Effective interactions between charged nanoparticles in water: What is left from the DLVO theory?

Current Opinion in Colloid & Interface Science 15 (2010) 2–7

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Current Opinion in Colloid & Interface Science j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o c i s

Effective interactions between charged nanoparticles in water: What is left from the DLVO theory? Vincent Dahirel, Marie Jardat ⁎ UPMC Univ Paris 06, UMR CNRS 7195, PECSA, 4 place Jussieu, F-75005 Paris, France

a r t i c l e

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Article history: Received 14 April 2009 Received in revised form 15 May 2009 Accepted 15 May 2009 Available online 22 May 2009 Keywords: Potential of mean force Multiscale modelling Numerical simulation Charged nanoparticles

a b s t r a c t The keystone of the modelling of complex systems is the potential of mean force (PMF) between particles. This review focuses on recent numerical simulation studies that concern the computation of the PMF between charged nanoparticles in solution. Such simulations explicitly sample the configurations of the microions or water molecules over which the potential is averaged out. The studies rely on different levels of modelling and permit to quantify the relative amplitude of the different factors governing the interaction, such as the structure of the nanoparticle, the polarisability of microions, or hydrophobic interactions. We discuss the conditions in which the potential of mean force can safely be expressed as a DLVO potential, and why in some cases such a simple analytical expression cannot be used. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Aqueous solutions of charged nanoparticles, i.e. charged particles of nanometric size, combine some properties of colloids with others typical of electrolyte solutions. They are often referred to as highly asymmetric electrolytes. Some examples of such nanoparticles are micelles, proteins, or metal oxide particles. In every case, interactions between these particles govern the static and dynamical properties of the whole sample, such as the phase diagram and the transport coefficients. In the presence of water molecules and microions (counterions and coions), the global interaction between nanoparticles has various origins, and one may conveniently distinguish between direct and induced contributions. In the case of proteins for example, direct interactions include short-range repulsive forces reflecting the shape of the protein, electrostatic forces associated with pH-dependent electric charges, and van der Waals interactions between the protein residues. Induced interactions are mediated by the surrounding molecules, such as water in the case of hydrophobic interactions. Although the mean force between large colloidal particles (of about one micrometer) may be measured directly, using optical means, interactions between particles of a few nanometers can only be inferred indirectly. From scattering measurements (light, X-ray or neutron scattering), one can derive the structure factor S(q) of the solution, or the second virial coefficient B2, that both give information on the mean interaction between scattering particles. Nevertheless, the resort to theoretical calculations is needed if one is interested in the precise ⁎ Corresponding author. E-mail addresses: [email protected] (V. Dahirel), [email protected] (M. Jardat). 1359-0294/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cocis.2009.05.006

analytical expression of the interaction potential, in order to deduce information on the nanoparticle itself (for instance to evaluate its size or its charge), or in order to compute the phase diagram or the transport coefficients. From the theoretical point of view, the issue of effective interactions between solute particles has a long story. The description of the properties of aqueous solutions and of colloids from statistical mechanics is difficult because of the very different length and time scales involved for the various species. Interactions are then commonly averaged out within a so-called coarse-graining procedure, whose formalism was introduced in 1945 by Mc Millan and Mayer [1]. An important ingredient of the McMillan–Mayer theory is the Potential of Mean Force (PMF), which is the interaction potential in the system where the degrees of freedom of the nanoparticles only are considered. In the case of aqueous electrolyte solutions, averaging over the configurations of the solvent molecules mainly results in a dielectric screening of Coulombic interactions. This effect may be accounted for by dividing the interactions by the dielectric constant of the pure solvent εr(T). The sum of this simple water-mediated potential plus a hard-sphere potential corresponds to the primitive model of electrolytes, which is known to describe efficiently the structural and dynamical properties of electrolyte solutions [2]. In the case of highly asymmetric electrolytes, averaging over the configuration of the solvent molecules and microions results in a dielectric screening, plus a Debye screening. Asymmetric electrolytes can a priori be described by the primitive model, but even with this simple model, the ion-averaged PMF between the nanoparticles cannot be analytically computed. The standard approximate theory for interactions in colloids is the DLVO one [3] (Derjaguin, Landau, Vervey and Overbeek). The DLVO potential is the sum of an effective electrostatic term and a direct van der

