral, planar dielectric slabs immersed in an asymmetric Coulomb fluid containing a mixture of mobile monovalent and multivalent ions. The presence of charge disorder on the apposed bounding surfaces of the slabs leads to substantial qualitative changes in the way they interact, as compared with the standard picture provided by the van der Waals and image-induced, ion-depletion interactions. While, the latter predict purely attractive interactions between strictly neutral slabs, we show that the combined effects from surface charge disorder, image depletion, Debye (or salt) screening, and also, in particular, their coupling with multivalent ions, give rise to a more diverse behavior for the effective interaction between net-neutral slabs at nano-scale separations. Disorder effects show large variation depending on the properly quantified strength of disorder, leading either to non-monotonic effective interaction with both repulsive and attractive branches when the surface charges are weakly disordered (small disorder variance) or to a dominating attractive interaction that is larger both in its range and magnitude than what is predicted from the van der Waals and image-induced, ion-depletion interactions, when the surfaces are strongly disordered (large disorder variance).

Electrolytes between dielectric charged surfaces: Simulations and theory

We present a simulation method to study electrolyte solutions in a dielectric slab geometry using a modified 3D Ewald summation. The method is fast and easy to implement, allowing us to rapidly resum an infinite series of image charges. In the weak coupling limit, we also develop a mean-field theory which allows us to predict the ionic distribution between the dielectric charged plates. The agreement between both approaches, theoretical and simulational, is very good, validating both methods. Examples of ionic density profiles in the strong electrostatic coupling limit are also presented. Finally, we explore the confinement of charge asymmetric electrolytes between neutral surfaces.

Ion specific effects: decoupling ion-ion and ion-water interactions

Ion-specific effects in aqueous solution, known as the Hofmeister effect, are prevalent in diverse systems ranging from pure ionic to complex protein solutions. The objective of this paper is to explicitly demonstrate how complex ion-ion and ion-water interactions manifest themselves in the Hofmeister effect based on a series of recent experimental observations. These effects are not considered in the classical descriptions of ion effects, such as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, and therefore they fail to describe the origin of the phenomenological Hofmeister effect. However, given that models considering the basic forces of electrostatic and van der Waals interactions can offer rationalization for the core experimental observations, a universal interaction model stands a chance of being developed. In this perspective, we separately derive the contribution from ion-ion electrostatic interactions and ion-water interactions from second harmonic generation (SHG) data at the air-ion solution interface, which yields an estimate of the ion-water interactions in solution. The Hofmeister ion effect observed for biological solutes in solution should be similarly influenced by contributions from ion-ion and ion-water interactions, where the same ion-water interaction parameters derived from SHG data at the air-ion solution interface could be applicable. A key experimental data set available from solution systems to probe ion-water interactions is the modulation of water diffusion dynamics near ions in a bulk ion solution, as well as near biological liposome surfaces. This is obtained from Overhauser dynamic nuclear polarization (ODNP), a nuclear magnetic resonance (NMR) relaxometry technique. The surface water diffusivity is influenced by the contribution from ion-water interactions, both from localized surface charges and adsorbed ions, although the relative contribution of the former is larger on liposome surfaces. In this perspective, ion-water interaction energy values derived from experimental data for various ions are compared with theoretical values in the literature. Ultimately, quantifying ion-induced changes in the surface energy for the purpose of developing valid theoretical models for ion-water interactions will be critical to rationalizing the Hofmeister effect.

On the theoretical description of weakly charged surfaces

It is widely accepted that the Poisson-Boltzmann (PB) theory provides a valid description for charged surfaces in the so-called weak coupling limit. Here, we show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. In the presence of dielectric discontinuity, there is no limiting condition for which the PB theory is valid.

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Effects of image charges on double layer structure and forces

The study of the electrical double layer lies at the heart of soft matter physics and biophysics. Here, we address the effects of the image charges on the double layer structure and forces. For electrolyte solutions between two neutral plates, we show that depletion of the salt ions by the image charge repulsion results in short-range attractive and long-range repulsive forces. If cations and anions are of different valency, the asymmetric depletion leads to the formation of an induced electrical double layer. In comparison to a 1:1 electrolyte solution, both the attractive and the repulsive parts of the interaction are stronger for the 2:1 electrolyte solution. For two charged plates, the competition between the surface charge and the image charge effect can give rise to like-charge attraction and charge inversion. These results are in stark contrast with predictions from the Poisson-Boltzmann theory.

