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Electrophoretic mobility of model colloids and overcharging: theory and experiment. M. QUESADA-PEREZ1, A. MARTIN-MOLINA2, F. GALISTEO-. GONZALEZ.
MOLECULAR PHYSICS, 2002, VOL. 100, N O. 18, 3029 ±3039

Electrophoretic mobility of model colloids and overcharging: theory and experiment M. QUESADA-PEREZ1 , A. MARTIN-MOLINA2 , F. GALISTEO2 2 GONZALEZ and R. HIDALGO-ALVAREZ * 1

Departamento de Fõ sica, Universidad de Jae n, Escuela Universitaria PoliteÂcnica de Linares, 23700 Linares, JaeÂn, Spain 2 Grupo de Fõ sica de Fluidos y Biocoloides, Departamento de Fõ sica Aplicada, Facultad de Ciencias, Universidad de Granada, Granada 18071, Spain (Received 31 August 2001; revised version accepted 11 January 2002)

Several theories claim that ion±ion correlations play an important role in the electric double layer of colloids. One of the most outstanding predictions is overcharging, which would take place at high electrolyte concentrations and surface charge densities. The counterion concentration next to the surface can become so large that the particle charge is overcompensated and the overcharging occurs. Sometimes this would also involve a reversal of the ± potential, but this phenomenon has been observed rarely through mobility measurements. This study explores the matter further. The electrophoretic mobility is measured for latex particles with moderate and extremely large surface charge densities at high ionic strengths (up to 2 M) in solutions of symmetric electrolytes. The results are analysed within the so-called hypernetted chain/mean-spherical approximation (HNC/MSA) and a Poisson±Boltzmann approach. In this way, the relevance of ion±ion correlations in practice and the occurrence of overcharging are probed experimentally.

1. Introduction Colloidal dispersions of electrically charged particles are often found in commercial products (e.g. paints, glues, inks and pharmaceuticals) as well as in natural systems (such as blood or milk). The particle charge can arise from the dissociation of chemical groups on the surface, but also from chemical binding or physical adsorption of ions from the electrolyte solution (salts are usually present in these complex ¯uids). Although small ions are attracted to or repelled from particles, they will remain dispersed and mobile in the solvent due to thermal agitation. The structure of the electrolyte around the charged particles, termed the electric double layer (EDL), plays an important role in colloid science (and biophysics). For instance, it is well known in the theory of colloidal stability that the intersection potential between two colloidal particles is determined by their EDLs. The ability to control this and other phenomena in which colloids are involved (e.g. electrophoresis, sedimentation, phase behaviour, rheology, diffusion) relies upon a theoretical understanding of the EDL.

* Author for correspondence. e-mail: [email protected]

A rudimentary representation of a double layer consists of uniformly charged surfaces (either planar or with constant curvature) immersed in an electrolyte solution comprised of mobile ions, which are treated as small charged hard spheres in a dielectric continuum. Although this simple picture, known as the primitive model (PM) double layer, ignores the discreteness of the surface charge distribution and the molecular nature of the solvent, it is expected to encompass the essential features of reality. The traditional approach was developed by Gouy and Chapman 90 years ago using a model of point charges next to a planar surface and Poisson±Boltzmann (PB) theory. This takes the density of ions to be proportional to the Boltzmann factor of the average electrostatic potential, which is related to the ion density by Poisson’s equation. However, it should be stressed that PB theory is a mean-®eld approach in which correlations between ions are partly neglected. This may be reasonable when the electrolyte concentration and the surface charge are su ciently low, but could lead to quantitatively and qualitatively wrong results in other cases. Since the early 1980s, this matter has been the subject of a great amount of theoretical work [1±22], which has been based on modern methods of statistical mechanics, particularly computer simulations, density functional theory, and especially,

Molecular Physics ISSN 0026±8976 print/ISSN 1362±3028 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0026897021012479 2

