Effective pair potentials for charged colloidal particles

Modeling the structure and thermodynamics of multicomponent and polydisperse hard-sphere dispersions with continuous potentials

A model system of identical particles interacting via a hard-sphere potential is essential in condensed matter physics; it helps to understand in and out of equilibrium phenomena in complex fluids, such as colloidal dispersions. Yet, most of the fixed time-step algorithms to study the transport properties of those systems have drawbacks due to the mathematical nature of the interparticle potential. Because of this, mapping a hard-sphere potential onto a soft potential has been recently proposed [Báez et al., J. Chem. Phys. 149, 164907 (2018)]. More specifically, using the second virial coefficient criterion, one can set a route to estimate the parameters of the soft potential that accurately reproduces the thermodynamic properties of a monocomponent hard-sphere system. However, real colloidal dispersions are multicomponent or polydisperse, making it important to find an efficient way to extend the potential model for dealing with such kind of many-body systems. In this paper, we report on the extension and applicability of the second virial coefficient criterion to build a description that correctly captures the phenomenology of both multicomponent and polydisperse hard-sphere dispersions. To assess the accuracy of the continuous potentials, we compare the structure of soft polydisperse systems with their hard-core counterpart. We also contrast the structural and thermodynamic properties of soft binary mixtures with those obtained through mean-field approximations and the Ornstein-Zernike equation for the two-component hard-sphere dispersion.

A self-consistent Ornstein-Zernike jellium for highly charged colloids (microgels) in suspensions with added salt

This work discusses a jellium scheme, built within the framework of the multicomponent Ornstein-Zernike (OZ) equation, which is capable of describing the collective structure of suspensions of highly charged colloids with added salt, even in the presence of finite-size multivalent microions. This approach uses a suitable approximation to decouple the microion-microion correlations from the macroion-microion profiles, which in combination with the methodology from the dressed ion theory (DIT) gives a full account of the electrostatic effective potential among the colloids. The main advantages of the present contribution reside in its ability to manage the short-range potentials and non-linear correlations among the microions, as well as its realistic characterization of the ionic clouds surrounding each macroion. The structure factors predicted by this jellium scheme are contrasted with previously reported experimental results for microgel suspensions with monovalent salts (2019Phys. Rev. E100032602), thus validating its high accuracy in these situations. The present theoretical analysis is then extended to microgel suspensions with multivalent salts, which reveals the prominent influence of the counterion valence on the makeup of the effective potentials. Although the induced differences may be difficult to identify through the mesoscopic structure, our results suggest that the microgel collapsing transition may be used to enhance these distinct effects, thus giving a feasible experimental probe for these phenomena.

The solvent mediated interaction potential between solute particles: theory and applications

In this paper we develop a theory to calculate the solvent mediated interaction potential between solute particles dispersed in a solvent. The potential is a functional of the instantaneous distribution of solute particles and is expressed in terms of the solute-solvent direct pair correlation function and the density-density correlation function of the bulk solvent. The dependence of the direct pair correlation function on multi-point correlations of the solute distribution is simplified with a mean field approximation. A self consistent approach is developed to calculate the effective potential between solute particles, the solute-solvent and the solute-solute correlation functions. The significance of the solvent fluctuations on the range of the effective potential is elucidated. The theory is applied to calculate equilibrium properties of the Asakura-Oosawa (AO) model for several values of solute and solvent densities and for several values of the particles size ratio. The results give a quantitative description of many-body effect on the effective potential and on the pair correlation functions.