Repulsion between oppositely charged planar macroions

The repulsive interaction between oppositely charged macroions is investigated using Grand Canonical Monte Carlo simulations of an unrestricted primitive model, including the effect of inhomogeneous surface charge and its density, the depth of surface charge, the cation size, and the dielectric permittivity of solvent and macroions, and their contrast. The origin of the repulsion is a combination of osmotic pressure and ionic screening resulting from excess salt between the macroions. The excess charge over-reduces the electrostatic attraction between macroions and raises the entropic repulsion. The magnitude of the repulsion increases when the dielectric constant of the solvent is lowered (below that of water) and/or the surface charge density is increased, in good agreement with experiment. Smaller size of surface charge and the cation, their discreteness and mobility are other factors that enhance the repulsion and charge inversion phenomenons.

Attraction between neutral dielectrics mediated by multivalent ions in an asymmetric ionic fluid

We study the interaction between two neutral plane-parallel dielectric bodies in the presence of a highly asymmetric ionic fluid, containing multivalent as well as monovalent (salt) ions. Image charge interactions, due to dielectric discontinuities at the boundaries, as well as effects from ion confinement in the slit region between the surfaces are taken fully into account, leading to image-generated depletion attraction, ion correlation attraction, and steric-like repulsive interactions. We investigate these effects by employing a combination of Monte Carlo simulation methods, including explicit-ion simulations (where all electrostatic interactions are simulated explicitly) and implicit-ion simulations (where monovalent ions are replaced by an effective screened electrostatic potential between multivalent ions), as well as an approximate analytical theory. The latter incorporates strong ion-image charge correlations, which develop in the presence of high valency ions in the mixture. We show that the implicit-ion simulations and the proposed analytical theory can describe the explicit simulation results on a qualitative level, while excellent quantitative agreement can be obtained for sufficiently large monovalent salt concentrations. The resultant attractive interaction between the neutral surfaces is shown to be significant, as compared with the usual van der Waals interactions between semi-infinite dielectrics, and can thus play an important role at the nano scale.

Electrostatic interactions mediated by polarizable counterions: weak and strong coupling limits

We investigate the statistical mechanics of an inhomogeneous Coulomb fluid composed of charged particles with static polarizability. We derive the weak- and the strong-coupling approximations and evaluate the partition function in a planar dielectric slab geometry with charged boundaries. We investigate the density profiles and the disjoining pressure for both approximations. Comparison to the case of non-polarizable counterions shows that polarizability brings important differences in the counterion density distribution as well as the counterion mediated electrostatic interactions between charged dielectric interfaces.

Tuning ion correlations at an electrified soft interface

Ion distributions play a central role in various settings-from biology, where they mediate the electrostatic interactions between charged biomolecules in solution, to energy storage devices, where they influence the charging properties of supercapacitors. These distributions are determined by interactions dictated by the chemical properties of the ions and their environment as well as the long-range nature of the electrostatic force. Recent theoretical and computational studies have explored the role of correlations between ions, which have been suggested to underlie a number of counterintuitive results, such as like-charge attraction. However, the interdependency between ion correlations and other interactions that ions experience in solution complicates the connection between physical models of ion correlations and the experimental investigation of ion distributions. We exploit the properties of the liquid/liquid interface to vary the coupling strength of ion-ion correlations from weak to strong while monitoring their influence on ion distributions at the nanometer scale with X-ray reflectivity and the macroscopic scale with interfacial tension measurements. These data are in agreement with the predictions of a parameter-free density functional theory that includes ion-ion correlations and ion-solvent interactions over the entire range of experimentally tunable correlation coupling strengths (from 0.8 to 3.7). This study provides evidence for a sharply defined electrical double layer for large coupling strengths in contrast to the diffuse distributions predicted by mean field theory, thereby confirming a common prediction of many ion correlation models. The reported findings represent a significant advance in elucidating the nature and role of ion correlations in charged soft matter.