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integral equations. Oscillations in the ion concentration pro®les and unexpected changes in the sign of the electrostatic potential were reported and interpreted in terms of an `overcharging’ phenomenon. Furthermore, it is remarkable that the behaviour of the di€ use potential as a function of the surface charge density (for a ®xed salt concentration) is non-monotonic. These predictions became illustrative examples of the failure of PB (GC) theory, since the ion densities and the electrostatic potential must be monotonic (as functions of the distance from the surface) according to this approach. Likewise, the magnitude of the di€ use potential should always increase with increasing magnitude of the surface charge density. However, this burst of theoretical activity about overcharging contrasts strongly with the considerable lack of experimental evidence, particularly for planar and spherical surfaces (with symmetric electrolytes). The PMs including ion±ion correlation predict a reversed di€ use potential at high (but physically acceptable) salt concentrations and surface charges for 2:2 electrolytes, and this should be corroborated by electrophoretic mobility experiments. Up to now, nevertheless, the reversal of mobility (for spherical particles) has been observed almost exclusively for solutions with trivalent counterions [23±26]. In such cases, the inversion has been attributed to the speci®c counterion adsorption. The aim of the present study is to revise some of the abovementioned theoretical treatments and show the extent to which the overcharging e€ ect predicted by them is con®rmed by electrophoretic mobility experiments carried out with di€ erent latex particles and electrolytes. 2. Theory As mentioned before, ionic size correlations are ignored by the classical PB approach. In reality, this is not correct. Each ion is surrounded by an exclusion volume (due to their ®nite size) where other ions cannot enter. Moreover, ions of opposite charge are attracted to its vicinity, whereas like charged ions are repelled from it, giving rise to concentration gradients near the reference ion, i.e. the ion±ion pair distribution function g ij …r† (indexes run over species) is not unity. The consequence of the two e€ ects can be taken into account in the PM context using integral equation theories. For instance, in 1982 Lozada-Cassou et al. applied the so-called hypernetted chain/mean-spherical approximation (HNC/MSA) to the planar EDL. The electrolyte was supposed to be a ¯uid of hard spheres with charge zi e (e is the elementary charge) and radius a (identical for coions and counterions) immersed in a dielectric continuum whose permitivity was "r "0 ("0 is the vacuum permitivity). Furthermore, binary and sym-

metric (z1 ˆ ¡z2 ) electrolytes were considered. The interface was assumed to be a hard wall with a surface charge density ¼0 (< 0). The HNC integral equations for the wall±ion distribution function of species i, g i …x†, turns out to be [27, 28, 3] g i …x† ˆ

"

exp ¡­ zi eÁ…x† ‡ 2p

X j

»j

…1

¡1

0

0

0

#

hj …x †Cij …x; x † dx ; …1 †

where ­ ˆ 1=kBT (T is the absolute temperature), hi …x† ˆ g i …x† ¡ 1 is the wall±ion total correlation function, Á…x† is the electrostatic potential at a distance x from the wall, »i is the bulk density of ions i and Cij …x; x 0 † are integrals depending on the direct correlation functions of the bulk species. These functions can be calculated using the MSA, which has the advantage of being analytical [29] and leads to accurate results. The electrostatic potential can be related to the correlation functions through …1 e X …x† ˆ z …x ¡ t†hj …t† dt: …2 † j »j Á 1 " r "0 j 0 It should be emphasized that the Cij …x; x † functions include ionic size correlations, and in fact they depend on certain parameters characterizing the electrolyte solution, such as its salt molar concentration (csalt ), and the size and charge of ions (see the closed expressions for them in [29] and [5]). If ionic size correlations between ionic species i and j are neglected, the Cij vanish. Consequently, equation (1) becomes the widely known Boltzmann exponential expression (relating the electrostatic potential and the ion distribution pro®le) used in GC theory. Under certain conditions, such correlations are almost negligible, and fair agreement between the HNC/MSA and the PB approaches has been reported [1, 5]. This is the case for solutions of low (and sometimes moderate) concentrations of 1:1 electrolytes whose ions are not too large and in presence of slightly charged surfaces. However, quantitative and qualitative discrepancies can be found if these conditions do not hold. This can be illustrated with a concrete example. Following the prescriptions given in [5], the ion concentration pro®les were calculated, and have been plotted in ®gure 1, for a representative case of high electrolyte concentrations and surface charge densities (¼0 ˆ ¡27:85 ¡ mC cm 2 , 2:2 electrolyte, csalt ˆ 0:2 M, a ˆ 0:2125 nm). According to the PB approach, the counterion concentration should increase monotonically with decreasing the distance x, whereas the coion density should decrease. This is not what the HNC/MSA predicts.

Overcharging in model colloids

Figure 1. Wall±ion distribution functions gi …x† obtained by HNC/MSA (solid lines) and PB (dashed lines) for a 0.2 M 2:2 electrolyte solution in the presence of a charged wall ¡2 (¼0 ˆ ¡27:85 mC cm ). The ion diameter (2a) is 0.425 nm.

The g i …x† functions are no longer monotonic and a test charge moving from a distant point towards the wall would experience an increasing coion concentration and a decreasing counterion density, as if the surface charge would have the opposite sign. Only in the immediate neighbourhood of the wall would the expected trends for these pro®les be found. This apparent charge reversal (`seen’ from a far point) is a consequence of an overcharging caused by a dense layer of counterions next to the wall. Having calculated the gi …x† functions, we can estimate the net charge enclosed by this layer, assuming a thickness a little greater than the ion diameter (0.425 nm), e.g. 0.51 nm. In the PB approach, this ¡2 quantity is found to be 23.26 mC cm . By contrast, ¡2 HNC/MSA theory gives 28.39 mC cm . This value is higher than the magnitude of the surface charge density. Obviously this overcharging must be compensated for by the distribution charge beyond this layer. Consequently, the coion density exceeds the counterion density in this region, which brings out the reversal in the roles of coions and counterions. The concept of overcharging is elucidated readily in this way, but the electrostatic potential pro®le, calculated from equation (2) and shown in ®gure 2, also reveals it. As can be seen, HNC/MSA theory allows for the occurrence of positive electrostatic potentials for a negatively charged surface. Our test charge would therefore feel an electrostatic potential whose sign is opposite to the expected one, except in the vicinity of the wall, where the sign does not su€ er reversal. This is a key point in our discussion, since our experimental technique is electrophoresis and, consequently, we are interested in the ± potential. This quantity is de®ned as the shear plane (SP), which may be up to 2 or 3 water molecule diameters away from the surface