Volume transition effects on the correlations and effective interactions among highly charged microgels

Recent experimental studies have demonstrated the huge influence that the volume phase transition (VPT) has on the collective structure of highly charged thermo-responsive microgels in aqueous solution with low concentrations of added monovalent salt, thus opening a promising new route for controlling the overall properties of practical colloidal suspensions. We present here an analysis of this structure based on the effective electrostatic potential obtained with the exact methodology of the dressed ion theory (DIT). Starting with a description at the primitive model level, we determine the correlations among the components of our model system (macroions plus monovalent anions and cations) by utilizing the two-density integral equation theory, thus allowing us to consider realistic values for the microgel charges. The resulting microgel structure factors show a good agreement with the reported light scattering measurements, whereas the microscopic pair distributions reveal that in this regime the shrunken states promote an enhanced counterion absorption into the microgels. This packing of counterions inside the microgels induces strongly non-linear correlations among the microions, and in turn provokes a substantial weakening of the microgel-microgel correlations. The ensuing effective interactions are then obtained by contracting the description to the level in which only the macroions are present. We find not only that the magnitude and reach of the corresponding pair potentials are markedly inhibited in the shrunken states, but also that their general form diverges from the conventional screened Coulomb shape. This makes it necessary to rethink the concepts of effective charge and screening length.

Accounting for effective interactions among charged microgels

We introduce a theoretical approach to describe structural correlations among charged permeable spheres at finite particle concentrations. This theory explicitly accounts for correlations among microions and between microions and macroions and allows for the proposal of an effective interaction among macroions that successfully captures structural correlations observed in poly-N-isopropyl acrylamide microgel systems. In our description the bare charge is fixed and independent of the microgel size, the microgel concentration, and the ionic strength, which contrasts with results obtained using linear response approximations, where the bare charge needs to be adapted to properly account for microgel correlations obtained at different conditions.

Long-range forces and charge inversions in model charged colloidal dispersions at finite concentration

In charged colloidal dispersion systems the interest is in finding their stability conditions, phase transitions, and transport properties, either in bulk or confinement, among other physicochemical quantities, for which the knowledge of the dispersions' molecular structure and the associated macroion-macroion forces is crucial. To investigate these phenomena simple models have been proposed. Most of the theoretical and simulation studies on charged particles suspensions are at infinite dilution conditions. Hence, these studies have been focused on the electrolyte structure around one or two isolated central particle(s), where phenomena as charge reversal, charge inversion and surface charge amplification have been shown to be relevant. However, experimental studies at finite volume fraction exhibit interesting phenomenology which imply very long-range correlations. A simple, yet useful, model is the Colloidal Primitive Model, in which the colloidal dispersion is modeled as a mixture of size (and charge) asymmetrical hard spheres, at finite volume fraction. In this paper we review recent integral equations solutions for this model, where very long-range attractive-repulsive forces, as well as new long-range, giant charge inversions are reported. The calculated macroions radial distribution functions, charge distributions, and macroion-macroion forces are qualitatively consistent with existing experimental results, and Monte Carlo and molecular dynamics simulations.

Evidence of electrostatic-enhanced depletion attraction in the structural properties and phase behavior of binary charged colloidal suspensions

In this paper we study the structure and phase behavior of binary mixtures of charged particles at low ionic strength. Due to the large size asymmetry between both species, light scattering measurements give us access only to the partial static structure factor that corresponds to the big particles. We observe that the addition of small charged colloids produces a decrease of the main peak of the measured static structure factor and a shift to larger scattering vector values. This finding is in agreement with theory based on integral equations with the Hypernetted-Chain Closure (HNC) relation. The effective interaction between two big particles due to the presence of small particles is obtained by a HNC inversion scheme and used in numerical simulations that adequately reproduce the experiments. We find that the presence of small particles induces an electrostatic depletion screening among the big colloids, creating around them an exclusion zone for the small charged colloids greater than that caused in the case of neutral small colloids, which in turn augments the depletion effect.

