Ion size correlations and charge reversal in real colloids

Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 24–30

Ion size correlations and charge reversal in real colloids M. Quesada-P´erez a , E. Gonz´alez-Tovar b,d , A. Mart´ın-Molina c , e,∗ ´ M. Lozada-Cassou d , R. Hidalgo-Alvarez a

Departamento de F´ısica, Universidad de Ja´en, Escuela Universitaria Polit´ecnica de Linares, 23700 Linares, Ja´en, Spain ´ Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı, Alvaro Obreg´on 64, 78000 San Luis Potos´ı, S.L.P., Mexico c Laboratoire de Physique Statistique de l’Ecole Normale Sup´ erieure Associ´ee au CNRS et aux Universit´es Paris VI et Paris VII, 24 Rue Lhomond, 75231 Paris Cedex 05, France Programa de Ingenier´ıa Molecular, Instituto Mexicano del Petr´oleo, Eje Central L´azaro C´ardenas 152, 07730 M´exico, D.F., Mexico e Grupo de F´ısica de Fluidos y Biocoloides, Departamento de F´ısica Aplicada, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, Granada 18071, Spain b

d

Available online 2 August 2005

Abstract For many decades, the Gouy–Chapman model, whose cornerstone is the Poisson–Boltzmann equation, has been the traditional approach to describing the electric double layer (EDL). Since the early 1980s, a great amount of theoretical work (mostly computer simulations and integral equation theories) has proved that this classical picture of the EDL presents severe failures in the case of electrolytes with multivalent ions, as a result of neglecting ion size correlations. The overlooking of the phenomenon of charge reversal is probably one of the most representative examples of such deficiencies. This work is a critical survey on the relevance of ion size correlations in real colloidal systems (focused mainly on solutions with multivalent counterions). A sophisticated electrophoresis theory (in which ionic steric correlations are taken into account) will be applied to analyze experimental data, which will be also compared with predictions of the classical approach. In addition, we will discuss to what extent ion size correlations contribute to charge reversal in colloids of biological nature and other real colloids. Unlike the classical Poisson–Boltzmann approach, the presented theory describes the charge inversion that occurs within aqueous latexes when increasing the trivalent aqueous electrolyte concentration well above the mmolar range. © 2005 Elsevier B.V. All rights reserved. Keywords: Colloids; Electric double layer; Charge reversal; Electrophoretic mobility

1. Introduction Nowadays, there is a growing and renewed interest in studying colloidal dispersions with multivalent counterions, mostly because we can find fascinating phenomena in the case of biological macromolecules in the presence of such multivalent charges. A representative example is the DNA condensation, which takes place in the presence of trivalent and tetravalent counterions [1]. Another instance related to the previous one is the appearance of attractive forces of electrostatic nature between highly charged DNA molecules, which occurs if a critical trivalent ion concentration is exceeded ∗

Corresponding author. Fax: +34 958 243214. ´ E-mail address: [email protected] (R. Hidalgo-Alvarez).

0927-7757/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2005.06.034

[1]. These attractive electrostatic forces appear also between other many kinds of colloidal particles (for example, mica surfaces [2,3]). Whichever the case, the DNA condensation and the existence of attractive electrostatic forces between likely charged particles are possible because there is a great concentration of counterions near the charged surface. This concentration can be so large that the surface charge is neutralized and even overcompensated. We will refer to this situation as charge reversal (in the past, however, the term overcharging has been incorrectly applied to name this effect) [4]. Such overcompensation of charge causes, in turn, that the effective electrical field produced by the particle (or electrode) plus the counterions reverses its direction with respect to the unscreened electrical field. Then, the effective reversed electrical field produces a concomitant layer of coions, next