Biomolecular electrostatics and solvation: a computational perspective

An understanding of molecular interactions is essential for insight into biological systems at the molecular scale. Among the various components of molecular interactions, electrostatics are of special importance because of their long-range nature and their influence on polar or charged molecules, including water, aqueous ions, proteins, nucleic acids, carbohydrates, and membrane lipids. In particular, robust models of electrostatic interactions are essential for understanding the solvation properties of biomolecules and the effects of solvation upon biomolecular folding, binding, enzyme catalysis, and dynamics. Electrostatics, therefore, are of central importance to understanding biomolecular structure and modeling interactions within and among biological molecules. This review discusses the solvation of biomolecules with a computational biophysics view toward describing the phenomenon. While our main focus lies on the computational aspect of the models, we provide an overview of the basic elements of biomolecular solvation (e.g. solvent structure, polarization, ion binding, and non-polar behavior) in order to provide a background to understand the different types of solvation models.

Computational evidence of two driving mechanisms for overcharging in an electric double layer near the point of zero charge

We have adopted an ensemble Monte Carlo simulation method to systematically verify two physical driving mechanisms responsible for overcharging which refers to the adsorption of an effective charge onto a like-charged planar surface around the point of zero charge within the primitive model of mixed electrolytes with varying salt concentrations. One is electrostatic in character dominated by dielectric images and the other is purely entropic in origin by ionic size asymmetry effects, of which the former has never been reported both theoretically and experimentally and the latter could be interpreted satisfactorily in terms of available theoretical approaches. The electrostatically driven mechanism is found to critically depend on the ionic sizes while the entropically driven mechanism occurs with almost the same efficiency in a relative wide range of surface charge density. Depending on the delicate interplay between charge and steric correlations, the two distinct driving mechanisms may cooperatively give rise to a more pronounced overcharging process.

Distribution of counterions and interaction between two similarly charged dielectric slabs: roles of charge discreteness and dielectric inhomogeneity

The distribution of counterions and the electrostatic interaction between two similarly charged dielectric slabs is studied in the strong coupling limit. Dielectric inhomogeneities and discreteness of charge on the slabs have been taken into account. It is found that the amount of dielectric constant difference between the slabs and the environment, and the discreteness of charge on the slabs have opposing effects on the equilibrium distribution of the counterions. At small interslab separations, increasing the amount of dielectric constant difference increases the tendency of the counterions toward the middle of the intersurface space between the slabs and the discreteness of charge pushes them to the surfaces of the slabs. In the limit of point charges, independent of the strength of dielectric inhomogeneity, counterions distribute near the surfaces of the slabs. The interaction between the slabs is attractive at low temperatures and its strength increases with the dielectric constant difference. At room temperature, the slabs may completely attract each other, reach to an equilibrium separation, or have two equilibrium separations with a barrier in between, depending on the system parameters.

Macroion solutions in the cell model studied by field theory and Monte Carlo simulations

Aqueous solutions of charged spherical macroions with variable dielectric permittivity and their associated counterions are examined within the cell model using a field theory and Monte Carlo simulations. The field theory is based on separation of fields into short- and long-wavelength terms, which are subjected to different statistical-mechanical treatments. The simulations were performed by using a new, accurate, and fast algorithm for numerical evaluation of the electrostatic polarization interaction. The field theory provides counterion distributions outside a macroion in good agreement with the simulation results over the full range from weak to strong electrostatic coupling. A low-dielectric macroion leads to a displacement of the counterions away from the macroion.