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Figure 2. Mean electrostatic potential Á…x† obtained by HNC/MSA (solid lines) and PB (dashed lines) for the same case as in ®gure 1. The inner ®gure is a logarithmic plot in which the exponential decay ®tting the electrostatic potential at large distances is shown as the dotted line.

[30, 31]. Hereafter, however, the approximation ± º Ád ˆ Á…a† will be applied, where Ád is the potential at the closest approach of the hydrated ions to the wall (known as the di€ use potential). In this particular example, the di€ use potential is negative (like the surface charge). However, if electrolyte concentration or surface charge density or ionic size were increased, positive electrostatic potentials would be obtained for distances from the wall smaller than a. A reversed di€ use potential would be therefore observed. The HNC/MSA predictions for the Ád -¼0 curves in 2:2 electrolytes must be emphasized as well. Besides numerical discrepancies and the potential reversal, it is remarkable again to have non-monotoni c behaviour (contrasting with the PB results). With increasing ¼0 (up to physically reasonable values) a maximum in the di€ use potential is clearly observed (see [5], ®gure 10), even for solutions with moderately low amounts of salt. The abovementioned new features derived from HNC/MSA theory for planar EDLs were also obtained by means of Monte Carlo (MC) simulations for j¼0 j < 30 mC cm¡2 two decades ago [4]. Overcharging phenomena have been reported also using di€ erent versions of modi®ed PB theories [32, 2], density functional approaches [9] and other approximations for integral equations (for comparisons between di€ erent approaches see [33, 34]). For instance, Attard and Kjellander (and their coworkers) have looked into this matter with the HNC and the anisotropic HNC (AHNC) approximation. The latter is based on the socalled inhomogeneous Ornstein±Zernike equation, and attempts to include the anisotropy at the level of ion±ion correlations induced by the charged surface. At any rate, the two studies are mainly focused on the asymptotic

M. Quesada-Perez et al.

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behaviour of the correlation functions and the electrostatic potential, which gives a rigorous procedure of de®ning an e€ ective surface charge for a double layer. We can gain some insight recalling the example that was discussed previously. Particularly, we shall pay attention to the inner plot in ®gure 2, which shows how the tail of the electrostatic potential decays exponentially for this case. This tail could actually be ®tted using the function Á…x† º A exp…¡bx† (where º denotes the asymptotic behaviour), which has the functional form of the solution for the linear PB equation, Á…x† ˆ ¼0 =…"0 "r µ† exp…¡µx† (where µ is the Debye screening parameter). In fact, an asymptotic analysis shows that the mean electrostatic potential behaves as [14, 17, 19] Á…x† º

¼~0 exp…¡µ~x† "0 "~r µ~

x ! 1;

…3†

where the screening parameter µ~, the surface charge density ¼~0 and the dielectric constant "~r are expressed in terms of the correlation functions of the bulk electrolyte. Here, "~r , µ~ and ¼~0 are considered as e€ ective parameters that allow us to apply Debye±HuÈckel theory (at least for large distances) including nonlinearity as well as many-body correlation e€ ects. Whichever the case, it should be noted that ¼~0 must be distinguished from the net charge density inside some ®nite region close to the wall, since the former gives information about the decay of the mean electrostatic potential far from the wall. This is also the reason why its sign does not have to be coincident with the sign of the di€ use potential. In our example, clearly ¼~0 is positive, unlike Ád . It should also be stressed that the exponential decay of the average electrostatic potential (given by equation (3)) is encountered for a large range of conditions, but if the electrolyte concentration exceeds a certain value, the decay parameter µ~ becomes a complex number. Physically, this means the appearance of exponentially damped oscillatory behaviour for Á…x† as well as the correlation functions, which is interpreted as the occurrence of alternating layers with positive and negative charge even far from the surface. According to some authors [14, 17, 19], this phenomenon is found irrespective of the magnitude of the surface charge. Attard has also put forward the extended PB approximation, which is essentially the solution of the MSA to the inhomogeneous OZ equation for ions of zero size. Thus it includes the ion correlation e€ ects due to charge but not those of excluded volume. This approach makes possible the calculation of ¼~0 using an analytical expression but does not work well for 2:2 electrolytes. On the other hand, the steric e€ ect (due to the ion ®nite size) has been investigated theoretically [35±37]. For high surface charge densities the ion concentration at the surface saturates to a maximum value (given by its close