V. Dahirel, M. Jardat / Current Opinion in Colloid & Interface Science 15 (2010) 2–7

Waals term. The van der Waals term is calculated as an integral of interatomic dispersion interactions over the volume of both particles [4]. This term is neglected within the primitive model, but it is important for large colloidal particles. To compute the effective electrostatic component, microions are described by point charges and two approximations are made, the Poisson–Boltzmann (PB) approximation (i.e. a mean-field treatment of microions), and an expansion of the charge density to linear order in the electrostatic potential. This leads to a simple analytical expression, PMFel expðκa1 Þ expðκa2 Þ expð − κr Þ DLVO ðr Þ = LB Z1 Z2 ð1 + κa1 Þð1 + κa2 Þ kB T r

ð1Þ

where r is the distance between colloids, LB is the Bjerrum length, 2 LB = 4πee o er kB T, κ− 1 is the Debye length, with κ 2 = 4πLB∑2i = 1ρiZi2, where ρi is the concentration of microion i, Z1, Z2 and a1, a2 are the charges and radii of the colloids 1 and 2. The charges Z1 and Z2 are effective or renormalized charges. They may be derived analytically as presented in a review by Belloni [5] (see also [6]). The review by Löwen and Hansen [7] gives an overview of the theoretical descriptions of charged-stabilised colloids, including spherical, cylindrical and lammelar polyions. The authors discuss the DLVO theory and its limitations, and present the more general theoretical framework of Density Functional Theory. The reader is also referred to the review by Vlachy [8], which gives additional details on cell models and integral equations. The DLVO potential and its generic Yukawa expression (PMF(r) = A exp(−Br)/r) have been extensively used to describe interactions in colloidal suspensions. Even recently, many experimental works on solutions of nanoparticles like proteins (see e.g. refs. [9–11]) or iron oxides (see e.g. [12]) could be interpreted assuming a Yukawa potential. These results suggest that such solutions of nanoparticles keep the dominant features of colloidal suspensions, even if the size of solute particles corresponds to the lowest limit of the colloidal domain. Despite its success, the DLVO theory fails to predict some experimental behaviours. The attraction between like-charged particles in the presence of multivalent counterions is the most surprising one [13]. Numerical simulations within the primitive model have remarkably contributed to understand such failure. It has been proved that Poisson–Boltzmann theory cannot predict an attraction, while the PMF computed by simulations can be attractive. Therefore, the attraction can be explained by the correlations between microions, missed within the mean field PB treatment, but present in the simulations. The review by Dijkstra [13] devoted to the simulations of charged colloids summarizes a number of works on this issue. Our review focuses on recent numerical simulation studies that explain or predict other deviations from the DLVO approximations. We only present results which concern the computation of the PMF between charged nanoparticles in solution. These simulations explicitly sample the configurations of the microions or water molecules over which the PMF is averaged out. They take advantage of the huge growth of computer speed in the past ten years. We discuss the conditions in which the PMF can safely be expressed as a DLVO potential with suitably renormalized parameters, and why in some cases such a simple analytical expression cannot be used. In the first part of this review, we present results from numerical simulations where microions and nanoparticles are described by the primitive model. In the second part, the effects of charge distribution and of nanoparticle's shape on the PMF are discussed. In the three other parts, simulation results that go beyond the primitive model itself are summarized. Indeed, the limitations of the DLVO theory do not only come from microion correlations, but also from the inaccuracy of the water-mediated potential between ions. In the third part, we review articles that report simulations using continuous solvent models beyond the primitive model, including for instance charge regulation and dielectric discontinuities. Then, in the fourth part, we discuss results from explicit solvent calculations. In the final

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Table 1 Range of the characteristic model parameters of some simulated systems: Radius R of the nanoparticles, absolute value of their charge Z and salt concentration. Ref.