Wigner-crystal formulation of strong-coupling theory for counterions near planar charged interfaces

We present a new analytical approach to the strong electrostatic coupling regime (SC) that can be achieved equivalently at low temperatures, high charges, low dielectric permittivity, etc. Two geometries are analyzed in detail: one charged wall first, and then two parallel walls at small distances that can be likely or oppositely charged. In all cases, only one type of mobile counterions is present, and ensures electroneutrality (salt-free case). The method is based on a systematic expansion around the ground state formed by the two-dimensional Wigner crystal(s) of counterions at the plate(s). The leading SC order stems from a single-particle theory, and coincides with the virial SC approach that has been much studied in the last 10 years. The first correction has the functional form of the virial SC prediction, but the prefactor is different. The present theory is free of divergences and the obtained results, both for symmetrically and asymmetrically charged plates, are in excellent agreement with available data of Monte Carlo simulations under strong and intermediate Coulombic couplings. All results obtained represent relevant improvements over the virial SC estimates. The present SC theory starting from the Wigner crystal and therefore coined Wigner SC, sheds light on anomalous phenomena like the counterion mediated like-charge attraction, and the opposite-charge repulsion.

Simulation of an electrical double layer model with a low dielectric layer between the electrode and the electrolyte

We report Monte Carlo simulation results for double layers of 1:1 and 2:1 electrolytes near an electrode with an inner layer that has a dielectric constant, ε(2), smaller than that of the electrolyte, ε(3). The electrolyte is modeled in the implicit solvent framework (primitive model), while the electrode is a metal electrode in this study (ε(1) → ∞). The charged hard sphere ions are not allowed to enter into the inner layer. We show that the capacitance of the inner layer is C(δ) = ε(0)(ε(2) + ε(3))/2δ, where δ is the thickness of the inner layer. This result is different from that obtained from solutions of the Poisson-Boltzmann equation (ε(0)ε(2)/δ), indicating that interpretation of experimental data with a fitted ε(2) dielectric constant of the inner layer must be done using a different equation. We also show that the properties of the diffuse layer are not independent of the value of ε(2), which is a usual assumption of the Poisson-Boltzmann theory. This is mainly because the repulsive image charges repel both the counterions and the co-ions, while the electrode charge attracts the counterions and repels the co-ions.

Weak and strong coupling theories for polarizable colloids and nanoparticles

A theory is presented which allows us to accurately calculate the density profile of monovalent and multivalent counterions in suspensions of polarizable colloids or nanoparticles. In the case of monovalent ions, we derive a weak-coupling theory that explicitly accounts for the ion-image interaction, leading to a modified Poisson-Boltzmann equation. For suspensions with multivalent counterions, a strong-coupling theory is used to calculate the density profile near the colloidal surface and a Poisson-Boltzmann equation with a renormalized boundary condition to account for the counterion distribution in the far field. All the results are compared with the Monte Carlo simulations, showing an excellent agreement between the theory and the simulations.

Interaction regimes for oppositely charged plates with multivalent counterions

Within a mean-field treatment of the interaction between two oppositely charged plates in a salt-free solution, the distance at which a transition from an attractive to a repulsive regime appears can be computed analytically. The mean-field description, however, breaks down under strong Coulombic couplings, which can be achieved at room temperature with multivalent counterions and highly charged surfaces. Making use of the contact theorem and simple physical arguments, we propose explicit expressions for the equation of state in several situations at short distances. The possibility of Bjerrum pair formation is addressed and is shown to have profound consequences on the interactions. To complete the picture, we consider the large-distance limit, from which schematic phase diagram discriminating attractive from repulsive regions can be proposed.

Dressed counterions: polyvalent and monovalent ions at charged dielectric interfaces

We investigate the ion distribution and overcharging at charged interfaces with dielectric inhomogeneities in the presence of asymmetric electrolytes containing polyvalent and monovalent ions. We formulate an effective "dressed counterion" approach by integrating out the monovalent salt degrees of freedom and show that it agrees with results of explicit Monte Carlo simulations. We then apply the dressed counterion approach within the framework of the generalized strong-coupling theory, valid for polyvalent ions at low concentrations, which enables an analytical description for salt effects as well as dielectric inhomogeneities in the limit of strong Coulomb interactions. Limitations and applicability of this theory are examined by comparing the results with simulations.