packing). It should be stressed that the sole incorporation of size e€ ects (leaving out ion charge correlations) leads to di€ use potentials greater (in absolute value) than those predicted by PB theory. This is a consequence of the ions interfering with each other due to their sizes. An asymptotic analysis and a computation of ¼~0 would be relevant if we were dealing, for instance, with the interaction of widely separated charged surfaces. In this case, the charge reversal might cause interesting situations. The force between two equally charged surfaces will always be repulsive, regardless of this phenomenon. Let us consider now the case of two surfaces whose charge densities have the same sign but di€ erent magnitudes. When varying the concentration of a divalent salt, the reversal will not occur simultaneously, so two like charged surfaces could attract each other as a result of their e€ ective charges having opposite signs. Moreover, it has been reported that ion± ion correlations can induce attractive forces in the short distance regime [10, 11, 17, 18]. The theoretical foundations of charge reversal in colloids have also been looked into by Shklovskii and coworkers assuming that (i) multivalent counterions form a 2-dimensional strongly correlated liquid (SCL) at the particle surface and (ii) the PB equation works well far from the surface. The SCL e€ ect is taken into account by deriving a new boundary condition for it. More speci®cally, these authors consider the problem of a z :1 electrolyte (z 5 2) alone [20] and in the presence of a monovalent salt the concentration of which must be rather large (compared with the z :1 electrolyte) [21]. For z ˆ 2 in the former case, however, this approach works only qualitatively. In this work we restrict ourselves to the HNC/MSA for planar geometry. MC simulations for planes have proved that this approximation is able to include the singular features that the ion±ion correlation brings out. Additionally, Lozada-Cassou and coworkers concluded that there were no signi®cant di€ erences in the Ád values obtained for particles with diameters larger than 8 nm and a planar wall [8]. However, the conversion of the ± potential into mobility invokes a rather complex hydrodynami c problem [22] that becomes trivial in the Helmholtz±Smoluchowski limit, µA ! 1 (where A is the particle radius). In fact, it can be shown readily that the electrophoretic mobility is given by ·e ˆ "0 "r ±=²:

…4 †

This result is formally identical to that obtained from the PB approach, because the entire mean electrostatic potential pro®le at equilibrium is not required in this limiting case. Ion±ion correlations are actually included in calculating ±. As our colloidal particles are quite large

Overcharging in model colloids and we are interested in the high electrolyte concentration regime, this approximation is reasonable. 3. Experimental Two polystyrene latexes were used, a sulphonated latex, SN10, and a carboxylated latex, CC2. The ®rst system was prepared by a two-stage `shot growth’ emulsion polymerization process in the absence of an emulsi®er. The whole synthesis method is described in [25]. CC2 was synthesized following another free emulsi®er polymerization in a two-step process. In the ®rst step, a latex analogous to the previous one was used as a core. Thereafter a shell of styrene and acrylic acid was put onto the core, using potassium persulphate as initiator. Finally, the latexes were cleaned by serum replacement until the conductivity of the waste was similar to that of the distilled water used in this work. The particle sizes, obtained by photon correlation spectroscopy (PCS), were 196 § 3 nm, and 180 § 6 nm for the SN10 and CC2 latexes, respectively. This technique also provides a polydispersity index de®ned as the quotient K 2 =G2 , where G and K2 are the ®rst and second moments, respectively, of the cumulant expansion of the logarithm of the so-called normalized ®eld autocorrelation function. These polydispersity indexes, which tend to zero for monodisperse samples, were about 0.05 for SN10 and 0.16 for CC2. Conductimetric and potentiometric titrations were used to determine their surface charge density. These experiments were performed with a Crison Instruments pH meter and conductimeter, at 25 8C in a stirred vessel ¯ushed with nitrogen. The titration agents were NaOH and HCl. In the case of the sulphonated latex, a surface charge density of ¡2 11:5 § 1:7 mC cm , which did not depend on pH, was found. Unlike the previous case, the surface charge of the latex CC2 is controlled by pH. As ®gure 3 shows, the particles are discharged at pH 2 (the surface groups are completely protonated) while ¡¼0 reaches a maximum

Figure 3.

Surface charge density versus pH for latex CC2.

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at pH 10. In particular, the electrophoretic mobility experiments that we shall describe in the next section ¡2 were performed at pH 5.8 where ¼0 º ¡40 mC cm . A new instrument, known as the Brookhaven ZetaPALS, based on the principles of phase analysis light scattering (PALS) is used to obtain electrophoretic mobilities (·e ). The setup is especially useful at high ionic strengths, where mobilities are usually low. In this sense, the PALS con®guration has been shown to be able to measure ·e at least two orders of magnitudes lower than traditional light scattering methods based on the shifted frequency spectrum (spectral analysis). Both techniques have in common the analysis of a mix of scattered light from a sample of a suspension of colloidal particles moving in an electric ®eld, with light directly from the source. However, phase analysis takes place over many cycles of the respective waveforms, whereas spectral analysis is sensitive to the period of one cycle. For that reason, when a small mobility is present, only phase shifts can be detected accurately [38]. Electrophoretic mobility measurements were performed at 25 8C. The particle concentrations (»p ) were 4:9 £ 109 particles per ml and 4:4 £ 109 particles per ml for the SN10 and CC2 latexes, respectively. We chose these values after plotting ·e versus »p curves. The electrolytes used to obtain the ·e against ionic strength curves were LiCl and MgSO4 , while the pH was approximately 5.8.