R/nm

|Z|

Csel/mM

[17,19••,24] [20,23•] [21,22] [44,45] [32] [33•] [35••]

1–2 1 2 1 1–2 1–2 1

0–20 1–10 6–15 3 0–20 30 0–10

100 120–220 0–2.103 125 6 0–25 0

All added salts are monovalent.

part, we address the question of ion-specific effects: The DLVO theory does not account for the chemical specificity of microions, which is mainly related to their polarisability and their solvation. It should be noticed that the PMF should be computed for each system under consideration, i.e. for each temperature, for each concentration of added salt, and for each type of salt. Moreover, the PMF can't be computed within a reasonable time if the dissymmetry between the size of the nanoparticles and that of the surrounding particles is too high. This is why the PMF has never been computed with explicit microions for typical colloidal particles of several thousand charges. The papers which are reviewed here focus on a finite number of systems at room temperature, where the radius of nanoparticles ranges between 1 and 2 nm, their charge ranges between 0 and 30 and the concentration of added salt is between 0 and 2 mol L− 1. Some of the parameters of these systems are summarized in Table 1. 2. The primitive model of electrolytes captures the essential features The computation of the PMF between charged nanoparticles in the framework of the primitive model may be achieved for example by Monte Carlo simulations of a system containing two fixed charged nanoparticles in the presence of microions. The PMF includes in this case two contributions in addition to the direct Coulombic interaction between the nanoparticles: The electrostatic interactions of nanoparticles with the surrounding microions, and a term arising from the collisions between microions and nanoparticles [14]. As already mentioned, simulations within the primitive model have provided the numerical evidence of a purely electrostatic attraction between like-charged colloids in the presence of multivalent counterions [14–16]. It is less known that, even in the presence of monovalent microions, deviations from the DLVO theory have been predicted by simulations. Using the DLVO potential as the PMF between nanoparticles in solution implies that (i) the PMF is monotonic, (ii) the interaction between a neutral and a charge particle vanishes, and (iii) the potential between two particles of same charge +Z is opposite to the potential between a particle of charge + Z and a particle of opposite charge −Z. These three statements fail even in a monovalent salt at a concentration of 0.1 mol L− 1. If two nanoparticles bear charges of opposite signs but unequal in absolute value, the interaction is attractive at long range and repulsive at short range, and therefore it cannot be expressed as a Yukawa potential [17]. In the limiting case where one of the two nanoparticles is neutral, a short-range repulsion exists [17,18]. This repulsion is mediated by the collisions between the microions and the neutral nanoparticle. This effective collision force may play a major role in the interaction between protein and DNA [19••]. In that case, the weakly charged positive protein experiences a strong effective repulsion when it approaches the highly charged negative DNA. Even when the potential can be fitted by a Yukawa potential, in the case of two particles of different charges or radii, the values of the effective DLVO parameters are not a characteristics of each nanoparticle considered separately [18]. In particular, comparisons between the interaction of a pair of like-charged particles +Z/+Z and the interaction of a pair of oppositely charged particles +Z/−Z reveal that the deviations from DLVO