Effective electrostatic interactions in colloid-nanoparticle mixtures

Interparticle interactions and bulk properties of colloidal suspensions can be substantially modified by the addition of nanoparticles. Extreme asymmetries in size and charge between colloidal particles and nanoparticles present severe computational challenges to molecular-scale modeling of such complex systems. We present a statistical mechanical theory of effective electrostatic interactions that can greatly ease large-scale modeling of charged colloid-nanoparticle mixtures. By applying a sequential coarse-graining procedure, we show that a multicomponent mixture of charged colloids, nanoparticles, counterions, and coions can be mapped first onto a binary mixture of colloids and nanoparticles and then onto a one-component model of colloids alone. In a linear-response approximation, the one-component model is governed by a single effective pair potential and a one-body volume energy, whose parameters depend nontrivially on nanoparticle size, charge, and concentration. To test the theory, we perform molecular dynamics simulations of the two-component and one-component models and compute structural properties. For moderate electrostatic couplings, colloid-colloid radial distribution functions and static structure factors agree closely between the two models, validating the sequential coarse-graining approach. Nanoparticles of sufficient charge and concentration enhance screening of electrostatic interactions, weakening correlations between charged colloids and destabilizing suspensions, consistent with experiments.

Adsorption isotherms of charged nanoparticles

We present theory and simulations which allow us to quantitatively calculate the amount of surface adsorption excess of charged nanoparticles onto a charged surface. The theory is very accurate for weakly charged nanoparticles and can be used at physiological concentrations of salt. We have also developed an efficient simulation algorithm which can be used for dilute suspensions of nanoparticles of any charge, even at very large salt concentrations. With the help of the new simulation method, we are able to efficiently calculate the adsorption isotherms of highly charged nanoparticles in suspensions containing multivalent ions, for which there are no accurate theoretical methods available.

RNA pseudo-knots simulated with a one-bead coarse-grained model

We present a revised version of a Monte Carlo simulation model for RNA molecules that was introduced in a previous communication [O. Taxilaga-Zetina, P. Pliego-Pastrana, and M. D. Carbajal-Tinoco, Phys. Rev. E 81, 041914 (2010)]. The basic model consists of a series of knowledge-based pair potentials that were obtained from the statistical analysis of large RNAs belonging to the Protein Data Bank. These effective interactions are then used to dress a polymeric chain that reproduces relatively simple secondary structures (e.g., small hairpins). In order to describe more complicated three-dimensional structures such as pseudo-knots, here we include orientational information for the interaction between nucleotides forming hydrogen bonds, as in the case of the Watson-Crick base pairs. As a result, the simulated molecules obtained through the modified model are now consistent with their corresponding experimental configurations.

A cell-model study on counterion fluctuations in macroionic systems: effect of non-extensiveness in entropy

Rigorously speaking, entropy is slightly non-extensive, and this non-extensiveness, which characterizes the degree of fluctuations, can contribute to effective interactions between mesoscopic objects. In this paper, we consider a pair of macroions, each accompanied by 1000 counterions, and with a cell-model description we demonstrate that the slow variation of non-extensiveness in counterion entropy over macroionic distance leads to an effective long-range attraction between the macroions. With the aid of Monte Carlo simulation and a Bragg-Williams theory including counterion number fluctuations, we find the depth of attraction in free energy to be approximately 0.2k(B)T. The observation in our cell-model study provides an insight for further understanding of effective interactions in real macroionic systems.

Large counterions boost the solubility and renormalized charge of suspended nanoparticles

Colloidal particles are ubiquitous in biology and in everyday products such as milk, cosmetics, lubricants, paints, or drugs. The stability and aggregation of colloidal suspensions are of paramount importance in nature and in diverse nanotechnological applications, including the fabrication of photonic materials and scaffolds for biological assemblies, gene therapy, diagnostics, targeted drug delivery, and molecular labeling. Electrolyte solutions have been extensively used to stabilize and direct the assembly of colloidal particles. In electrolytes, the effective electrostatic interactions among the suspended colloids can be changed over various length scales by tuning the ionic concentration. However, a major limitation is gelation or flocculation at high salt concentrations. This is explained by classical theories, which show that the electrostatic repulsion among charged colloids is significantly reduced at high electrolyte concentrations. As a result, these screened colloidal particles are expected to aggregate due to short-range attractive interactions or dispersion forces as the salt concentration increases. We discuss here a robust, tunable mechanism for colloidal stability by which large counterions prevent highly charged nanoparticles from aggregating in salt solutions with concentrations up to 1 M. Large counterions are shown to generate a thicker ionic cloud in the proximity of each charged colloid, which strengthens short-range repulsions among colloidal particles and also increases the corresponding renormalized colloidal charge perceived at larger separation distances. These effects thus provide a reliable stabilization mechanism in a broad range of biological and synthetic colloidal suspensions.