M. Quesada-P´erez et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 24–30

to the first counterion layer, which is referred to as charge inversion. In any case, charge reversal can be revealed by electrophoresis as a result of a possible reversal of the mobility. In fact, Ottewill and Shaw reported a mobility reversal for carboxylic latexes in the presence of trivalent counterions many years ago [5]. Such charge reversal phenomenon was justified by counterion adsorption. We should however point out that the DNA condensation, the electrostatic attraction between likely charged colloidal particles and charge reversal have another feature in common: all of them can also be explained with approaches that consider certain ion correlations, not included in the classical Poisson–Boltzmann (PB) theory. Correspondingly, it has been widely proved that this classical treatment fails in the case of EDLs with multivalent ions. The reader interested in these topics is referred to recent reviews addressing the notion of ion correlation in colloidal dispersions and related issues [6–9]. In this work, we will concentrate on the ionic correlations due to ion size. Such correlations began to be included in the electric double layer theory more than two decades ago (see Ref. [8] and many references cited therein). Because of the advent of fast computers, simulations from the early 1980s tested these new approaches. Only recently, however, they have been applied to experimental data [3,9–16] to investigate the relevance of ion size correlations in real colloids. In particular, a widely known integral equation theory, the hypernetted chain/mean spherical approximation (HNC/MSA) [17,18], has extensively been used to analyze the electrophoretic mobility of model colloids [10–16]. However, these studies were restricted generally to the case of large electrokinetic radii (i.e., for high salt concentrations and large particle sizes) so that the Helmholtz–Smoluchowski approximation could be applied to convert the ␨-potential provided by the HNC/MSA into electrophoretic mobility. Accordingly, the aim of this paper is twofold. On the one hand, we will show that the analysis of mobility data can be considerably improved for moderate and low salt concentrations by applying an electrokinetic conversion theory consistent with the HNC/MSA, which has recently been developed by Lozada-Cassou and Gonz´alez-Tovar [11,19] and extensively applied therein to study the electrophoresis of model macroions dissolved in uni- and multivalent electrolytes. In this way, the above-mentioned limitation of previous studies is therefore overcome. On the other hand, a brief discussion about the relevance of ion size correlations is derived from these results and the main conclusions of preceding papers. The remainder of this work is organized as follows. First, we review how ion size correlations are included in the model of EDL by integral equation theories (and, more specifically, by the HNC/MSA). The Primitive Model Electrophoresis (PME) theory for converting the ␨-potential into electrophoretic mobility is also summarized. Then, this theory is applied to predict mobility values, which are compared with experimental results. Finally, some conclusions are presented.

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2. The primitive model and ion size effects The primitive model is a simple representation of a double layer in which ions are considered as charged spheres, but the molecular nature of the solvent is not considered. According to the primitive model, the exclusion volume effect between ions due to their size is taken into account through a shortrange repulsion, usually a hard-sphere interaction potential. Thus the interionic potential can be given by  2  Zi Zj e , r ≥ (ai + aj ), uij (r) = 4πε0 εr r (1)  ∞, r < (ai + aj ), where e is the protonic charge, Zi and ai are the valence and the radius, respectively, of the i-ions, εr is the permittivity of the dielectric continuum (ε0 is the vacuum permittivity) and r is the distance between the ion centres. It should be noted, however, that the interionic exclusion volume term is not included in the Poisson–Boltzmann theory. In addition, we should also recall that in the primitive model used in this work (as well as in many others) the only interaction between ions and charged colloidal particles has also the form given by Eq. (1) (thereinafter the index i = 0 stands for colloidal particles). Therefore, specific interactions between small ions and macroparticles are not included usually in this model. In the following, we will restrict the primitive model to the case of identical size for counter- and coions, i.e., ai = a for i ≥ 1. The Ornstein–Zernike formalism for equilibrium systems can be a suitable starting point for our description of the EDL beyond the Poisson–Boltzmann scheme. Treating the macroparticle-ion correlations by means of the HNC approximation and the ion–ion correlations within the bulk MSA, the resulting HNC/MSA integral equations for the macrosphereion distribution functions, g0i (r), are [18]  g0i (r) = exp  − (u0i (r)/kB T )

+

 

 ρj [g0j (t) − 1]cij (|r − t |) d3 t  ,

(2)

j

where kB is Boltzmann’s constant, T is the absolute temperature, ρi is the bulk number density of species i (= +, −), and cij (|r − t |) are the MSA direct correlation functions accounting for electrostatic and ion size effects between two ions of species i and j, situated at r and t , respectively, i.e. cij (s) = cs (s) + Zi Zj cdsr (s) −

Z i Z j e2 , kB T 4πε0 εr s

(3)

such as cs (s) and cdsr (s) are known algebraic functions containing the ion size contributions. It must be noticed that in Eq. (2) the sum runs only for small ions because it is assumed a diluted dispersion where macroions are infinitely separated and the interaction between them is neglected. In