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theory are not similar in both cases [20], although the nanoparticles have the same radius and absolute charge. The PMF is stronger and of longer range in the case of oppositely charged particles. 3. Shape and charge distribution affect the potential of mean force The previous examples deal with spherical nanoparticles carrying a central charge. However, the detailed structure of the nanoparticles, beyond their mean radius and their mean charge, may significantly affect the PMF. Simulations offer the possibility to distinguish different influences of the nanoparticle structure. They have helped to rationalize the relative influence of charge discretisation and shape. Considering a central charge is a reasonable approximation for large spherical colloidal particles, but the discreteness of charge distributions of nanoparticles can no longer be ignored when the distance between two charged residues on the surface is not small compared to the diameter of the nanoparticle. Several authors have computed the PMF between nanoparticles bearing charge patterns, the microions being again described by the primitive model. Allahyarov and co-workers [21,22] used molecular dynamics (MD) to compute the angle-averaged PMF for various charge patterns at the surface of nanoparticles. In agreement with experimental data on protein solutions, the computed values of the virial coefficient B2 display a non-monotonic variation as a function of the microion concentration, while the Poisson–Boltzmann (PB) theory predicts a monotonically increasing curve. This behaviour is not observed when the charge of the nanoparticle is central, and is thus a typical effect of charge distribution. Other deviations from the PB and DLVO predictions can be seen if each nanoparticle bears a non-neutral charge pattern made of charges of opposite signs [23•]. In the presence of monovalent microions at a physiological concentration, there is a short-range attraction of a few kBT between like-charged (or neutral) model nanoparticles [23•]. Both the direct interaction between the nanoparticles and the ion-averaged effective interaction are attractive in that case. The direct Coulomb interaction is not screened, but rather amplified. Again, the potential of mean force is non-monotonic and therefore, it cannot be fitted by a Yukawa expression. Although the angle-averaged PMF is sufficient to evaluate thermodynamical quantities such as B2, the angle dependence of the PMF is required for the study of other properties, such as the orientational dynamics of nanoparticles with anisotropic charge distributions. We have systematically studied the influence of the charge distribution of model proteins on the effective interaction, as a function of the orientation of the proteins [24]. Two proteins were considered, one carrying no dipole but a net charge distributed on a pattern and another carrying a dipolar charge pattern. For orientations where the direct interaction between the two charge distributions is repulsive, the PMF can be more repulsive than this direct interaction at short distances. Here again, the direct Coulomb interaction is amplified. These effects are more pronounced in the presence of a divalent added salt than in a monovalent salt, which suggests that they are due to correlations between microions. The finite size of microions also influence the effective interaction. It leads to important depletion effects at short distances, when microions excluded volumes around the nanoparticles overlap [23•,24]. These depletion effects only appear when charges of the same sign approach each other (i.e. for repulsive orientations), and thus when ions make bridges between these charges. There is thus a dissymmetry between the PMF for attractive orientations, which may be fitted by a Yukawa potential, and the PMF for repulsive orientations which is non-monotonic and of shorter range [24]. Other works tackled the influence of the shape of the nanoparticle, which can be non-spherical in the case of proteins for example. For model proteins that only differ by their shape, we quantified the intensity of ion-mediated forces for like-charged and oppositely charged proteins, extending the primitive model to non spherical particles [25•]. The effects of shape are both quantitative and qualitative. In particular,

the sign of ion-induced collision forces may change, because shape complementarity amplifies microion depletion between the nanoparticles. For oppositely charged proteins, the attraction between proteins with complementary shapes is increased by several kBT relatively to spherical particles. In the meantime, the height of the potential wall between like-charged particles is enhanced by the same amount. Interestingly, these effects are not only present at short distances: Their range is similar to the Debye length κ− 1. In the case of DNA–protein interactions, the effective repulsion between DNA and protein also highly depends on the shape. The repulsion is more important for complementary shapes than for non-complementary ones [19••]. These works emphasize the necessity of including the specific distribution of charges and the shape to compute the PMF between nanoparticles in the framework of the primitive model. 4. Beyond the primitive model: effects of dielectric discontinuities and charge regulation In the primitive model, the solute particles have the same dielectric constant as the solvent. However, the dielectric constants of hydrocarbonated molecules like proteins and micelles or of metal oxide are different from that of water. The influence of dielectric discontinuities on electrostatic interactions has been the subject of a number of theoretical works. The vast majority of these works addresses the case of an electrolyte solution near a planar dielectric interface or near a spherical one. When the interface is planar, the method of image charges is straightforward: A charge experiences an electric field similar to the field created by an image charge at a mirror position across the discontinuity. Within this description, counterions are less attracted by the surface of the nanoparticle when the dielectric constant inside the nanoparticle is lower than the solvent one, while counterions are more attracted by the surface of the nanoparticle in the opposite situation. The recent review by Messina [26] summarizes the main works concerning these situations. The computation of the interaction between two charged nanoparticles embedded in a solvent of distinct dielectric constant, in the presence of microions, is a challenging issue. Very recently, Rescic and Linse [27•] computed the potential of mean force between two spherical like-charged nanoparticles of low dielectric constant (with a central charge) in the presence of their counterions (modelled as point charges), using Monte Carlo simulations. The polarisation surface charge densities occurring at the dielectric discontinuities are computed following the expressions proposed by Linse [28]. For systems with purely repulsive internanoparticles interactions, the PMF increases by a few kBT at contact distance when dielectric discontinuities are taken into account. The influence of dielectric discontinuities is short ranged, and becomes negligible when the surface-to-surface distance between nanoparticles is larger than one nanoparticle radius. For situations where the PMF is attractive at contact, the dielectric discontinuities induce a short-range repulsion and reduces the depth of the energy minimum. The computational method used by Rescic and Linse is appealing, but unfortunately, also very time consuming. The primitive model does not take into account the chemical exchange of protons between solvent molecules and the surface atoms of nanoparticles. The chemical equilibrium of constant Ka between acidic and basic groups leads to a solute charge which depends on the pH. It is reasonable to assume that this equilibrium does not depend on electrostatic interactions when pKa ≫ pH or pKa ≪ pH, and thus that the charge of the nanoparticles does not vary in these conditions. However, when pH≃ pKa — or more precisely pH ≃pI, the isoelectric point — the charge distribution of the nanoparticle may change under the influence of an electric field to reduce the free energy of the system. Several papers deal with the computation of titration curves which give the charge of the nanoparticle as a function of the pH [29,30•,31]. They use Monte Carlo simulations at a given pH. For each Monte Carlo step, a random site is chosen, whose charge is randomly changed. The acceptance rate depends