Micro-rheology on (polymer-grafted) colloids using optical tweezers

Optical tweezers are experimental tools with extraordinary resolution in positioning (± 1 nm) a micron-sized colloid and in the measurement of forces (± 50 fN) acting on it-without any mechanical contact. This enables one to carry out a multitude of novel experiments in nano- and microfluidics, of which the following will be presented in this review: (i) forces within single pairs of colloids in media of varying concentration and valency of the surrounding ionic solution, (ii) measurements of the electrophoretic mobility of single colloids in different solvents (concentration, valency of the ionic solution and pH), (iii) similar experiments as in (i) with DNA-grafted colloids, (iv) the nonlinear response of single DNA-grafted colloids in shear flow and (v) the drag force on single colloids pulled through a polymer solution. The experiments will be described in detail and their analysis discussed.

Studies on electrostatic interactions of colloidal particles in two dimensions: a modeling approach

We study the effective electrostatic interactions between a pair of charged colloidal particles without salt ions while the system is confined in two dimensions. In particular, we use a simplified model to elucidate the effects of rotational fluctuations in counterion distribution. The results exhibit effective colloidal attractions under appropriate conditions. Meanwhile, long-range repulsions persist over most of our studied cases. The repulsive forces arise from the fact that in two dimensions, the charged colloids cannot be perfectly screened by counterions, as the residual quadrupole moments contribute to the repulsions at longer range. By applying multiple expansions, we find that the attractive forces observed at short range are mainly contributed by electrostatic interactions among higher-order electric moments. We argue that the scenario for attractive interactions discussed in this work is applicable to systems of charged nanoparticles or colloidal solutions with macroions.

Three-dimensional structures of RNA obtained by means of knowledge-based interaction potentials

We derive a set of effective potentials describing the interaction between pairs of nucleotides that belong to an RNA molecule. Such interaction potentials are then used as the main constituents of a simplified simulation model, which is tested in the description of small secondary structure motifs. Our simulated RNA hairpins are consistent with the experimental structures obtained by NMR.

Asymmetric charge renormalization for nanoparticles in aqueous media

The effective renormalized charge of nanoparticles in an aqueous electrolyte is essential to determine their solubility. By using a molecular model for the supporting aqueous electrolyte, we find that the effective renormalized charge of the nanoparticles is strongly dependent on the sign of the bare charge. Negatively charged nanoparticles have a lower effective renormalized charge than positively charged nanoparticles. The degree of asymmetry is a nonmonotonic function of the bare charge of the nanoparticle. We show that the effect is due to the asymmetric charge distribution of the water molecules, which we model using a simple three-site molecular structure of point charges.

Multiscale modeling of emergent materials: biological and soft matter

In this review, we focus on four current related issues in multiscale modeling of soft and biological matter. First, we discuss how to use structural information from detailed models (or experiments) to construct coarse-grained ones in a hierarchical and systematic way. This is discussed in the context of the so-called Henderson theorem and the inverse Monte Carlo method of Lyubartsev and Laaksonen. In the second part, we take a different look at coarse graining by analyzing conformations of molecules. This is done by the application of self-organizing maps, i.e., a neural network type approach. Such an approach can be used to guide the selection of the relevant degrees of freedom. Then, we discuss technical issues related to the popular dissipative particle dynamics (DPD) method. Importantly, the potentials derived using the inverse Monte Carlo method can be used together with the DPD thermostat. In the final part we focus on solvent-free modeling which offers a different route to coarse graining by integrating out the degrees of freedom associated with solvent.