Counterions at highly charged interfaces: from one plate to like-charge attraction

We present an analytical approach for similarly and highly charged planar interfaces in the presence of counterions. The procedure is physically transparent and based on an exact low temperature expansion around the ground state formed by the two-dimensional Wigner crystal of counterions. The one plate problem is worked out, together with the two plates situation. Unlike previous approaches, the expansion is free of divergences, and is shown to be in excellent agreement with available data of Monte Carlo simulations under strong Coulombic couplings. In the two plates case, the present results shed light on the like-charge attraction regime.

Counter-ions at charged walls: two-dimensional systems

We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the vicinity of a single uniformly charged line (one single double layer), or between two such lines (interacting double layers). The weak-coupling Poisson-Boltzmann theory, which applies when the coupling constant [Formula: see text] is small, is briefly recapitulated (the coupling constant is defined as [Formula: see text] [Formula: see text] [Formula: see text] e (2) , where [Formula: see text] is the inverse temperature, and e the counter-ion charge). The opposite limit ( [Formula: see text] [Formula: see text] ∞ is treated by using a recent method based on an exact expansion around the ground-state Wigner crystal of counter-ions. These two limiting results are compared at intermediary values of the coupling constant [Formula: see text] = 2[Formula: see text] ([Formula: see text] = 1, 2, 3) , to exact results derived within a 1D lattice representation of 2D Coulomb systems in terms of anti-commuting field variables. The models (density profile, pressure) are solved exactly for any particles numbers N at [Formula: see text] = 2 and up to relatively large finite N at [Formula: see text] = 4 and 6. For the one-line geometry, the decay of the density profile at asymptotic distance from the line undergoes a fundamental change with respect to the mean-field behavior at [Formula: see text] = 6 . The like-charge attraction regime, possible for large [Formula: see text] but precluded at mean-field level, survives for [Formula: see text] = 4 and 6, but disappears at [Formula: see text] = 2 .

Insights from Monte Carlo simulations on charge inversion of planar electric double layers in mixtures of asymmetric electrolytes

Monte Carlo simulations of a planar negatively charged dielectric interface in contact with a mixture of 1:1 and 3:1 electrolytes are carried out using the unrestricted primitive model under more realistic hydrated ion sizes. Two typical surface charge densities are chosen to represent the systems from the weak to strong coupling regimes. Our goal is to determine the dependence of the degree of charge inversion on increasing concentration of both mono- and trivalent salts and to provide a systematic study on this peculiar effect between short-range and electrostatic correlations. The numerical results show that addition of monovalent salt diminishes the condensation of trivalent counterions due to either the favorable solvation energy or the available space constraints. As the concentration of trivalent salt increases, on the other hand, the inclusion of the ionic size and size asymmetry results in a damped oscillatory charge inversion at low enough surface charge and another counterintuitive surface charge amplification. It is proposed that both of the anomalous events in the weak coupling regime are thought to be entropic in origin which is completely different from the electrostatic driven charge inversion in the strong coupling regime. In addition, the electrostatic images arising from the dielectric mismatch lead to a decaying depletion effect on the structure of double layer with growing salt concentration in the case of low charged interface but have no effect at high surface charge values. The microscopic information obtained here points to the need for a more quantitative theoretical treatment in describing the charge inversion phenomenon of real colloidal systems.

Dressed counterions: strong electrostatic coupling in the presence of salt

We reformulate the theory of strong electrostatic coupling in order to describe an asymmetric electrolyte solution of monovalent salt ions and polyvalent counterions using field-theoretical techniques and Monte Carlo simulations. The theory is based on an asymmetric treatment of the different components of the electrolyte solution. The weak coupling Debye-Hückel approach is used in order to describe the monovalent salt ions while a strong coupling approach is used to tackle the polyvalent counterions. This combined weak-strong coupling approach effectively leads to dressed interactions between polyvalent counterions and thus directly affects the correlation attraction mediated by polyvalent counterions between like-charged objects. The general theory is specifically applied to a system composed of two uniformly charged plane-parallel surfaces in the presence of salt and polyvalent counterions. In the strong coupling limit for polyvalent counterions, the comparison with Monte Carlo simulations shows good agreement for large enough values of the electrostatic coupling parameter. We delineate two limiting laws that in fact encompass all the Monte Carlo data.