V. Dahirel, M. Jardat / Current Opinion in Colloid & Interface Science 15 (2010) 2–7

both on electrostatic interactions and on the pH. Seijo et al. [29] show that the distribution of charged sites at the surface of the nanoparticle plays an important role on the charging process. Dielectric discontinuities have some influence when patchy distributions of charges are considered [29] but are negligible for titration curves of small and hydrophilic proteins [30•]. The surface charge should also depend on the distance between charged nanoparticles and then the pH should influence the PMF between two nanoparticles. Lund and Jönsson [32] used Monte Carlo simulations with a pHdependent trial energy for proton exchange, and computed the free energy between a protein near its isoelectric point and a fully charged protein. At contact, there is a difference in the interaction energy of about 1 kBT between a model with fixed charges compared to a situation where the proteins are free to adjust their charges. This effect, which only appears at specific pH, is thus rather small and does not change the functional expression of the potential. 5. Direct influence of solvent molecules and hydrophobic effects The first simulation study including explicitly solvent molecules to compute the PMF between charged nanoparticles with added microions used the so-called solvent primitive model [33•]. The solvent is described by hard spheres, and the interactions between the charged species are similar to the primitive model. The authors obtain two results that merit to be stressed. The first one has important methodological consequences. The solvent-averaged potential between one ion and one nanoparticle is pairwise additive and transferable to high ionic densities. This justifies a multiscale approach splitting the PMF determination into two parts: First, averaging over the water degrees of freedom for a given distance between one microion and one nanoparticle, and secondly, averaging over the microion degrees of freedom for a given distance between two nanoparticles. As discussed in the following section, this multiscale approach can be used to understand ion-specific effects. The second result of ref. [33•] is that the effective depletion force induced by the solvent molecules on the nanoparticles is small compared to the direct Coulombic repulsion. However, the total effective interaction averaged over both the solvent and the microions is much less repulsive in the presence of the hard sphere solvent than with the primitive model. Indeed, the hard-sphere solvent particles induce an over-condensation of the counterions at the surface of the nanoparticle, which enhances screening. Moreover, in the presence of divalent counterions, the solvent can induce an attraction for a range parameters where the primitive model would predict a repulsion. Although these effects are quantitatively important, they are not inconsistent with a DLVO potential, since the authors were able to reproduce their results with a suitably renormalized Yukawa expression. Unfortunately, real solvents like water share few similarities with hard spheres. For instance, water-induced depletion forces should be closely related to electrostatic interactions between water molecules. When neutral particles approach each other, the water molecules between them escape into the bulk. This causes a short-range attraction between the solutes, the hydrophobic interaction [34]. It is generally assumed that charged particles are hydrophilic and thus do not interact through hydrophobic interactions. However, recent simulations have shown that this statement is not valid for nanoparticles [35••]. Dzubiella and Hansen performed explicit-water molecular dynamic simulations and computed effective interactions in the cases of like-charged and oppositely charged nanoparticles, with a single central charge or a pattern of charges. For distances above 0.5 nm the water-averaged potential between nanoparticles becomes similar to the PMF obtained within the primitive model. The amplitude of the hydrophobic interaction for shorter distances is much higher than kBT for charged solutes (with a nanoparticle charge of 10 for a radius of 1.1 nm). For instance, for two nanoparticles of radius 1.1 nm and central charge −4, the PMF is attractive and reaches −15 kBT at contact, and the attraction is