4. Results and discussion In section 2, it was shown that a considerable theoretical e€ ort has been devoted to the overcharging phenomenon, and the great interest may relate to its potential practical applications, such as gene therapy. As is well known, both bare DNA and a cell surface are negatively charged. Thus the former is not expected to approach to the latter. The goal is reversing the DNA charge while leaving the surface negative. As noted above, this would be feasible (according to these theories) as a consequence of the inversion depending on the surface charge density. In this way, the delivery of genes to a cell could be possible. In spite of the large number of these studies, experimental tests are rather scarce. It is worthwhile quoting them before presenting our results. Although HNC/ MSA theory predicts the reversal of mobility in the case of 2:2 electrolytes in the presence of planar and spherical surfaces (with physically acceptable charge densities), to the best of our knowledge this phenomenon has been reported almost exclusively for electrolytes with trivalent counterions [23±26], and it is justi®ed through speci®c adsorption. Reversed mobility in sol-

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utions of divalent symmetrical salts was reported only for certain (non-spherical) polyelectrolytes [39]. At this point, a study by Attard et al. must be mentioned [40]. These authors used the HNC and EPB approximations to analyse the ± potential observed in colloidal dispersions of silver iodide, titanium dioxide, aluminum oxide and silicon dioxide. This study does not deal with the mobility reversal itself, but it attempts to ®nd out the in¯uence of ion±ion correlation in practice. Considerable agreement is only reported for silver iodide dispersions. Its main conclusion is that the HNC does not resolve (by itself) the di€ erences between the di€ use potential calculated from the titrated charge density and the ± potential estimated from mobility experiments. It should be noted, however, that most of the experimental measurements are restricted to moderately concentrated solutions (1±12 mM) of monovalent salts. Ion correlation e€ ects are not too important under such conditions. Electrophoretic mobility measurements for DNA in solutions of di€ erent monovalent counterions have been analysed by GonzaÂlez-Tovar et al. [7], who considered the DNA molecules as cylindrical polyelectrolytes. Taking the ion correlation e€ ect into account, an improvement (compared with PB predictions) for high electrolyte concentrations is observed (see [7], ®gure 18), particularly in the case of large hydrated counterions ‡ (such as Li ). In relation to our results, the electrophoretic mobility measured for latex SN10 at moderate, high and considerably high electrolyte concentrations (0.01±2 M) of a monovalent salt (LiCl) is shown in ®gure 4 (error bars are the standard deviation of a set of measurements). As

Figure 4. Electrophoretic mobility versus LiCl molar concentration for latex SN10. The experimental results (squares) and the HNC/MSA (solid line) and GC (dashed line) approaches are shown. The prediction for ¼d ˆ ¡4:0 ¡ mC cm 2 is plotted as the dotted line.

can be seen, the magnitude of ·e decreases with increasing csalt , but is always negative (mobility reversal is not observed). In this ®gure, the mobility calculated from the GC model has also been plotted. This value can be obtained from equation (4) with the approximation ± º Ád and using the GC relationship for symmetrical electrolytes (z1 ˆ ¡z2 ˆ z): ³ ´ kT ze¼0 2 …5 † Ád ˆ ez a sinh ; 2"0 "r µkT which assumes a di€ use double layer just beyond the outer Helmholtz plane (OHP), located at the closest approach of the hydrated ions, x ˆ a. The GC mobility values are found to be much larger than the experimental ones. It is widely known that for low electrokinetic radii there exist notable discrepancies between ± and Ád and several theories (e.g. surface conductance) have been put forward to justify them [41]. However, these di€ erences seem to disappear with increasing electrolyte concentration. At least, this is what has been reported for low and moderate surface charge densities ([41], ®gure 2 is an illustrative example). Consequently, the disagreement found between GC and experimental ·e values (at high ionic strengths) is somewhat remarkable. The HNC/MSA predictions for ·e are also plotted in ®gure 4. In order to calculate them, the hydrated ion radius has to be speci®ed as input parameter in the theory. As will be discussed later, this is a key point that must be carefully considered. For this ®rst case, we have chosen a ˆ 0:36 nm, which is practically iden‡ tical to that reported for Li by certain authors [42]. As can be observed, although the HNC/MSA ·e values also exceed (in magnitude) those obtained from measurements, the di€ erences between theory and experiment become smaller than in the GC analysis. It should also be stressed that the approach including ionic size correlation does not seem to predict mobility reversal for this system, which is corroborated by the corresponding measurements. Concerning PB theory and electrokinetic properties, the EDL is supposed to consist of two parts, the Stern layer and the di€ use part. The border between them would just be the OHP. In this context, it should be noted that equation (5) assumes that the Stern layer does not contain adsorbed ions. If this is not ful®lled, ¼0 should be replaced in equation (5) by a di€ use surface charge density ¼d . This quantity can be expressed as the di€ erence ¼d ˆ ¼0 ¡ ¼i , where ¼i is the surface charge density of the ions in the inner (non-di€ use) part of the EDL. Obviously, this speci®c adsorption cannot be justi®ed in terms of the conventional electrostatic forces that control the di€ use layer. In other words, a knowledge of ¼i (and ¼d ) requires further information, which a priori is not usually available. Hence, several values for

