Weak- and strong-coupling electrostatic interactions between asymmetrically charged planar surfaces

Like-Charge Attraction at the Nanoscale: Ground-State Correlations and Water Destructuring

Like-charge attraction, driven by ionic correlations, challenges our understanding of electrostatics both in soft and hard matter. For two charged planar surfaces confining counterions and water, we prove that, even at relatively low correlation strength, the relevant physics is the ground-state one, oblivious of fluctuations. Based on this, we derive a simple and accurate interaction pressure that fulfills known exact requirements and can be used as an effective potential. We test this equation against implicit-solvent Monte Carlo simulations and against explicit-solvent simulations of cement and several types of clays. We argue that water destructuring under nanometric confinement drastically reduces dielectric screening, enhancing ionic correlations. Our equation of state at reduced permittivity therefore explains the exotic attractive regime reported for these materials, even in the absence of multivalent counterions.

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→∞.

Electroosmosis as a probe for electrostatic correlations

We study the role of ionic correlations on the electroosmotic flow in planar double-slit channels, without salt. We propose an analytical theory, based on recent advances in the understanding of correlated systems. We compare the theory with mean-field results and validate it by means of dissipative particle dynamics simulations. Interestingly, for some surface separations, correlated systems exhibit a larger flow than predicted by mean-field. We conclude that the electroosmotic properties of a charged system can be used, in general, to infer and weight the importance of electrostatic correlations therein.

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.

Poisson-Boltzmann equation with a random field for charged fluids

The classical Poisson-Boltzmann equation (CPBE), which is a mean field theory by averaging the ion fluctuation, has been widely used to study ion distributions in charged fluids. In this study, we derive a modified Poisson-Boltzmann equation with a random field from the field theory and recover the ion fluctuation through a multiplicative noise added in the CPBE. The Poisson-Boltzmann equation with a random field (RFPBE) captures the effect of the ion fluctuation and gives different ion distributions in the charged fluids compared to the CPBE. To solve the RFPBE, we propose a Monte Carlo method based on the path integral representation. Numerical results show that the effect of the ion fluctuation strengthens the ion diffusion into the domain and intends to distribute the ions in the fluid uniformly. The final ion distribution in the fluid is determined by the competition between the ion fluctuation and the electrostatic forces exerted by the boundaries. The RFPBE is general and feasible for high dimensional systems by taking the advantage of the Monte Carlo method. We use the RFPBE to study a two dimensional system as an example, in which the effect of ion fluctuation is clearly captured.

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.

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.

A Versatile Method for the Distance-Dependent Structural Characterization of Interacting Soft Interfaces by Neutron Reflectometry

Interactions between soft interfaces govern the behavior of emulsions and foams and crucially influence the functions of biological entities like membranes. To understand the character of these interactions, detailed insight into the interfaces' structural response in terms of molecular arrangements and conformations is often essential. This requires the realization of controlled interaction conditions and surface-sensitive techniques capable of resolving the structure of buried interfaces. Here, we present a new approach to determine the distance-dependent structure of interacting soft interfaces by neutron reflectometry. A solid/water interface and a water/oil interface are functionalized independently and initially macroscopically separated. They are then brought into contact and structurally characterized under interacting conditions. The nanometric distance between the two interfaces can be varied via the exertion of osmotic pressures. Our first experiments on lipid-anchored polymer brushes interacting across water with solid-grafted polyelectrolyte brushes and with bare silicon surfaces reveal qualitatively different interaction scenarios depending on the chemical composition of the two involved interfaces.

Potential of mean force between oppositely charged nanoparticles: A comprehensive comparison between Poisson-Boltzmann theory and Monte Carlo simulations

Ion-mediated interactions between like-charged polyelectrolytes have been paid much attention, and the Poisson-Boltzmann (PB) theory has been shown to fail in qualitatively predicting multivalent ion-mediated like-charge attraction. However, inadequate attention has been paid to the ion-mediated interactions between oppositely charged polyelectrolytes. In this work, the potentials of mean force (PMF) between oppositely charged nanoparticles in 1:1 and 2:2 salt solutions were investigated by Monte Carlo simulations and the PB theory. Our calculations show that the PMFs between oppositely charged nanoparticles are generally attractive in 1:1 and 2:2 salt solutions and that such attractive PMFs become weaker at higher 1:1 or 2:2 salt concentrations. The comprehensive comparisons show that the PB theory can quantitatively predict the PMFs between oppositely charged nanoparticles in 1:1 salt solutions, except for the slight deviation at very high 1:1 salt concentration. However, for 2:2 salt solutions, the PB theory generally overestimates the attractive PMF between oppositely charged nanoparticles, and this overestimation becomes more pronounced for nanoparticles with higher charge density and for higher 2:2 salt concentration. Our microscopic analyses suggest that the overestimation of the PB theory on the attractive PMFs for 2:2 salt solutions is attributed to the underestimation of divalent ions bound to nanoparticles.

