Forces between interfaces in concentrated nanoparticle suspensions and polyelectrolyte solutions

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Article

Reference

Forces between interfaces in concentrated nanoparticle suspensions and polyelectrolyte solutions

SCARRATT, Liam Ronald John, TREFALT, Gregor, BORKOVEC, Michal

Abstract

This article provides an overview of interactions between charged interfaces across concentrated suspensions of charged nanoparticles or solutions of polyelectrolytes. These systems bear many similarities. We distinguish the likecharged and oppositely charged situations. In the like-charged situation, a layered structure in the proximity of the interface is formed. This structure induces a strongly repulsive energy profile at shorter distances, which originates from a gap that is free of nanoparticles or polyelectrolytes. At larger distances, the profile becomes oscillatory. This energy profile can be quantified with a simple model, which distinguishes the nearfield region and the far-field region. The parameters entering the model show characteristic scaling relations. In the oppositely charged situation, a saturated, tightly bound layer at the interface forms. This layer leads to a charge reversal of the interface and induces a similar layered structure as in the likecharged case.

SCARRATT, Liam Ronald John, TREFALT, Gregor, BORKOVEC, Michal. Forces between interfaces in concentrated nanoparticle suspensions and polyelectrolyte solutions. Current Opinion in Colloid & Interface Science , 2021, vol. 55, no. 101482, p. 1-18

DOI : 10.1016/j.cocis.2021.101482

Available at:

http://archive-ouverte.unige.ch/unige:153631

Disclaimer: layout of this document may differ from the published version.

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Forces between interfaces in concentrated nanoparticle suspensions and polyelectrolyte solutions

Liam R. J. Scarratt, Gregor Trefalt and Michal Borkovec

Abstract

This article provides an overview of interactions between charged interfaces across concentrated suspensions of charged nanoparticles or solutions of polyelectrolytes. These systems bear many similarities. We distinguish the like- charged and oppositely charged situations. In the like-charged situation, a layered structure in the proximity of the interface is formed. This structure induces a strongly repulsive energy profile at shorter distances, which originates from a gap that is free of nanoparticles or polyelectrolytes. At larger distances, the profile becomes oscillatory. This energy profile can be quantified with a simple model, which distinguishes the near- field region and the far-field region. The parameters entering the model show characteristic scaling relations. In the oppositely charged situation, a saturated, tightly bound layer at the interface forms. This layer leads to a charge reversal of the interface and induces a similar layered structure as in the like- charged case.

Addresses

Department of Inorganic and Analytical Chemistry, University of Geneva, Sciences II, 30 Quai Ernest-Ansermet, 1205 Geneva, Switzerland

Corresponding author: Borkovec, Michal (michal.borkovec@unige.ch)

Current Opinion in Colloid & Interface Science2021,55:101482 This review comes from a themed issue onMemorial volume for Peter Kralchevsky

Edited byNicolai Denkov,Elena Mileva,Slavka Tcholakovaand Krassimir D. Danov

For a complete overview see theIssueand theEditorial https://doi.org/10.1016/j.cocis.2021.101482

org/licenses/by/4.0/).

Keywords

Nanoparticle, Polyelectrolyte, Interaction, Water–solid interface.

Introduction

Charged nanoparticle suspensions and solutions of polyelectrolytes have long been in focus in colloid, polymer, and soft matter research [1e11]. Such systems are relevant in numerous industrial applications,

including water purification, papermaking, concrete processing, chemical-mechanical polishing, mineral separation, and oil recovery [1e6]. They are equally used as protective coatings, either in the form of monolayers [7,8] or to build multilayered structures with the layer- by-layer technique [9e11].

Due to many similarities between charged nanoparticles and polyelectrolytes, we shall approach these systems from a common point of view and jointly refer to them as macroions. We shall thus mostly use the terms solution of macroions and their adsorption. While normally, these terms are only used for polyelectrolytes, their use is less common for nanoparticles, where one would rather refer to suspension, dispersion, and deposition. However, this nomenclature will simplify our discussion.

Much research effort has been invested in elucidating the solutions of macroions. While this article does not deal with this topic in detail, let us recall some relevant aspects and especially stress the distinction between dilute and concentrated systems.

Dilute solutions of macroions are normally salty, meaning that their ionic composition is dominated by counterions and co-ions originating from dissolved salts. The stability of nanoparticle suspensions was much investigated in the past [12e14]. As predicted by the theory of Derjaguin, Verwey, Landau, and Overbeek (DLVO) [15,16], such systems are stable at low salt concentrations, while they become unstable at higher concentrations. In the latter situation, nanoparticles form aggregates, basically irreversibly, which grow in size with time, and finally may sediment or form gels [13]. The transition between stable and unstable regimes is governed by double-layer forces, which are repulsive at low salt concentrations but become screened at higher concentrations. According to the Schulze-Hardy rule, the valence of the counterions is critical, as with increasing valence, the unstable regime is shifted toward lower salt concentrations [14]. A similar scenario is expected to apply to dilute polyelectrolyte solutions, although its precise nature remains nebulous. Aggre- gates of polyelectrolyte molecules might form at higher salt conditions, and their presence is probably related to Colloid & Interface Science

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the presence of a slow mode that can be observed in dynamic light scattering [17].

Interactions between wateresolid interfaces with solutions of macroions were investigated by numerous re- searchers [12,13,18e20]. Two different scenarios must be distinguished, namely that the interface is like- charged or oppositely charged to the macroions, see Figure 1. Let us first recall the relevant aspects of these interactions in dilute solutions of macroions.

In the like-charged situation, the macroions interact with the interface weakly or not at all. At low salt concentrations, strong electrostatic repulsion between the interface and the macroions typically precludes adsorption and leads to a macroion-free layer next to the interface. In the presence of salt, the double-layer forces are progressively screened, and thus some deposition or adsorption phenomena might be present. Strong adsorption only occurs in the presence of multivalent counterions or when hy- drophobic forces are operational [21,22]. Overall, the like-charged situation remains poorly investigated at low concentrations of macroions.

In the oppositely charged situation, the macroions are strongly attracted to the interface, leading to tightly bound adsorbed layers. Such adsorbed layers were investigated in detail for a wide variety of systems [18e 20,23e25]. Nanoparticles and polyelectrolytes behave similarly. In both systems, a tightly bound monolayer is formed at the interface, which saturates at a well- defined adsorbed amount [19,20]. Saturated monolayers of macroions normally induce a charge reversal of the substrate, which means that the double layer charge (or the electrokinetic charge) is opposite to one of the substrates [18,19,23,26,27]. Furthermore, adsorption of the macroions is not an equilibrium process but is irreversible and kinetically limited [18,19,28,29].