Overcharging in model colloids ¼d have been tried and, ®nally, we have found that mobility data can be ®tted fairly well if ¼d º ¡4:0 ¡ mC cm 2 , about 35% of ¼0 . This percentage turns out to be rather small if it is compared with others reported in certain works. Particularly, it has been found that the di€ use charge tends to the surface charge at higher ionic strengths [43, 44]. At any rate, there exists another important problem with this usual explanation: it should not be invoked for ions that do not interact speci®cally with the surface (indi€ erent electrolytes), as Attard et al. [40] pointed out. At this point, we should bear in mind that the theoretical predictions are obtained with the approximation ± º Ád . Obviously, one could assume that the OHP and the SP (where the ± potential is evaluated) are separated by a certain distance. In ®tting experimental data, the HNC/MSA would require a shorter distance than the PB approach (because its predictions are closer to reality). For instance, 0.12 nm and 0.40 nm would be enough (for the former and the latter, respectively) to ®t the mobility data shown in ®gure 4. In both cases, nevertheless, we would be dealing with an additional phenomenological parameter that could hardly support only one of the theories. The results for much more charged latex particles in a solution of monovalent ions are now presented and analysed. In ®gure 5 the electrophoretic mobility measured for CC2 as a function of the salt concentration is plotted. The magnitude of the surface charge density is ¡ signi®cantly higher (40 mC cm 2 in round numbers) for this latex, but mobility measurements seem to be insen-

Figure 5. Electrophoretic mobility versus LiCl molar concentration for latex CC2. The experimental results (squares) and the HNC/MSA (solid lines) and GC (dashed line) approaches are shown. The prediction for ¼d ˆ ¡1:5 ¡ mC cm 2 is plotted as the dotted line. In the HNC/MSA analysis, two di€ erent radii have been used (as indicated next to the corresponding lines).

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sitive to this fact, as can be concluded by comparing ®gures 4 and 5. Mobility values for csalt > 0:1 M are quite similar. Thus the di€ erences between the GC predictions (dashed line) and the experimental values are greater than in the previously discussed case. In applying the HNC/MSA, a ˆ 0:36 nm was used again. The mobility values so obtained decrease rapidly in magnitude, and even a reversal, which is not observed experimentally, should be expected beyond 1 M. Nevertheless, these results must be interpreted carefully. As mentioned in section 2, counterions form a dense layer in the vicinity of the charged wall. There exist di€ erent estimates of the volume fraction of this layer. For 3 instance, it could be given by 8p· »w a =6, where »·w would be the average counterion concentration for that area … 1 2a ˆ …6 † »·w »1 g1 …x† dx: 2a 0 For latex CC2 at 1 M and a ˆ 0:36 nm, the volume fraction next to the surface is almost identical to the value corresponding to close packing. Thus the predictions for salt concentrations exceeding 1 M could hardly have physical meaning. Unfortunately, the MC simulations that support the validity of the HNC/MSA approach have been carried out only for a particular (and moderate) ionic radius (2a ˆ 0:425 nm), so the e€ ects due to the saturation of large ions near highly charged walls are still unknown. Apart from that, the choice of the hydrated ion radius must be carefully considered. In ®gure 6, the Ád -¼0 plots for solutions 1 M of monovalent ions with di€ erent hydrated radii are shown. For large surface charge den-

Figure 6. Di€ use potential versus surface charge density for a 1:1 electrolyte at 1 M for di€ erent hydrated ion radii (indicated in the graph) according to HNC/MSA theory (solid lines). The prediction of the PB approach is plotted as the dashed line.