Repulsion between oppositely charged rod-shaped macromolecules: Role of overcharging and ionic confinement

The interaction between two oppositely charged rod-shaped macro-ions in a micro-ion solution is investigated via Monte Carlo simulations of the primitive model. The focus is on the asymmetry in rod and/or ion charge, i.e., conditions where oppositely charged objects can repel one another. For equally and oppositely charged rods with asymmetric z:1 micro-ions, repulsion may be induced by overcharging one of the rods with the z valent ions. For asymmetrically charged rods in a symmetric z:z micro-ion solution, a repulsive interaction-at separation of the order of one ion diameter-can arise via an unbalanced osmotic pressure contribution from the ionic atmosphere in the inter-rod space, and an attractive interaction-at a smaller separation-may occur due to a "squeezing out" of the micro-ions from the space between the rods (with a consequent gain in entropy). The thermodynamics of each mechanism is investigated in terms of rod charge and size and micro-ion valence, size, and concentration. Our findings contribute to the understanding of the complex role of charge asymmetry on the interaction of, for example, oppositely charged polyelectrolytes, functionalized nanotubes, and rod-like biomolecules, e.g., viruses.

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.

Interaction modes between asymmetrically and oppositely charged rods

The interaction of oppositely and asymmetrically charged rods in salt-a simple model of (bio)macromolecular assembly-is observed via simulation to exhibit two free energy minima, separated by a repulsive barrier. In contrast to similar minima in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the governing mechanism includes electrostatic attraction at large separation, osmotic repulsion at close range, and depletion attraction near contact. A model accounting for ion condensation and excluded volume is shown to be superior to a mean-field treatment in predicting the effect of charge asymmetry on the free-energy profile.

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).

Screening like charges in one-dimensional Coulomb systems: Exact results

The possibility that like charges can attract each other under the mediation of mobile counterions is by now well documented experimentally, numerically, and analytically. Yet, obtaining exact results is in general impossible, or restricted to some limiting cases. We work out here in detail a one-dimensional model that retains the essence of the phenomena present in higher-dimensional systems. The partition function is obtained explicitly, from which a wealth of relevant quantities follow, such as the effective force between the charges or the counterion profile in their vicinity. Isobaric and canonical ensembles are distinguished. The case of two equal charges screened by an arbitrary number N of counterions is first studied, before the more general asymmetric situation is addressed. It is shown that the parity of N plays a key role in the long-range physics.

Strong coupling electrostatics for randomly charged surfaces: antifragility and effective interactions

We study the effective interaction mediated by strongly coupled Coulomb fluids between dielectric surfaces carrying quenched, random monopolar charges with equal mean and variance, both when the Coulomb fluid consists only of mobile multivalent counterions and when it consists of an asymmetric ionic mixture containing multivalent and monovalent (salt) ions in equilibrium with an aqueous bulk reservoir. We analyze the consequences that follow from the interplay between surface charge disorder, dielectric and salt image effects, and the strong electrostatic coupling that results from multivalent counterions on the distribution of these ions and the effective interaction pressure they mediate between the surfaces. In a dielectrically homogeneous system, we show that the multivalent counterions are attracted towards the surfaces with a singular, disorder-induced potential that diverges logarithmically on approach to the surfaces, creating a singular but integrable counterion density profile that exhibits an algebraic divergence at the surfaces with an exponent that depends on the surface charge (disorder) variance. This effect drives the system towards a state of lower thermal 'disorder', one that can be described by a renormalized temperature, exhibiting thus a remarkable antifragility. In the presence of an interfacial dielectric discontinuity, the singular behavior of counterion density at the surfaces is removed but multivalent counterions are still accumulated much more strongly close to randomly charged surfaces as compared with uniformly charged ones. The interaction pressure acting on the surfaces displays in general a highly non-monotonic behavior as a function of the inter-surface separation with a prominent regime of attraction at small to intermediate separations. This attraction is caused directly by the combined effects from charge disorder and strong coupling electrostatics of multivalent counterions, which dominate the surface-surface repulsion due to the (equal) mean charges on the two surfaces and the osmotic pressure of monovalent ions residing between them. These effects can be quite significant even with a small degree of surface charge disorder relative to the mean surface charge. The strong coupling, disorder-induced attraction is typically much stronger than the van der Waals interaction between the surfaces, especially within a range of several nanometers for the inter-surface separation, where such effects are predicted to be most pronounced.