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order to analyze the EDL for a given surface charge density, σ0 = Z0 e/4πa02 , these integral equations are solved numerically along with the boundary condition:  ∞ Z0 = − Zj ρj g0j (t)4πt 2 dt. (4) a0

j

Once the g0i (r) profiles are obtained, the diffuse or mean electrostatic potential, ψ(r), can be calculated via the Poisson equation written in an integral form (for a system with spherical symmetry)

  ∞ t2 e Zj ρj g0j (t) t − ψ(r) = dt. (5) ε0 εr r r j

Combining Eqs. (2), (3) and (5), it has been shown [20,21] that the HNC/MSA integral equations can be suitably recast as

g0i (r) = exp −(Zi eψ(r)/kB T ) − (Ji (r)/kB T ) , (6) where

    Ji (r) = −kB T ρj [g0j (t) − 1]cs (|r − t |) d3 t j

+Zi

 



Zj ρj [g0j (t) − 1]cdsr (|r − t |) d3 t  .

(7)

equilibrium, all systems are in a state of maximum entropy, hence, it is expected that, at the interface, charged hard-sphere ions would be more adsorbed than point ions, thus increasing the counterion density next to the colloid. For a thorough discussion of this point see reference [4]. Under certain conditions, the total charge of the counterions near the surface can even exceed the charge of the colloidal particle and give rise to charge reversal, as mentioned above. Similarly to the phenomenon of charge reversal, the related effects of charge inversion and the recently reported overcharging can also be understood in terms of an energy/entropy balance in the system, which has been extensively studied by some of us in the past [4,8,22–24]. Concerning the diffuse potential, ψ(r), this EDL property can decrease and even undergo a reversal with increasing the surface charge density as a result of the large concentration of counterions near the particle surface [8,17–19]. However, recent simulations suggest that this decrease (and the subsequent reversal) will not occur for concentrations of counterions close to the steric saturation in the case of electrolytes with divalent counterions [25]. At any rate, if the diffuse potential changes its sign, a mobility reversal can also be expected. To illustrate the notable contrast between the PB and HNC/MSA descriptions of the EDL for multivalent electrolytes, in Figs. 1(a) and 2(a) we portray the ionic

j

It should be stressed that Eq. (6) resembles the Poisson–Boltzmann expression of the ionic distributions, but includes additional integral terms taking into account ion size or short-range correlations (see Eq. (7)). With these integral terms, new situations appear. For instance, the concentration of counterions near the particle surface could be larger than in the PB case. Even more, in some cases, this effect could involve an overcompensation of the surface charge, which cannot be predicted by the PB theory. Why is the concentration of non-punctual counterions near the particle surface larger than in the PB case? Imagine a test counterion near an oppositely charged colloidal particle. This counterion experiences an electrostatic attraction towards the macroparticle. In the PB picture, other counterions behave like a cloud (or swarm) of points, which can interpose between the above-mentioned counterion and the colloid, screening the attraction between them. However, when the ionic size is considered, the space between the macroparticle and a test ion cannot be occupied by others (as a result of their exclusion volumes). The effective attraction will therefore be stronger than in the PB case. Consequently, the counterion concentration near the particle surface will be larger than that predicted by a PB approach. In other words, the inclusion of the ionic size increases the system excluded volume, which, in turn, by elementary statistical mechanics, decreases the bulk entropy, as compared to a system with point ions. On the other hand, by adsorbing particles to the wall the system augments the accessible volume and, then, gains entropy. At

Fig. 1. (a) Ionic distribution profiles (gi (r)) obtained for counterions (upper lines) and coions (lower lines) from the HNC/MSA (solid lines) and the PB approach (dotted lines) for an electrolyte 2:1 (0.0332 M) and σ 0 = −0.115 C/m2 . (b) Mean electrostatic potential (ψ(r)) as a function of the distance from the macroparticle centre calculated from the HNC/MSA (solid lines) and the PB approach (dotted lines). Here a = 0.35 nm, a0 = 98 nm, T = 298 K and ε = 78.5. The PB results are with the Stern correction.