Closure-based density functional theory applied to interfacial colloidal fluids

The behavior of dense colloidal fluids near surfaces can now be probed in great detail with experimental techniques like confocal microscopy. In fact, we are approaching a point where quantitative comparisons of experiment with particle-level theory, such as classical density functional theory (DFT), are appropriate. In a forward sense, we may use a known surface potential to predict a particle density distribution function from DFT; in an inverse sense, we may use an experimentally measured particle density distribution function to predict the underlying surface potential from DFT. In this paper, we tested the ability of the closure-based DFT of Zhou and Ruckenstein (J. Chem. Phys. 2000, 112, 8079-8082) to perform forward and inverse calculations on potential models commonly employed for colloidal particles and surfaces. To reduce sources of uncertainty in this initial study, Monte Carlo simulation results played the role of experimental data. The combination of Rogers-Young and modified-Verlet closures consistently performed well across the different potential models. For a reasonable range of choices of the density, temperature, and potential parameters, the inversion procedure yielded particle-surface potentials to an accuracy on the order of 0.1kT.

Forces between single pairs of charged colloids in aqueous salt solutions

Forces between single pairs of negatively charged micrometer-sized colloids in aqueous solutions of monovalent, divalent, or trivalent counter-ions at varying concentrations have been measured by employing optical tweezers. The experimental data have been analyzed by using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory and a numerical solution of the Poisson-Boltzmann (PB) equation. With monovalent counterions, the data are well described by the DLVO and PB theories, suggesting that the DLVO theory is adequate to describe the colloidal forces at these conditions. At higher counter-ion valence, the approximations within the two theories become evident.

On attractive interaction of a colloid pair of like charge at infinite dilution

Numerical data on the potential of mean force W(r) at infinite dilution of a highly charged colloid pair embedded in a 1:1 electrolyte are reported. The authors obtain attractive minima (W<0) at short interparticle distance in these potential functions in hypernetted chain (HNC) approximation, as salt concentration is increased. These minima, however, disappear in all system sets studied when a self-consistent Zerah-Hansen (ZH) closure is used. The authors infer that the attractive minima obtained in a HNC closure are spurious and result from the neglect of bridge diagrams in HNC approximation. An expression of bridge function, which the ZH closure in effect incorporates in W(r) to remove attractive minima, is derived in terms of modification of correlation functions. Features of repulsive pair potentials obtained using the ZH closure, their dependence on particle charge and salt concentration, and their agreement with those of the Derajguin-Landau-Verwey-Overbeek theory are investigated.

A simple phenomenological fix for the dielectric constant within the reference interaction site model approach

The effective interactions among ions immersed in water are studied by means of the effective pair potentials (EPPs) [J. Chem. Phys. 2002, 117, 6133] obtained after contracting (integrating out) the degrees of freedom of the solvent molecules. The dressed interaction site theory (DIST) leads to a simple way of adjusting the effective dielectric constant of the model solvent to its experimental value at standard conditions. The molecular structure of the solvent is mirrored in the structure of the short-ranged component of the induced EPPs, with noticeable differences between the cases with trivial (ideal gas) and nontrivial (experimental) values of the dielectric constant. The shape of these EPPs remains almost invariant over the whole range of salt concentrations considered here. The asymptotic behavior of the EPP between two macroions obtained after contracting the supporting electrolyte (water molecules plus small ions) is also briefly discussed.

Two-Component Polypeptides Modeled with Effective Pair Potentials

We present Monte Carlo simulations performed within a model based on a set of distance-dependent effective potentials which are used to describe the interactions between a pair of distinct amino acids. These effective potentials are extracted from experimental correlation functions through the Ornstein-Zernike equations and adequate closure approximations. We focus our attention on the sequences of two specific residues, namely, alanine and glycine. The studied sequences are (a) (Ala)(12)-(Gly)(4)-(Ala)(12) and (b) three interacting chains of alternating alanines and glycines (with five residues per chain). The resulting structures are combinations of known secondary structures. More importantly, we verify that our simulated structures are in thermodynamic equilibrium by means of an estimation of the density of states.