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even amplified when the charge is distributed on a pattern. We think however that the effects observed here could be smaller with a more attractive interaction between water and the nanoparticle, which could reduce water depletion, for instance including van der Waals interactions. Moreover, for highly charged nanoparticles, this effect should be strongly reduced and at much shorter range than the screened Coulomb repulsion. Nevertheless, this important work has demonstrated the unavoidable contribution of hydrophobic interactions to effective interactions between weakly charged nanoparticles in water, like proteins. This result suggests that explicit water calculations cannot be avoided, unless the hydrophobic attraction can be approximated by an analytical formula. Unfortunately, simple analytic expressions depending on the surface tension are not valid for charged nanoparticles. Dzubiella and co-workers [36] have proposed an implicit solvent model accounting for these effects, relating the solvent dewetting at the interface to the electric field created by the nanoparticle charges and by the microions. However, their method is not easy to implement and also computationally demanding. 6. Accounting for the chemical specificity of microions In the late 1800s, Hofmeister reported that the interactions between proteins are affected at different levels by salt with common cation but different anions [37,38]. This behaviour cannot be accounted for by point microions as modeled in the DLVO theory, and is poorly reproduced by simple size effects within the primitive model. After Hofmeister's pioneering work, many experimentalists helped to quantify ion-specific effects, and recently, SAXS experiments have been used to investigate the influence of salt type on the structure of protein solutions [10,39–41]. These studies also suggest that large monovalent ions can induce attraction between likecharged proteins. For instance, a study by Liu et al. [41] revealed an attraction between proteins that depends on the type of added salt, and whose range is several times the protein diameter. Ninham et al. have proposed that dispersion forces between ions, related to ionic polarisability, provide the key to interpret measured ion-specific effects [37]. Semi-quantitative estimates of these effects have been provided via mean-field statistical mechanical theories similar to the PB theory, but where the van der Waals (dispersion) attraction is added to the interaction Hamiltonian whatever the ion type (nanoparticles and microions) [37,42•]. The dispersion interactions are calculated from Lifshitz's theory [43]. These works predict a long-range attraction between the nanoparticles although the range of van der Waals interactions is short. Tavares and co-workers [44,45] have performed several simulation studies, where microion–microion and microion–nanoparticle dispersion interactions are added to the primitive model potential. Results from these calculations are qualitatively consistent with these from mean-field theories and show a qualitative effect of ionic polarisability. For instance, in a NaI electrolyte of ionic strength 0.125 M, the PMF between like-charged nanoparticles is attractive for distances larger than 1.2 times the nanoparticle's diameter [44]. Another route to account for ion-specificity is to compute wateraveraged potential between microions and nanoparticles using explicit-solvent simulations, following the method of Allahyarov and Löwen [33•]. The purpose of recent works has been to define solventaveraged potentials which are specific of both the microion type and the chemical nature of the nanoparticle's surface, and to rationalize these potentials to become more predictive. Zhang and Cremer have reviewed works on the interactions between ions and nanoparticles [46]. They stress that ion-specific effects are not due to the ion's influence on water structure, except at the surface of the nanoparticle. Recent MD simulation studies have shown that interactions between microions and the protein surfaces are specific of both microions and amino-acids [47•,48]. To compute the ion-averaged PMF between proteins, one should therefore