M. Quesada-P´erez et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 24–30

distributions for two representative cases of 2:1 and 3:1 systems, respectively. In these examples the parameters were chosen to make them relevant to the mobility results reported below. It should be pointed out that in our PB calculations we have included the Stern or free ion region, i.e., in the point ion model the ionic size is not considered in the ion–ion interaction, but it is considered in the macroion interaction. Clearly, the ionic profiles in Figs. 1 and 2 exhibit the wellknown quantitative and qualitative differences occurring in high-coupled EDL systems (i.e. with high surface charges, valences, concentrations and/or ionic diameters) when the PB predictions are compared to those of non-punctual treatments (e.g. HNC/MSA). Such discrepancies have been largely documented in the EDL literature since the 1980s [26] and, correspondingly, have been corroborated by many computer simulations and experiments [1,8,27]. In particular, the structural results presented here show that the EDL of HNC/MSA is more compact than that of PB and has larger values of the counterion radial distribution function at HNC/MSA PB (a + a)). Besides, the contact (i.e. g− (a0 + a) > g− 0 PB profiles are always monotonic, a characteristic behavior of this point-ion theory, whereas the HNC/MSA ones can be oscillatory and display charge inversion (see Fig. 2(a)). Finally, a straightforward integration of the 3:1 ionic distributions for HNC/MSA confirms the existence of the charge reversal phenomenon [4,22]. A complementary analysis can now be performed in terms of the PB and HNC/MSA mean

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electrostatic potentials included also in Figs. 1(b) and 2(b). Again, important quantitative separations are observed between the potential curves of the two theories, being the PB potentials always larger than the HNC/MSA ones for all distances. One the other hand, whereas the PB potential functions, ψPB (r), are monotonic and one-signed, those of HNC/MSA, ψHNC/MSA (r), can wave and experience a change of sign (see Fig. 2(b)). Interestingly, it has also been demonstrated by EDL simulations that the importance of the ionic size correlations for the mean electrostatic potential increases with σ 0 , Zi , ρi and/or ai [25], and that, unlike the PB equation, the HNC/MSA formalism copes adequately with such ionic size effects. As can be foreseen, all the related distinctions between the PB and HNC/MSA EDLs will have a decisive effect on the ensuing electrophoretic mobilities calculated from that equilibrium information. Particularly, the marked numerical differences between the diffuse potentials of HNC/MSA and PB near the macroparticle should lead to significant disagreements among their ␨-potentials, which, at last, will be reflected in the mobilities. However, it should be noted that the HNC/MSA only provides an equilibrium description of the EDL. Even admitting that in many cases the ␨-potential is practically identical to the diffuse potential, a conversion theory is required, in general, to calculate the electrophoretic mobility (µe ) from ζ. With such purpose in mind, the PME theory has been lately developed from general principles of momentum, fluid mass and ionic flux balance, and solved by dint of a perturbation method [19] analogous to that employed by O’Brien and White (OW) [28]. In Refs. [11,19] a full account of this novel PME approach and a discussion of its differences with respect to the usual OW theory are presented, however, for the benefit of the reader, a summary of the PME scheme is offered in the following. Starting from the Liouville equation and a molecular expression of the stress tensor, the stationary momentum balance equation for our macroion + electrolyte system, in the  is presence of an external electrical field, E,  (NE)  (NE) η∇ 2 v(r ) − ρi (r )∇uTOT (r ) − kB T ∇ρi (r ) i −

 

i

i (NE)

ρij

 (r , r )∇uij (r , r ) d3 r  = 0,

(8)

i,j

Fig. 2. (a) Ionic distribution profiles (gi (r)) obtained for counterions (upper lines) and coions (lower lines) from the HNC/MSA (solid lines) and the PB approach (dotted lines) for an electrolyte 3:1 (0.0166 M) and σ 0 = −0.115 C/m2 . (b) Mean electrostatic potential as a function of the distance from the macroparticle centre calculated from the HNC/MSA (solid lines) and the PB approach (dotted lines). Here a = 0.44 nm, a0 = 98 nm, T = 298 K and ε = 78.5. The PB results are with the Stern correction.