Electrostatic correlations near charged planar surfaces

Electrostatic correlation effects near charged planar surfaces immersed in a symmetric electrolytes solution are systematically studied by numerically solving the nonlinear six-dimensional electrostatic self-consistent equations. We compare our numerical results with widely accepted mean-field (MF) theory results, and find that the MF theory remains quantitatively accurate only in weakly charged regimes, whereas in strongly charged regimes, the MF predictions deviate drastically due to the electrostatic correlation effects. We also observe a first-order like phase-transition corresponding to the counterion condensation phenomenon in strongly charged regimes, and compute the phase diagram numerically within a wide parameter range. Finally, we investigate the interactions between two likely-charged planar surfaces, which repulse each other as MF theory predicts in weakly charged regimes. However, our results show that they attract each other above a certain distance in strongly charged regimes due to significant electrostatic correlations.

Disjoining pressure of an electrolyte film confined between semipermeable membranes

We consider an electrolyte solution confined by infinitesimally thin semipermeable membranes in contact with a salt-free solvent. Membranes are uncharged, but since small counter-ions leak-out into infinite salt-free reservoirs, we observe a distance-dependent membrane potential, which generates a repulsive electrostatic disjoining pressure. We obtain the distribution of the potential and of ions, and derive explicit formulas for the disjoining pressure, which are validated by computer simulations. We predict a strong short-range power-law repulsion, and a weaker long-range exponential decay. Our results also demonstrate that an interaction between membranes does strongly depend on the screening lengths, valency of an electrolyte solution, and an inter-membrane film thickness. Finally, our analysis can be directly extended to the study of more complex situations and some biological problems.

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.

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.

Lateral dynamics of charged lipids and peripheral proteins in spatially heterogeneous membranes: comparison of continuous and Monte Carlo approaches

Biological membranes are complex environments whose physico-chemical properties are of utmost importance for the understanding of many crucial biological processes. Much attention has been given in the literature to the description of membranes along the z-axis perpendicular to the membrane. Here, we instead consider the lateral dynamics of lipids and peripheral proteins due to their electrostatic interaction. Previously, we constructed a Monte Carlo automaton capable of simulating mutual diffusive dynamics of charged lipids and associated positively charged peptides. Here, we derive and numerically analyze a system of Poisson-Boltzmann-Nernst-Planck (PBNP) equations that provide a mean-field approximation compatible with our Monte Carlo model. The thorough comparison between the mean-field PBNP equations and Monte Carlo simulations demonstrates that both the approaches are in a good qualitative agreement in all tested scenarios. We find that the two methods quantitatively deviate when the local charge density is high, presumably because the Poisson-Boltzmann formalism is applicable in the so-called weak coupling limit, whose validity is restricted to low charge densities. Nevertheless, we conclude that the mean-field PBNP approach provides a good approximation for the considerably more detailed Monte Carlo model at only a fraction of the associated computational cost and allows simulation of the membrane lateral dynamics on the space and time scales relevant for the realistic biological problems.

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.

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.

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 .

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.

One-dimensional counterion gas between charged surfaces: exact results compared with weak- and strong-coupling analyses

We evaluate exactly the statistical integral for an inhomogeneous one-dimensional (1D) counterion-only Coulomb gas between two charged boundaries and from this compute the effective interaction, or disjoining pressure, between the bounding surfaces. Our exact results are compared to the limiting cases of weak and strong couplings which are the same for 1D and three-dimensional (3D) systems. For systems with a large number of counterions it is found that the weak-coupling (mean-field) approximation for the disjoining pressure works perfectly and that fluctuations around the mean-field in 1D are much smaller than in 3D. In the case of few counterions it works less well and strong-coupling approximation performs much better as it takes into account properly the discreteness of the counterion charges.