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V. Dahirel, M. Jardat / Current Opinion in Colloid & Interface Science 15 (2010) 2–7

evaluate all the water-averaged potentials between ions and aminoacids. In the case of hydrophobic amino-acids, a possible approximation is to use the potential between ion and hydrophobic surfaces. Such potentials have been computed by Horinek and Netz [49] for sodium and halide anions, using MD simulations with polarisable force fields. Interestingly, the water-averaged dispersion term between a microion and a hydrophobic surface is repulsive, which seems to disqualify a simple addition of van der Waals attractions to the primitive model for this kind of systems. The interactions between ions and charged headgroups (including carboxylates, phosphates, sulfates, and sulfonates) have also been studied by MD [50•]. The simulation results were combined with experimental data to order the binding affinities of headgroups and ions. Once the water-averaged potentials are computed, they can be used to compute the ion-averaged interactions between nanoparticles. Lund and et al. [51••] used the potentials between microions and a hydrophobic surface derived in Ref. [49] to compute the PMF between lyzozyme molecules in NaCl and NaI salts, at concentrations between 0.05 M and 0.3 M. In their model, specificity is thus included via an ion-specific interaction with hydrophobic amino-acids, but not through electronic polarisability. The PMFs display an attractive range of nearly 1 nm, that only disappears in NaCl for the lowest concentration. This, again, cannot be accounted for by the DLVO theory with reasonable values of the dispersion interaction between the proteins. Moreover, the hydrophobic interaction of ions with the protein is stronger for iodide than for chloride, which leads to a stronger attraction between the proteins. The results of this method are encouraging: The computed values of B2 agree quantitatively with experimental data, without adjusting the parameters. Very recently, the PMF between tetraalkylammonium ions in the presence of bromide or chloride ions has been computed from molecular dynamics simulations [52] and used to deduce the osmotic coefficients of the solutions at several concentrations. This approach permitted to account for the differences observed experimentally between chloride and bromide salts. 7. Conclusive discussion The studies reviewed here are focused on aqueous suspensions of charged particles of mean radius of about 1 nm. Although the systems are comparable, the studies rely on different levels of modelling and thus enable to quantify the relative amplitude of the different factors governing the interaction. The aim of modelling is to explain experimental observations from a microscopic description of the system. In the case of charge-stabilised colloids, experimental results may be split into two classes, whether they can or cannot be interpreted using a Yukawa potential between nanoparticles. In suspensions of large charged colloidal particles, it is well known that the DLVO picture fails in the presence of multivalent microions. In the case of nanoparticles, there are additional effects that lead to deviations from the DLVO theory. These deviations can be computed exactly by simulations with explicit microion or explicit water models. Most works are devoted to protein solutions, whose experimental properties are often interpreted thanks to the DLVO theory or related theories. Although the DLVO theory seems valid in some cases, there is no obvious relationship between the parameters of the potential and the microscopic forces at play. Simulations show that distinct effects due for example to the charge distribution or the polarisability of ions are hidden in these parameters. In some conditions, an attractive component of the force adds up to the Yukawa potential. This can occur even at moderate concentration of monovalent microions. In the case of a long-range attraction, ionic polarisability may be the cause of the attraction. In the case of short-range attraction, strong microion adsorption due to chemical effects or hydrophobic interactions are explanations to explore. The success of DLVO theory for colloidal particles, and in particular for nanoparticles, comes from the simplicity of the analytical expression of the potential. There is now a need to extract from simulations new

analytical expressions for an easy experimental use, or to propose simple criteria to evaluate whether a phenomenon is important or not. A first step in this direction is to quantify the range and intensity of the solventmediated phenomena at play. This should help to make the right hypothesis when interpreting data. For instance, hydrophobic interactions between micelles or iron oxide species may be negligible, because micelles composed of charged surfactants and iron oxides are usually more charged than proteins, and thus the hydrophobic interactions should be weak and short ranged. As computers become ever faster, it should be soon possible to have a complete understanding of the interactions between nanoparticles. The multiscale route is promising and should contribute to this enterprise. There is still to check that force-fields are transferable from one system to another. If one determines the potential between a microion and a surface, this potential may depend on the microion concentration (through many-body interactions between microions), it may depend on the nanoparticle concentration (many-body interactions between nanoparticles), it may change if the surface curvature changes (in the case of micellar systems for instance), and so on. Moreover, the wateraveraged potential between nanoparticles may depend on the microion concentration, and therefore the accuracy of the multiscale procedure — splitting the PMF determination into water averaging and microion averaging — should be checked. More simulations can help to elucidate these questions in the case of charged nanoparticles.

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