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M. Quesada-Perez et al.

sities, the di€ use potential depends strongly on the ion size. Additionally, it should be mentioned that this quantity depends on how it is measured (as pointed out in [42]), and some authors have even reported that the hydrated ion size could be reduced in the presence of highly charged surfaces [45]. Taking these facts into account, we eventually decided to use a ˆ 0:28 nm, a reduced value that does not lead to volume fractions without physical meaning (even up to 2 M and ¡40 mC cm¡2 ). Furthermore, it is close to the values given by neutron di€ raction experiments [46]. The result is shown in ®gure 5. Comparing with the GC predictions, the agreement between theory and experiment improves signi®cantly. Again, an adjustable di€ use charge density can be used in the GC framework. In ¡2 doing so (dotted line), the value ¼d º ¡ 1:5 mC cm is obtained in this case, which is an extremely small fraction of the surface charge density (about 4%). Concerning the HNC/MSA analysis, an additional e€ ect due to the large surface charge of CC2 should be brie¯y mentioned. As a result of the ®nite size of surface groups, large charge densities cannot be positioned on a planar layer, which leads to the existence of a certain roughness. Since one of the main assumptions of the primitive model is not ful®lled, discrepancies between experimental and theoretical data are expected to some extent. Before discussing the results for divalent electrolytes, it is advisable to examine theoretical Ád -¼0 plots for several ionic radii, as shown in ®gure 7 for a high electrolyte concentration (0.5 M). The measurements were carried out in MgSO4 solutions, and we have included 2‡ the radius corresponding to the hydrated Mg ion

(given in [42]). In relation to this case, it is striking that the HNC/MSA (within the approximation ± º Ád ) yields a reversal of the electrostatic potential even for low surface charge densities. At any rate, these plots are not monotonic (as stated in sections 1 and 2), and the magnitude of the electrostatic potential can decrease with increasing magnitude of the surface charge density. In fact, potential reversals are predicted for moderate (and low) surface charge densities, depending strongly on the ionic radius. We also can say that given a ®xed charge density, this parameter a€ ects considerably both the magnitude and the sign of the electrostatic potential. The electrophoretic mobility measured for latex SN10 in MgSO 4 solutions at high ionic strength is plotted in ®gure 8, together with the GC prediction (dashed line) obtained from the di€ use potential. Again, signi®cant discrepancies between them are found. In this case the phenomenological di€ use charge density that ®ts the ¡2 experimental data turns out to be ¡3:5 mC cm (dotted line), which is quite close to that found for LiCl solutions (and the same latex). This is rather sur‡ prising, since it means that di€ erent counterions (Li 2‡ and Mg ) would neutralize practically the same quantity of surface charge. Concerning the computation of the HNC/MSA results for this system, the hydrated ion radius given in [42] (0.43 nm) cannot be used any longer since it leads to a mobility reversal (which is not experimentally con®rmed). After several trials, it has been shown that the measurements are matched using a ˆ 0:30 nm, which is somewhat smaller than the reference value (0.43 nm), but acceptable.

Figure 7. Di€ use potential versus charge density for a 2: 2 electrolyte at 0.5 M for di€ erent hydrated ion radii (indicated in the graph) according to HNC/MSA theory (solid lines). The prediction of the PB approach is plotted as the dashed line.

Figure 8. Electrophoretic mobility versus MgSO4 molar concentration for latex SN10. The experimental results (squares) and the HNC/MSA (solid line) and GC (dashed line) approaches are shown. The prediction for ¡2 ¼d ˆ ¡3:5 mC cm is plotted as the dotted line.

Overcharging in model colloids

Figure 9. Electrophoretic mobility versus MgSO4 molar concentration for latex CC2. The experimental results (squares) and the HNC/MSA (solid line) and GC (dashed line) approaches are shown. The prediction for ¼d ˆ ¡1:0 mC cm¡2 is plotted as the dotted line.

At any rate, it should also be stressed that the overcharging phenomenon (revealed by mobility reversal) is not observed for SN10. Thus experiments with CC2 were carried out also, despite the fact that this latex can hardly be a model system to test the theory rigorously (as the previous discussion pointed out). Figure 9 shows mobility as a function of the salt (MgSO4 ) concentration. The most noticeable ®nding in this ®gure is the considerably low mobility in spite of the extremely large surface charge density. In fact, at high ionic strengths both positive and negative values in the series of measurements were obtained and mobility was determined by averaging. The appearance of small positive values should be interpreted as a quite modest mobility reversal rather than as convincing evidence of its existence. Apart from that, the small ·e magnitudes contrast strongly with the GC predictions (dashed line) although a reasonable ®t can be achieved using ¡2 ¼d º ¡1:0 mC cm , a really small fraction of the surface charge (2.5%). What is more, this quantity is (again) ‡ ¡2 almost identical to that found for Li (¡1:5 mC cm ). Certainly, the speci®c adsorption should be expected to exhibit singular features for each ionic species. Consequently, the reason why it causes the same e€ ect for both ions is not obvious. It is also remarkable that the mobility measurements are even signi®cantly smaller (in absolute value) than those found for SN10, which suggests that the di€ use potential could decrease with increasing surface charge, and the ion correlation e€ ect might play an important role. Consequently, these results were also examined within HNC/MSA theory. In order to ®t the experimental data, the hydrated ion radius was reduced to

3037

Figure 10. Di€ use potential versus surface charge density predicted by the HNC/MSA and the AHNC approaches (solid and dashed lines, respectively) for a 0.125 M solution of a 2:2 electrolyte (a ˆ 0:23 nm).