The present article focuses on the interesting phenomena occurring in concentrated solutions of

macroions, both for nanoparticles and polyelectrolytes [30e35]. A frequent situation for such concentrated systems is that they often are (close to) salt-free, meaning that the charge originating from the ionized groups of the macroions is neutralized by small, monovalent counterions. The reason why salt-free conditions are more frequently encountered for concentrated systems is that the concentration of the counterions dissociating from the macroions can be relatively high (say mM), and may easily exceed the concentrations of the ions originating from trace amounts of salt. For dilute macroion solutions, salt-free conditions are more difficult to realize since the concentration of the counter ions originating from the macroions is low. Attempts to obtain salt-free conditions with multivalent counterions typically lead to phase separations and/or the formation of precipitates [36]. The present article mainly focuses on salt-free systems. The effects of salt will be addressed only superficially, mainly since experimental studies of such systems are scarce.

Concentrated nanoparticle suspensions or polyelectrolyte solutions form liquid-like structures under salt-free conditions. This structuring is best illustrated for charged nanoparticle suspensions with the so-called

‘pair correlation function’ shown in Figure 2a. The function shown was calculated for a suspension of charged nanoparticles under salt-free conditions theoretically by Gonzalez-Mozuelos et al. [37]. Similar results were obtained computationally and experimentally, the latter especially with small-angle neutron or X-ray scattering [34,38e40]. This function represents the local concentration of macroions around a given macroion placed at the origin, which is normalized to the bulk concentration. The double-layer repulsion between the charged nanoparticles induces a nanoparticle-free shell in its immediate vicinity. As a consequence, the pair correlation function vanishes at short distances. As the distance increases, this function becomes oscillatory and typically features several peaks. These peaks reflect the formation of spherical shells of macroions around the central macroion. For larger distances, this structuring disappears, and the concentration attains the bulk value.

In the presence of salt, the extent of structuring di- minishes, and with increasing concentration of the macroions or the salt, the systems may be destabilized.

Polyelectrolytes behave similarly to nanoparticles at lower concentrations, as long as the polymer chains do not interpenetrate. These conditions are referred to as the dilute regime. At higher concentrations, the chains interpenetrate, which then marks the onset of the semi- dilute regime. Such liquid-like structuring may already occur for relatively low concentrations, sometimes even at fractions of g/L. These low concentrations are caused by repulsive double-layer forces, which lead to appre- ciable interactions between the macroions already at distances that are much larger than the macroionic

Figure 1

Scheme of an aqueous solution of macroions near a charged water–solid

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macroions and is most pronounced in salt-free systems.

When the macroions would interact with short-ranged forces, for example, as hard spheres, liquid-like structuring occurs only at substantially larger concentrations, typically a few tenths of g/L. At higher concentrations nanoparticle suspensions may form colloidal crystals [13,41]. Interesting size selection phenomena may occur in polydisperse systems [42].

Interactions between wateresolid interfaces in concentrated solutions of macroions were also investigated to some extent [30,32e35]. In a way, the state of research is opposite to the situation in dilute macroion solutions. We are currently starting to understand the like-charged situation relatively well, while there is still little known concerning the oppositely charged case.

The relevant aspects of the like-charged situation can be again nicely illustrated with the calculations of Gonzalez-Mozuelos et al. [37,43]. These studies sug- gest that the concentration profile of the nanoparticles next to an isolated interface feature pronounced layering, see Figure 2b. Next to an isolated interface, this layering leads to an oscillatory concentration profile,

which resembles the pair correlation function in the bulk, as recently demonstrated by reflectivity experiments [39,44]. From these experiments, the respective concentration profiles could be extracted, and they resemble the one shown inFigure 2b closely. For the slit geometry, the layering develops between the two interfaces, and the concentration profile can be thought as the superposition of the deviations of the respective profiles from their bulk values, which originate from each interface, see Figure 2c. While measurements of the concentration profiles in the slit geometry are now becoming possible [45], the slit geometry is best investigated by direct force measurements. In these experiments, which were carried out with nanoparticles, as well as with polyelectrolytes, the layering manifests itself by the appearance of characteristic oscillatory force profiles [30,32e35]. In the presence of salt, the extent of this layering decreases, but the salty regime was not investigated in much detail yet [46].

Direct force measurements provide access to the free energy per unit areaWversus the separation distanceh, or to the corresponding pressureP. These two quantities are related as [12,13].

Figure 2

Concentration profiles involving charged nanoparticle suspensions and like-charged interfaces as calculated by Gonzalez-Mozuelos et al. [37,43]. The schemes above illustrate the structuring pictorially and define the distance coordinates used. The concentrations are normalized to the bulk value, while the distancesrandxto the particle radiusa.(a)Pair-correlation function in bulk. Concentration profiles(b)near a like-charged interface and(c)in a slit between two like-charged interfaces with a surface separation of 90a.

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P ¼ dW

dh (1)

Figure 3a and c shows the typical distance dependence of these two quantities for two charged interfaces in a concentrated salt-free solution of like-charged macroions.

Let us briefly discuss the qualitative features of these profiles. At small distances, the macroions are expelled from the gap between the interfaces, and the pressure is

large and positive due to the electrostatic repulsion of the charged interfaces. At larger distances, the pressure becomes oscillatory, which indicates the layering of the macroions in between the interfaces. The first minimum of the free energy, which corresponds to the first zero in the pressure profile, reflects the formation of a stable monolayer of the macroions. The second minimum in the free energy, which corresponds to the third zero in the pressure, indicates the formation of a stable bilayer of macroions. The situation depicted inFigure 2c would correspond to the fourth minimum of the free energy, but this minimum is not shown on the scale of

Figure 3

Interactions between two identical charged plates across a solution of like-charged macroions.(a)Free energy per unit area and(c)pressure versus separation are indicated as the solid line while the near-field and far-field profiles are dashed. Comparison of the exact solution of the PB equation with CC boundary conditions with the near-field approximation given by eq.(14)and the salt-free contribution given by eq.(12)in(b)linear and(d)semi- logarithmic representation. The osmotic contribution given by eq.(12)is shown in(b), and the DH limit given by eq.(11)in(d). The schemes indicate (i) the macroion-free gap, (ii) the monolayer, and (iii) the bilayer of macroions in the gap. The calculations were carried out with the model described in Sect.

3 for a 1:Zelectrolyte with a concentration of 10mM and a macroion valenceZ= 150. The far-field behavior only features a damped oscillatory function, which is characterized with a wavelengthl¼40 nm and a decay lengthx¼35 nm. The interface has a charge density of−10 mC/m². The merging point is located athm ¼36 nm and indicated with (B). The values of the zero of the free energyh0and of the double layer thicknesshdlare also indicated with (△) and (,).