Entropic forces in dilute colloidal systems

Depletion forces can be accounted for by a contraction of the description in the framework of the integral equations theory of simple liquids. This approach includes, in a natural way, the effects of the concentration on the depletion forces, as well as energetic contributions. In this paper we systematically study this approach in a large variety of dilute colloidal systems composed of spherical and nonspherical hard particles, in two and in three dimensions, in the bulk and in front of a hard wall with a relief pattern. We show by this way the form in which concentration and geometry determine the entropic interaction between colloidal particles. The accuracy of our results is corroborated by comparison with computer simulations.

A reference interaction site model approach to depletion forces induced by hard rodlike particles

We study the effective interactions among large hard spherical colloidal particles induced by small hard rodlike particles and compare them with those induced by small hard spherical particles to highlight the specific effects due to the anisotropic shape of the former. This is done by determining the effective pair potentials within the framework of the reference interaction site model approach. The rodlike particles are modeled as N nonoverlapping spherical units arranged in a straight line, so that their total length is N times their transversal diameter. These results are compared against those obtained in the Asakura-Oosawa limit.

Molecular multivalent electrolytes: microstructure and screening lengths

We study small rod-like molecular electrolytes solutions with their corresponding atomic counterions. The asymptotic length scales (decay length and wavelength) of the structural correlations are analyzed using the formalism of the dressed interaction site theory (DIST). The correlation functions are determined using the reference interaction site model equation complemented with a mixed approach in which the hypernetted-chain closure is used for the repulsive interactions, and the mean spherical approximation is used for the attractive interactions. The results from this scheme are in good agreement with the Monte Carlo computer simulations reported here. The asymptotic properties of the correlation functions of this molecular system are compared against those corresponding to two related simple (atomic) electrolyte models. The main conclusion is that the molecular structure of the ions lowers by two orders of magnitude the concentration at which the transition from monotonic to oscillatory decay occurs.

Dipolar effective interaction in a fluid of charged spheres near a dielectric plate

Static correlations in a classical fluid of charged spheres at equilibrium are studied in the vicinity of an insulating wall characterized by its dielectric constant. It is well known that the deformations of screening clouds induced by the presence of the wall result into an effective f(alphaalpha('))(x,x('))/y(3) interaction in the pair distribution function between two charges e(alpha) and e(alpha(')) located at distances x and x(') from the wall and separated by a large distance y along the wall. We investigate the structure of f(alphaalpha('))(x,x(')). The method is based on systematic resummations in the Mayer diagrammatics, which are valid both in the bulk and in an inhomogeneous situation. The screened potential phi arising in the formalism happens to coincide with the linearized mean-field approximation for the immersion free energy of two external unit charges. Phi is shown to decay as a repulsive f(phi)(x,x('))/y(3) interaction, whatever the density profiles may be. f(phi)(x,x(')) takes a factorized dipolar structure f(phi)(x,x('))=D(phi)(x)D(phi)(x(')) for distances x and x(') larger than the maximum of the closest approach distances b(alpha)'s to the wall for every species alpha. Moreover, we devise a reorganization of resummed diagrammatics, which is adequate for the determination of the large-distance behavior of correlations, and we prove that, when all species have the same approach distance b to the wall, f(alphaalpha('))(x,x(');b)=D(alpha)(x)D(alpha('))(x(')). In this case, the leading tail of the effective electrostatic interaction between two like charges at the same distance x from a single wall is repulsive. Results are independent of charge magnitudes, of excluded-volume sphere sizes, and of the existence of a surface charge on the wall. It holds whether charges are concentrated at sphere centers or uniformly spread over their surfaces. Comparison is made with an experiment about dilute colloids where the linearized mean-field approximation proves to be relevant. At equilibrium attraction between like charges in confined geometry might arise from purely electrostatic charge-charge interactions only through correlation effects not taken into account in the latter approximation.

