A broad range of manufactured products and biological fluids are colliods. The ability to understand and control the processes (of scientific, technological and industrial interest) in which such colloids are involved relies upon a precise knowledge of the electrical double layer. The traditional approach to describing this ion cloud around colloidal particles has been the Gouy-Chapman model developed on the basis of the Poisson-Boltzmann equation. Since the early 1980s, however, more sophisticated theoretical treatments have revealed both quantitative and qualitative deficiencies in the Poisson-Boltzmann theory, particularly at high ionic strengths and/or high surface charge densities. This review deals with these novel approaches, which are mostly computer simulations and approximate integral equation theories based on the so-called primitive model. Special attention is paid to phenomena that cannot be accounted for by the classic theory as a result of neglecting ion size correlations, such as overcharging, namely, the counterion concentration in the immediate neighborhood of the surface is so large that the particle surface is overcompensated. Other illustrative examples are the nonmonotonic behavior of the electrostatic potential and attractive interactions between equally charged surfaces. These predictions are certainly remarkable and, on paper, they can have an effect on experimentally measurable quantities (for instance, electrophoretic mobility). Even so, these new approaches have scarcely been applied in practice. Thus a critical survey on the relevance of ion size correlation in real systems is also included. Overcharging of macroions can also be brought about by adsorption of oppositely charged polyelectrolytes. Noteworthy examples and theoretical approaches for them are also briefly reviewed.

PACS. 61.20.Qg -Structure of associated liquids: electrolytes, molten salts, etc.. PACS. 82.70.Dd -Colloids. PACS. 87.15.Aa -Theory and modeling; computer simulation.Abstract. -In this Letter we investigate the mechanism for overcharging of a single spherical colloid in the presence of aqueous salts within the framework of the primitive model by molecular dynamics (MD) simulations as well as integral-equation theory. We find that the occurrence and strength of overcharging strongly depends on the salt-ion size, and the available volume in the fluid. To understand the role of the excluded volume of the microions, we first consider an uncharged system. For a fixed bulk concentration we find that upon increasing the fluid particle size one strongly increases the local concentration nearby the colloidal surface and that the particles become laterally ordered. For a charged system the first surface layer is built up predominantly by strongly correlated counterions. We argue that this a key mechanism to produce overcharging with a low electrostatic coupling, and as a more practical consequence, to account for charge inversion with monovalent aqueous salt ions.

The reduced electrophoretic mobility-reduced zeta potential relationship for a charged macroparticle is shown to be nonuniversal and to be highly nonlinear. In agreement with experimental results, a mobility reversal due to the macroion's charge inversion and a nonlinear dependence of the mobility on salt concentration is obtained.

The structure of an electrolyte around a spherical, charged colloid particle is studied, using a simple model in which the diameter of the ions in the soution is considered. The hypernetted chain mean spherical approximation (HNC/MSA) equation is derived and solved numerically. As a result the ionic distribution around the colloid particle is obtained. Calculations are made for 1:l and 2:2 electrolytes for various values of the concentration, colloid radius, and electrical charge. By use of these ionic distributions, excess charge adsorption isotherms and zeta potentials are calculated. These quantities are compared with the nonlinear Poisson-Boltzmann (PB) results. Important quantitative and qualitative differences between the PB and HNC/MSA are found. The zeta potential is found to be a nonmonotonic function of various parameters, for example, the charge density. Qualitative agreement with experimental results for the zeta potential is found.

Results from Monte Carlo (MC) simulations for a restricted primitive model symmetrical electrolyte next to an isolated spherical macroion are reported. Calculations were made for various 1:1 and 2:2 electrolyte concentrations and macroion’s radii and charges. The MC results are compared to those from the hypernetted chain/mean spherical approximation (HNC/MSA) integral equation and the fifth version of the modified Poisson–Boltzmann (MPB5) and Poisson–Boltzmann (PB) differential equations. An overall good agreement of the HNC/MSA and MPB5 results with the MC data is found. On the other hand, the widely used PB theory is found to have important quantitative and qualitative disagreements with the MC results.

We study the consistent inclusion of ionic size-asymmetry for a wide range of macroparticle charges in the primitive model of an electrical double layer around a spherical colloid using (1) Monte-Carlo simulations, (2) the hybrid integral-equation formalism of hypernetted-chain (HNC) and mean-spherical approximation (MSA), and (3) the Gouy-Chapman theory modified for unequal ionic radii. In our simulations, for a weakly charged macroion, we observe surface charge amplification from adsorption of like-charged ions, as well as charge reversal due to overcompensation of the bare nanoparticle charge by counterions. When the nanoparticle charge increases, we detect both asymmetric neutralization and asymmetric electrostatic screening that depend on the sign of the macroion's valence. Specifically, there exists a higher reduction of the original bare charge and a smaller electrostatic potential for the case of negative nanoparticles with positive small counterions, versus the case of positive nanoparticles with negative large counterions. These coarse-grained results are in agreement with the predictions of asymmetric charge renormalization (P. Gonz alez-Mozuelos and M. Olvera de la Cruz, Phys. Rev. E, 2009, 79, 031901), in which the aqueous solvent is explicitly taken into account. Results from the Gouy-Chapman theory modified for unequal ionic radii differ notably from our obtained Monte-Carlo data, while good agreement exists between simulation results and HNC/MSA-treatment findings.

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

The ionic adsorption around a weakly charged spherical colloid, immersed in size-asymmetric 1:1 and 2:2 salts, is studied. We use the primitive model (PM) of an electrolyte to perform Monte Carlo simulations as well as theoretical calculations by means of the hypernetted chain/mean spherical approximation (HNC/MSA) and the unequal-radius modified Gouy-Chapman (URMGC) integral equations. Structural quantities such as the radial distribution functions, the integrated charge, and the mean electrostatic potential are reported. Our Monte Carlo "experiments" evidence that near the point of zero charge, the smallest ionic species is preferentially adsorbed onto the macroparticle, independently of the sign of the charge carried by this tiniest electrolytic component, giving rise to the appearance of the phenomena of charge reversal (CR) and overcharging (OC). Accordingly, colloidal CR, due to an excessive attachment of counterions, is observed when the macroion is slightly charged and the coions are larger than the counterions. In the opposite situation, i.e., if the counterions are larger than the coions, the central macroion acquires additional like-charge (coions) and hence becomes "overcharged," a feature theoretically predicted in the past [F. Jiménez-Angeles and M. Lozada-Cassou, J. Phys. Chem. B 108, 7286 (2004)]. In other words, here we present the first simulation data on OC in the PM electrical double layer, showing that close to the point of zero charge, this novel effect surges as a consequence of the ionic size asymmetry. We also find that the HNC/MSA theory captures well the CR and OC phenomena exhibited by the computer experiments, especially as the macroion's charge increases. On the contrary, even if URMGC also displays CR and OC, its predictions do not compare favorably with the Monte Carlo data, evidencing that the inclusion of hard-core correlations in Monte Carlo and HNC/MSA enhances and extends those effects. We explain our findings in terms of the energy-entropy balance. In the field of electrophoresis, it has been generally agreed that the charge of a colloid in motion is partially decreased by counterion adsorption. Depending on the location of the macroion's slipping surface, the OC results of this paper could imply an increase in the expected electrophoretic mobility. These observations aware about the interpretation of electrokinetic measurements using the standard Poisson-Boltzmann approximation beyond its validity region.