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Figure 3c. The progressive decrease of the oscillatory amplitude with distance is due to the increasing disor- der within these layers.

The oppositely charged situation in concentrated solutions of macroions is poorly investigated [31,47]. It seems clear, however, that the adsorbing macroions induce a charge reversal, and one is then faced with a similar situation as in the like-charged one, seeFigure 4a and b. The correctness of this scenario was confirmed with cationic polyelectrolytes interacting with a negatively charged silica substrate [31]. This study has indeed shown that oscillatory forces also occur in these systems, which indicates that the adsorbed polyelectrolyte layer is sufficiently smooth such that the oscillatory structuring occurs. The situation appears similar for nanoparticles [47], but in that situation, various aspects remain to be clarified.

An interesting but poorly investigated phenomenon occurs at higher electrolyte concentrations. In these systems, the nanoparticles will start to aggregate, but they may not only do that in bulk but also at the interface, see Figure 4c. In some situations, such aggregates form at the interface first, and this phenomenon is referred to as ripening [48]. A similar scenario is expected for polyelectrolytes, but it has been hardly investigated.

The layer-by-layer deposition technique provides the possibility to build thicker macroion films. The principle of this technique is simple. One immerses a charged substrate into a solution of oppositely charged macroions, and as discussed above, a tightly bound monolayer of the macroions will adsorb to this substrate. The charge of this substrate is now reversed, and this substrate is now used again to absorb another type of macroion of opposite charge. This sequence of adsorption steps can be repeated many times, and relatively

thick surface films can be fabricated in this fashion. This approach was initially proposed to build multilayer films with two types of oppositely charged nanoparticles [49].

This technique was later adapted to create layers with oppositely charged polyelectrolytes [50e52] and layers with polyelectrolytes and nanoparticles, see Figure 4d and e [53].

The purpose of the present article is to review our current understanding of interactions between charged interfaces in concentrated solutions of macroions. We shall mainly focus on salt-free systems in the like- charged situation and the symmetric slit geometry as realized in the direct force measurements. Numerous high-quality experiments are available for these systems, and a quantitative understanding of the interaction forces in these systems is emerging. Some remarks on the oppositely charged systems will be equally made, but these will remain more superficial, as only a few experiments have been realized in these situations so far.

Debye-Hückel and Poisson-Boltzmann approach

The classical approach to quantify the structure and thermodynamic properties of electrolyte solutions relies on Debye-Hu¨ckel (DH) and Poisson-Boltzmann (PB) theories. These theories can also be adapted to solutions of macroions, whereby the macroions are treated as highly charged multivalent ions in the presence of small monovalent counterions. This situation is also referred to as a highly asymmetric electrolyte.

For an isolated interface or a slit between two parallel interfaces in contact with a bulk electrolyte solution, which contains ions of chargez_iexpressed in units of the elementary chargeq, the electric potentialjobeys the Poisson equation [12,13,54,55].

Figure 4

Illustrations of structuring in solutions of macroions near a charged interface.(a)Suspension of oppositely charged concentrated nanoparticles,(b) solution of oppositely charged polyelectrolytes,(c)ripening in a nanoparticle suspension,(d)polyelectrolyte multilayer, and(e)a multilayer fabricated with polyelectrolytes and nanoparticles.

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d²j

dx² ¼ q ε0ε

X

i

z_ibc_i (2) where x is the coordinate normal to the interface,ε0 the permittivity of vacuum,ε is the dielectric constant of the solvent, andbciis the local number concentration of typei. These local concentrations are given by the Boltzmann law bci¼ cie^zⁱ^bqj (3) whereci is the number concentration in the bulk electrolyte solution andbis the inverse thermal energy, namely b ¼ 1=ðkTÞwhereby k is the Boltzmann constant andT the absolute temperature. Thereby, the ionic energies are approximated with the electrostatic energies of the respective ions. The combination of eqs. (2) and (3) is referred to as the Poisson-Boltzmann (PB) equation, from which the potential and concentration profiles can be evaluated. The advantage of the PB equation is that it can be solved analytically for various situations, and otherwise, the numerical solution is also rather straightforward.

Obviously, the PB equation is an approximation, whereby the effect of other ions on a given ion is approximated in a mean-field like fashion. This approach ignores several effects. First, ioneion correlations are neglected, meaning that the structuring of the ions is not considered. Second, the dielectric response of the medium, which varies with the ionic concentration, is ignored.

Finally, all ions are assumed to be point-like, whereby all excluded volume interactions and packing effects are neglected. These effects may become important for concentrated systems, high surface charge densities, or in the presence of multivalent ions.

For low electric potentials, the PB equation can be linearized and reduces to the DH equation [13,54].

d²j

dx² ¼k²j (4)

where k¹ is the Debye length defined by the following relation

k²¼ bq² ε0ε

X

i

z²_ici (5)

The DH equation, given in eq.(4), can be solved for a charged interface, where one finds an exponential potential profile

jðxÞ ¼j_De^kx (6) where j_D is the diffuse layer potential. In this situation, the concentration profiles also vary exponentially at larger distances, namely

bciðxÞ

ci ¼ 1zibqj_De^kxþ::: (7)

Based on this solution, the surface charge densityscan be expressed in terms of the diffuse layer potential by means of the linear charge-potential relationship s ¼

εε0kjD.

The exponential dependence given in eqs.(6) and (7)is also preserved for the full PB equation. However, this dependence is only found at larger distances, referred to as the far-field region. The reason for this behavior is easily understood. At larger distances, the electric potential is low, which is precisely the situation where the DH approximation is valid.

Within the PB description, another relevant regime is encountered at short distances, also referred to as the near-field region. When the surface is highly charged, the magnitude of the electric potential becomes high as well, and all co-ions, which have the same sign of charge as the interface, are expelled from its vicinity. In this case, only the counterions remain, which leads to the formation of a salt-free layer close to the interface. For simplicity, we assume that these counterions are monovalent, and we thus, suppress the indexi. In this case, the sum in eq.(2)reduces to a single term, and the PB equation can then be solved analytically with the result [56,57].

jðxÞ ¼ 2

bqlnðxþLGCÞ þconst: (8) whereLGC ¼ 2ε0εkT=ðqjsjÞis the Gouy-Chapman length, and refers to a negatively and positively charged interface, respectively. The concentration profile then follows from eq.(3)as [56,57].

b

cðxÞ ¼2εε0

bq²$ 1

ðxþLGCÞ² (9)

The ionic profile is now a slowly decaying hyperbolic function, which is markedly different from the classical exponential decay given in eq.(7).

Let us now focus on the slit geometry in more detail and discuss the interactions acting between such interfaces. We consider two parallel interfaces separated by a distance h. When one considers the pressure acting between interfaces, which is the force per unit area, this quantity can be evaluated within the PB approximation from the concentration and potential profiles as [13,55].