where η is the viscosity of the incompressible fluid, v(r) is the velocity of the electrolytic solution surrounding the colloid,  · r ], ρij(NE) (r , r ) and uij (r , r ) uTOT (r ) = eZi [ψ(NE) (r ) − E i are the two-particle distribution function and interaction potential for a pair of ions of species i and j, respectively, and the superscript NE means that the quantities correspond to the non-equilibrium state. Eq. (8) is combined with the stationary ionic mass balance (NE)

∇ · [fi ρi

(NE)

(r )v(r ) − ρi

(NE)

(r )∇µi

(r )] = 0,

(9)

M. Quesada-P´erez et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 24–30

28 (NE)

with µi (r ) and fi being the chemical potential and drag coefficient of the ionic species i, respectively. The totality of PME equations is then completed with a generalized form of the chemical potential for an inhomogeneous fluid [29]. In order to solve the PME integro-differential equations, it is assumed that any of the non-equilibrium quantities can be written as A(NE) (r ) = A(EQ) (r) + δA(r ), such as A(EQ) (r) stands for the equilibrium or static value of the quantity and δA(r ) is a small out-of-equilibrium perturbation. The theoretical procedure continues with an expansion, up to linear terms, of Eqs. (8) and (9) to obtain first-order equations for the deviations δA(r )s. Once the supposedly known equilib(EQ) (EQ) rium densities, ρi (r), and chemical potentials, µi (r), HNC/MSA (in our case, furnished by the g0i (r)) are inserted into the linearized equations, they are numerically worked out to finally get the electrophoretic mobility. It must be emphasized that the PME theory incorporates consistently the ionic size effects in the deduction of its fundamental equations and, also, through the use of the HNC/MSA radial distribution functions. Is in this way that the appreciable differences between the PB and HNC/MSA structural results will be mirrored in the electrokinetics of macroions in suspension. Recently, the PME formulation has been employed to analyze in depth the electrophoresis of uni- and multivalent model systems [11,19] and the adequacy of its predictions has been confirmed by modern computer experiments [30,31]. In the present work, the test of PME is extended to case of adjusting mobility data for true colloidal solutions.

3. Results Usually, the electrophoretic mobility data obtained from experiments are analyzed as follows: µe is converted into ζ applying a classical theory (for instance, OW). Then, one can compare this quantity with the diffuse potential (estimated from a classical EDL theory, e.g. PB). Also, the charge enclosed under the shear plane can be estimated from ζ and compared with the surface charge density determined by titration. Before discussing the results, we would like to stress that we have preferred to follow rather an inverse way. From the titrated surface charge density, the diffuse potential, ψd , is obtained applying an EDL theory (the HNC/MSA or the PB approach). Then the ␨-potential is supposed to be the diffuse potential, and it is converted into a theoretical mobility applying an electrokinetic conversion theory. For HNC/MSA data we have applied the PME and for PB we used OW. Finally, we compare the theoretical mobility with the experimental one. First, we will analyze the results obtained for a sulfonated latex (SN10) of σ 0 = −0.115 C/m2 and a diameter of 196 nm in the presence of 2:1 aqueous electrolytes at T = 298 K. The experimental details about the synthesis and characterization of this latex, as well as the procedure for measuring µe are given elsewhere [12,14]. In Fig. 3 we show the experimen-

Fig. 3. Electrophoretic mobility (µe ) as a function of the salt concentration (csalt ). The solid and open circles stand for experimental data with Mg(NO3 )2 and Ca(NO3 )2 , respectively. The solid ad dotted lines are the predictions obtained from the PME and classical (PB + OW) approaches, respectively (with a = 0.35 nm in the former case). The dashed line stands for the PB + OW results incorporating an ion-free region (of width 0.35 nm) adjacent to the macroparticle surface.