0.18 nm. This value seems to be rather small but, surprisingly, it matches with moderate accuracy the mobility measurements (without requiring additional adjustable parameters). However, this ionic radius must be analysed very carefully. We should always keep in mind that the HNC/MSA is an approximate theory that has not been tested (using simulations) for j¼0 j > 30 mC cm¡2 . One assumption is that the ion±ion correlation functions are practically identical to the bulk ion±ion correlation function [3], which is actually applied to derive equation (1). This approximation works properly provided that the external ®eld does not become too strong. In ®gure 10 the Ád ¡ ¼0 predictions of two approximate integral equation approaches (HNC/MSA and AHNC) are plotted for a particular case (2:2 electrolyte, 0.125 M, a ˆ 0:23 nm). The AHNC values have been estimated from [19], ®gure 6. As concluded, there are no remarkable di€ erences between the two theories within the range of surface ¡ charge densities studied (0±25 mC cm 2 ) but this ®gure also suggests that discrepancies may emerge for j¼0 j > 30 mC cm¡2 . Thus, these approximations do not work in the same manner for heavily charged surfaces. In any case, it is conceivable that the theory might be able to ®t the experimental data even if the assumptions underlying its derivation did break down. In this sense, the values obtained for the adjustable parameters should be considered as e€ ective rather than actual. 5. Conclusion We should like to complete this work by summing up the results and discussing several conclusions. Regarding monovalent ion solutions, neither the HNC/MSA approach nor the GC model can explain

3038

M. Quesada-Perez et al.

the mobility measurements found for these charged latexes considering exclusively the surface charge density determined by titration, although it should be stressed that the predictions including ion correlations are better than those obtained without them. In the PB framework, the discrepancies could be reduced, including e€ ects such as the speci®c adsorption, which is also considered to explain experimental results at moderate and low ionic strengths. However, this procedure requires additional parameters that are not known a priori. This is equivalent to the use of a phenomenological di€ use charge density, the magnitude of which turns out to be remarkably small (in particular for the higher charged latex) and, inexplicably, almost the ‡ 2‡ same for two di€ erent counterions (Li and Mg ). In divalent electrolytes, the role of ion correlations is more relevant. This conclusion is supported by the fact that the HNC/MSA can ®t mobility measurements acceptably without requiring additional e€ ects or parameters. However, this does not mean that this theory can predict on its own electrokinetic properties (at high electrolyte concentrations and surface charge densities) with outstanding precision. In addition to the in¯uence of several e€ ects (speci®c adsorption, surface conduction, roughness), certain features should be considered with regard to this. For instance, ®gure 7 illustrates that the di€ use potential depends strongly on the hydrated ion radius for this kind of electrolyte, but quite di€ erent values for this property can be found in literature. Hence, it is advisable to use this quantity simply as an adjustable parameter. In doing so, a fairly reasonable value has been obtained for the latex with the smaller surface charge density. The result for the extremely charged colloid is somewhat questionable, since the validity of the HNC/MSA under such conditions has not been con®rmed (through simulations). At any rate, the HNC/MSA approach (and other integral theories) can be a helpful tool in any attempt to justify certain experimental ®ndings in divalent salts. For instance, studying two latexes we found that the one having the higher charge presents lower mobility, whereas the HNC/MSA predicts that the magnitude of Ád can decrease on increasing the absolute value of ¼0 . Ion correlations may therefore contribute to justify this behaviour. However, our main concern was to look into whether overcharging is revealed by mobility reversal. For the moderately charged latex, experiments show clearly that this phenomenon does not take place. For solutions of monovalent ions, the reversal is not actually predicted, whereas the sensitivity of the di€ use potential to the ion size could be argued as a likely cause in the 2‡ case of divalent salts. However, if the Mg radius obtained for SN10 is assumed to be valid (0.30 nm)

and ®gure 7 is reviewed, the reversal should be expected for latex CC2. Although the mobility measurements obtained at high electrolyte concentrations are not conclusive evidence, this reversal is somewhat suggested since these are practically zero. In relation to this, it should be kept in mind that large surface charge densities could reduce the size of the hydrated ions. Consequently, the overcharging would not be as intense as expected, and the mobility reversal might fade away. In spite of this, the electrostatic potential could remain reversed far from the particle surface (as noted in } 2), which might be revealed experimentally in other ways. We should like to thank Dr D. Bastos for providing us with one of the latexes used in this work, and M. Lozada-Cassou (SimulacioÂn Molecular, IMP, MeÂxico) for valuable discussions on theoretical aspects. The authors also acknowledge the ®nancial support from `Ministerio de Ciencia y TecnologõÂ a, Plan Nacional de InvestigacioÂn, Desarrollo e InnovacioÂn TechnoloÂgica (I+D+I)’. M.Q.P. wishes to express his gratitude to Project MAT2000-1550-C03-0 3 and A.M.M., F.G.G. and R.H.A. to Project MAT1999-0662-C03-02 .

References

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