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P¼ kTX

i

bcici

εε0

2 dj

dx ₂

(10)

The first term corresponds to an osmotic pressure contribution, while the second term is the contribution from the electric field. This relation applies for any positionxand can be used to derive the PB equation. For two identical interfaces, the potential profile is symmetric and has a minimum at the midplane. At this position, the second term in eq.(10)vanishes, and one can evaluate the pressure just by knowing the electric potential at the midplane. When the interfaces are infinitely far apart, the pressure vanishes, and this situation becomes equivalent to a pair of two isolated interfaces.

For sufficiently large separations, the potential at the midplane can be evaluated by superposing the electric potential profiles of the two respective interfaces. When this argument is applied to the exponential profile given in eq. (6), one finds that the pressure profile obeys for sufficiently large distances [13,55,58].

P ¼2εε0k²j²_De^kh (11)

The pressure profile follows the same distance dependence as the concentration profiles since this pressure results from concentration differences within the slit and the bulk, see eq.(10). At larger distances, one also finds the same pressure profile for the PB situation, but the diffuse layer potential jD must be replaced by the appropriate, effective potential. The pressure dependence is thus always exponential in the far-field region.

At shorter distances, the pressure profiles become more complicated, especially since charge regulation effects come into play. In that case, the surface charge density of the interface may vary with the distance due to ionic adsorption equilibrium. In this context, the simplest situations to consider are the constant charge (CC) and constant potential (CP) conditions. For the CC conditions, the interfaces maintain a constant charge upon approach, while for CP conditions, the double layer potential remains constant [13,55,58].

In the salt-free situation, the long-distance behavior of the pressure profile can also be evaluated analytically.

While the superposition approximation yields the proper functional dependence, the correct prefactor can only be found by solving the PB equation in the slit geometry.

The corresponding result is due to Langmuir and reads [32,59].

Psf ¼2p²ε0ε b²q² $ 1

ðhþ2LGCÞ² (12)

The pressure now also follows a hyperbolic function.

The salt-free near-field region is often considered to be somewhat academic since in many situations, this region is very thin. However, this region becomes very promi- nent in highly asymmetric 1:Z electrolytes when the charge of the multivalent ions has the same sign as the charge of the interface. As we shall see, this situation is relevant for solutions of charged macroions in contact with a like-charged interface, whereby one mimics the macroions as point-like ions.

The multivalent ions are excluded from the vicinity of the interface through their strong electrostatic repulsion, which results in a salt-free layer containing only monovalent counterions. However, the important dif- ference to the salt-free situation is that this layer has a finite thickness, and beyond this layer, the decay follows the regular DH decay. This effect can be considered by including an attractive osmotic pressure term [32].

Pos¼ kTð1þZÞc (13) where c is the concentration of 1:Z electrolyte. The approximate pressure profile in the near-field region then becomes

Pnf ¼ PsfþPos (14) wherePsfis the salt-free result given by eq.(12)andPosis the attractive osmotic pressure given by eq.(13).

This result is compared with an exact numerical PB calculation inFigure 3b. One observes that the approximation given in eq.(14)describes the exact result very accurately. The latter result was obtained by solving the PB equation numerically and by invoking CC boundary conditions [55,60]. In this situation, the boundary conditions are unimportant since they influence this profile at very short distances only. The different contributions to eq.(14) are also illustrated in Figure 3b. Note that this approximation is only valid for sufficiently high va- lences, namely forZ[1.

At larger distances, the pressure should decay exponentially according to eq.(11). This regime can indeed be identified in the logarithmic representation given in Figure 3d. This DH decay is very steep since the corresponding Debye length is very small.

The pressure predicted by eq. (13) vanishes at a distance h ¼ hdl. When the surface charge is sufficiently

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large, we have LGC/0, and the respective distance becomes

hdl ¼

2p²εε0

bq²Zc 1=2

fc¹⁼² (15)

For larger distancesh>hdl, the pressure is small, and can be neglected.

More recently, an interesting approach was put forward regarding how the ionic concentration profiles are modified at higher concentrations by ioneion correlation and polarization effects [61,62]. At larger distances, the exponential dependence, as suggested by eq.(7)of the concentration is preserved, and the far-field region is typically dominated by three exponentially decaying terms, namely

bciðxÞ

ci ¼ 1þA^ðiÞ₁ e^k¹^xþA^ðiÞ₂ e^k²^xþA^ðiÞ₃ e^k³^xþ:::

(16) whereby the decay lengthsk₁¹,k₂¹andk₃¹are different from the classical Debye length given by eq.(5). Note that the prefactorsA^ðiÞ₁ ,A^ðiÞ₂ , andA^ðiÞ₃ depend of the type of ion, while the different decay lengths are identical for all ions.

At low concentrations, only the first term survives, and the decay length k¹₁ reduces to the classical Debye length.

The deviations at higher concentrations are caused by the microscopic dielectric response of the medium. Thereby, the dielectric constantε and the valencieszi entering eq.

(5) become functions of k, and this equation may have multiple solutions fork. This approach has been referred to as the extended Debye-Hu¨ckel (EDH) theory.

Kirkwood [63] has already pointed out that at higher concentrations, these solutions may also become complex, and recent work has reiterated on this aspect in more detail [61,62,64]. In that case, two of the exponential functions can be combined into an oscillatory component, namely

bciðxÞ

ci ¼1þD^ðiÞe^x=xcos 2px

l þq^ðiÞ

þA^ðiÞ₃ e^k³^xþ:::

(17) wherebylandxare the wavelength and the decay length, andD^ðiÞandq^ðiÞrepresent the prefactor and the phase shift.

All these parameters are now real numbers. Since these concentration profiles enter into the pressure relation given by eq. (10), we surmise the same dependencies for the pressure profile. Particularly, for the concentration profile given in eq.(17)one expects a far-field pressure

Pff ¼A e^x=xcos 2ph

l þq

þBe^x=z (18)

where z ¼ k₃¹ is the decay length of the exponential contribution. One of the main arguments of the present article is that the oscillatory forces observed in colloidal particle suspensions represent another manifestation of this situation. One should note that eq.(18)was initially proposed for charged hard-spheres by Kralchevsky and Denkov [65]. These authors have argued that the first term originates from packing effects, while the second one reflects the decay of charge density within the diffuse layer. A more recent but related interpretation suggests that the oscillatory term indicates the decay of the concentrations, while the exponential one the decay of the charge density [66].

The present simplified EDH or PB theories should be contrasted to more rigorous approaches to study the interactions between interfaces in solutions of macroions. One normally assumes that macroions interact with a screened Coulomb potential and that their interaction with the interface can be described with an exponential law [34,37,43,67]. The fact that polyelectrolyte solutions induce oscillatory forces between interfaces was equally established theoretically [68].