tal electrophoretic mobility for latex SN10 as a function of the salt concentration for two electrolytes: Mg(NO3 )2 and Ca(NO3 )2 . As can be seen, the results for these two electrolytes are almost identical and exhibit a minimum at moderate salt concentrations (0.01 M approximately). As usual, the magnitude of the mobility decreases with increasing the salt concentration beyond this minimum. At very high electrolyte concentration, both positive and negative µe -values are obtained in a series of measurements, so a mobility reversal is not conclusively observed. Anyhow, it is worthwhile to compare these results with the mobility predictions obtained starting from the titrated surface charge density and EDL theories, the PB and the HNC/MSA approaches. For the integral equation theory, a hydrated ion radius of 0.35 nm has been used. This value is quite close to that found in the scientific literature for Ca2+ , Mg2+ and NO3 − [2]. The PB mobilities (dotted line) were derived as follows: first a diffuse potential, at the surface of the particle (i.e., we are neglecting the Stern or free ion region correction), was obtained from σ 0 applying approximate surface charge density-electrostatic potential relationships that take the particle curvature into account. Such relationships have been reported by Ohshima and Furusawa [32]. The charge enclosed by the outer Helmholtz plane was assumed to be identical to the surface charge, since the specific adsorption is neglected. Then, ζ (≈ψd ) is converted into µe with the help of the OW theory. As can be seen, considerable disagreement between the PB results and the experiment is found. From a numerical viewpoint, the theoretical mobilities are significantly larger (in magnitude) than the experimental ones. Previous studies have shown that such discrepancies grow with increasing the surface charge density [15]. Qualitatively speaking, one can also observe that the PB model does not reproduce the minimum reported from experiments at 0.01 M. The explanation of this minimum is a major issue in electrokinetics (at least, in the framework of the PB approach) (see, for

M. Quesada-P´erez et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 24–30

instance, reference [33] and others cited therein). At any rate, we could imagine that the Stern layer contains adsorbed ions in order to find better agreement between experiment and theory. In such a case, the surface charge density (σ 0 ) is not an adequate input parameter for the EDL model. Instead, the diffuse charge (which is not usually available a priori) should be used. Despite this, in order to better the operation of the standard electrokinetic model (PB + OW), it is a normal practice in the literature to allow the possibility of a displaced slipping plane, i.e. to consider the hydrodynamical radius of the colloid as an additional parameter in the model. Accordingly, in Fig. 3 we also report (with dashed line) the mobility obtained via the PB + OW scheme but, this time, incorporating an ion-free region (of width 0.35 nm) adjacent to the macroparticle surface. Judging from the two versions of the PB + OW formalism (dotted and dashed lines), the addition of an ion-free zone improves only qualitatively the achievements of the classical electrophoretic theory. On the other hand, we find that the HNC/MSA predicts the experimental values fairly well, both quantitative and qualitatively. In particular, this theory is able to justify the minimum of mobility at 0.01 M, which is not controversial within this new approach. This clearly suggests an important role of ion size correlations in the presence of divalent counterions. However, it should be noted that the mobility reversal predicted by the integral equation theory for high concentrations is not corroborated in this experiment. This absence of mobility reversal has been studied recently by means of computer simulations [25]. To finish the analysis of results for 2:1 electrolytes, we would like to draw the reader’s attention to the coincidence of the experimental mobilities in Mg(NO3 )2 and Ca(NO3 )2 solutions. In certain sense, this supports clearly the relevance of ion size correlations since the hydrated ionic radii of both are supposed to be quite similar [2]. In relation to the more demanding case of 3:1 aqueous electrolytes, the electrophoretic mobility (for the same latex) in the presence of La(NO3 )3 at T = 298 K is now plotted in Fig. 4. The predictions of the classical approach (PB + OW) and the PME theory are also shown. In the latter case, a hydrated ion radius of 0.44 nm was used, which is of the order of trivalent counterions such as Al3+ [2]. Again, to appraise the effect of a displaced slipping plane in the PB + OW treatment, we display its results for two different widths of the ion-free region: 0 nm (dotted line) and 0.44 nm (dashed line). Similarly to the 2:1 situation, we find considerable numerical disagreement between the 3:1 mobility data and the predictions of the two PB + OW versions. In this case, the addition of an ion-free zone does not improve the results obtained from the classical electrophoretic theory. Moreover, the classical approach cannot account for the mobility reversal found experimentally unless a mechanism of specific adsorption is included in the model. Ottewill and Shaw [5] argued that La3+ ions could be electrostatically bound to carboxylic groups, but these groups are not present on the surface of this latex. On the contrary, the PME theory does justify the mobility rever-

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Fig. 4. Electrophoretic mobility (µe ) as a function of the salt concentration (csalt ). The solid circles stand for experimental data with La(NO3 )3 . The solid and dotted lines are the predictions obtained from the PME and classical (PB + OW) approaches, respectively (with a = 0.44 nm in the former case). The dashed line stands for the PB + OW results incorporating an ion-free region (of width 0.44 nm) adjacent to the macroparticle surface.