Effective pair potentials between protein amino acids

We present an effective potential describing the interaction between pairs of residues (alanines, in this case) that belong to a protein. The effective potential is extracted from an experimental correlation function, by means of the Ornstein-Zernike equation together with a closure approximation. It is found that the most relevant features of the effective potential are consistent with the formation of two different secondary structures of proteins.

Assessment of Small-Angle and Angle-Averaged Structure Factor for Monitoring Electrostatic Colloidal Interactions Using Multiply Scattered Light

The isotropic scattering coefficients of 143-nm diameter polystyrene latex suspensions were measured using frequency-domain photon migration (FDPM) at 687 and 828 nm as a function of volume fraction (0.05-0.3) and ionic strength (1.0 to 120 mM NaCl equivalents) in order to derive the angle-integrated structure factor, S(q), and structure factor at zero wave vector, S(0). The effective surface charges of the dispersions were estimated by fitting the measured isotropic scattering coefficients at each wavelength as a function of volume fraction to the solution of the Orstein-Zernike integral equation using the hard sphere Yukawa potential model and mean spherical approximation as a closure relation. The estimates of surface charges were comparable at both wavelengths, but decreased with ionic strength. At 120 mM NaCl equivalents, the values of S(0) obtained from FDPM matched those predicted by the Percus-Yevick model, and decreased with volume fraction, consistent with prediction by the Carnahan-Starling equation.

Interaction potentials, structural ordering and effective charges in dispersions of charged colloidal particles

As colloidal dispersions of charged particles exhibit a wide variety of commercial, technological and scientific applications, a considerable theoretical effort has been devoted to finding an effective interaction potential from primitive models. The forces derived from this potential should justify the spatial ordering experimentally observed under certain conditions. This paper reviews the advances in these theoretical studies as well as some experiments (based on the mentioned order) that try to corroborate them. Special attention has been paid to the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential. Nowadays, many of these theoretical investigations suggest that it could be applied if some of its parameters are renormalized. Nevertheless, to achieve a renormalization procedure in a strict way (from a primitive model) is a difficult task as a result of the size and charge asymmetries between small ions and macroions. Thus, several procedures for computing renormalized charges in a simple way have been developed. However, the notion of effective charge has also been widely used (as a adjustable parameter) in order to justify results found for several kinds of colloids (like solid particle dispersions or micellar systems) by means of quite different experimental techniques. Renormalization (as well as ion condensation) approaches, experiments and the controversial relationship between theoretical and phenomenological effective charges are also reviewed in this work.

Energetic contributions to wall-particle depletion forces

Recently, depletion forces were accounted for by a contraction of the description based on the integral equations theory of simple liquids [Phys. Rev. E 61, 4095 (2000)]. The extension of those results to the case of inhomogeneous systems is reported here. Besides, the energetic contributions to the wall-particle depletion forces are studied, as they arise as soon as charge is put on some of the components of a binary mixture of hard spheres on the front of a hard wall. By charging the small particles the amplitude of the depletion attraction between wall and large particles is reduced, and can even become a repulsion. A similar effect is observed if an attractive interaction between wall and small particles is present.

Effective macroion-macroion potentials in asymmetric electrolytes

Effective macroion-macroion potentials in solutions of macroions carrying 60 elementary charges and either monovalent or divalent counterions have been calculated at different concentrations by means of Monte Carlo simulations with a consequent inversion of radial distribution functions according to Lyubartsev and Laaksonen [Phys. Rev. E 52, 3730 (1995)]. With monovalent counterions, the effective potentials are essentially of a Yukawa type, whereas with divalent ones, an attractive region appears at short separation. A charge renormalization scheme invoking the cell model and the assumption of a Yukawa-type potential works favorably only in the case of monovalent counterions.

Effective interactions between electric double layers

This review summarizes and assesses recent theoretical and experimental advances, with special emphasis on the effective interaction between charge-stabilized colloids, in the bulk or in confined geometries, and on the ambiguities of defining an effective charge of the colloidal particles. Some consideration is given to the often neglected discrete solvent effects.