The most straightforward way to obtain exact distribution functions and pressures is by Monte Carlo simula- tion techniques. This approach was used to calculate these quantities for polyelectrolyte solutions [68], as well as nanoparticle suspensions [34,37,43,68,69].

However, this approach remains computationally expensive, and for this reason, alternative but approximate schemes have been developed. One technique of this type is based on integral equation theories, whereby the pair distribution function is related to the direct correlation function, which is then calculated through an approximate closure relation. Such an integral equation theory was used by Gonzalez-Mozuelos et al. [37,43] to evaluate the correlation functions and concentration profiles, which were already highlighted in Figure 2.

Integral equation theories were also used to analyze the asymptotic decay of concentration profiles through the pole structure in the complex plane, and the emergence of oscillatory concentration profiles could be demonstrated as well [34,69].

Another approach is based on density functional theories, whereby a suitable but approximate functional for the free energy of the system is devised. The quantities of interest are obtained through the minimization of this functional, which is again carried out numerically. This computational scheme was further used to evaluate pressure profiles in nanoparticle suspensions [67,70].

While integral equation and density functional theories remain approximate, they can yield very accurate results, as can be verified by comparing with respective Monte Carlo simulations [34,37,43,67]. While these schemes are less computationally demanding than full

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Monte Carlo simulations, they still require quite some computational effort and are not yet practical for quantitative comparison with experimental data. For this reason, the focus here will be on simplified EDH/PB models for data interpretation. But due to continuously increasing computer power, direct comparison with more sophisticated theories and experimental data should become feasible in the near future.

Free energy profiles

Forces between two interfaces across solutions of charged macroions can be measured in various ways. The most popular technique is the colloidal probe technique based on the atomic force microscope (AFM) [35,58,71]. Nevertheless, other techniques were used to measure forces in such systems too. The surface forces apparatus (SFA) was used in one of the early reports of oscillatory forces [72]. Oscillatory potential energy profiles were further established in charged nanoparticle suspensions with optical tweezers [73]. Total internal reflection microscopy (TIRM) was also used to measure such oscillatory profiles in polyelectrolyte solutions [74]. Interferometric techniques have further demonstrated the existence of discrete stable states occurring in thin liquid films made with nanoparticle suspensions [40,75,76]. But the use of these techniques remained sporadic, and it seems difficult to surpass the data quality that can be obtained by the colloidal probe technique. For this reason, we will only discuss force measurements that were obtained with that technique.

The colloidal probe technique employs microparticles as probes, and these are substantially larger than the range of the interaction forces. In this situation, the Derjaguin approximation can be used to relate the measured force F to the interaction free energy per unit area W as [12,13,58].

F ¼ 2pReffW (19)

where R¹_eff ¼ R¹1 þR¹2 whereR1andR2 are the curva- ture radii of the two interfaces involved. This relation in- cludes the two relevant experimental geometries. The sphereesphere geometry involves two identical (or very similar) microparticles, where R1 ¼ R2. Here, we will present several examples of force profiles measured in this geometry. The sphere-plane geometry involves a probe microparticle of radius R1 and an planar interface with R2/N, which are chosen such that the two surfaces are as similar as possible. Such force measurements thus provide access to free energy per unit area W in the plateeplate geometry.

Let us first discuss the qualitative features of measured free energy profiles in suspensions of charged nanoparticles or polyelectrolyte solutions. We will first focus on the by now well-understood like-charged situation, meaning that the nanoparticles or polyelectrolyte have

the same sign of charge as the interfaces. We will further consider the symmetric geometry only, meaning that the two interfaces are (close to) identical.

Figure 5 summarizes typical experimental free energy profiles measured in the sphereesphere geometry with a pair of very similar silica microparticles with a radius of about 2

m

m. The top row shows the profiles on the linear scale while the second row shows their magnitude on a semi-logarithmic scale. Vanishing surface separation refers to hard contact between the silica microparticles.

All measurements were carried out in systems without added salt.

Measurements in a suspension of silica nanoparticles with a radius of 6 nm for two different concentrations are shown inFigure 5a and b [77]. Results of similar measurements in solutions anionic polyelectrolytes, namely with sodium poly(styrene sulfonate) (PSS) of molecular mass of 200 kg/mol and 2300 kg/mol, are presented in Figure 5c and d [32]. These measurements refer to the like-charged systems and will be discussed in detail in the following. Figure 5e shows results for a cationic polyelectrolyte, namely for poly(L-lysine) (PLL) bromide [31]. The latter data involves an oppositely charged system and will be discussed later in Sect. 5.

In spite of the differences between the various systems, one observes that the free energy profiles are similar. At short distances, the interaction energy is strongly repulsive. With increasing distance, the free energy weakens substantially and starts to oscillate. One further observes that the amplitude of these oscillations decreases with distance, and they typically disappear in the experimental noise after few oscillations. Clearly, there is a close similarity between the experimental data and the free energy shown in Figure 3c. The strongly repulsive region at short distances corresponds to the macroion-free gap. The first minimum signals the formation of a stable monolayer of macroions in the slit, while the second minimum signifies a bilayer.

Let us now quantify the free energy profiles with a simple model over the entire distance range. This model follows directly from the arguments presented in Sect.

2, where the distinction between near-field and the far- field regions of the diffuse layer was made. In simple electrolytes, these are limiting regions at short and large distances, and they are separated by a wide and nontrivial intermediate region. In the like-charged situation and for highly asymmetric electrolytes, we surmise that this transition region is narrow, and can be effectively approximated by joining the near-field and far-field profiles. The near-field pressure Pnf given by eq.(14)and the far-field pressurePffgiven by eq.(18) cross, thusPnf ¼ Pff ath ¼ hm. From this condition, the merging distancehmis found, seeFigure 3a. We thus approximate the entire pressure profile as

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P¼

Pnf for hhm

Pff for hhm (20)

The present merging condition further implies that the near-field region is free of macroions, and thathmis the largest thickness of the macroion-free gap. Similar merging procedures were already proposed earlier for the pressure profile of hard-spheres [65,78] and for the free energy profile of soft-spheres [79].

The present merging condition implies the continuity of the pressure profile ath ¼ hm. This assumption differs from the expected behavior for hard-spheres, which features a discontinuity in the pressure at a distance corresponding to the hard-sphere diameter [65,78].

This discontinuity signals the presence of a first-order phase transition [80]. We suspect, however, that this discontinuity only occurs in the hard-sphere system and that in a soft-sphere system, the pressure profile remains continuous. This view is also supported by computer simulations and density functional theories for charged nanoparticles between charged interfaces since the calculated pressure profiles are continuous [67,81].