sal, which takes place at 10–20 mM, approximately. What is more, it is able to fit reasonably well µe -values for salt concentrations below the inversion point. This result improves the one obtained in a previous paper, in which the HNC/MSA together with the Helmholtz–Smoluchowski approximation was applied (instead of the PME theory) [12–15]. This obviously means that this sophisticated approach is suitable for analyzing results concerning colloidal particles with finite curvature and/or low salt concentrations. It must be noted, however, that the PME theory cannot match the experimental results at high electrolyte concentrations. In particular, it is observed a plateau that is not captured by the theoretical approach. The existence of such a plateau was also reported from computer simulations [30] and suggests that the HNC/MSA might fail at very high salt concentrations of trivalent counterions [14]. Expectedly, the substitution of the HNC/MSA, as an input of the hydrodynamical equations, for a better equilibrium description of the spherical EDL (not available at present!) should enhance appreciably the performance of the PME theory for highly concentrated suspensions. Even admitting this fact, it must be stressed that ion size correlations can induce charge reversal and mobility reversals by themselves, i.e., without requiring specific interactions between colloidal particles and small ions (different from those given by Eq. (1)). It should be noted, nevertheless, that the role of the surface groups is very important. Presumably, the PME could justify mobility reversals found for latexes with strong acid groups (sulfate and sulfonate [34,35]), whose inversion points are of the order of millimolar (like the one reported here). Notwithstanding, the PME would hardly justify the mobility reversals reported by Ottewill and Shaw, because they take place at very low salt concentrations (of the order of 10−5 M). Mobility reversals like these have also been reported for biologic systems (such as retroviruses [36] and unicellular algae [37]), and even in the presence of divalent counterions (for liposomes) [38]. In these cases, the existence

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M. Quesada-P´erez et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 267 (2005) 24–30

of specific adsorption is quite reasonable, since simulations and integral equation theories predict much larger inversion concentrations. Unfortunately, the parameters characterizing these effects are usually unknown and are estimated from fitting procedures. However, this should not be done in the framework of PB approach, since, in this way, ion size correlation effects would be wrongly attributed to other phenomena (e.g., counterion binding), yielding erroneous estimations for the corresponding adjustable parameters. Finally, we mention that mixtures of 3:1 and 1:1 electrolytes have also been studied applying the HNC/MSA (but not the hydrodynamic equation formalism that involves the PME theory) [16]. In this case, the HNC/MSA succeeds in predicting certain behaviors (at least, qualitatively), such as the augmentation in the magnitude of µe with increasing the content of monovalent salt. 4. Conclusions To sum up, we highlight the following conclusions. Integral equation theories can explain to a great extent the behavior of sulfonated latexes in the presence of di- and trivalent counterions. This clearly means that ion size correlations play an important role in these cases and they can induce charge and mobility reversals (with trivalent counterions). On the contrary, the standard PB + OW formulation is unable to fit the experimental data, even if a displaced slipping plane is introduced in the model. However, integral equation approaches are much more complex from a conceptual and mathematical viewpoint than the classical PB theory. In addition, an accurate knowledge of the ion size is required to obtain good predictions of the electrokinetic behavior and specific interactions must be taken into account in some cases. Anyhow, their complexity should not be an excuse for applying PB approaches indiscriminately, particularly in the presence of divalent and trivalent counterions. Acknowledgements M.Q.-P. and R.H.-A. are grateful to “Ministerio de Ciencia y Tecnolog´ıa, Plan Nacional de Investigaci´on, Desarrollo e Innovaci´on Tecnol´ogica (I + D + I)” for financial support, projects MAT2003-08356-C04-01 and MAT200301257, respectively. A.M.-M. thanks the grant from “Plan Propio de la Universidad de Granada”. E.G.-T. thanks the financial support from CONACYT, PROMEP and FAIUASLP. References [1] W.M. Gelbart, R.F. Bruinsma, P.A. Pincus, V.A. Parsegian, Phys. Today 53 (September issue) (2000) 38. [2] J. Israelachvili, Intermolecular and Surface Forces, 2nd ed., Academic Press, London, 1992.

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