The free energy per unit area can then be obtained by integrating the pressure profile, see eq.(1)

WðhÞ ¼ Z^N

h

Pðh⁰Þdh⁰ (21)

In the present situation, integration can be carried out analytically. In the far-field region, integration of the pressure given in eq. (18) yields the same functional dependence

Wff ¼ A⁰e^x=xcos 2ph

l þq⁰

þB⁰e^x=z for hmh (22) but the prefactors A⁰ andB⁰ and the phase shiftq⁰ differ from the respective coefficients given in eq. (18). The respective phase shifts are related by

q⁰ ¼qþatanð2px=lÞ (23)

On the other hand, the lengthsx,l, andzare the same as in eq.(18). One should note that eq.(22)was already used to interpret experimental force profiles in nanoparticle suspensions earlier [82]. The integration of the near-field pressure given by eq.(13)yields

Figure 5

Comparison of model calculations with experimental free energy per unit area versus the separation distance in salt-free systems. Linear representation (top row) and semi-logarithmic representation (bottom row) of the magnitude. The merging distancehmis shown as a dotted line. Suspensions of silica nanoparticles with a radius of about 6 nm at a concentration of(a)53 g/L and (b) 218 g/L [77].(b)Solutions of sodium polystyrene sulfonate (PSS) of(c) molecular mass 200 kg/mol and concentration of 100 g/L and(d)of molecular mass 2300 kg/mol and concentration of 10 g/L [32]. Poly(L-lysine) (PLL) bromide of molecular mass of 167 kg/mol and a concentration of 35 g/L [31]. The used sphere–sphere geometry for the direct force measurements with the AFM is depicted in the inset(e).

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Wnf ¼2p²ε0ε b²q² $ 1

ðhþ2LGCÞþkTð1þZÞc h

þconst: for hhm (24) where the constant must be chosen such that the free energyWgiven by eqs.(22) and (24)is continuous ath ¼

hm, seeFigure 3b.

The present matching procedure leads to a continuous free energy profile with a continuous derivative but a discontinuous second derivative. In principle, such a discontinuity signals the presence of a second-order phase transition, but this feature is likely an artifact of the present model. While second-order phase transitions may occur at larger distances, for example, during the transition between a stable monolayer and a bilayer [81,83e86], such transitions are not included in the present model. Even when actually present, detecting such second-order phase transitions with force measurements will be difficult, and they could be masked by the finite size or kinetic effects.

The force profiles have been fitted with this model, and the results are shown inFigure 5. This simple model is indeed capable of representing the oscillatory free energy profile extremely well. All parameters entering eqs.(22) and (24)can be extracted reliably. The surface charge densitysof the substrate can be found from the Gouy-Chapman lengthLGCentering eq.(24).

This model was further used to interpret numerous experimental force profiles from Refs. [31,32,77]. We have observed that those parameters that were already reported in the cited references are very similar to the newly calculated ones, and for this reason, we mostly focus on the already published values in the following.

The merging distancehm turns out to be quite close to the largest thickness of the diffuse layer in the near-field region hdl and to the smallest distance where the free energy crosses zeroh0, namely whereWðh0Þ ¼ 0. The latter distance h0 can be easily extracted from the experimental force profile. We found that hm=h0 ¼ 0.960.11 andhdl=h0 ¼1.110.12, whereby our error bars always represent standard deviations. Indeed, the first zero in the free energy profile h0 is a very good estimate of the merging distance hm and of the largest thickness of the macroion-free gap.

Parameters and scaling laws

Let us now discuss the dependence of the parameters entering the free energy on the concentration of the macroions in salt-free systems. We find it helpful to work with normalized variables, and therefore, we introduce a normalized concentration~c ¼ c=cwhere c is the number concentration and c the corresponding

crossover concentration. For polyelectrolytes, values~c<

1 indicate the dilute regime, while~c>1 the semi-dilute one. The characteristic distance corresponding to the crossover concentration d ¼ c¹⁼³ approximately reflects the size of the polyelectrolyte chain in solution [31]. For nanoparticles, the characteristic distance is defined by d ¼ 2a where ais the particle radius. The crossover concentration is defined accordingly and corresponds to the number concentration in a closed- packed cubic lattice. One always has ~c<1 for solid nanoparticles.

The dependence of the wavelength l on the concentrations has been studied by numerous authors. Here we consider the normalized wavelength ~l ¼ l=d, which scales with the normalized concentration as

~l ¼ ~c^a (25)

where the exponent a ¼ 1=3 below the crossover concentration (~c<1) and a ¼ 1=2 above (~c>1) [30,32,40,87]. Figure 6a confronts this scaling law with experimental literature data for nanoparticles and polyelectrolytes [31,32,73,77,87e92]. Besides PSS, we also include data for poly(acrylic acid) (PAA) [93,94] and poly(2-acrylamido-2-methylpropanesulfonate) (LPAMPS) [95]. One observes that these scaling laws describe the experiments well, although minor but systematic deviations remain. Note that this wavelength is very similar to the wavelength extracted from concentration profile of an isolated interface [44] and reflects the position of the scattering peak in bulk solutions [34].

The decay length x of the damped oscillatory contribution shows a similar behavior as the wavelength, see Figure 6b. For particles, the decay length appears to be concentration independent, but this conclusion is un- certain, since the experimentally accessible concentration range is limited [77]. For polyelectrolytes, the observed behavior suggests thatx=lxconst. [31,32]. For high molecular mass PSS, this ratio is close to unity, while it decreases to about 1/4 for lower molecular masses. These numbers indicate that with increasing molecular mass, the oscillations persist toward larger distances. The amplitude A⁰ of the oscillatory term in eq.(22) increases approximately linearly with the concentration, but no further conclusions are possible due to substantial scatter.

The oscillatory free energy profiles are further somewhat modified by the additional exponential function entering eq. (22). We have investigated the corresponding decay length z, and in spite of substantial scatter, we find that z=x ¼ 0.8 0.3. The fact that these two decay lengths are comparable suggests that both originate from the DH-like screening. The ratio of the two amplitudesB⁰=A⁰ entering eq.(22)has

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typically values around 0.6 0.3. The contribution of the exponential term is typically larger in silica suspensions, while it is lower for polyelectrolytes, especially for the ones of high molecular mass.

The free energy profile in the near-field region is given

strongly repulsive nature and the corresponding thickness. We will express this thickness with the first zeroh0

of the free energy profile, seeFigure 3a. In this region, the actual free energy profile is further influenced by the particle concentration, the valence of the multivalent co-ionsZ, and the charge density of the substrates.

Figure 6

Relevant parameters entering the modeled force profile plotted versus the normalized concentration~c¼c=cin salt-free systems. Normalized(a) wavelengthl=d,(b)decay lengthx=d,(c)thickness of the particle free layerh0=d, whereby the spacing is given byd ¼c¹⁼³. The experimental data points are taken from Refs. [31,32,77], while additional data in(a)are taken from these references and from Refs. [73,87–95]. The solid and dashed lines correspond to eq.(25)witha¼1=2 anda ¼1=3, respectively. The dotted line in(c)is the scaling law given in eq.(26).(d)Phase shiftq⁰entering the free energy. The black solid and dashed lines in(d)correspond to the scaling laws, while the solid lines in color are predictions of the model described in the text.

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charge regulation effects, but these effects are weak and play a role at short distances only.

The thickness of the largest macroion-free gap h0 is shown versus the normalized concentration inFigure 6c.

One observes that this parameter is comparable to the wavelength but typically smaller. The expected scaling behavior suggested by eq. (15) can be expressed in terms of the normalized variables as

h~0 ¼ L~c¹⁼² (26) whereh~0 ¼ h0=dis the normalized thickness. For particles and polyelectrolytes, the dimensionless prefactors L are very close, namely 0.72 and 0.79, respectively. These prefactors follow from the saturation valence Zof 6a= LB

[96,97] and L=LB [32,98], respectively, whereby LB ¼ bq²=ð4pε0εÞis the Bjerrum length andLis the length of the stretched polyelectrolyte. Note that one expects a rod-like configuration in salt-free solutions of highly charged polyelectrolytes [99]. Due to the similarity of these prefactors, we only show eq.(26)with the average valueL ¼ 0:76 in Figure 6c. The overall dependence is rather well satisfied for the different systems investigated, but the actual prefactors do not necessarily correspond to the theoretical one. The main reason for these differences is that the saturation charges used do not always represent an accurate estimate of the actual charge [32,77]. For particles, the measured valence can be larger than the theoretical saturation valence, even up to a factor of two, probably due to remaining traces of salt in the suspensions or polydispersity effects [77,100]. For polyelectrolytes, the experimental valence is lower, sometimes even by a factor of two [31,32].

The surface charge density sof the interface entering the free energy profile depends on the substrate. The interfaces used here are negatively charged silica substrates, and the charge densities observed in salt-free solutions of macroions are similar to the ones in simple electrolytes, featuring a magnitude around 5 mC/m² [32,77,101]. This value refers to mildly acidic conditions for pH 4.0 and increases with pH to the additional dissociation of the surface silanol groups. At pH 10.0, the charge densities typically have a magnitude of 15e 25 mC/m².

The phase shift q⁰ entering the oscillatory part of the interaction free energy eq.(22)in the far-field region is influenced by the wavelength, but also the thickness of the near-field region. This phase shift can be estimated from the condition that the pressure profile in the far- field region has its maximum at the onset of the near- field region, namely approximately at h0. In other words, the phaseqof the pressure can be approximated byqx2ph0=l, and by virtue of eq.(23)provided that the decay lengthxis not too small, one hasq⁰xp=2 2ph0=lþ2pk, wherekis an integer chosen such thatq⁰ is between 0 and 2p. Based on the scaling relations eqs.

(25) and (26)one would expect thath0=lf~c¹⁼⁶for~c<

1 while for ~c>1 one should have h0=lxconst.

Inspecting Figure 6d, one indeed observes that these relations are approximately satisfied, as for ~c<1 the phase increases with the concentration strongly, while for~c>1 the phase remains approximately constant. To obtain more precise predictions of the phase, one must realize that the actual value of the phase shift sensitively depends on the precise value of the wavelength l and the thickness h0. For this reason, it is essential to consider the accurate values of the charges and to include the effect of the finite charge density of the substrate. With these corrections, quite satisfactory predictions of the fitted observed phase shifts can be obtained, seeFigure 6d.

Like-charged versus oppositely charged systems

The above discussion summarized our understanding of interactions between highly charged interfaces in solutions of like-charged macroions. While this situation is realized in numerous systems, the charge density of the interface is an independent parameter. The question then arises as to how these systems behave when the charge of this interface is varied.

Let us first discuss the situation of a highly and oppositely charged interface, see Figure 1b. As has been shown in numerous experimental studies involving nanoparticles, polyelectrolytes, or proteins, such charged macroions adsorb to oppositely charged interfaces strongly [18,19,26]. This coating induces a charge reversal of the interface, leading to an inversion of the double layer potential, also referred to as overcharging.

The charged macroions in solution are now like-charged as this coated interface, and this situation will resemble the like-charged scenario discussed above. At higher concentrations, one thus expects that the solution of macroions will structure in a layered fashion near the interface, as depicted in Figure 4a. Given this structuring, the forces acting between such coated surfaces should resemble the ones in the like-charged situation.

This scenario is supported by the theoretical studies of nanoparticle suspensions by Gonzalez-Mozuelos et al.

[37,43]. These authors have also investigated the oppositely charged situation for nanoparticle suspensions, and they have found that a layer of tightly bound nanoparticles is formed. Next to this layer, an oscillatory concentration profile develops, which is very similar to the like-charged case, seeFigure 2b. To stress this point, the resulting concentration profiles were further compared with the ones near a like-charged interface.

When an appropriate effective charge density for the interface was introduced, the concentration profiles could be reproduced in the oppositely charged situation rather well.

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The oppositely charged scenario was recently realized experimentally with cationic polyelectrolytes and negatively charged silica substrates by Kubiak et al. [31].

One of the measured force profiles for poly(L-lysine) (PLL) is shown in Figure 5e, and fully confirms the scenario described above. The wavelength also shows the expected behavior, see Figure 6a. Similar behavior was also observed for other cationic polyelectrolytes, namely for poly(2-vinylpyridine) (P2VP) and linear polyethylenimine (LPEI). The silica substrate reverses its charge due to the adsorption of the polyelectrolyte, which forms a thin and tightly bound layer at the interface. The interface is then positively charged, and at higher polyelectrolyte concentrations, oscillatory forces were observed, which were similar to the ones in the like-charged case. Indeed, the parameters extracted from these force profiles show analogous trends as the ones for the like-charged systems, seeFigure 6.

While the study by Kubiak et al. [31] nicely confirms the validity of the presented oppositely charged scenario, the corresponding situation in nanoparticle suspensions was poorly studied so far. We are only aware of one study of this kind [47]. These authors prepare positively charged mica substrates by coating them with a monolayer of ceria nanoparticles with a diameter of about 5 nm in diameter, and subsequently, they study the forces between such substrates in suspensions of silica nanoparticles of about 25 nm in diameter and a concentration of 5 g/L. These authors concluded that the silica nanoparticles form a subsequent monolayer on the ceria-coated mica substrate, and they further report an attractive force at larger distances, which they attribute to depletion attraction. This system might have shown oscillatory forces at higher concentrations of silica nanoparticles, but this regime was not studied.

The formation of the tightly bound layer of adsorbed macroions at an oppositely charged substrate is relatively well understood [20,23e25]. This layer has two main characteristics. First, this layer normally leads to a charge reversal of the interface [18,19,23,26,27].

Second, this layer is close to irreversibly bound, and there is hardly any lateral mobility of the adsorbed macroions [18,19,28,29,102,103]. These features can be understood as follows. The main driving force for adsorption surely is the attractive double-layer interaction between the interface and the macroions, but other attractive forces also come into play. These may include the omnipresent van der Waals interactions or hydro- phobic forces, but nonclassical electrostatic forces, such as patch-charge or ioneion correlation forces, may equally contribute. As the adsorption of macroions is basically irreversible, the formation of the tightly bound layer is governed by a sequential, kinetic process [18,19,28,29]. Such a process can be well described by the random sequential adsorption (RSA) model, which

assumes that the interface is progressively covered by irreversibly deposited disks.

Adsorption of macroions normally stops when the adsorbed layer becomes saturated. This saturation occurs due to the lateral repulsion between the adsorbed macroions. The adsorbed amount of macroions at saturation strongly depends on solution conditions. At low salt levels, the adsorbed amount is low since the electrostatic repulsion between the adsorbing macroions is only weakly screened. The adsorbed amount increases with increasing salt level since this electrostatic repulsion weakens due to increased screening. At higher salt levels, one normally reaches saturation around half coverage, which corresponds to the jamming limit for hard disks within the RSA model.

This adsorption scenario is applicable to nanoparticles, as well as to polyelectrolytes. The additional aspect concerning polyelectrolytes is that they substantially flatten during adsorption, leading to very thin films, with typical thicknesses of a few nanometers only [18,25].

This flattening is also observed for other flexible macroions, such as dendrimers or proteins [97,104].

When adsorbed from concentrated solutions, polyelectrolytes may lead to higher adsorbed amounts and thicker films [25,31]. For example, Kubiak et al. [31]

reported that the surface charge density of saturated films of cationic polyelectrolytes substantially increases when adsorbed from more concentrated polyelectrolyte solutions. These effects may be related to poor relaxation of the adsorbing polyelectrolyte or the formation of multilayers. However, the mechanism leading to such thicker polyelectrolyte films is poorly understood.

One should realize that the tightly bound layer may be rough and will have a laterally heterogeneous charge distribution. These features are obvious for deposited nanoparticles, but lateral charge heterogeneities were also reported for adsorbed polyelectrolytes [18]. Such heterogeneities may weaken (or even suppress) the formation of layered structures near such interfaces [105,106]. While Kubiak et al. [31] have demonstrated that for certain polyelectrolyte systems, these layers are sufficiently smooth such that oscillatory forces and the respective layering occur, and this situation may be specific to (some) polyelectrolytes. The presence of tightly bound layers of nanoparticles on interfaces may inhibit the formation of layered structures in concentrated suspensions, since saturated tightly bound layers of nanoparticles are expected to be more heterogeneous.

While an interesting attempt to address this question has been presented [47], such situations have not been studied experimentally in sufficient detail so far. The available theoretical studies are possibly of limited use to address such questions since lateral relaxation within such films is normally assumed. In real systems,

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however, existing lateral surface heterogeneities lead to frozen lateral structures, where such relaxation pro- cesses are basically inexistent.

The transition from the like-charged to the oppositely charged case is poorly understood. In the oppositely charged case, it has been experimentally observed that a tightly bound layer forms even for almost neutral interfaces, but the saturated amount remains low [97].

With increasing surface charge density, the adsorbed amount increases, especially at low salt levels. The reason for this increase is that the ions within the diffuse layer of the interface also contribute to the screening of the repulsive interaction between the macroions. In the like-charged situation, no tightly bound layer seems to be present, provided the interface is sufficiently highly charged. When the magnitude of its charge density decreases, a tightly bound layer should form at one point.

However, this transition has only been documented in theoretical studies so far [107e109].

Conclusion and outlook

The present article provides an overview of our current understanding of interactions between charged interfaces in concentrated solutions of charged macroions.

These systems include polyelectrolyte solutions, as well as nanoparticle suspensions, and they bear many similarities. One must distinguish the like-charged and the oppositely charged situation, see Figure 1. The like- charged situation is relatively well understood. In this case, macroions form a layered structure in the proximity of the interface, which induces an oscillatory pressure profile between two interfaces. At shorter distances, the force becomes strongly repulsive, which is caused by the formation of a macroion-free gap where only counterions are present. These forces can be described quantita- tively over the whole distance range by a relatively simple model, and the relevant parameters scale with the concentration in a characteristic fashion.

In the oppositely charged situation, macroions form a saturated, tightly bound layer at the interface. Such a layer does not form in the like-charged case. This layer leads to a charge reversal of the interface and induces the formation of similar layered structures as in the like- charged case. This scenario has been confirmed for polyelectrolytes near an oppositely charged interface [31]. The structuring probably also occurs in nanoparticle suspensions, but this aspect has not been demonstrated experimentally so far. While the presence of charge reversal is well established, the role of roughness and/or charge heterogeneities of the tightly bound layer is un- known. These effects may suppress the formation of the layered structure and the presence of oscillatory forces.

The transition between the like-charged and oppositely charged situation is poorly understood. The essential

question concerns the onset of the formation of the tightly bound layer of macroions near a weakly charged interface. One theoretical study suggests that this onset occurs within the oppositely charged situation [107], but that study does not consider the role of additional short- ranged attractive forces. For this reason, we suspect that this onset rather occurs for a weakly charged interface within the like-charged situation, as also theoretically proposed for polyelectrolytes [109]. However, systematic experimental studies under such conditions are missing, and they would be essential to settle this question.

Another open issue concerns the interactions between different types of interfaces in an asymmetric slit. When the two interfaces are oppositely and highly charged, one expects that a tightly bound layer forms at one of these interfaces, whereby a charge reversal is induced.

One is then faced again with the like-charged situation, where one expects the formation of a layered structure and oscillatory forces. However, the charge densities of the two interfaces will be different, and it is unclear how these interactions will be affected by this asymmetry. A similar situation is expected to occur in the asymmetric like-charged case. While the formation of a layered structure is expected, the details of the interaction forces might differ substantially from the ones in the symmetric slit. Recently, it was demonstrated that charge regulation effects become very important in such asymmetric systems, especially when one of the interfaces is neutral or weakly charged [60]. Whether charge regulation also influences interactions between unequally charged interfaces in concentrated suspensions macroions remains to be seen.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research was supported by the Swiss National Science Foundation and University of Geneva.

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